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      "title": "A compact S4 anomaly channel for the cosmological constant",
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        ],
        "qualifier": "Class-relative uniqueness: local, linear constant-Weyl response on round conformally flat S⁴ under C1 and C2; not uniqueness across all nonlocal observables or backgrounds.",
        "limits_and_open_issues": "The theorem identifies the protected matter coordinate. It does not by itself produce the cosmological curvature.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained from v1; current theorem wording follows the cosmological-constant paper v2.0."
        }
      },
      "EXT-02": {
        "title": "Euler projector isolates the protected coordinate",
        "display_order": 5,
        "sector": "matter",
        "statement": {
          "public": "A constant-Weyl projector removes local counterterm contamination and isolates the Euler response.",
          "technical": "At a conformal fixed point, A[Γ]=−(1/4) Π_H0(dΓ/d ln H) and Π_H0(dΓ/d ln H)=−4a. Away from a fixed point, the corresponding local-RG Euler-cocycle projection is applied after beta-function and operator-mixing responses are separated."
        },
        "status": "derived",
        "type": "identity",
        "depends_on": [
          "EXT-01"
        ],
        "equation_ids": [
          "eq:calA",
          "eq:weylvar"
        ],
        "evidence": [
          {
            "reference_id": "REF-025",
            "role": "foundation"
          },
          {
            "reference_id": "REF-026",
            "role": "foundation"
          },
          {
            "reference_id": "REF-027",
            "role": "foundation"
          },
          {
            "reference_id": "REF-033",
            "role": "foundation"
          },
          {
            "reference_id": "REF-049",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Euler-extraction theorem, Eq. defining A[Γ] and proof.",
            "structure_id": "sec:extraction"
          }
        ],
        "qualifier": "The running-theory statement requires separation of beta-function and operator-mixing terms.",
        "limits_and_open_issues": "The projector extracts the protected coordinate; it does not assert that the rest of the effective action vanishes.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained from v1; sign and running-theory scope aligned to v2.0."
        }
      },
      "EXT-03": {
        "title": "Standalone Type-A theorem across two de Sitter realisations",
        "display_order": 6,
        "sector": "matter",
        "statement": {
          "public": "A standalone theorem publication finds that the protected local logarithmic anomaly response common to the closed-sphere and round static-patch descriptions of de Sitter is uniquely type A, with microscopic coordinate a.",
          "technical": "Within the parity-even fixed-point local four-derivative logarithmic anomaly sector of exact round de Sitter, the universal paired closed-sphere/static-patch quotient has one-dimensional image generated by the Euler/Wess-Zumino class, and the transmitted QFT coordinate is a(Q)."
        },
        "status": "derived",
        "type": "theorem",
        "depends_on": [],
        "evidence": [
          {
            "reference_id": "REF-018",
            "role": "foundation"
          },
          {
            "reference_id": "REF-019",
            "role": "foundation"
          },
          {
            "reference_id": "REF-025",
            "role": "foundation"
          },
          {
            "reference_id": "REF-090",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-091",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-092",
            "role": "structural_convergence"
          }
        ],
        "locations": [
          {
            "work_id": "type-a-paper",
            "locator_text": "Standalone Type-A theorem publication, current public record; exact theorem and scope on its own paper page."
          }
        ],
        "qualifier": "Independent cross-realisation theorem. It is not an incoming dependency of the cosmological numerical chain.",
        "limits_and_open_issues": "Does not establish the QCD state-preparation premise, the boundary-member prescription, u_max=1 UV-domain normalization, contour choice, quantum dressing, global source pairing or numerical cosmological result.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained as an independent v2.0 programme record."
        }
      },
      "SM-01": {
        "title": "Minimal Standard Model Euler coefficient",
        "display_order": 7,
        "sector": "matter",
        "statement": {
          "public": "For the minimal Standard Model free-field census, the protected type-A coefficient is a_SM = 1991/720.",
          "technical": "For four real Higgs scalars, 45 Weyl fermions and 12 vector bosons, a_SM=4/360+45(11/720)+12(31/180)=1991/720=2.76527778."
        },
        "status": "computed",
        "type": "computed_value",
        "depends_on": [
          "EXT-01"
        ],
        "equation_ids": [
          "eq:aSMvalue"
        ],
        "evidence": [
          {
            "reference_id": "REF-004",
            "role": "observational_input"
          },
          {
            "reference_id": "REF-010",
            "role": "foundation"
          },
          {
            "reference_id": "REF-011",
            "role": "foundation"
          },
          {
            "reference_id": "REF-035",
            "role": "foundation"
          },
          {
            "reference_id": "REF-036",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Standard Model evaluation of the extracted datum”.",
            "structure_id": "sec:anomaly",
            "section_title": "Standard Model evaluation of the extracted datum"
          }
        ],
        "qualifier": "Leading free-field Standard Model value; the full interacting curved-space coefficient is CORR-01.",
        "limits_and_open_issues": "Not a statement that no additional ultraviolet degrees of freedom exist, and not a calculation of the full interacting a_eff.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained from v1; old provisional 0.5% uncertainty removed."
        }
      },
      "IR-01": {
        "title": "Direct Planck-scale mechanisms tested do not supply the linear IR factor",
        "display_order": 8,
        "sector": "matter",
        "statement": {
          "public": "A distinct infrared readout is required; the direct Planck-scale mechanisms tested do not generate the needed linear QCD hierarchy.",
          "technical": "Within the stated compact treatment, the local analytic heat-kernel mass expansion begins in even powers of dimensionless mass and perturbative Planck-scale Yang-Mills instanton weights are far too suppressed; neither supplies a term linear in m_p/M_P. The protected Euler coefficient itself is order one."
        },
        "status": "derived",
        "type": "no_go",
        "depends_on": [
          "EXT-01"
        ],
        "evidence": [
          {
            "reference_id": "REF-015",
            "role": "foundation"
          },
          {
            "reference_id": "REF-044",
            "role": "foundation"
          },
          {
            "reference_id": "REF-045",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, end of Sec. “Standard Model evaluation of the extracted datum”.",
            "structure_id": "sec:anomaly",
            "section_title": "Standard Model evaluation of the extracted datum"
          }
        ],
        "qualifier": "Limited exclusion of the tested direct/local routes, not a no-go theorem for every nonlocal or UV-complete effect.",
        "limits_and_open_issues": "The construction obtains the infrared factor through the prepared QCD+QED sector instead.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained and simplified from v1; no midpoint-scale machinery remains."
        }
      },
      "TOP-01": {
        "title": "Normalized Euler-Chern unit on S⁴",
        "display_order": 9,
        "sector": "geometry",
        "statement": {
          "public": "The oriented round spin sphere carries one intrinsic normalized Euler-Chern unit: N_E = χ(S⁴)/2 = c₂(Σ⁻) = 1.",
          "technical": "For the chosen orientation on S⁴, N_E=(1/64π²)∫√g E₄=χ(S⁴)/2=1, while ⟨c₂(Σ⁻),[S⁴]⟩=+1 and ⟨c₂(Σ⁺),[S⁴]⟩=−1."
        },
        "status": "derived",
        "type": "identity",
        "depends_on": [],
        "equation_ids": [
          "eq:Eulerunit",
          "eq:intrinsicunit",
          "eq:spinChern",
          "eq:spinclasses"
        ],
        "evidence": [
          {
            "reference_id": "REF-021",
            "role": "foundation"
          },
          {
            "reference_id": "REF-023",
            "role": "foundation"
          },
          {
            "reference_id": "REF-078",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “One intrinsic Euler-Chern unit on the round sphere”.",
            "structure_id": "loc-015",
            "section_title": "One intrinsic Euler-Chern unit on the round sphere"
          }
        ],
        "qualifier": "Purely geometric; it does not identify spin SU(2) with QCD flavour SU(2).",
        "limits_and_open_issues": "Fixes the primitive geometric integer, not its physical QCD state preparation.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0."
        }
      },
      "TOP-02": {
        "title": "Degree-one equatorial clutching map",
        "display_order": 10,
        "sector": "geometry",
        "statement": {
          "public": "The chiral spin bundle’s equatorial transition map is a primitive degree-one map S³→SU(2).",
          "technical": "With Σ⁻ trivialised on the two hemispheres of S⁴, its transition function g₋:S³→SU(2) satisfies deg g₋=(1/24π²)∫Tr(g₋⁻¹dg₋)³=1 and N_E=⟨c₂(Σ⁻),[S⁴]⟩=deg g₋=1."
        },
        "status": "derived",
        "type": "identity",
        "depends_on": [
          "TOP-01"
        ],
        "equation_ids": [
          "eq:clutchdegree",
          "eq:clutchmap",
          "eq:intrinsicunit"
        ],
        "evidence": [
          {
            "reference_id": "REF-022",
            "role": "foundation"
          },
          {
            "reference_id": "REF-023",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “The equatorial clutching map has degree one”.",
            "structure_id": "loc-016",
            "section_title": "The equatorial clutching map has degree one"
          }
        ],
        "qualifier": "Relative Chern class is represented by the boundary clutching/Chern-Simons transgression; the half-sphere curvature integral need not itself be integer.",
        "limits_and_open_issues": "A geometric spin-bundle statement, not yet a QCD claim.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0."
        }
      },
      "TOP-03": {
        "title": "Explicit degree-one spin-holonomy representative",
        "display_order": 11,
        "sector": "geometry",
        "statement": {
          "public": "The same primitive spin class has an explicit group-valued representative U_spin with deg U_spin=1 and [U_spin]=[g₋].",
          "technical": "The Levi-Civita-induced charge-one SU(2) connection on Σ⁻ has Atiyah-Manton holonomy U_spin:S³→SU(2)_spin satisfying deg U_spin=⟨c₂(Σ⁻),[S⁴]⟩=1 and [U_spin]=[g₋]."
        },
        "status": "derived",
        "type": "identity",
        "depends_on": [
          "TOP-02"
        ],
        "equation_ids": [
          "eq:spinholonomydegree"
        ],
        "evidence": [
          {
            "reference_id": "REF-082",
            "role": "direct_construction_input"
          },
          {
            "reference_id": "REF-085",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, paragraph “An explicit degree-one spin-holonomy representative”.",
            "structure_id": "loc-017",
            "paragraph_title": "An explicit degree-one spin-holonomy representative"
          }
        ],
        "qualifier": "Connection-level representative of the geometric spin class. No SU(2)_spin=SU(2)_flavour identification is asserted.",
        "limits_and_open_issues": "The physical class transfer to QCD remains QCD-03.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0."
        }
      },
      "QCD-01": {
        "title": "QCD supplies a physical infrared mass scale",
        "display_order": 12,
        "sector": "qcd",
        "statement": {
          "public": "QCD dimensional transmutation generates a hadronic infrared scale, and the measured proton-to-Planck ratio is m_p/M_P = 7.685×10⁻²⁰.",
          "technical": "QCD dimensional transmutation generates the confined hadronic spectrum. Using the physical proton mass and unreduced Newton mass M_P=G⁻¹/² gives m_p/M_P=7.68514844×10⁻²⁰."
        },
        "status": "standard_input",
        "type": "standard_input",
        "depends_on": [],
        "equation_ids": [
          "eq:hierarchy"
        ],
        "evidence": [
          {
            "reference_id": "REF-004",
            "role": "observational_input"
          },
          {
            "reference_id": "REF-046",
            "role": "foundation"
          },
          {
            "reference_id": "REF-047",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, QCD section and numerical audit.",
            "structure_id": "sec:gauge"
          }
        ],
        "qualifier": "This record supplies the standard QCD scale and measured ratio only; it performs no state selection or spin-to-QCD matching.",
        "limits_and_open_issues": "The role of the B=1 sector is established only after QCD-03.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained as a standard foundation; the v1 selected-endpoint claim QCD-02 is superseded."
        }
      },
      "QCD-03": {
        "title": "QCD state-preparation class identification",
        "display_order": 13,
        "sector": "qcd",
        "statement": {
          "public": "The compact channel assumes that the infrared QCD chiral boundary state is prepared in the homotopy class of the intrinsic degree-one spin field.",
          "technical": "At confinement the physical premise is [U_IR]=φ_*[U_spin]=φ_*[g₋] in π₃(SU(2)_flavour), with φ orientation preserving at the level of the abstract SU(2) groups."
        },
        "status": "structural_premise",
        "type": "structural_premise",
        "depends_on": [
          "TOP-03"
        ],
        "equation_ids": [
          "eq:qcd_state_preparation",
          "eq:skyrmionB"
        ],
        "evidence": [
          {
            "reference_id": "REF-079",
            "role": "foundation"
          },
          {
            "reference_id": "REF-080",
            "role": "foundation"
          },
          {
            "reference_id": "REF-086",
            "role": "foundation"
          },
          {
            "reference_id": "REF-087",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “One class-level QCD boundary premise”.",
            "structure_id": "loc-018",
            "section_title": "One class-level QCD boundary premise"
          }
        ],
        "qualifier": "The paper’s single new matter-side physical premise. It is class-level only, not a connection-level spin-flavour identification.",
        "limits_and_open_issues": "A microscopic derivation from Standard Model plus gravity remains open.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0; supersedes the v1 endpoint/fold matching architecture on the matter side."
        }
      },
      "QCD-04": {
        "title": "Prepared B=1 QCD+QED sector has proton spectral floor",
        "display_order": 14,
        "sector": "qcd",
        "statement": {
          "public": "Given the QCD state-preparation premise, the degree-one sector is B=1 and its gauge-consistent QCD+QED spectral floor is the proton.",
          "technical": "QCD-03 transfers the primitive winding to the infrared chiral field, giving B=+1 (and B=−1 for conjugate orientation). In the compatible electromagnetic superselection sector, the regulated long-time energy difference tends to m_p; the conjugate sector gives the antiproton."
        },
        "status": "derived_given_premise",
        "type": "conditional_result",
        "depends_on": [
          "QCD-03",
          "QCD-01"
        ],
        "equation_ids": [
          "eq:Bfromclutch",
          "eq:chargedfloor",
          "eq:gaussedge"
        ],
        "evidence": [
          {
            "reference_id": "REF-004",
            "role": "observational_input"
          },
          {
            "reference_id": "REF-072",
            "role": "methodological_foundation"
          },
          {
            "reference_id": "REF-073",
            "role": "methodological_foundation"
          },
          {
            "reference_id": "REF-074",
            "role": "methodological_foundation"
          },
          {
            "reference_id": "REF-075",
            "role": "methodological_foundation"
          },
          {
            "reference_id": "REF-081",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, “From the B=1 sector to the proton channel” and “Gauge-consistent proton floor”.",
            "structure_id": "loc-020"
          }
        ],
        "qualifier": "The proton is not chosen from a scored candidate list; it follows as the spectral floor once B=1 is prepared.",
        "limits_and_open_issues": "Depends on QCD-03. Electromagnetic edge flux dresses the charged subsystem but does not change baryon winding.",
        "relations": [
          {
            "type": "supersedes",
            "target_id": "QCD-02"
          }
        ],
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0; replaces the selected proton-endpoint record QCD-02."
        }
      },
      "TRC-01": {
        "title": "Hamiltonian trace-source normalization",
        "display_order": 15,
        "sector": "qcd",
        "statement": {
          "public": "The proton response is read with a trace source whose normalization is fixed by H(σ)=H(0)+σΘ+O(σ²), so H′(0)=Θ.",
          "technical": "Define Θ=∫_Σ√h T^μ_μ d³x and H(σ)=H(0)+σΘ+O(σ²), hence H′(0)=Θ. This Hamiltonian source convention is distinct from the geometric Weyl parameter used in the Euler extraction."
        },
        "status": "defined_source_normalization",
        "type": "defined_observable",
        "depends_on": [],
        "equation_ids": [
          "eq:tracesource"
        ],
        "evidence": [
          {
            "reference_id": "REF-069",
            "role": "methodological_foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Eq. defining the trace source.",
            "structure_id": "loc-022"
          }
        ],
        "qualifier": "Definition of the source coordinate used by the proton response theorem; not a new dynamical assumption.",
        "limits_and_open_issues": "The source normalization fixes how the derivative is read; it does not prepare the B=1 sector.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0."
        }
      },
      "QCD-05": {
        "title": "Proton Weyl-slope theorem",
        "display_order": 16,
        "sector": "qcd",
        "statement": {
          "public": "For the prepared proton channel, the normalized long-time trace-source slope is m_p/M_P.",
          "technical": "With TRC-01 and the regulated proton correlator, −lim_{s→∞}(1/s)∂_σ ln[C_p(s;σ)/C_p(s;0)]|_{σ=0}=m_p√G=m_p/M_P. The result follows from spectral projection, field-theoretic Feynman-Hellmann and the exact forward energy-momentum-tensor normalization."
        },
        "status": "derived_given_premise",
        "type": "theorem",
        "depends_on": [
          "QCD-04",
          "TRC-01"
        ],
        "equation_ids": [
          "eq:dimensionlessslope",
          "eq:weylslope"
        ],
        "evidence": [
          {
            "reference_id": "REF-069",
            "role": "methodological_foundation"
          },
          {
            "reference_id": "REF-070",
            "role": "methodological_foundation"
          },
          {
            "reference_id": "REF-076",
            "role": "foundation"
          },
          {
            "reference_id": "REF-077",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Theorem “Proton Weyl-slope theorem”.",
            "structure_id": "loc-022",
            "theorem_title": "Proton Weyl-slope theorem"
          }
        ],
        "qualifier": "Derived once the prepared B=1 proton channel exists; endpoint overlaps and subextensive edge dressings drop out of the normalized long-time slope.",
        "limits_and_open_issues": "Does not derive QCD-03.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0."
        }
      },
      "UNIT-01": {
        "title": "Planck-unit normalization covariance",
        "display_order": 17,
        "sector": "qcd",
        "statement": {
          "public": "Reduced and unreduced Planck conventions are algebraically equivalent when every normalization is transformed consistently; changing notation does not add a physical choice.",
          "technical": "With M_P=G⁻¹/² and M̄_P=(8πG)⁻¹/², m_p/M_P=(1/√(8π))(m_p/M̄_P) and GΛ=(1/8π)(Λ/M̄_P²). Consistent conversion transforms both response and curvature observable."
        },
        "status": "normalization_covariance",
        "type": "normalization_covariance",
        "depends_on": [],
        "equation_ids": [
          "eq:Planckcovariance"
        ],
        "evidence": [
          {
            "reference_id": "REF-004",
            "role": "observational_input"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Eq. “Planck covariance”.",
            "structure_id": "loc-022"
          }
        ],
        "qualifier": "Algebraic covariance of conventions, not a dynamical selection.",
        "limits_and_open_issues": "Changing only one denominator while holding a coefficient-one source law fixed would define a different normalization.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0."
        }
      },
      "MAT-01": {
        "title": "Normalized matter/IR composite",
        "display_order": 18,
        "sector": "qcd",
        "statement": {
          "public": "The strict channel defines the matter/IR composite as q_matter = a_eff·(m_p/M_P), with coefficient one between two separately normalized response coordinates.",
          "technical": "Define A_E≡a_eff and Q_p≡m_p/M_P, then q_matter^(1)≡A_E Q_p=a_eff(m_p/M_P). This is the channel’s definition of the normalized matter/IR composite; any coefficient coupling the composite to curvature belongs to the separate global source map."
        },
        "status": "defined_strict_channel_composite",
        "type": "defined_observable",
        "depends_on": [
          "EXT-01",
          "QCD-05",
          "UNIT-01"
        ],
        "equation_ids": [
          "eq:qmatter"
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Normalized matter/IR composite”.",
            "structure_id": "loc-023",
            "section_title": "Normalized matter/IR composite"
          }
        ],
        "qualifier": "Defined observable, not a separately derived dynamical interaction.",
        "limits_and_open_issues": "The leading numerical evaluation substitutes a_eff→a_SM; the full a_eff remains CORR-01.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0."
        }
      },
      "SAD-01": {
        "title": "Round-S⁴ Einstein action and Euler-normalized form",
        "display_order": 19,
        "sector": "gravity",
        "statement": {
          "public": "For round Euclidean de Sitter, the Einstein-Hilbert action has magnitude |S_EH|=24π²/u, and the same result can be written (24π²/u)N_E.",
          "technical": "For R_{μν}=Λ_UVg_{μν} on round S⁴, S_EH=−24π²/(κ²Λ_UV). With u=κ²Λ_UV and N_E=1, |S_EH|=(24π²/u)N_E. The magnitude also equals the de Sitter entropy."
        },
        "status": "derived",
        "type": "identity",
        "depends_on": [],
        "equation_ids": [
          "eq:SEH",
          "eq:SEHNE",
          "eq:geometric_identity"
        ],
        "evidence": [
          {
            "reference_id": "REF-021",
            "role": "foundation"
          },
          {
            "reference_id": "REF-024",
            "role": "foundation"
          },
          {
            "reference_id": "REF-038",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Round-sphere action”.",
            "structure_id": "loc-024",
            "section_title": "Round-sphere action"
          }
        ],
        "qualifier": "Classical on-shell Einstein result. It does not by itself select u=1 or a path-integral contour.",
        "limits_and_open_issues": "Quantum dressing and higher-curvature shifts are CORR-02.",
        "relations": [
          {
            "type": "structural_parallel",
            "target_id": "REF-055",
            "target_type": "reference",
            "note": "Independent Λ>0 gravitational-EFT analysis finds round S⁴ leading-order dominant",
            "evidence_role": "direct_architectural_support"
          },
          {
            "type": "structural_parallel",
            "target_id": "REF-059",
            "target_type": "reference",
            "note": "Exact canonical gravity independently realises primitive SU(2) winding with inverse dimensionless cosmological coupling",
            "evidence_role": "direct_architectural_support"
          }
        ],
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained/reframed from v1; direct v2.0 support relations added. v2.1 (2026-09-07): saddle notation aligned to the paper's Λ_UV; no content change."
        }
      },
      "SAD-02": {
        "title": "Decaying compact thimble",
        "display_order": 20,
        "sector": "gravity",
        "statement": {
          "public": "The strict channel selects the decaying orientation of the compact round-S⁴ thimble.",
          "technical": "The cosmological compact observable is defined so that the round-S⁴ thimble contributes with the damped orientation e^{−|S_EH|}. The opposite sign would replace suppression by an enormous enhancement and is not the channel analysed."
        },
        "status": "selected_contour",
        "type": "selected_sector",
        "depends_on": [
          "SAD-01"
        ],
        "equation_ids": [
          "eq:instanton_value"
        ],
        "evidence": [
          {
            "reference_id": "REF-039",
            "role": "foundation"
          },
          {
            "reference_id": "REF-040",
            "role": "foundation"
          },
          {
            "reference_id": "REF-053",
            "role": "foundation"
          },
          {
            "reference_id": "REF-065",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, paragraph “Suppressed-thimble clause”.",
            "structure_id": "loc-027",
            "paragraph_title": "Suppressed-thimble clause"
          }
        ],
        "qualifier": "Contour inclusion/orientation is selected, not derived by the compact action formula.",
        "limits_and_open_issues": "Broader contour selection depends on Stokes data, boundary conditions and observable.",
        "relations": [
          {
            "type": "structural_parallel",
            "target_id": "REF-060",
            "target_type": "reference",
            "note": "Explicit de Sitter Lefschetz thimble with damped anisotropic fluctuations",
            "evidence_role": "direct_architectural_support"
          }
        ],
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained from v1; v2.0 adds direct thimble support."
        }
      },
      "SAD-04": {
        "title": "Exact reduced-Planck-density UV endpoint",
        "display_order": 21,
        "sector": "gravity",
        "statement": {
          "public": "The strict compact channel declares the reduced-Planck-density domain 0<u≤1 and fixes its exact upper endpoint at u_max=1.",
          "technical": "With u=κ²Λ_UV=ρ_UV/M̄_P⁴, the declared ultraviolet domain is 0<ρ_UV≤M̄_P⁴, equivalently 0<u≤1, with u_max=1 ⇔ ρ_UV=M̄_P⁴."
        },
        "status": "selected_uv_domain",
        "type": "selected_sector",
        "depends_on": [
          "SAD-01"
        ],
        "equation_ids": [
          "eq:u_endpoint"
        ],
        "evidence": [
          {
            "reference_id": "REF-006",
            "role": "foundation"
          },
          {
            "reference_id": "REF-008",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, reduced-Planck-density UV-domain paragraph."
          }
        ],
        "qualifier": "The exact numerical boundary is a selected UV-domain normalization, not a theorem that every ultraviolet completion breaks down at the same coefficient.",
        "limits_and_open_issues": "Microscopic selection of this exact endpoint remains a central gravity target.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained ID but scientifically sharpened in v2.0. v2.1 (2026-09-07): saddle notation aligned to the paper's Λ_UV/ρ_UV; no content change."
        }
      },
      "SAD-03": {
        "title": "Monotonic ordering within the declared domain",
        "display_order": 22,
        "sector": "gravity",
        "statement": {
          "public": "Within the declared 0<u≤1 domain, B(u)=24π²/u decreases monotonically as u increases; the largest allowed u therefore has the smallest classical action.",
          "technical": "Because B(u)=24π²/u decreases monotonically as u increases, the mathematical ordering is fixed once the domain is declared. For the separately selected decaying factor e^{−B(u)}, this makes u=u_max the least-suppressed member, but the ordering does not itself select evaluation there."
        },
        "status": "derived_given_strict_domain",
        "type": "conditional_result",
        "depends_on": [
          "SAD-01",
          "SAD-04"
        ],
        "equation_ids": [
          "eq:SEHNE",
          "eq:u_domain",
          "eq:u_endpoint"
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, compact de Sitter saddle section."
          }
        ],
        "qualifier": "Derived ordering only. Selection of a single boundary member is priced separately in SAD-07; the exact domain endpoint remains SAD-04.",
        "limits_and_open_issues": "Does not define a modulus measure or select the member at which the compact observable is evaluated.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained ID; v2.0 finalization separates derived monotonic ordering from the selected boundary-member prescription."
        }
      },
      "SAD-07": {
        "title": "Boundary-member prescription (no modulus integration)",
        "display_order": 23,
        "sector": "gravity",
        "statement": {
          "public": "The strict compact observable is evaluated at the least-suppressed boundary member u_b=u_max; no integration over u is part of the defined compact observable.",
          "technical": "After SAD-03 establishes the monotonic ordering of the declared domain, the compact channel selects single-member evaluation at u_b=u_max rather than introducing a modulus integral over u. Combined with the selected UV endpoint SAD-04, this gives u_b=u_max=1."
        },
        "status": "selected_compact_sector_prescription",
        "type": "selected_sector",
        "depends_on": [
          "SAD-03"
        ],
        "equation_ids": [
          "eq:ubone"
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, compact de Sitter saddle section, boundary-member paragraph."
          }
        ],
        "qualifier": "This is a prescription for the compact observable, not a consequence of monotonic endpoint ordering and not a claim that u is a dynamical modulus with a preferred measure.",
        "limits_and_open_issues": "A microscopic gravitational derivation may reproduce, replace or falsify this prescription. No modulus integration over u is asserted within the defined compact observable.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0 finalization to price separately the single-member evaluation step that had previously been folded into endpoint dominance."
        }
      },
      "SAD-06": {
        "title": "Classical strict-channel gravitational factor",
        "display_order": 24,
        "sector": "gravity",
        "statement": {
          "public": "Given the selected u_max=1 UV endpoint, selected boundary-member prescription u_b=u_max and decaying contour, the classical compact gravitational factor is e⁻²⁴π².",
          "technical": "At the selected member u_b=u_max=1, B_1=|S_EH|=24π². With the selected damped orientation, the strict leading Einstein-Hilbert saddle contribution is e^{−B_1}=e^{−24π²}=1.34414611×10⁻¹⁰³."
        },
        "status": "derived_given_boundary_member_uv_domain_and_contour_selections",
        "type": "conditional_result",
        "depends_on": [
          "SAD-01",
          "SAD-02",
          "SAD-04",
          "SAD-07"
        ],
        "equation_ids": [
          "eq:B1",
          "eq:instanton_value"
        ],
        "evidence": [
          {
            "reference_id": "REF-038",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, “Classical strict-channel factor”.",
            "structure_id": "loc-028"
          }
        ],
        "qualifier": "Classical Einstein-Hilbert factor only; the member prescription, UV endpoint and contour are selected, while quantum/source-dependent dressing is open.",
        "limits_and_open_issues": "Does not assert a universal absolute one-loop prefactor.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0 to price the factor explicitly."
        }
      },
      "DET-01": {
        "title": "No canonical absolute determinant constant on round S⁴",
        "display_order": 25,
        "sector": "gravity",
        "statement": {
          "public": "A raw absolute sphere determinant does not carry a universal scheme-independent constant prefactor that can simply be set to one.",
          "technical": "Finite R² and Euler counterterms shift the absolute round-S⁴ effective action by arbitrary constants without changing the local closed-sphere equations. Therefore a raw finite one-loop constant C_grav is not scheme independent without an additional renormalisation and measure prescription."
        },
        "status": "derived",
        "type": "theorem",
        "depends_on": [
          "SAD-01"
        ],
        "equation_ids": [
          "eq:app_detcts",
          "eq:app_rawdet"
        ],
        "evidence": [
          {
            "reference_id": "REF-014",
            "role": "foundation"
          },
          {
            "reference_id": "REF-035",
            "role": "foundation"
          },
          {
            "reference_id": "REF-061",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Appendix “Determinant status of the strict compact weight”.",
            "structure_id": "loc-049"
          }
        ],
        "qualifier": "No unit-prefactor or determinant-cancellation claim is used; source-dependent quantum-gravity corrections remain open.",
        "limits_and_open_issues": "Source-dependent determinants, Stokes changes and higher-curvature shifts remain CORR-02.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0; replaces the old paired-cap determinant-normalisation story."
        }
      },
      "ASM-01": {
        "title": "Factorized compact assembly",
        "display_order": 26,
        "sector": "assembly",
        "statement": {
          "public": "The leading compact observable is defined by multiplying the normalized matter/IR composite by the selected classical compact factor.",
          "technical": "Define q_comp≡q_matter^(1)e^{−24π²}=a_eff(m_p/M_P)e^{−24π²}. The coefficient-one factorization is the strict-channel observable definition, not a separately derived interaction vertex."
        },
        "status": "defined_strict_channel_composite",
        "type": "defined_observable",
        "depends_on": [
          "MAT-01",
          "SAD-06"
        ],
        "equation_ids": [
          "eq:qcomp_general"
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Leading compact-channel result”.",
            "structure_id": "sec:assembly",
            "section_title": "Leading compact-channel result"
          }
        ],
        "qualifier": "Definition of the compact response. The later map from q_comp to cosmological curvature is separate.",
        "limits_and_open_issues": "Does not upgrade the QCD premise or gravitational selections to derived status.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0."
        }
      },
      "NUM-01": {
        "title": "Leading strict-channel compact benchmark",
        "display_order": 27,
        "sector": "assembly",
        "statement": {
          "public": "Within the stated leading strict channel, the compact response is q_comp = 2.85652154×10⁻¹²², with no continuous parameter fitted to the cosmological value.",
          "technical": "Using a_eff→a_SM=1991/720 in ASM-01 together with the physical m_p/M_P and e^{−24π²}, q_comp=(1991/720)(7.68514844×10⁻²⁰)(1.34414611×10⁻¹⁰³)=2.85652154×10⁻¹²²."
        },
        "status": "derived_given_stated_premise_selections",
        "type": "computed_value",
        "depends_on": [
          "ASM-01",
          "SM-01"
        ],
        "equation_ids": [
          "eq:qcomp_numeric_audit",
          "eq:qcomp_result"
        ],
        "evidence": [
          {
            "reference_id": "REF-004",
            "role": "observational_input"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, numerical audit and compact-status ledger.",
            "structure_id": "loc-032"
          }
        ],
        "qualifier": "Compact benchmark only. Its interpretation as cosmological curvature is MAP-01. The QCD premise and gravitational selections enter through upstream dependencies and are not fitted.",
        "limits_and_open_issues": "Full a_eff and gravitational dressing remain open; the formula cannot be retuned to their outcome.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained ID but completely rebased onto the v2.0 geometric/QCD/gravity chain."
        }
      },
      "GR-01": {
        "title": "Ordinary-GR no-go for a linear compact source",
        "display_order": 28,
        "sector": "global",
        "statement": {
          "public": "Ordinary local type-A anomaly backreaction has the wrong scaling: it produces an H⁴ stress and no small branch linear in the compact response.",
          "technical": "On a maximally symmetric branch the type-A anomaly stress scales as Λ_g². In ordinary semiclassical Einstein gravity, Λ_g−Λ_0=(a/3π)(Λ_g²/M_P²); multiplying the anomaly sector by a small ε places ε in the denominator of the nonzero root rather than producing Λ_g/M_P²∝εa."
        },
        "status": "derived",
        "type": "theorem",
        "depends_on": [
          "EXT-01"
        ],
        "equation_ids": [
          "eq:H4obstruction"
        ],
        "evidence": [
          {
            "reference_id": "REF-018",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Proposition “Ordinary-GR no-go for a linear compact source”.",
            "structure_id": "loc-034"
          }
        ],
        "qualifier": "Applies to the ordinary local metric-variation route.",
        "limits_and_open_issues": "Motivates a separate scalar zero-mode equation; it does not rule out all nonlocal/global completions.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained from v1."
        }
      },
      "GR-02": {
        "title": "Trace-free quotient leaves one scalar mode",
        "display_order": 29,
        "sector": "global",
        "statement": {
          "public": "Removing the volume-counterterm representative leaves a unique trace-free local Einstein equation and one undetermined spacetime-constant scalar mode.",
          "technical": "The unique algebraic projection invariant under E_{μν}→E_{μν}+cg_{μν} and equal to the identity on traceless tensors is E_{μν}−Eg_{μν}/4, yielding R_{μν}−Rg_{μν}/4=(8π/M_P²)(T_{μν}−Tg_{μν}/4). Conservation leaves R+(8π/M_P²)T=4Λ_int."
        },
        "status": "derived",
        "type": "lemma",
        "depends_on": [
          "VAC-01"
        ],
        "equation_ids": [
          "eq:integrationconstant",
          "eq:tracefree"
        ],
        "evidence": [
          {
            "reference_id": "REF-028",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-029",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Lemma “Trace-free quotient projector”.",
            "structure_id": "loc-035"
          }
        ],
        "qualifier": "Determines the representative-independent local equation shape, not a complete physical law or the value of Λ_int.",
        "limits_and_open_issues": "The missing scalar equation is supplied only by the proposed completion.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained from v1."
        }
      },
      "GS-01": {
        "title": "Four-form global-source completion",
        "display_order": 30,
        "sector": "global",
        "statement": {
          "public": "A Henneaux-Teitelboim/sequestering-type four-form action is proposed to supply the scalar-curvature mode omitted by the trace-free local equation.",
          "technical": "The proposed action S_gs[q] contains a rigid Newton variable η, a volume four-form constraint and a source term proportional to η²M_P⁴ q F₄. The compact calculation supplies q=q_comp; the action is varied at fixed rigid q."
        },
        "status": "proposed_completion",
        "type": "proposed_model",
        "depends_on": [
          "GR-02"
        ],
        "equation_ids": [
          "eq:globalaction"
        ],
        "evidence": [
          {
            "reference_id": "REF-028",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-030",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-031",
            "role": "structural_convergence"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “From compact response to cosmological curvature: global-source completion”.",
            "structure_id": "sec:globalsource",
            "section_title": "From compact response to cosmological curvature: global-source completion"
          }
        ],
        "qualifier": "Gravity-side completion proposed independently of the extraction theorem.",
        "limits_and_open_issues": "Microscopic selection of this action remains open.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained ID; action rewritten and status sharpened in v2.0."
        }
      },
      "GS-05": {
        "title": "Branch-unit covariance fixes η² source dependence",
        "display_order": 31,
        "sector": "global",
        "statement": {
          "public": "Within the proposed completion, covariance under the arbitrary branch Newton-unit parametrization fixes the source dependence to η² up to one overall compact-to-global coefficient.",
          "technical": "Replacing η² by f(η), branch-unit independence of Λ/M_N² relative to q requires f′(η)/(2η)=c, hence f(η)=cη²+f₀. The minimal one-source action sets f₀=0; covariance fixes the η² form but not the overall c."
        },
        "status": "derived_from_branch_unit_covariance",
        "type": "identity",
        "depends_on": [
          "GS-01"
        ],
        "equation_ids": [
          "eq:source_covariance"
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Appendix “Off-shell variation of the global-source action”.",
            "structure_id": "loc-050"
          }
        ],
        "qualifier": "Derived within the proposed completion; it does not fix the unit coefficient c=1.",
        "limits_and_open_issues": "The remaining coefficient is GS-04.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0."
        }
      },
      "GS-04": {
        "title": "Unit compact-to-global source pairing",
        "display_order": 32,
        "sector": "global",
        "statement": {
          "public": "The proposed completion uses unit pairing between the normalized compact response and the global four-form source.",
          "technical": "After GS-05 fixes the η² dependence, the overall source coefficient c remains undetermined by branch-unit covariance. The paper adopts the unit choice c=1 in S_gs[q]."
        },
        "status": "proposed_unit_pairing",
        "type": "proposed_model",
        "depends_on": [
          "GS-01",
          "GS-05"
        ],
        "equation_ids": [
          "eq:globalaction",
          "eq:source_covariance"
        ],
        "evidence": [
          {
            "reference_id": "REF-028",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-030",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-031",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-032",
            "role": "structural_convergence"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, global-source action and Appendix on branch-unit covariance.",
            "structure_id": "sec:globalsource"
          }
        ],
        "qualifier": "A proposed physical normalization, not fitted after comparison and not derived by the compact anomaly calculation.",
        "limits_and_open_issues": "A microscopic parent could derive a different coefficient and thereby change the curvature map.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained ID but fundamentally revised in v2.0; old v1 source-charge convention is superseded."
        }
      },
      "GS-02": {
        "title": "Field equations of the proposed global-source action",
        "display_order": 33,
        "sector": "global",
        "statement": {
          "public": "Once the stated global-source action and unit pairing are adopted, its variations cancel homogeneous matter shifts and fix the residual vacuum relation Λ/M_N²=q.",
          "technical": "Variation gives F₄=*1, ⟨R⟩=4ηM_P²q and G_{μν}+ηM_P²q g_{μν}=(8π/(ηM_P²))(T_{μν}−⟨T⟩g_{μν}/4). On a maximally symmetric vacuum with M_N²=ηM_P², Λ/M_N²=q. Constant shifts of L_m cancel exactly."
        },
        "status": "derived_once_action_and_unit_pairing_are_adopted",
        "type": "theorem",
        "depends_on": [
          "GS-01",
          "GS-04"
        ],
        "equation_ids": [
          "eq:compactsumrule",
          "eq:completedEinstein",
          "eq:vacuumq"
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Theorem “Global-source field equations”.",
            "structure_id": "sec:globalsource",
            "theorem_title": "Global-source field equations"
          }
        ],
        "qualifier": "The equations are a theorem of the stated action; the action and unit pairing remain proposed.",
        "limits_and_open_issues": "Does not establish microscopic selection of GS-01/GS-04.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained ID; equations and normalization updated to v2.0."
        }
      },
      "GS-03": {
        "title": "Regulator-independent late-time trace-average condition",
        "display_order": 34,
        "sector": "global",
        "statement": {
          "public": "Direct late-time identification of the compact response with the asymptotic cosmological amplitude additionally requires the regulated non-vacuum trace average to vanish.",
          "technical": "For connected future-directed exhaustions M_τ with subextensive boundary/edge terms and regulator-independent limit, require ⟨T_nonvac⟩_reg=lim_{τ→∞}[∫_{Mτ}√−g T_nonvac]/[∫_{Mτ}√−g]=0. Future-eternal asymptotically de Sitter dilution satisfies this standard sufficient regime."
        },
        "status": "amplitude_condition",
        "type": "amplitude_condition",
        "depends_on": [
          "GS-02"
        ],
        "equation_ids": [
          "eq:traceaveragecondition",
          "eq:traceaveragedef"
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, late-time exhaustion/trace-average subsection.",
            "structure_id": "sec:globalsource"
          }
        ],
        "qualifier": "Amplitude condition. It fixes the late-time amplitude, not the equation of state, which follows from the spacetime-constant residual source.",
        "limits_and_open_issues": "Not asserted for every recollapsing or non-asymptotic cosmological history.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained ID but reclassified from selected branch to explicit amplitude condition in v2.0."
        }
      },
      "MAP-01": {
        "title": "Compact response to cosmological curvature map",
        "display_order": 35,
        "sector": "global",
        "statement": {
          "public": "The proposed gravity-side completion identifies the calculated compact response with the residual cosmological curvature; the late-time amplitude comparison additionally uses GS-03.",
          "technical": "With q=q_comp inserted into the proposed action, GS-02 gives Λ/M_N²=q on a maximally symmetric vacuum. Matching to the measured branch Newton unit yields the residual curvature map; direct asymptotic identification Λ_∞/M_P²=q_comp additionally assumes GS-03."
        },
        "status": "proposed_cosmological_identification",
        "type": "proposed_model",
        "depends_on": [
          "NUM-01",
          "GS-02"
        ],
        "equation_ids": [
          "eq:asymptoticq",
          "eq:vacuumq"
        ],
        "evidence": [
          {
            "reference_id": "REF-028",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-030",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-031",
            "role": "structural_convergence"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, global-source completion and asymptotic relation.",
            "structure_id": "sec:globalsource"
          }
        ],
        "qualifier": "Separate from the compact calculation. The identification inherits the proposed action and unit pairing; the observational late-time amplitude inherits GS-03.",
        "limits_and_open_issues": "A different microscopic global source law or pairing would change the cosmological reading without undoing NUM-01.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained ID; v2.0 cleanly separates compact response from curvature map."
        }
      },
      "OBS-01": {
        "title": "Planck 2018 flat-ΛCDM inferred benchmark",
        "display_order": 36,
        "sector": "observations",
        "statement": {
          "public": "Planck 2018 flat-ΛCDM-inferred value: (2.846±0.057)×10⁻¹²² in the paper’s GΛ convention.",
          "technical": "The comparison benchmark is (Λ/M_P²)_obs=(2.846±0.057)×10⁻¹²² inferred from Planck 2018 under flat ΛCDM."
        },
        "status": "observational_input",
        "type": "observational_input",
        "depends_on": [],
        "evidence": [
          {
            "reference_id": "REF-003",
            "role": "observational_input"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, abstract/numerical comparison.",
            "structure_id": "loc-032"
          }
        ],
        "qualifier": "Model-dependent cosmological inference, not a direct measurement of an asymptotic constant and not a theory error bar.",
        "limits_and_open_issues": "If evolving dark energy is established, this flat-ΛCDM inference is not the direct asymptotic quantity predicted by the completion.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained from v1."
        }
      },
      "OBS-02": {
        "title": "Numerical comparison with the Planck benchmark",
        "display_order": 37,
        "sector": "observations",
        "statement": {
          "public": "Under the proposed cosmological identification and late-time amplitude condition, the calculated central value differs from the Planck 2018 flat-ΛCDM inference by about 0.4%.",
          "technical": "The leading value 2.85652154×10⁻¹²² differs from 2.846×10⁻¹²² by about 0.4%, equal to about 0.18 times the quoted observational standard deviation. The 0.18 figure is not a combined significance because theory uncertainty is not assigned."
        },
        "status": "derived_comparison",
        "type": "comparison",
        "depends_on": [
          "NUM-01",
          "MAP-01",
          "GS-03",
          "OBS-01"
        ],
        "equation_ids": [
          "eq:mainresult"
        ],
        "evidence": [
          {
            "reference_id": "REF-003",
            "role": "observational_input"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, global-source comparison and correction-budget section.",
            "structure_id": "sec:errorbudget"
          }
        ],
        "qualifier": "Plain benchmark comparison, not a likelihood ratio or sigma-level theory confirmation.",
        "limits_and_open_issues": "Open a_eff and gravitational corrections determine how the benchmark ultimately moves.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained ID; dependency chain updated to v2.0."
        }
      },
      "CORR-01": {
        "title": "Full interacting curved-space Standard Model coefficient",
        "display_order": 38,
        "sector": "corrections",
        "statement": {
          "public": "The full interacting curved-space Standard Model coefficient a_eff in the precise compact channel remains to be calculated.",
          "technical": "The leading free-field value a_SM is known, and local-RG/interacting-QFT results constrain the structure of corrections, but the paper does not assign a numerical uncertainty to a_eff without a complete curved-space Standard Model calculation including running, beta-function/operator mixing and the relevant compact projection."
        },
        "status": "open_calculation",
        "type": "open_calculation",
        "depends_on": [
          "EXT-01",
          "SM-01"
        ],
        "equation_ids": [
          "eq:aSMvalue"
        ],
        "evidence": [
          {
            "reference_id": "REF-012",
            "role": "foundation"
          },
          {
            "reference_id": "REF-026",
            "role": "foundation"
          },
          {
            "reference_id": "REF-037",
            "role": "foundation"
          },
          {
            "reference_id": "REF-049",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Standard Model evaluation and correction budget.",
            "structure_id": "sec:anomaly"
          }
        ],
        "qualifier": "Open matter calculation; no numerical theory uncertainty is assigned without the complete curved-space Standard Model calculation.",
        "limits_and_open_issues": "No theory error bar is inferred from partial perturbative information in the released paper.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained ID but completely revised in v2.0; old 0.5% estimate removed."
        }
      },
      "CORR-02": {
        "title": "Quantum gravitational dressing and higher-curvature action shift",
        "display_order": 39,
        "sector": "corrections",
        "statement": {
          "public": "The principal open gravity calculation is the complete source-dependent gravitational dressing and higher-curvature shift of the primitive compact action beyond the Einstein-Hilbert benchmark.",
          "technical": "The strict leading benchmark is the classical Einstein-Hilbert truncation δB=0. This does not assert a universal canonical determinant value C_grav=1. Source-dependent determinants, higher-curvature contributions changing the Planck-boundary action, Stokes changes or nonperturbative changes of the selected compact sector can shift the complete primitive exponent; these effects are not fixed by DET-01."
        },
        "status": "open_calculation",
        "type": "open_calculation",
        "depends_on": [
          "SAD-01",
          "DET-01"
        ],
        "equation_ids": [
          "eq:app_detcts",
          "eq:app_rawdet"
        ],
        "evidence": [
          {
            "reference_id": "REF-061",
            "role": "foundation"
          },
          {
            "reference_id": "REF-091",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-092",
            "role": "structural_convergence"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, determinant/higher-curvature status and correction budget.",
            "structure_id": "loc-029"
          }
        ],
        "qualifier": "The leading benchmark sets the open net correction δB to zero at Einstein-Hilbert order; the physical quantum/higher-curvature shift remains to be calculated.",
        "limits_and_open_issues": "Their size is not inferred from the 0.4% central-value agreement.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained ID but rebased from the old paired-cap cancellation framework."
        }
      },
      "CORR-03": {
        "title": "Gravity-only δB tolerance",
        "display_order": 40,
        "sector": "corrections",
        "statement": {
          "public": "At fixed a_eff=a_SM, the Planck benchmark allows only a few×10⁻² shift in the complete primitive action.",
          "technical": "For q(δB)=q₀e^{−δB} at a_eff=a_SM, remaining within the quoted Planck 2018 1σ interval requires −0.0161≲δB≲0.0239; the 2σ interval is −0.0356≲δB≲0.0446."
        },
        "status": "derived_sensitivity_bound",
        "type": "test",
        "depends_on": [
          "NUM-01",
          "OBS-01"
        ],
        "equation_ids": [
          "eq:dB1sigma",
          "eq:dB2sigma"
        ],
        "evidence": [
          {
            "reference_id": "REF-003",
            "role": "observational_input"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Eqs. for δB 1σ and 2σ intervals.",
            "structure_id": "sec:errorbudget"
          }
        ],
        "qualifier": "Gravity-only slice of the joint open correction space, not a prior distribution or fitted error bar.",
        "limits_and_open_issues": "Matter and gravity corrections can both move the benchmark; the general combination is CORR-04.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0."
        }
      },
      "CORR-04": {
        "title": "Joint matter-gravity correction relation",
        "display_order": 41,
        "sector": "corrections",
        "statement": {
          "public": "The leading joint correction depends on the combination δB−ln(a_eff/a_SM), not on two independently fitted offsets.",
          "technical": "Relative to the leading benchmark, q/q₀=(a_eff/a_SM)e^{−δB}; equivalently ln(q/q₀)=ln(a_eff/a_SM)−δB. The constant-q contours are sensitivity relations between two independently open calculations, not a fit space."
        },
        "status": "derived_correction_relation",
        "type": "identity",
        "depends_on": [
          "NUM-01",
          "CORR-01",
          "CORR-02"
        ],
        "equation_ids": [
          "eq:joint_correction"
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, correction-budget paragraph on joint matter-gravity space.",
            "structure_id": "sec:errorbudget"
          }
        ],
        "qualifier": "Sensitivity relation only. The open calculations determine the point; observation does not select it.",
        "limits_and_open_issues": "No cancellation between open sectors is assumed or encouraged.",
        "history": {
          "introduced_in": "v2.0",
          "public_version_history": "Introduced in v2.0."
        }
      },
      "TEST-01": {
        "title": "Exact w=-1 observational falsifier of the completion",
        "display_order": 42,
        "sector": "tests",
        "statement": {
          "public": "The proposed completion gives a spacetime-constant residual source with w=-1; persistent cross-dataset, systematics-robust evidence for evolving dark energy would falsify that cosmological identification while leaving the compact extraction intact.",
          "technical": "On any fixed branch of GS-02 the residual source is proportional to g_{μν} and has w=-1. DESI/DES evidence for evolution is currently dataset dependent; decisive robust evolution would falsify the proposed global-source identification, not EXT-01 or NUM-01 as compact statements."
        },
        "status": "test",
        "type": "test",
        "depends_on": [
          "GS-02"
        ],
        "equation_ids": [
          "eq:completedEinstein",
          "eq:residuallambda"
        ],
        "evidence": [
          {
            "reference_id": "REF-042",
            "role": "observational_input"
          },
          {
            "reference_id": "REF-043",
            "role": "observational_input"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, observational-tests paragraph.",
            "structure_id": "loc-043"
          }
        ],
        "qualifier": "Falsifier of the proposed completion/observational identification, not of the upstream compact theorem.",
        "limits_and_open_issues": "A preference under one dataset combination/parameterisation is not by itself the stated falsification threshold.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained from v1; status and separation from GS-03 preserved."
        }
      },
      "TEST-02": {
        "title": "Microscopic field-content sensitivity",
        "display_order": 43,
        "sector": "tests",
        "statement": {
          "public": "Changes in microscopic field content can move the benchmark through a_eff and, if they alter the spin-to-QCD implementation, through new finite interface terms; that sensitivity remains qualitative until the relevant calculations are done.",
          "technical": "New microscopic degrees of freedom change the protected matter coordinate through the full a_eff calculation, and may also change a microscopic realization of QCD-03 or introduce finite cap/interface terms that belong in δB. The paper does not publish a percentage exclusion table as a current result."
        },
        "status": "test",
        "type": "test",
        "depends_on": [
          "SM-01",
          "CORR-01",
          "CORR-02",
          "QCD-03"
        ],
        "evidence": [
          {
            "reference_id": "REF-004",
            "role": "observational_input"
          },
          {
            "reference_id": "REF-012",
            "role": "foundation"
          },
          {
            "reference_id": "REF-026",
            "role": "foundation"
          },
          {
            "reference_id": "REF-049",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, final paragraph of correction budget/observational tests.",
            "structure_id": "loc-043"
          }
        ],
        "qualifier": "Conditional field-content test, not a present BSM exclusion.",
        "limits_and_open_issues": "Quantitative exclusion requires the full matter and gravity calculations.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained ID but revised in v2.0; legacy percentage scenarios removed from canonical copy."
        }
      },
      "TEST-03": {
        "title": "Primitive-action/UV-boundary precision test",
        "display_order": 44,
        "sector": "tests",
        "statement": {
          "public": "The strict gravitational boundary is sharply testable because any microscopic change of the primitive action moves the prediction exponentially.",
          "technical": "A parent gravitational theory must reproduce, replace or falsify the boundary-member prescription, exact u_max=1 UV-domain normalization and decaying compact contour, and control the complete primitive exponent within the CORR-03 budget if the leading Planck comparison is to remain near its quoted interval. Larger shifts change or falsify the leading benchmark rather than becoming fit parameters."
        },
        "status": "test",
        "type": "test",
        "depends_on": [
          "SAD-04",
          "SAD-07",
          "CORR-02",
          "CORR-03"
        ],
        "equation_ids": [
          "eq:dB1sigma",
          "eq:dB2sigma"
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, “Strict compact gravitational channel” and observational tests.",
            "structure_id": "loc-040"
          }
        ],
        "qualifier": "Test of the selected UV model and quantum completion, not a fitted uncertainty.",
        "limits_and_open_issues": "Alternative boundary conventions or higher-curvature actions are different microscopic outcomes, not statistical variations of one fixed model.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained ID but recast around the v2.0 δB budget."
        }
      },
      "INT-01": {
        "title": "Residual-curvature interpretation",
        "display_order": 45,
        "sector": "interpretation",
        "statement": {
          "public": "In the completed model, the cosmological constant is the residual vacuum curvature fixed by the global-source equation, not a separately observable absolute zero-point-energy sum.",
          "technical": "Given GS-01/GS-04 and the derived GS-02 field equations, the compact response enters the rigid scalar source while homogeneous matter shifts cancel from the trace-free local equation. The resulting residual vacuum term is metric proportional."
        },
        "status": "derived_within_proposed_completion",
        "type": "interpretation",
        "depends_on": [
          "GS-02",
          "MAP-01"
        ],
        "equation_ids": [
          "eq:completedEinstein",
          "eq:residuallambda"
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, global-source section and Discussion.",
            "structure_id": "sec:globalsource"
          }
        ],
        "qualifier": "Interpretation of the proposed completion; not required for the extraction theorem.",
        "limits_and_open_issues": "Inherits Proposed status through MAP-01.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained from v1, updated to the v2.0 action."
        }
      },
      "INT-02": {
        "title": "103+19 decade decomposition",
        "display_order": 46,
        "sector": "interpretation",
        "statement": {
          "public": "The 122-decade hierarchy separates into about 103 decades from compact gravity and 19 from the QCD-to-Planck ratio, multiplied by an order-one Standard Model coefficient.",
          "technical": "From NUM-01, −log10(e^{−24π²})≈102.87 and −log10(m_p/M_P)≈19.11; a_SM is O(1). The hierarchy is multiplicative rather than a cancellation among large vacuum-energy terms."
        },
        "status": "derived_arithmetic",
        "type": "interpretation",
        "depends_on": [
          "NUM-01"
        ],
        "equation_ids": [
          "eq:qcomp_numeric_audit"
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, numerical audit discussion.",
            "structure_id": "loc-032"
          }
        ],
        "qualifier": "Arithmetic decomposition of the leading strict result.",
        "limits_and_open_issues": "Inherits all upstream conditions of NUM-01.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained from v1."
        }
      },
      "INT-03": {
        "title": "Combined-exponent rewriting",
        "display_order": 47,
        "sector": "interpretation",
        "statement": {
          "public": "The two suppressions can be rewritten as a single exponential of the sum of two distinct exponents.",
          "technical": "e^{−24π²}(m_p/M_P)=exp[−24π²−ln(M_P/m_p)]. This is an algebraic rewriting of the gravitational and QCD hierarchies, not evidence that they arise from one thermal or microscopic process."
        },
        "status": "derived_rewriting",
        "type": "interpretation",
        "depends_on": [
          "NUM-01"
        ],
        "equation_ids": [
          "eq:qcomp_general"
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, numerical audit/interpretive discussion.",
            "structure_id": "loc-032"
          }
        ],
        "qualifier": "Rewriting only.",
        "limits_and_open_issues": "Does not establish a common dynamical origin of the exponents.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained from v1 as a non-load-bearing interpretation."
        }
      },
      "INT-04": {
        "title": "Compact action-de Sitter entropy identity",
        "display_order": 48,
        "sector": "interpretation",
        "statement": {
          "public": "For the round compact saddle, the magnitude of the classical Euclidean action equals its de Sitter entropy; with the selected damped orientation the weight is e^(−S_dS).",
          "technical": "For the round de Sitter saddle, |S_EH|=3πM_P²/Λ=S_dS. Under SAD-02 and the selected SAD-04/SAD-07 endpoint-member prescription, S_dS=24π² and the classical weight is e^{−S_dS}=e^{−24π²}."
        },
        "status": "standard_identity_applied_to_selected_sector",
        "type": "identity",
        "depends_on": [
          "SAD-01",
          "SAD-02",
          "SAD-04",
          "SAD-07"
        ],
        "equation_ids": [
          "eq:BandS",
          "eq:instanton_value"
        ],
        "evidence": [
          {
            "reference_id": "REF-038",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, action-entropy identity in compact gravity section.",
            "structure_id": "loc-024"
          }
        ],
        "qualifier": "Classical on-shell identity applied to the selected sector, not an all-orders quantum-gravity identity.",
        "limits_and_open_issues": "The entropy belongs to the ultraviolet compact saddle, not the observed late-time horizon.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained from v1."
        }
      },
      "CMP-01": {
        "title": "Specified compact channel closes the leading numerical chain",
        "display_order": 49,
        "sector": "comparison",
        "statement": {
          "public": "The construction closes the leading compact numerical chain while isolating, rather than hiding, the premise and gravitational selections on which it depends.",
          "technical": "The chain supplies a protected matter coordinate, a derived primitive geometric unit, an explicit QCD state-preparation premise with downstream proton response, a specified compact gravitational factor, and a defined factorized compact observable. The cosmological curvature map remains a separate proposed completion."
        },
        "status": "comparative_synthesis",
        "type": "comparison",
        "depends_on": [
          "VAC-02",
          "EXT-01",
          "TOP-01",
          "QCD-03",
          "SAD-01",
          "NUM-01"
        ],
        "equation_ids": [
          "eq:qcomp_result"
        ],
        "evidence": [
          {
            "reference_id": "REF-005",
            "role": "foundation"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Introduction, Discussion and Conclusion.",
            "structure_id": "sec:intro"
          }
        ],
        "qualifier": "Synthesis, not a claim of unique microscopic selection or completed first-principles derivation.",
        "limits_and_open_issues": "Five remaining programme targets are exposed explicitly.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained ID; v2.0 wording reflects the new architecture."
        }
      },
      "CMP-02": {
        "title": "Relation to trace-free and sequestering approaches",
        "display_order": 50,
        "sector": "comparison",
        "statement": {
          "public": "The compact response acts as a candidate selector for the scalar residual that trace-free and sequestering formulations leave undetermined.",
          "technical": "GR-02 gives the trace-free local equation and an undetermined scalar mode; within the proposed GS-01/GS-04 completion, GS-02 assigns that mode through q_comp. This is structurally close to global-variable/sequestering approaches but uses a different source value."
        },
        "status": "comparative_synthesis",
        "type": "comparison",
        "depends_on": [
          "GR-02",
          "GS-02",
          "MAP-01"
        ],
        "equation_ids": [
          "eq:completedEinstein",
          "eq:tracefree",
          "eq:vacuumq"
        ],
        "evidence": [
          {
            "reference_id": "REF-028",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-030",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-032",
            "role": "structural_convergence"
          }
        ],
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Discussion “Independent structural convergence”.",
            "structure_id": "loc-045"
          }
        ],
        "qualifier": "Comparison conditional on the proposed completion.",
        "limits_and_open_issues": "Does not establish that existing sequestering models derive q_comp or the unit pairing.",
        "history": {
          "introduced_in": "v1",
          "public_version_history": "Retained from v1, updated to v2.0."
        }
      },
      "CMP-03": {
        "title": "Microscopic parent-theory derivation-or-falsification test",
        "display_order": 51,
        "sector": "comparison",
        "statement": {
          "public": "The construction exposes a finite set of quantities for a microscopic parent theory to compute; the parent can derive the channel or rule it out.",
          "technical": "A microscopic completion must address the full interacting a_eff; derive or replace QCD-03; derive, replace or falsify the boundary-member prescription, exact u_max=1 UV-domain normalization and decaying compact contour; compute source-dependent quantum/higher-curvature corrections within the δB budget; and derive the global source law/unit pairing together with the late-time amplitude condition. Different results change or falsify the channel rather than provide fit freedom."
        },
        "status": "test",
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        }
      },
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        "title": "Independent support and structural convergence",
        "display_order": 52,
        "sector": "comparison",
        "statement": {
          "public": "Several pieces of the architecture have independent support or close structural precedents, but none of those literature relations is a dependency of the numerical result.",
          "technical": "Direct architectural support includes leading-order S⁴ dominance in Λ>0 gravitational EFT, primitive-SU(2)/inverse-cosmological-coupling structure in exact canonical gravity and an explicit de Sitter Lefschetz thimble. Broader convergence includes sequestering/global-variable, Euler/four-form, pregeometry/topological-Λ and gravitational-instanton/SM-charge constructions. All are encoded as non-dependency relations."
        },
        "status": "comparison_non_dependency",
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        ],
        "qualifier": "Comparative context only; cannot upgrade Derived/Selected/Premise/Proposed/Open manuscript statuses.",
        "limits_and_open_issues": "Structural convergence is not validation of the numerical cosmological result.",
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      "METH-01": {
        "title": "Human-AI research-method disclosure",
        "display_order": 53,
        "sector": "method",
        "statement": {
          "public": "The research was developed through human-AI collaboration; the author remains responsible for every published claim.",
          "technical": "AI systems contributed synthesis, mathematical exploration, consistency testing, literature mapping and rapid iteration. Human judgement set research direction, assumptions, interpretations and publication decisions; AI participation is not scientific evidence for the claims."
        },
        "status": "author_disclosure",
        "type": "author_disclosure",
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          {
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        "limits_and_open_issues": "Operational prompts and internal transcripts are not part of the scientific claim graph.",
        "history": {
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    },
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        "display_order": 2,
        "sector": "qcd"
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        "name": "Strict-channel round-S4 action exponent at u=1",
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          "equation_ids": [
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        "display_order": 3,
        "sector": "gravity"
      },
      "Q-exp-minus-B1": {
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        "sector": "gravity"
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        "name": "Leading compact response",
        "symbol_latex": "q_{\\rm comp}",
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        "calculation_id": "CALC-CC-001",
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        "sector": "assembly"
      },
      "Q-planck2018-lambda": {
        "name": "Planck 2018 flat-LambdaCDM comparison value",
        "symbol_latex": "(\\Lambda/M_P^2)_{\\rm Planck\\ 2018}",
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        "value": {
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        "sector": "observations"
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        "latex": "\\Gamma[g] \\;\\to\\; \\Gamma[g]\n  &+ \\alpha\\!\\int\\!d^4x\\sqrt{g}\n  + \\beta\\!\\int\\!d^4x\\sqrt{g}\\,R \\nonumber\\\\\n  &+ \\gamma\\!\\int\\!d^4x\\sqrt{g}\\,R^2\n  + \\delta\\!\\int\\!d^4x\\sqrt{g}\\,W^2\n  + \\epsilon\\!\\int\\!d^4x\\sqrt{g}\\,E_4\n  + \\cdots\\,,",
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          {
            "symbol": "Γ[g]",
            "definition": "renormalized effective action (dimensionless in ℏ=c=1 conventions)"
          },
          {
            "symbol": "E₄",
            "definition": "Euler density (mass dimension 4)"
          },
          {
            "symbol": "W",
            "definition": "Weyl tensor / W² invariant"
          }
        ],
        "depends_on_equations": [],
        "status": "standard_input",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Extraction theorem on conformally flat S⁴”, paragraph “Local counterterm ambiguities.”",
            "section_title": "Extraction theorem on conformally flat S⁴",
            "paragraph_title": "Local counterterm ambiguities."
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        ],
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      },
      "eq:calA": {
        "title": "Euler extraction projector",
        "display_order": 2,
        "latex": "\\mathcal A[\\Gamma]\n=-\\frac14\\Pi_{H^0}\\!\\left(\\frac{d\\Gamma(H)}{d\\ln H}\\right).",
        "paper_number": 2,
        "variables": [
          {
            "symbol": "Γ[g]",
            "definition": "renormalized effective action (dimensionless in ℏ=c=1 conventions)"
          },
          {
            "symbol": "H",
            "definition": "inverse sphere radius / curvature scale (mass dimension 1)"
          },
          {
            "symbol": "a",
            "definition": "type-A Euler anomaly coefficient (dimensionless)"
          }
        ],
        "depends_on_equations": [
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          "eq:anomaly",
          "eq:weylvar"
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        "status": "derived",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Extraction theorem on conformally flat S⁴”, paragraph “Extraction criteria.”",
            "section_title": "Extraction theorem on conformally flat S⁴",
            "paragraph_title": "Extraction criteria."
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
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          {
            "reference_id": "REF-025",
            "role": "foundation"
          },
          {
            "reference_id": "REF-026",
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          {
            "reference_id": "REF-027",
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          {
            "reference_id": "REF-033",
            "role": "foundation"
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          {
            "reference_id": "REF-049",
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        ]
      },
      "eq:weylvar": {
        "title": "Constant-Weyl variation of the effective action",
        "display_order": 3,
        "latex": "\\delta_\\sigma\\Gamma[g]\n&=-\\int d^4x\\sqrt g\\,\\sigma\\langle T^\\mu{}_{\\mu}\\rangle\\nonumber\\\\\n&=+\\frac{a}{(4\\pi)^2}\\int d^4x\\sqrt g\\,\\sigma E_4",
        "paper_number": 3,
        "variables": [
          {
            "symbol": "Γ[g]",
            "definition": "renormalized effective action (dimensionless in ℏ=c=1 conventions)"
          },
          {
            "symbol": "σ",
            "definition": "constant Hamiltonian trace source (dimensionless)"
          },
          {
            "symbol": "a",
            "definition": "type-A Euler anomaly coefficient (dimensionless)"
          },
          {
            "symbol": "E₄",
            "definition": "Euler density (mass dimension 4)"
          }
        ],
        "depends_on_equations": [],
        "status": "derived",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Extraction theorem on conformally flat S⁴”, paragraph “Extraction criteria.”",
            "section_title": "Extraction theorem on conformally flat S⁴",
            "paragraph_title": "Extraction criteria."
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            "reference_id": "REF-033",
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      "eq:anomaly": {
        "title": "Four-dimensional trace anomaly",
        "display_order": 4,
        "latex": "\\langle T^\\mu{}_{\\mu}\\rangle\n=&\\;\\frac{c}{16\\pi^2}W_{\\mu\\nu\\rho\\sigma}W^{\\mu\\nu\\rho\\sigma}\n-\\frac{a}{16\\pi^2}E_4\n+(\\text{total derivatives}).",
        "paper_number": 4,
        "variables": [
          {
            "symbol": "a",
            "definition": "type-A Euler anomaly coefficient (dimensionless)"
          },
          {
            "symbol": "c",
            "definition": "type-B Weyl anomaly coefficient (dimensionless)"
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          {
            "symbol": "E₄",
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          {
            "symbol": "W",
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            "section_title": "Standard Model evaluation of the extracted datum"
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      },
      "eq:aSMvalue": {
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        "latex": "\\boxed{a_{\\mathrm{SM}}\n =\\frac4{360}+\\frac{45\\times11}{720}+\\frac{12\\times31}{180}\n =\\frac{1991}{720}=2.76528.}",
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            "section_title": "Standard Model evaluation of the extracted datum"
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            "reference_id": "REF-010",
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      "eq:Eulerunit": {
        "title": "Normalized Euler unit",
        "display_order": 6,
        "latex": "\\boxed{\\mathcal N_E\\equiv\\frac{1}{64\\pi^2}\n \\int_{S^4}\\sqrt g\\,E_4=\\frac{\\chi(S^4)}2=1.}",
        "paper_number": 6,
        "variables": [
          {
            "symbol": "𝒩_E",
            "definition": "normalized Euler unit (dimensionless)"
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          {
            "symbol": "E₄",
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        ],
        "depends_on_equations": [],
        "status": "derived",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “One intrinsic Euler–Chern unit on the round sphere”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
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        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-021",
            "role": "foundation"
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          {
            "reference_id": "REF-023",
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          {
            "reference_id": "REF-078",
            "role": "foundation"
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        ]
      },
      "eq:spinclasses": {
        "title": "Chiral spin-bundle characteristic-class identities",
        "display_order": 7,
        "latex": "e(TM)=c_2(\\Sigma^-)-c_2(\\Sigma^+),\\qquad\n p_1(TM)=-2\\bigl(c_2(\\Sigma^-)+c_2(\\Sigma^+)\\bigr).",
        "paper_number": 7,
        "variables": [
          {
            "symbol": "Σ±",
            "definition": "chiral spin bundles"
          }
        ],
        "depends_on_equations": [],
        "status": "standard_identity",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “One intrinsic Euler–Chern unit on the round sphere”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-021",
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          {
            "reference_id": "REF-023",
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          {
            "reference_id": "REF-078",
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        ]
      },
      "eq:spinChern": {
        "title": "Chiral spin-bundle Chern numbers on S⁴",
        "display_order": 8,
        "latex": "\\boxed{\\langle c_2(\\Sigma^-),[S^4]\\rangle=+1,\\qquad\n \\langle c_2(\\Sigma^+),[S^4]\\rangle=-1.}",
        "paper_number": 8,
        "variables": [
          {
            "symbol": "Σ±",
            "definition": "chiral spin bundles"
          }
        ],
        "depends_on_equations": [
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          "eq:Eulerunit"
        ],
        "status": "derived",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “One intrinsic Euler–Chern unit on the round sphere”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
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            "reference_id": "REF-021",
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            "reference_id": "REF-023",
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          {
            "reference_id": "REF-078",
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      },
      "eq:clutchmap": {
        "title": "Equatorial clutching map",
        "display_order": 9,
        "latex": "g_-(x)=x_4\\mathbf1+i x_j\\sigma_j,\n \\qquad x_4^2+\\sum_jx_j^2=1.",
        "paper_number": 9,
        "variables": [
          {
            "symbol": "g₋",
            "definition": "equatorial SU(2) clutching map"
          }
        ],
        "depends_on_equations": [],
        "status": "derived_representative",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “The equatorial clutching map has degree one”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-022",
            "role": "foundation"
          },
          {
            "reference_id": "REF-023",
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          }
        ]
      },
      "eq:clutchdegree": {
        "title": "Degree of the equatorial clutching map",
        "display_order": 10,
        "latex": "\\boxed{\\deg g_-\n =\\frac{1}{24\\pi^2}\\int_{S^3}\\operatorname{Tr}(g_-^{-1}\\mathrm d g_-)^3=1.}",
        "paper_number": 10,
        "variables": [
          {
            "symbol": "g₋",
            "definition": "equatorial SU(2) clutching map"
          }
        ],
        "depends_on_equations": [
          "eq:clutchmap",
          "eq:spinChern"
        ],
        "status": "derived",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “The equatorial clutching map has degree one”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
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            "reference_id": "REF-022",
            "role": "foundation"
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          {
            "reference_id": "REF-023",
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          }
        ]
      },
      "eq:intrinsicunit": {
        "title": "Intrinsic Euler-Chern-clutching identity",
        "display_order": 11,
        "latex": "\\boxed{\\mathcal N_E\n =\\langle c_2(\\Sigma^-),[S^4]\\rangle\n =\\deg g_-=1.}",
        "paper_number": 11,
        "variables": [
          {
            "symbol": "𝒩_E",
            "definition": "normalized Euler unit (dimensionless)"
          },
          {
            "symbol": "Σ±",
            "definition": "chiral spin bundles"
          },
          {
            "symbol": "g₋",
            "definition": "equatorial SU(2) clutching map"
          }
        ],
        "depends_on_equations": [
          "eq:Eulerunit",
          "eq:spinChern",
          "eq:clutchdegree"
        ],
        "status": "derived",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “The equatorial clutching map has degree one”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
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          {
            "reference_id": "REF-022",
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          {
            "reference_id": "REF-023",
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            "reference_id": "REF-078",
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          }
        ]
      },
      "eq:spinholonomydegree": {
        "title": "Explicit degree-one spin-holonomy representative",
        "display_order": 12,
        "latex": "\\boxed{\\deg U_{\\rm spin}\n =\\langle c_2(\\Sigma^-),[S^4]\\rangle\n =1,\\qquad [U_{\\rm spin}]=[g_-].}",
        "paper_number": 12,
        "variables": [
          {
            "symbol": "U_spin",
            "definition": "SU(2)_spin holonomy field on S³"
          },
          {
            "symbol": "Σ±",
            "definition": "chiral spin bundles"
          },
          {
            "symbol": "g₋",
            "definition": "equatorial SU(2) clutching map"
          }
        ],
        "depends_on_equations": [
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          "eq:clutchdegree"
        ],
        "status": "derived",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “The equatorial clutching map has degree one”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-082",
            "role": "direct_construction_input"
          },
          {
            "reference_id": "REF-085",
            "role": "foundation"
          }
        ]
      },
      "eq:skyrmionB": {
        "title": "QCD baryon winding",
        "display_order": 13,
        "latex": "B[U]=\\frac{1}{24\\pi^2}\\int_{S^3}\\operatorname{Tr}(U^{-1}\\mathrm d U)^3\\in\\mathbb Z,",
        "paper_number": 13,
        "variables": [
          {
            "symbol": "U",
            "definition": "two-flavour QCD chiral field"
          },
          {
            "symbol": "B",
            "definition": "baryon/skyrmion winding number (integer)"
          }
        ],
        "depends_on_equations": [],
        "status": "standard_qcd_identity",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “One class-level QCD boundary premise”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-079",
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          },
          {
            "reference_id": "REF-080",
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          {
            "reference_id": "REF-086",
            "role": "foundation"
          },
          {
            "reference_id": "REF-087",
            "role": "foundation"
          }
        ]
      },
      "eq:qcd_state_preparation": {
        "title": "QCD state-preparation class identification",
        "display_order": 14,
        "latex": "\\boxed{[U_{\\mathrm{IR}}]=\\varphi_*[U_{\\mathrm{spin}}]=\\varphi_*[g_-]\\in\\pi_3(SU(2)_{\\mathrm{flavour}})}",
        "variables": [
          {
            "symbol": "U",
            "definition": "two-flavour QCD chiral field"
          },
          {
            "symbol": "U_spin",
            "definition": "SU(2)_spin holonomy field on S³"
          },
          {
            "symbol": "g₋",
            "definition": "equatorial SU(2) clutching map"
          }
        ],
        "depends_on_equations": [
          "eq:spinholonomydegree",
          "eq:clutchdegree",
          "eq:skyrmionB"
        ],
        "status": "structural_premise",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “One class-level QCD boundary premise”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Stable equation record for a load-bearing displayed relation that is unnumbered in the formal manuscript.",
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            "reference_id": "REF-080",
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          {
            "reference_id": "REF-086",
            "role": "foundation"
          },
          {
            "reference_id": "REF-087",
            "role": "foundation"
          }
        ]
      },
      "eq:Bfromclutch": {
        "title": "Baryon number from the prepared primitive class",
        "display_order": 15,
        "latex": "\\boxed{B=+1,\\qquad B=-1\\ \\text{for the conjugate orientation}.}",
        "paper_number": 14,
        "variables": [
          {
            "symbol": "B",
            "definition": "baryon/skyrmion winding number (integer)"
          }
        ],
        "depends_on_equations": [
          "eq:qcd_state_preparation",
          "eq:clutchdegree",
          "eq:skyrmionB"
        ],
        "status": "derived_given_premise",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “One class-level QCD boundary premise”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-004",
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          },
          {
            "reference_id": "REF-072",
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          },
          {
            "reference_id": "REF-073",
            "role": "methodological_foundation"
          },
          {
            "reference_id": "REF-074",
            "role": "methodological_foundation"
          },
          {
            "reference_id": "REF-075",
            "role": "methodological_foundation"
          },
          {
            "reference_id": "REF-081",
            "role": "foundation"
          }
        ]
      },
      "eq:gaussedge": {
        "title": "Electric-flux edge label",
        "display_order": 16,
        "latex": "\\int_{\\partial\\mathcal R} *F=e,",
        "paper_number": 15,
        "variables": [
          {
            "symbol": "𝓡",
            "definition": "factorization region"
          },
          {
            "symbol": "∂𝓡 its boundary",
            "definition": ""
          },
          {
            "symbol": "F",
            "definition": "electromagnetic field strength"
          }
        ],
        "depends_on_equations": [],
        "status": "derived_given_premise",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “Gauge-consistent proton floor”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction."
      },
      "eq:chargedfloor": {
        "title": "Gauge-consistent B=1 spectral floor",
        "display_order": 17,
        "latex": "\\lim_{L_{\\rm IR}\\to\\infty}\\min_{\\mathcal E\\,\\mathrm{compatible}}\n \\left[E_{B=1,\\mathcal E}(L_{\\rm IR})-E_0(L_{\\rm IR})\\right]=m_p.",
        "paper_number": 16,
        "variables": [
          {
            "symbol": "L_IR",
            "definition": "charged-state infrared regulator length"
          },
          {
            "symbol": "B",
            "definition": "baryon/skyrmion winding number (integer)"
          },
          {
            "symbol": "m_p",
            "definition": "physical proton mass"
          }
        ],
        "depends_on_equations": [
          "eq:Bfromclutch",
          "eq:gaussedge"
        ],
        "status": "derived_given_premise",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “Gauge-consistent proton floor”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-004",
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          },
          {
            "reference_id": "REF-072",
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          },
          {
            "reference_id": "REF-073",
            "role": "methodological_foundation"
          },
          {
            "reference_id": "REF-074",
            "role": "methodological_foundation"
          },
          {
            "reference_id": "REF-075",
            "role": "methodological_foundation"
          },
          {
            "reference_id": "REF-081",
            "role": "foundation"
          }
        ]
      },
      "eq:tracesource": {
        "title": "Hamiltonian trace-source deformation",
        "display_order": 18,
        "latex": "H(\\sigma)=H(0)+\\sigma\\Theta+O(\\sigma^2),\n \\qquad\n \\Theta\\equiv\\int_{\\Sigma}\\sqrt h\\,T^\\mu{}_{\\mu}\\,d^3x,",
        "paper_number": 17,
        "variables": [
          {
            "symbol": "σ",
            "definition": "constant Hamiltonian trace source (dimensionless)"
          },
          {
            "symbol": "Θ",
            "definition": "spatial integral of T^μ_μ (energy dimension 1)"
          },
          {
            "symbol": "T_{μν}",
            "definition": "stress-energy tensor (mass dimension 4)"
          }
        ],
        "depends_on_equations": [],
        "status": "defined_source_normalization",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “Normalized Weyl slope and Newton-unit covariance”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-069",
            "role": "methodological_foundation"
          }
        ]
      },
      "eq:spectralCp": {
        "title": "Long-time proton correlator",
        "display_order": 19,
        "latex": "C_{p,\\mathcal E}(T,L_{\\rm IR};\\sigma)\n =Z_p(\\sigma)e^{-E_p(\\sigma)T}\\left[1+O(e^{-\\Delta T})\\right].",
        "paper_number": 18,
        "variables": [
          {
            "symbol": "C_p",
            "definition": "gauge-invariant Euclidean proton correlator"
          },
          {
            "symbol": "σ",
            "definition": "constant Hamiltonian trace source (dimensionless)"
          },
          {
            "symbol": "L_IR",
            "definition": "charged-state infrared regulator length"
          },
          {
            "symbol": "m_p",
            "definition": "physical proton mass"
          }
        ],
        "depends_on_equations": [
          "eq:chargedfloor",
          "eq:tracesource"
        ],
        "status": "derived_spectral_form",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “Normalized Weyl slope and Newton-unit covariance”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction."
      },
      "eq:weylslope": {
        "title": "Proton Weyl-slope theorem",
        "display_order": 20,
        "latex": "\\boxed{\n -\\lim_{T\\to\\infty}\\frac1T\n \\left.\\partial_\\sigma\\ln\n \\frac{C_{p,\\mathcal E}(T,L_{\\rm IR};\\sigma)}\n {C_{p,\\mathcal E}(T,L_{\\rm IR};0)}\\right|_{\\sigma=0}=m_p.}",
        "paper_number": 19,
        "variables": [
          {
            "symbol": "C_p",
            "definition": "gauge-invariant Euclidean proton correlator"
          },
          {
            "symbol": "σ",
            "definition": "constant Hamiltonian trace source (dimensionless)"
          },
          {
            "symbol": "L_IR",
            "definition": "charged-state infrared regulator length"
          },
          {
            "symbol": "m_p",
            "definition": "physical proton mass"
          }
        ],
        "depends_on_equations": [
          "eq:spectralCp",
          "eq:tracesource"
        ],
        "status": "derived_given_premise",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “Normalized Weyl slope and Newton-unit covariance”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-069",
            "role": "methodological_foundation"
          },
          {
            "reference_id": "REF-070",
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          },
          {
            "reference_id": "REF-076",
            "role": "foundation"
          },
          {
            "reference_id": "REF-077",
            "role": "foundation"
          }
        ]
      },
      "eq:dimensionlessslope": {
        "title": "Dimensionless proton response",
        "display_order": 21,
        "latex": "\\boxed{\n -\\lim_{s\\to\\infty}\\frac1s\n \\left.\\partial_\\sigma\\ln\\frac{C_p(s;\\sigma)}{C_p(s;0)}\\right|_{\\sigma=0}\n =m_p\\sqrt G=\\frac{m_p}{M_P},\n \\qquad s=M_PT.}",
        "paper_number": 20,
        "variables": [
          {
            "symbol": "C_p",
            "definition": "gauge-invariant Euclidean proton correlator"
          },
          {
            "symbol": "σ",
            "definition": "constant Hamiltonian trace source (dimensionless)"
          },
          {
            "symbol": "m_p",
            "definition": "physical proton mass"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          },
          {
            "symbol": "G",
            "definition": "Newton constant (mass dimension −2)"
          }
        ],
        "depends_on_equations": [
          "eq:weylslope"
        ],
        "status": "derived_given_premise",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “Normalized Weyl slope and Newton-unit covariance”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-069",
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          },
          {
            "reference_id": "REF-070",
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          },
          {
            "reference_id": "REF-076",
            "role": "foundation"
          },
          {
            "reference_id": "REF-077",
            "role": "foundation"
          }
        ]
      },
      "eq:Planckcovariance": {
        "title": "Newton/reduced-Planck normalization covariance",
        "display_order": 22,
        "latex": "\\frac{m_p}{M_P}=\\frac1{\\sqrt{8\\pi}}\\frac{m_p}{\\bar M_P},\n \\qquad\n G\\Lambda=\\frac1{8\\pi}\\frac{\\Lambda}{\\bar M_P^2}.",
        "paper_number": 21,
        "variables": [
          {
            "symbol": "m_p",
            "definition": "physical proton mass"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          },
          {
            "symbol": "M̄_P=κ^{-1}",
            "definition": "reduced Planck mass (mass dimension 1)"
          },
          {
            "symbol": "G",
            "definition": "Newton constant (mass dimension −2)"
          },
          {
            "symbol": "Λ",
            "definition": "geometric cosmological constant (mass dimension 2)"
          }
        ],
        "depends_on_equations": [],
        "status": "normalization_covariance",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “Normalized Weyl slope and Newton-unit covariance”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
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          {
            "reference_id": "REF-004",
            "role": "observational_input"
          }
        ]
      },
      "eq:qmatter": {
        "title": "Defined matter/IR composite",
        "display_order": 23,
        "latex": "\\boxed{q_{\\mathrm{matter}}^{(1)}\\equiv\\mathcal A_E\\mathcal Q_p\n =a_{\\mathrm{eff}}\\frac{m_p}{M_{\\mathrm P}}.}",
        "paper_number": 22,
        "variables": [
          {
            "symbol": "a_eff",
            "definition": "interacting compact-channel Euler coefficient (dimensionless)"
          },
          {
            "symbol": "m_p",
            "definition": "physical proton mass"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          }
        ],
        "depends_on_equations": [
          "eq:dimensionlessslope"
        ],
        "status": "defined_strict_channel_composite",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “Normalized matter/IR composite”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction."
      },
      "eq:hierarchy": {
        "title": "Proton-to-Planck hierarchy",
        "display_order": 24,
        "latex": "\\frac{m_p}{M_{\\mathrm P}}=7.685\\times10^{-20},",
        "paper_number": 23,
        "variables": [
          {
            "symbol": "m_p",
            "definition": "physical proton mass"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          }
        ],
        "depends_on_equations": [],
        "status": "standard_physical_input",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “Normalized matter/IR composite”",
            "section_title": "Intrinsic chiral-spin clutching and the QCD infrared response"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-004",
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          },
          {
            "reference_id": "REF-046",
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          },
          {
            "reference_id": "REF-047",
            "role": "foundation"
          }
        ]
      },
      "eq:SEH": {
        "title": "Round-S⁴ Einstein-Hilbert action",
        "display_order": 25,
        "latex": "S_{\\rm EH}\n =-\\frac{1}{2\\kappa^2}\\int_{S^4}(R-2\\Lambda)\\sqrt g\\,d^4x\n =-\\frac{24\\pi^2}{\\kappa^2\\Lambda}.",
        "paper_number": 24,
        "variables": [
          {
            "symbol": "S_EH",
            "definition": "Euclidean Einstein-Hilbert action (dimensionless)"
          },
          {
            "symbol": "Λ",
            "definition": "geometric cosmological constant (mass dimension 2)"
          }
        ],
        "depends_on_equations": [],
        "status": "derived",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact de Sitter saddle weight e^{-24π²}”, paragraph “Round-sphere action.”",
            "section_title": "The compact de Sitter saddle weight e^{-24π²}",
            "paragraph_title": "Round-sphere action."
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-021",
            "role": "foundation"
          },
          {
            "reference_id": "REF-024",
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          },
          {
            "reference_id": "REF-038",
            "role": "foundation"
          }
        ]
      },
      "eq:SEHNE": {
        "title": "Euler-normalized round-sphere action",
        "display_order": 26,
        "latex": "\\boxed{|S_{\\rm EH}|=\\frac{24\\pi^2}{u}\\,\\mathcal N_E.}",
        "paper_number": 25,
        "variables": [
          {
            "symbol": "S_EH",
            "definition": "Euclidean Einstein-Hilbert action (dimensionless)"
          },
          {
            "symbol": "u=κ²Λ",
            "definition": "dimensionless compact-saddle coordinate"
          },
          {
            "symbol": "𝒩_E",
            "definition": "normalized Euler unit (dimensionless)"
          }
        ],
        "depends_on_equations": [
          "eq:SEH",
          "eq:Eulerunit",
          "eq:u_coordinate"
        ],
        "status": "derived",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact de Sitter saddle weight e^{-24π²}”, paragraph “Round-sphere action.”",
            "section_title": "The compact de Sitter saddle weight e^{-24π²}",
            "paragraph_title": "Round-sphere action."
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-021",
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          },
          {
            "reference_id": "REF-024",
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          },
          {
            "reference_id": "REF-038",
            "role": "foundation"
          }
        ]
      },
      "eq:u_coordinate": {
        "title": "Reduced-Planck-density saddle coordinate",
        "display_order": 27,
        "latex": "u\\equiv\\kappa^2\\Lambda=\\frac{\\Lambda}{\\bar M_P^2}=\\frac{\\rho_\\Lambda}{\\bar M_P^4},\\qquad \\rho_\\Lambda\\equiv\\frac{\\Lambda}{\\kappa^2}",
        "variables": [
          {
            "symbol": "u=κ²Λ",
            "definition": "dimensionless compact-saddle coordinate"
          },
          {
            "symbol": "Λ",
            "definition": "geometric cosmological constant (mass dimension 2)"
          },
          {
            "symbol": "M̄_P=κ^{-1}",
            "definition": "reduced Planck mass (mass dimension 1)"
          },
          {
            "symbol": "ρ_Λ=Λ/κ²",
            "definition": "vacuum-energy density (mass dimension 4)"
          }
        ],
        "depends_on_equations": [],
        "status": "defined_saddle_coordinate",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact de Sitter saddle weight e^{-24π²}”, paragraph “Round-sphere action.”",
            "section_title": "The compact de Sitter saddle weight e^{-24π²}",
            "paragraph_title": "Round-sphere action."
          }
        ],
        "qualifier": "Stable equation record for a load-bearing displayed relation that is unnumbered in the formal manuscript."
      },
      "eq:u_domain": {
        "title": "Declared reduced-Planck-density UV domain",
        "display_order": 28,
        "latex": "0<\\rho_\\Lambda\\le\\bar M_P^4\\quad\\Longleftrightarrow\\quad 0<u\\le1",
        "variables": [
          {
            "symbol": "u=κ²Λ",
            "definition": "dimensionless compact-saddle coordinate"
          },
          {
            "symbol": "ρ_Λ=Λ/κ²",
            "definition": "vacuum-energy density (mass dimension 4)"
          },
          {
            "symbol": "M̄_P=κ^{-1}",
            "definition": "reduced Planck mass (mass dimension 1)"
          }
        ],
        "depends_on_equations": [
          "eq:u_coordinate"
        ],
        "status": "selected_uv_domain",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact de Sitter saddle weight e^{-24π²}”, paragraph “Canonical topological cross-check.”",
            "section_title": "The compact de Sitter saddle weight e^{-24π²}",
            "paragraph_title": "Canonical topological cross-check."
          }
        ],
        "qualifier": "Stable equation record for a load-bearing displayed relation that is unnumbered in the formal manuscript."
      },
      "eq:u_endpoint": {
        "title": "Exact UV-endpoint normalization",
        "display_order": 29,
        "latex": "\\boxed{u_{\\max}=1\\quad\\Longleftrightarrow\\quad\\rho_\\Lambda=\\bar M_P^4}",
        "variables": [
          {
            "symbol": "u=κ²Λ",
            "definition": "dimensionless compact-saddle coordinate"
          },
          {
            "symbol": "ρ_Λ=Λ/κ²",
            "definition": "vacuum-energy density (mass dimension 4)"
          },
          {
            "symbol": "M̄_P=κ^{-1}",
            "definition": "reduced Planck mass (mass dimension 1)"
          }
        ],
        "depends_on_equations": [
          "eq:u_domain"
        ],
        "status": "selected_uv_domain",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact de Sitter saddle weight e^{-24π²}”, paragraph “Canonical topological cross-check.”",
            "section_title": "The compact de Sitter saddle weight e^{-24π²}",
            "paragraph_title": "Canonical topological cross-check."
          }
        ],
        "qualifier": "Stable equation record for a load-bearing displayed relation that is unnumbered in the formal manuscript.",
        "evidence": [
          {
            "reference_id": "REF-006",
            "role": "foundation"
          },
          {
            "reference_id": "REF-008",
            "role": "foundation"
          }
        ]
      },
      "eq:ubone": {
        "title": "Boundary-member prescription at the selected UV endpoint",
        "display_order": 30,
        "latex": "\\boxed{u_b=u_{\\max}=1.}",
        "paper_number": 26,
        "variables": [
          {
            "symbol": "u=κ²Λ",
            "definition": "dimensionless compact-saddle coordinate"
          }
        ],
        "depends_on_equations": [
          "eq:SEHNE",
          "eq:u_domain",
          "eq:u_endpoint"
        ],
        "status": "selected_boundary_member_prescription_at_selected_uv_endpoint",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact de Sitter saddle weight e^{-24π²}”, paragraph “Canonical topological cross-check.”",
            "section_title": "The compact de Sitter saddle weight e^{-24π²}",
            "paragraph_title": "Canonical topological cross-check."
          }
        ],
        "qualifier": "The monotonic ordering is derived separately in SAD-03. The equality u_b=u_max is a selected single-member prescription, while u_max=1 is the selected UV-domain endpoint SAD-04.",
        "evidence": [
          {
            "reference_id": "REF-006",
            "role": "foundation"
          },
          {
            "reference_id": "REF-008",
            "role": "foundation"
          }
        ]
      },
      "eq:B1": {
        "title": "Primitive boundary action",
        "display_order": 31,
        "latex": "\\boxed{B_1\\equiv |S_{\\rm EH}|_{u_b=1}=24\\pi^2.}",
        "paper_number": 27,
        "variables": [
          {
            "symbol": "B₁",
            "definition": "positive primitive Euclidean action magnitude (dimensionless)"
          },
          {
            "symbol": "S_EH",
            "definition": "Euclidean Einstein-Hilbert action (dimensionless)"
          }
        ],
        "depends_on_equations": [
          "eq:SEHNE",
          "eq:ubone"
        ],
        "status": "derived_given_selected_boundary_member_and_uv_endpoint",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact de Sitter saddle weight e^{-24π²}”, paragraph “Canonical topological cross-check.”",
            "section_title": "The compact de Sitter saddle weight e^{-24π²}",
            "paragraph_title": "Canonical topological cross-check."
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-038",
            "role": "foundation"
          }
        ]
      },
      "eq:BandS": {
        "title": "de Sitter action-entropy identity",
        "display_order": 32,
        "latex": "B=S_{\\rm dS}=\\frac{3\\pi M_{\\mathrm P}^2}{\\Lambda},",
        "paper_number": 28,
        "variables": [
          {
            "symbol": "B₁",
            "definition": "positive primitive Euclidean action magnitude (dimensionless)"
          },
          {
            "symbol": "S_dS",
            "definition": "de Sitter entropy (dimensionless)"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          },
          {
            "symbol": "Λ",
            "definition": "geometric cosmological constant (mass dimension 2)"
          }
        ],
        "depends_on_equations": [
          "eq:SEH"
        ],
        "status": "derived_identity",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact de Sitter saddle weight e^{-24π²}”, paragraph “Canonical topological cross-check.”",
            "section_title": "The compact de Sitter saddle weight e^{-24π²}",
            "paragraph_title": "Canonical topological cross-check."
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-038",
            "role": "foundation"
          }
        ]
      },
      "eq:geometric_identity": {
        "title": "Chern-Gauss-Bonnet action identity",
        "display_order": 33,
        "latex": "|S_{\\rm EH}|=\\frac32\\,\\chi(M)\\,\\frac{8\\pi^2}{\\kappa^2\\Lambda}.",
        "paper_number": 29,
        "variables": [
          {
            "symbol": "S_EH",
            "definition": "Euclidean Einstein-Hilbert action (dimensionless)"
          },
          {
            "symbol": "Λ",
            "definition": "geometric cosmological constant (mass dimension 2)"
          }
        ],
        "depends_on_equations": [
          "eq:SEH",
          "eq:Eulerunit"
        ],
        "status": "derived_identity",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact de Sitter saddle weight e^{-24π²}”, paragraph “Canonical topological cross-check.”",
            "section_title": "The compact de Sitter saddle weight e^{-24π²}",
            "paragraph_title": "Canonical topological cross-check."
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-021",
            "role": "foundation"
          },
          {
            "reference_id": "REF-024",
            "role": "foundation"
          },
          {
            "reference_id": "REF-038",
            "role": "foundation"
          }
        ]
      },
      "eq:instanton_value": {
        "title": "Classical strict-channel gravitational factor",
        "display_order": 34,
        "latex": "\\boxed{e^{-B_1}=e^{-24\\pi^2}=1.344\\times10^{-103}.}",
        "paper_number": 30,
        "variables": [
          {
            "symbol": "B₁",
            "definition": "positive primitive Euclidean action magnitude (dimensionless)"
          }
        ],
        "depends_on_equations": [
          "eq:B1"
        ],
        "status": "derived_given_boundary_member_uv_domain_and_contour_selections",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact de Sitter saddle weight e^{-24π²}”, paragraph “Classical strict-channel factor.”",
            "section_title": "The compact de Sitter saddle weight e^{-24π²}",
            "paragraph_title": "Classical strict-channel factor."
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-038",
            "role": "foundation"
          },
          {
            "reference_id": "REF-039",
            "role": "foundation"
          },
          {
            "reference_id": "REF-040",
            "role": "foundation"
          },
          {
            "reference_id": "REF-053",
            "role": "foundation"
          },
          {
            "reference_id": "REF-065",
            "role": "foundation"
          }
        ]
      },
      "eq:qcomp_general": {
        "title": "Defined compact-channel composite",
        "display_order": 35,
        "latex": "\\boxed{q_{\\mathrm{comp}}\\equiv q_{\\mathrm{matter}}^{(1)} e^{-24\\pi^2}\n =a_{\\mathrm{eff}}\\frac{m_p}{M_{\\mathrm P}}e^{-24\\pi^2}.}",
        "paper_number": 31,
        "variables": [
          {
            "symbol": "q_comp",
            "definition": "defined compact response (dimensionless)"
          },
          {
            "symbol": "a_eff",
            "definition": "interacting compact-channel Euler coefficient (dimensionless)"
          },
          {
            "symbol": "m_p",
            "definition": "physical proton mass"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          }
        ],
        "depends_on_equations": [
          "eq:qmatter",
          "eq:instanton_value"
        ],
        "status": "defined_strict_channel_composite",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Leading compact-channel result”",
            "section_title": "Leading compact-channel result"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction."
      },
      "eq:qcomp_result": {
        "title": "Leading strict-channel compact benchmark",
        "display_order": 36,
        "latex": "\\boxed{\n q_{\\mathrm{comp}}=a_{\\mathrm{SM}}\\frac{m_p}{M_{\\mathrm P}}e^{-24\\pi^2}\n =2.856\\times10^{-122}.}",
        "paper_number": 32,
        "variables": [
          {
            "symbol": "q_comp",
            "definition": "defined compact response (dimensionless)"
          },
          {
            "symbol": "a",
            "definition": "type-A Euler anomaly coefficient (dimensionless)"
          },
          {
            "symbol": "m_p",
            "definition": "physical proton mass"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          }
        ],
        "depends_on_equations": [
          "eq:qcomp_general",
          "eq:aSMvalue",
          "eq:hierarchy"
        ],
        "status": "derived_given_stated_premise_selections",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Leading compact-channel result”, paragraph “Leading strict-channel benchmark.”",
            "section_title": "Leading compact-channel result",
            "paragraph_title": "Leading strict-channel benchmark."
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-004",
            "role": "observational_input"
          },
          {
            "reference_id": "REF-005",
            "role": "foundation"
          }
        ]
      },
      "eq:qcomp_numeric_audit": {
        "title": "Numerical audit of the compact benchmark",
        "display_order": 37,
        "latex": "a_{\\mathrm{SM}} &= \\frac{1991}{720}=2.76527778,\\\\\n \\frac{m_p}{M_{\\mathrm P}} &= 7.68514844\\times10^{-20},\\\\\n e^{-24\\pi^2} &= 1.34414611\\times10^{-103},\\\\[2pt]\n q_{\\mathrm{comp}}\n &= (2.76527778)(7.68514844\\times10^{-20})(1.34414611\\times10^{-103})\\\\\n &=2.85652154\\times10^{-122}.",
        "paper_number": 33,
        "variables": [
          {
            "symbol": "q_comp",
            "definition": "defined compact response (dimensionless)"
          },
          {
            "symbol": "a",
            "definition": "type-A Euler anomaly coefficient (dimensionless)"
          },
          {
            "symbol": "m_p",
            "definition": "physical proton mass"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          }
        ],
        "depends_on_equations": [
          "eq:aSMvalue",
          "eq:hierarchy",
          "eq:instanton_value",
          "eq:qcomp_result"
        ],
        "status": "computed_numerical_audit",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Leading compact-channel result”, paragraph “Numerical audit.”",
            "section_title": "Leading compact-channel result",
            "paragraph_title": "Numerical audit."
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-004",
            "role": "observational_input"
          }
        ]
      },
      "eq:H4obstruction": {
        "title": "Ordinary-GR anomaly obstruction",
        "display_order": 38,
        "latex": "\\Lambda_g-\\Lambda_0\n =\\frac{a}{3\\pi}\\frac{\\Lambda_g^2}{M_{\\mathrm P}^2}.",
        "paper_number": 34,
        "variables": [
          {
            "symbol": "Λ",
            "definition": "geometric cosmological constant (mass dimension 2)"
          },
          {
            "symbol": "a",
            "definition": "type-A Euler anomaly coefficient (dimensionless)"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          }
        ],
        "depends_on_equations": [
          "eq:anomaly"
        ],
        "status": "derived",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Ordinary-GR obstruction and the trace-free quotient”, paragraph “The ordinary-GR obstruction.”",
            "section_title": "Ordinary-GR obstruction and the trace-free quotient",
            "paragraph_title": "The ordinary-GR obstruction."
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-018",
            "role": "foundation"
          }
        ]
      },
      "eq:tracefree": {
        "title": "Trace-free quotient equation",
        "display_order": 39,
        "latex": "R_{\\mu\\nu}-\\frac14Rg_{\\mu\\nu}\n =\\frac{8\\pi}{M_{\\mathrm P}^2}\n \\left(T_{\\mu\\nu}-\\frac14Tg_{\\mu\\nu}\\right).",
        "paper_number": 35,
        "variables": [
          {
            "symbol": "T_{μν}",
            "definition": "stress-energy tensor (mass dimension 4)"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          }
        ],
        "depends_on_equations": [],
        "status": "derived",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Ordinary-GR obstruction and the trace-free quotient”, paragraph “The local equation on the volume-counterterm quotient.”",
            "section_title": "Ordinary-GR obstruction and the trace-free quotient",
            "paragraph_title": "The local equation on the volume-counterterm quotient."
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-028",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-029",
            "role": "foundation"
          },
          {
            "reference_id": "REF-030",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-032",
            "role": "structural_convergence"
          }
        ]
      },
      "eq:integrationconstant": {
        "title": "Scalar integration constant",
        "display_order": 40,
        "latex": "R+\\frac{8\\pi}{M_{\\mathrm P}^2}T=4\\Lambda_{\\rm int}.",
        "paper_number": 36,
        "variables": [
          {
            "symbol": "T_{μν}",
            "definition": "stress-energy tensor (mass dimension 4)"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          },
          {
            "symbol": "Λ",
            "definition": "geometric cosmological constant (mass dimension 2)"
          }
        ],
        "depends_on_equations": [
          "eq:tracefree"
        ],
        "status": "derived",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Ordinary-GR obstruction and the trace-free quotient”, paragraph “The local equation on the volume-counterterm quotient.”",
            "section_title": "Ordinary-GR obstruction and the trace-free quotient",
            "paragraph_title": "The local equation on the volume-counterterm quotient."
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-028",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-029",
            "role": "foundation"
          }
        ]
      },
      "eq:globalaction": {
        "title": "Proposed four-form global-source action",
        "display_order": 41,
        "latex": "\\boxed{\n S_{\\rm gs}[q]\n =\\int_M\\left[\n \\left(\\frac{\\eta M_{\\mathrm P}^2}{16\\pi}R-\\mathcal L_m\\right)\\star1\n +\\Lambda_b(F_4-\\star1)\n -\\frac{\\eta^2M_{\\mathrm P}^4}{8\\pi}qF_4\n \\right]\n +\\frac1{2\\pi}\\int_M C_\\eta\\wedge d\\eta.}",
        "paper_number": 37,
        "variables": [
          {
            "symbol": "η",
            "definition": "rigid dimensionless branch-Newton coordinate"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          },
          {
            "symbol": "q=GΛ",
            "definition": "dimensionless compact/cosmological response coordinate"
          },
          {
            "symbol": "F₄=dA₃",
            "definition": "four-form field strength"
          },
          {
            "symbol": "Λ_b",
            "definition": "four-form constraint multiplier"
          }
        ],
        "depends_on_equations": [],
        "status": "proposed_completion",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “From compact response to cosmological curvature: global-source completion”",
            "section_title": "From compact response to cosmological curvature: global-source completion"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-028",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-030",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-031",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-032",
            "role": "structural_convergence"
          }
        ]
      },
      "eq:source_covariance": {
        "title": "Branch-unit covariance of the source function",
        "display_order": 42,
        "latex": "f'(\\eta)=2c\\eta,\\qquad f(\\eta)=c\\eta^2+f_0",
        "variables": [
          {
            "symbol": "η",
            "definition": "rigid dimensionless branch-Newton coordinate"
          },
          {
            "symbol": "c",
            "definition": "overall compact-to-global pairing coefficient (dimensionless)"
          },
          {
            "symbol": "q=GΛ",
            "definition": "dimensionless compact/cosmological response coordinate"
          }
        ],
        "depends_on_equations": [
          "eq:globalaction"
        ],
        "status": "derived_from_branch_unit_covariance",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Appendix: “Off-shell variation of the global-source action”"
          }
        ],
        "qualifier": "Stable equation record for a load-bearing displayed relation that is unnumbered in the formal manuscript.",
        "evidence": [
          {
            "reference_id": "REF-028",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-030",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-031",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-032",
            "role": "structural_convergence"
          }
        ]
      },
      "eq:global_variations": {
        "title": "Off-shell variations of the global-source action",
        "display_order": 43,
        "latex": "\\delta\\Lambda_b:\\ F_4=\\star1;\\quad \\delta A_3:\\ d\\!\\left[\\Lambda_b-\\frac{\\eta^2M_P^4}{8\\pi}q\\right]=0;\\quad \\delta\\eta:\\ \\langle R\\rangle=4\\eta M_P^2q;\\quad \\delta g^{\\mu\\nu}:\\ \\frac{\\eta M_P^2}{8\\pi}G_{\\mu\\nu}+\\Lambda_b g_{\\mu\\nu}=T_{\\mu\\nu}",
        "variables": [
          {
            "symbol": "η",
            "definition": "rigid dimensionless branch-Newton coordinate"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          },
          {
            "symbol": "q=GΛ",
            "definition": "dimensionless compact/cosmological response coordinate"
          },
          {
            "symbol": "F₄=dA₃",
            "definition": "four-form field strength"
          },
          {
            "symbol": "Λ_b",
            "definition": "four-form constraint multiplier"
          },
          {
            "symbol": "T_{μν}",
            "definition": "stress-energy tensor (mass dimension 4)"
          }
        ],
        "depends_on_equations": [
          "eq:globalaction"
        ],
        "status": "derived_within_proposed_action",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Appendix: “Off-shell variation of the global-source action”"
          }
        ],
        "qualifier": "Stable equation record for a load-bearing displayed relation that is unnumbered in the formal manuscript."
      },
      "eq:compactsumrule": {
        "title": "Global curvature sum rule",
        "display_order": 44,
        "latex": "\\langle R\\rangle=4\\eta M_{\\mathrm P}^2q,",
        "paper_number": 38,
        "variables": [
          {
            "symbol": "η",
            "definition": "rigid dimensionless branch-Newton coordinate"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          },
          {
            "symbol": "q=GΛ",
            "definition": "dimensionless compact/cosmological response coordinate"
          }
        ],
        "depends_on_equations": [
          "eq:global_variations"
        ],
        "status": "derived_within_proposed_action",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “From compact response to cosmological curvature: global-source completion”",
            "section_title": "From compact response to cosmological curvature: global-source completion"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction."
      },
      "eq:completedEinstein": {
        "title": "Global-source Einstein equation",
        "display_order": 45,
        "latex": "\\boxed{\n G_{\\mu\\nu}+\\eta M_{\\mathrm P}^2q\\,g_{\\mu\\nu}\n =\\frac{8\\pi}{\\eta M_{\\mathrm P}^2}\n \\left(T_{\\mu\\nu}-\\frac14\\langle T\\rangle g_{\\mu\\nu}\\right).}",
        "paper_number": 39,
        "variables": [
          {
            "symbol": "η",
            "definition": "rigid dimensionless branch-Newton coordinate"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          },
          {
            "symbol": "q=GΛ",
            "definition": "dimensionless compact/cosmological response coordinate"
          },
          {
            "symbol": "T_{μν}",
            "definition": "stress-energy tensor (mass dimension 4)"
          }
        ],
        "depends_on_equations": [
          "eq:global_variations",
          "eq:compactsumrule"
        ],
        "status": "derived_once_action_and_unit_pairing_are_adopted",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “From compact response to cosmological curvature: global-source completion”",
            "section_title": "From compact response to cosmological curvature: global-source completion"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-028",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-030",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-032",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-042",
            "role": "observational_input"
          },
          {
            "reference_id": "REF-043",
            "role": "observational_input"
          }
        ]
      },
      "eq:vacuumq": {
        "title": "Vacuum-branch curvature map",
        "display_order": 46,
        "latex": "\\boxed{\\frac{\\Lambda}{M_N^2}=q,\\qquad M_N^2=\\eta M_{\\mathrm P}^2.}",
        "paper_number": 40,
        "variables": [
          {
            "symbol": "Λ",
            "definition": "geometric cosmological constant (mass dimension 2)"
          },
          {
            "symbol": "M_N²=ηM_P²",
            "definition": "branch Newton mass squared"
          },
          {
            "symbol": "η",
            "definition": "rigid dimensionless branch-Newton coordinate"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          },
          {
            "symbol": "q=GΛ",
            "definition": "dimensionless compact/cosmological response coordinate"
          }
        ],
        "depends_on_equations": [
          "eq:completedEinstein"
        ],
        "status": "derived_once_action_and_unit_pairing_are_adopted",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “From compact response to cosmological curvature: global-source completion”",
            "section_title": "From compact response to cosmological curvature: global-source completion"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-028",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-030",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-031",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-032",
            "role": "structural_convergence"
          }
        ]
      },
      "eq:residuallambda": {
        "title": "Residual cosmological term",
        "display_order": 47,
        "latex": "G_{\\mu\\nu}+\\Lambda_{\\rm res}g_{\\mu\\nu}\n =\\frac{8\\pi}{M_{\\mathrm P}^2}T_{\\mu\\nu},\n \\qquad\n \\Lambda_{\\rm res}=M_{\\mathrm P}^2q+\\frac{2\\pi}{M_{\\mathrm P}^2}\\langle T\\rangle.",
        "paper_number": 41,
        "variables": [
          {
            "symbol": "Λ_res",
            "definition": "residual cosmological term (mass dimension 2)"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          },
          {
            "symbol": "q=GΛ",
            "definition": "dimensionless compact/cosmological response coordinate"
          },
          {
            "symbol": "T_{μν}",
            "definition": "stress-energy tensor (mass dimension 4)"
          }
        ],
        "depends_on_equations": [
          "eq:completedEinstein"
        ],
        "status": "derived_within_proposed_completion",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “From compact response to cosmological curvature: global-source completion”",
            "section_title": "From compact response to cosmological curvature: global-source completion"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-042",
            "role": "observational_input"
          },
          {
            "reference_id": "REF-043",
            "role": "observational_input"
          }
        ]
      },
      "eq:traceaveragedef": {
        "title": "Regulated late-time trace average",
        "display_order": 48,
        "latex": "\\langle T_{\\rm nonvac}\\rangle_{\\rm reg}\n \\equiv\\lim_{\\tau\\to\\infty}\n \\frac{\\int_{M_\\tau}\\sqrt{-g}\\,T_{\\rm nonvac}\\,d^4x}\n      {\\int_{M_\\tau}\\sqrt{-g}\\,d^4x}.",
        "paper_number": 42,
        "variables": [
          {
            "symbol": "T_{μν}",
            "definition": "stress-energy tensor (mass dimension 4)"
          }
        ],
        "depends_on_equations": [],
        "status": "defined_regulator_prescription",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “From compact response to cosmological curvature: global-source completion”",
            "section_title": "From compact response to cosmological curvature: global-source completion"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction."
      },
      "eq:traceaveragecondition": {
        "title": "Late-time amplitude condition",
        "display_order": 49,
        "latex": "\\boxed{\\langle T_{\\rm nonvac}\\rangle_{\\rm reg}=0.}",
        "paper_number": 43,
        "variables": [
          {
            "symbol": "T_{μν}",
            "definition": "stress-energy tensor (mass dimension 4)"
          }
        ],
        "depends_on_equations": [
          "eq:traceaveragedef"
        ],
        "status": "amplitude_condition",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “From compact response to cosmological curvature: global-source completion”",
            "section_title": "From compact response to cosmological curvature: global-source completion"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction."
      },
      "eq:asymptoticq": {
        "title": "Asymptotic curvature identification",
        "display_order": 50,
        "latex": "\\frac{\\Lambda_\\infty}{M_{\\mathrm P}^2}=q_{\\mathrm{comp}}.",
        "paper_number": 44,
        "variables": [
          {
            "symbol": "Λ",
            "definition": "geometric cosmological constant (mass dimension 2)"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          },
          {
            "symbol": "q_comp",
            "definition": "defined compact response (dimensionless)"
          }
        ],
        "depends_on_equations": [
          "eq:vacuumq",
          "eq:traceaveragecondition",
          "eq:qcomp_result"
        ],
        "status": "derived_given_completion_and_amplitude_condition",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “From compact response to cosmological curvature: global-source completion”",
            "section_title": "From compact response to cosmological curvature: global-source completion"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-028",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-030",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-031",
            "role": "structural_convergence"
          }
        ]
      },
      "eq:mainresult": {
        "title": "Cosmological benchmark",
        "display_order": 51,
        "latex": "\\boxed{\n \\frac{\\Lambda_\\infty}{M_{\\mathrm P}^2}\n =a_{\\mathrm{SM}}\\frac{m_p}{M_{\\mathrm P}}e^{-24\\pi^2}\n =2.856\\times10^{-122}.}",
        "paper_number": 45,
        "variables": [
          {
            "symbol": "Λ",
            "definition": "geometric cosmological constant (mass dimension 2)"
          },
          {
            "symbol": "M_P=G^{-1/2}",
            "definition": "unreduced Newton mass (mass dimension 1)"
          },
          {
            "symbol": "a",
            "definition": "type-A Euler anomaly coefficient (dimensionless)"
          },
          {
            "symbol": "m_p",
            "definition": "physical proton mass"
          }
        ],
        "depends_on_equations": [
          "eq:asymptoticq",
          "eq:qcomp_result"
        ],
        "status": "derived_cosmological_benchmark_given_completion",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “From compact response to cosmological curvature: global-source completion”",
            "section_title": "From compact response to cosmological curvature: global-source completion"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-003",
            "role": "observational_input"
          }
        ]
      },
      "eq:dB1sigma": {
        "title": "One-sigma gravity-only action-shift tolerance",
        "display_order": 52,
        "latex": "\\boxed{-0.0161\\lesssim\\delta B\\lesssim0.0239,}",
        "paper_number": 46,
        "variables": [
          {
            "symbol": "δB",
            "definition": "net source-dependent quantum/higher-curvature shift of the primitive action (dimensionless)"
          }
        ],
        "depends_on_equations": [
          "eq:qcomp_numeric_audit"
        ],
        "status": "derived_sensitivity_bound",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Correction budget, predictions, and tests”, paragraph “Strict compact gravitational channel.”",
            "section_title": "Correction budget, predictions, and tests",
            "paragraph_title": "Strict compact gravitational channel."
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-003",
            "role": "observational_input"
          }
        ]
      },
      "eq:dB2sigma": {
        "title": "Two-sigma gravity-only action-shift tolerance",
        "display_order": 53,
        "latex": "\\boxed{-0.0356\\lesssim\\delta B\\lesssim0.0446.}",
        "paper_number": 47,
        "variables": [
          {
            "symbol": "δB",
            "definition": "net source-dependent quantum/higher-curvature shift of the primitive action (dimensionless)"
          }
        ],
        "depends_on_equations": [
          "eq:qcomp_numeric_audit"
        ],
        "status": "derived_sensitivity_bound",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Correction budget, predictions, and tests”, paragraph “Strict compact gravitational channel.”",
            "section_title": "Correction budget, predictions, and tests",
            "paragraph_title": "Strict compact gravitational channel."
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-003",
            "role": "observational_input"
          }
        ]
      },
      "eq:joint_correction": {
        "title": "Joint matter-gravity correction relation",
        "display_order": 54,
        "latex": "\\frac{q}{q_0}=\\frac{a_{\\mathrm{eff}}}{a_{\\mathrm{SM}}}e^{-\\delta B}\\quad\\Longleftrightarrow\\quad \\ln\\frac{q}{q_0}=\\ln\\frac{a_{\\mathrm{eff}}}{a_{\\mathrm{SM}}}-\\delta B",
        "variables": [
          {
            "symbol": "q=GΛ",
            "definition": "dimensionless compact/cosmological response coordinate"
          },
          {
            "symbol": "a_eff",
            "definition": "interacting compact-channel Euler coefficient (dimensionless)"
          },
          {
            "symbol": "a",
            "definition": "type-A Euler anomaly coefficient (dimensionless)"
          },
          {
            "symbol": "δB",
            "definition": "net source-dependent quantum/higher-curvature shift of the primitive action (dimensionless)"
          }
        ],
        "depends_on_equations": [
          "eq:qcomp_general"
        ],
        "status": "derived_correction_relation",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Sec. “Correction budget, predictions, and tests”, paragraph “Strict compact gravitational channel.”",
            "section_title": "Correction budget, predictions, and tests",
            "paragraph_title": "Strict compact gravitational channel."
          }
        ],
        "qualifier": "Stable equation record for a load-bearing displayed relation that is unnumbered in the formal manuscript."
      },
      "eq:app_rawdet": {
        "title": "Generic one-saddle determinant form",
        "display_order": 55,
        "latex": "Z_{S^4}=C_{\\rm grav}e^{-B_{\\rm cl}},",
        "paper_number": 48,
        "variables": [
          {
            "symbol": "S_EH",
            "definition": "Euclidean Einstein-Hilbert action (dimensionless)"
          }
        ],
        "depends_on_equations": [
          "eq:instanton_value"
        ],
        "status": "standard_semiclassical_form",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Appendix: “Determinant status of the strict compact weight”"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-014",
            "role": "foundation"
          },
          {
            "reference_id": "REF-035",
            "role": "foundation"
          },
          {
            "reference_id": "REF-061",
            "role": "foundation"
          },
          {
            "reference_id": "REF-091",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-092",
            "role": "structural_convergence"
          }
        ]
      },
      "eq:app_detcts": {
        "title": "Finite gravitational counterterm ambiguity",
        "display_order": 56,
        "latex": "\\Gamma[g]\\mapsto\\Gamma[g]+\\int d^4x\\sqrt g\\,\n \\left(\\alpha+\\beta R+\\gamma R^2+\\delta E_4+\\zeta W^2+\\xi\\Box R\\right).",
        "paper_number": 49,
        "variables": [
          {
            "symbol": "Γ[g]",
            "definition": "renormalized effective action (dimensionless in ℏ=c=1 conventions)"
          },
          {
            "symbol": "E₄",
            "definition": "Euler density (mass dimension 4)"
          },
          {
            "symbol": "W",
            "definition": "Weyl tensor / W² invariant"
          }
        ],
        "depends_on_equations": [],
        "status": "standard_renormalization_freedom",
        "locations": [
          {
            "work_id": "cc-paper",
            "locator_text": "the cosmological-constant paper v2.0, Appendix: “Determinant status of the strict compact weight”"
          }
        ],
        "qualifier": "Equation status follows the scientific status of the step it represents; the equation record itself does not upgrade a premise, selected sector, proposed completion or open correction.",
        "evidence": [
          {
            "reference_id": "REF-014",
            "role": "foundation"
          },
          {
            "reference_id": "REF-035",
            "role": "foundation"
          },
          {
            "reference_id": "REF-061",
            "role": "foundation"
          },
          {
            "reference_id": "REF-091",
            "role": "structural_convergence"
          },
          {
            "reference_id": "REF-092",
            "role": "structural_convergence"
          }
        ]
      }
    },
    "calculations": {
      "CALC-CC-001": {
        "title": "Leading compact cosmological-constant benchmark",
        "formula_latex": "q_{\\rm comp}=a_{\\rm SM}\\frac{m_p}{M_P}e^{-24\\pi^2}",
        "inputs": [
          {
            "quantity_id": "Q-aSM"
          },
          {
            "quantity_id": "Q-mp-over-MP"
          },
          {
            "quantity_id": "Q-exp-minus-B1"
          }
        ],
        "output_quantity_id": "Q-qcomp",
        "comparison": {
          "quantity_id": "Q-planck2018-lambda",
          "central_difference_percent": 0.3696957133,
          "difference_in_quoted_sigma": 0.1845884211,
          "display": "0.4% (0.18σ); benchmark comparison, not combined significance"
        },
        "correction_budget": {
          "gravity_only_at_a_eff_equals_a_SM": {
            "one_sigma_deltaB": [
              -0.0161,
              0.0239
            ],
            "two_sigma_deltaB": [
              -0.0356,
              0.0446
            ]
          },
          "joint_relation": "ln(q/q0)=ln(a_eff/a_SM)-deltaB"
        },
        "source_note": "the cosmological-constant paper numerical audit and correction-budget equations; no continuous parameter fitted to the cosmological value."
      }
    },
    "references": {
      "REF-001": {
        "citation_key": "Riess1998",
        "display_order": 1,
        "authors_display": "A. G. Riess et al.",
        "title": "Observational evidence from supernovae for an accelerating universe and a cosmological constant",
        "publication_display": "Astron. J. 116, 1009–1038 (1998) [astro-ph/9805201].",
        "bibliographic_record": "A. G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116, 1009–1038 (1998) [astro-ph/9805201].",
        "source_type": "journal_article",
        "identifiers": {
          "doi": "10.1086/300499",
          "arxiv": "astro-ph/9805201"
        },
        "verification": {
          "doi_status": "verified",
          "metadata_checked_date": "2026-08-28"
        },
        "citation_contexts": [
          {
            "work_id": "cc-paper",
            "role": "observational_input",
            "why_cited": "Cited in the cosmological-constant paper v2.0, Sec. “Introduction”.",
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            "locations": [
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Introduction”"
              }
            ]
          }
        ]
      },
      "REF-002": {
        "citation_key": "Perlmutter1999",
        "display_order": 2,
        "authors_display": "S. Perlmutter et al.",
        "title": "Measurements of Ω and Λ from 42 high-redshift supernovae",
        "publication_display": "Astrophys. J. 517, 565–586 (1999) [astro-ph/9812133].",
        "bibliographic_record": "S. Perlmutter et al., Measurements of Ω and Λ from 42 high-redshift supernovae, Astrophys. J. 517, 565–586 (1999) [astro-ph/9812133].",
        "source_type": "journal_article",
        "identifiers": {
          "doi": "10.1086/307221",
          "arxiv": "astro-ph/9812133"
        },
        "verification": {
          "doi_status": "verified",
          "metadata_checked_date": "2026-08-28"
        },
        "citation_contexts": [
          {
            "work_id": "cc-paper",
            "role": "observational_input",
            "why_cited": "Cited in the cosmological-constant paper v2.0, Sec. “Introduction”.",
            "scope": "Provides the empirical input, convention or comparison used at the cited step.",
            "locations": [
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Introduction”"
              }
            ]
          }
        ]
      },
      "REF-003": {
        "citation_key": "Planck2018",
        "display_order": 3,
        "authors_display": "Planck Collaboration",
        "title": "Planck 2018 results. VI. Cosmological parameters",
        "publication_display": "Astron. Astrophys. 641, A6 (2020) [arXiv:1807.06209].",
        "bibliographic_record": "Planck Collaboration, Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020) [arXiv:1807.06209].",
        "source_type": "journal_article",
        "identifiers": {
          "doi": "10.1051/0004-6361/201833910",
          "arxiv": "1807.06209"
        },
        "verification": {
          "doi_status": "verified",
          "metadata_checked_date": "2026-08-28"
        },
        "citation_contexts": [
          {
            "work_id": "cc-paper",
            "role": "observational_input",
            "why_cited": "Supports or contextualizes the following controlled claim records: OBS-01 — Planck 2018 flat-ΛCDM inferred benchmark; OBS-02 — Numerical comparison with the Planck benchmark; CORR-03 — Gravity-only δB tolerance.",
            "scope": "Provides the empirical input, convention or comparison used at the cited step.",
            "locations": [
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Introduction”"
              },
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “From compact response to cosmological curvature: global-source completion”"
              },
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Correction budget, predictions, and tests”, paragraph “Strict compact gravitational channel.”"
              },
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Correction budget, predictions, and tests”, paragraph “Observational tests.”"
              },
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Conclusion”"
              }
            ]
          }
        ]
      },
      "REF-004": {
        "citation_key": "PDG2026",
        "display_order": 4,
        "authors_display": "F. Takahashi et al. (Particle Data Group)",
        "title": "Review of Particle Physics",
        "publication_display": "Int. J. Mod. Phys. A 41, 2630011 (2026).",
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          "doi": "10.1142/S0217751X26300115"
        },
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          "doi_status": "verified",
          "metadata_checked_date": "2026-08-28"
        },
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          {
            "work_id": "cc-paper",
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            "why_cited": "Supports or contextualizes the following controlled claim records: SM-01 — Minimal Standard Model Euler coefficient; QCD-01 — QCD supplies a physical infrared mass scale; QCD-04 — Prepared B=1 QCD+QED sector has proton spectral floor; UNIT-01 — Planck-unit normalization covariance; NUM-01 — Leading strict-channel compact benchmark; plus 1 additional claim backlink(s).",
            "scope": "Provides the empirical input, convention or comparison used at the cited step.",
            "locations": [
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Introduction”, paragraph “Conventions.”"
              },
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Standard Model evaluation of the extracted datum”"
              },
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “Gauge-consistent proton floor”"
              },
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “Normalized matter/IR composite”"
              }
            ]
          }
        ]
      },
      "REF-005": {
        "citation_key": "Weinberg1989",
        "display_order": 5,
        "authors_display": "S. Weinberg",
        "title": "The cosmological constant problem",
        "publication_display": "Rev. Mod. Phys. 61, 1 (1989).",
        "bibliographic_record": "S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. 61, 1 (1989).",
        "source_type": "journal_article",
        "identifiers": {
          "doi": "10.1103/RevModPhys.61.1"
        },
        "verification": {
          "doi_status": "verified",
          "metadata_checked_date": "2026-08-28"
        },
        "citation_contexts": [
          {
            "work_id": "cc-paper",
            "role": "foundation",
            "why_cited": "Supports or contextualizes the following controlled claim records: CMP-01 — Specified compact channel closes the leading numerical chain.",
            "scope": "Supports the standard result or background statement for which it is cited.",
            "locations": [
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Introduction”"
              }
            ]
          }
        ]
      },
      "REF-006": {
        "citation_key": "Carroll2001",
        "display_order": 6,
        "authors_display": "S. M. Carroll",
        "title": "The cosmological constant",
        "publication_display": "Living Rev. Relativ. 4, 1 (2001).",
        "bibliographic_record": "S. M. Carroll, The cosmological constant, Living Rev. Relativ. 4, 1 (2001).",
        "source_type": "journal_article",
        "identifiers": {
          "doi": "10.12942/lrr-2001-1"
        },
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              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Introduction”"
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              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact-S⁴ vacuum-energy ambiguity”"
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              {
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              {
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            "locations": [
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Extraction theorem on conformally flat S⁴”, paragraph “Free-scalar sign audit.”"
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                "locator_text": "the cosmological-constant paper v2.0, Sec. “Standard Model evaluation of the extracted datum”"
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              {
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              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact-S⁴ vacuum-energy ambiguity”"
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              {
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              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Introduction”, paragraph “Conventions.”"
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              {
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              {
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              {
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        "publication_display": "arXiv:2605.27221 [astro-ph.CO].",
        "bibliographic_record": "DES Collaboration, T. M. C. Abbott et al., Constraints on dynamical dark energy from multiple probes in the full Dark Energy Survey, arXiv:2605.27221 [astro-ph.CO].",
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              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Correction budget, predictions, and tests”, paragraph “Observational tests.”"
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            "scope": "Supports the standard result or background statement for which it is cited.",
            "locations": [
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “One class-level QCD boundary premise”"
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        "citation_contexts": [
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            "scope": "Supports the standard result or background statement for which it is cited.",
            "locations": [
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “One class-level QCD boundary premise”"
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            "scope": "Supports the standard result or background statement for which it is cited.",
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              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “One intrinsic Euler–Chern unit on the round sphere”"
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            "locations": [
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Standard Model evaluation of the extracted datum”"
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              {
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              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Extraction theorem on conformally flat S⁴”, paragraph “Interpretation of the extracted datum.”"
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              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Discussion”, subsection “Independent structural convergence”"
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              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Discussion”, subsection “Independent structural convergence”"
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              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Discussion”, subsection “Independent structural convergence”"
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              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Discussion”, subsection “Independent structural convergence”"
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                "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact de Sitter saddle weight e^{-24π²}”, paragraph “Independent saddle support.”"
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                "locator_text": "the cosmological-constant paper v2.0, Sec. “Discussion”, subsection “Independent structural convergence”"
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                "locator_text": "the cosmological-constant paper v2.0, Sec. “Discussion”, subsection “Independent structural convergence”"
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              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Discussion”, subsection “Independent structural convergence”"
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        "authors_display": "M. M. Anber",
        "title": "Gauging the Standard Model 1-form symmetry via gravitational instantons",
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              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Discussion”, subsection “Independent structural convergence”"
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        "authors_display": "S. Alexander, H. Bernardo, and A. Hui",
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        "publication_display": "Phys. Rev. Lett. 136, 151501 (2026) [arXiv:2506.14886].",
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            "scope": "Directly supports the primitive large-SU(2)/inverse-cosmological-coupling architecture in exact nonperturbative canonical gravity. It does not independently derive this paper's compact action weight, select the boundary-member prescription, u_max=1 or this observable's contour, establish the QCD state-preparation premise, or validate the numerical benchmark.",
            "locations": [
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact de Sitter saddle weight e^{-24π²}”, paragraph “Canonical topological cross-check.”"
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                "locator_text": "the cosmological-constant paper v2.0, Sec. “Discussion”, subsection “Independent structural convergence”"
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        "authors_display": "S. Alexander and K. Blakey",
        "title": "Quantum de Sitter and Analytically Continued Chern Simons Theory",
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              {
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        "citation_key": "Witten1983",
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        "authors_display": "E. Witten",
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        "bibliographic_record": "E. Witten, Current algebra, baryons, and quark confinement, Nucl. Phys. B 223, 433–444 (1983), doi:10.1016/0550-3213(83)90064-0.",
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                "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “One class-level QCD boundary premise”"
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        "citation_key": "AdkinsNappiWitten1983",
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        "authors_display": "G. S. Adkins, C. R. Nappi, and E. Witten",
        "title": "Static properties of nucleons in the Skyrme model",
        "publication_display": "Nucl. Phys. B 228, 552–566 (1983), doi:10.1016/0550-3213(83)90559-X.",
        "bibliographic_record": "G. S. Adkins, C. R. Nappi, and E. Witten, Static properties of nucleons in the Skyrme model, Nucl. Phys. B 228, 552–566 (1983), doi:10.1016/0550-3213(83)90559-X.",
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        "title": "Skyrmions from instantons",
        "publication_display": "Phys. Lett. B 222, 438–442 (1989), doi:10.1016/0370-2693(89)90340-7.",
        "bibliographic_record": "M. F. Atiyah and N. S. Manton, Skyrmions from instantons, Phys. Lett. B 222, 438–442 (1989), doi:10.1016/0370-2693(89)90340-7.",
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            "scope": "Directly supports the spin-holonomy to Skyrme representative and degree relation used to construct U_spin. It does not independently establish the physical QCD flavour-state identification or the downstream cosmological map.",
            "locations": [
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                "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “The equatorial clutching map has degree one”"
              }
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        ]
      },
      "REF-083": {
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        "title": "Baryons from instantons in holographic QCD",
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        "bibliographic_record": "H. Hata, T. Sakai, S. Sugimoto, and S. Yamato, Baryons from instantons in holographic QCD, Prog. Theor. Phys. 117, 1157 (2007) [arXiv:hep-th/0701280].",
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            "locations": [
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            "locations": [
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                "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “The equatorial clutching map has degree one”"
              },
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “One class-level QCD boundary premise”"
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      "REF-086": {
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        "title": "Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking",
        "publication_display": "JHEP 08, 171 (2018) [arXiv:1807.07666].",
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                "locator_text": "the cosmological-constant paper v2.0, Sec. “Intrinsic chiral-spin clutching and the QCD infrared response”, subsection “One class-level QCD boundary premise”"
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          }
        ]
      },
      "REF-088": {
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        "authors_display": "T. Padmanabhan",
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        "bibliographic_record": "T. Padmanabhan, The physical principle that determines the value of the cosmological constant, arXiv:1210.4174 (2012).",
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          "arxiv": "1210.4174"
        },
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          {
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            "why_cited": "Cited in the cosmological-constant paper v2.0, Sec. “The compact de Sitter saddle weight e^{-24π²}”, paragraph “Canonical topological cross-check.”; the cosmological-constant paper v2.0, Sec. “Discussion”, subsection “Independent structural convergence”.",
            "scope": "Supports the standard result or background statement for which it is cited.",
            "locations": [
              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “The compact de Sitter saddle weight e^{-24π²}”, paragraph “Canonical topological cross-check.”"
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              {
                "locator_text": "the cosmological-constant paper v2.0, Sec. “Discussion”, subsection “Independent structural convergence”"
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      "REF-089": {
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        "publication_display": "Phys. Rev. D 72, 021301(R) (2005), arXiv:hep-th/0503158, doi:10.1103/PhysRevD.72.021301.",
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        "citation_key": "HerzogHuangJensen2016",
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        "authors_display": "C. P. Herzog, K.-W. Huang, and K. Jensen",
        "title": "Universal entanglement and boundary geometry in conformal field theory",
        "publication_display": "JHEP 01, 162 (2016), arXiv:1510.00021, doi:10.1007/JHEP01(2016)162.",
        "bibliographic_record": "C. P. Herzog, K.-W. Huang, and K. Jensen, “Universal entanglement and boundary geometry in conformal field theory,” JHEP 01, 162 (2016), arXiv:1510.00021, doi:10.1007/JHEP01(2016)162.",
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        "citation_key": "AnninosDenefLawSun2022",
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        "title": "Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions",
        "publication_display": "JHEP 01, 088 (2022), arXiv:2009.12464, doi:10.1007/JHEP01(2022)088.",
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        "title": "A compendium of sphere path integrals",
        "publication_display": "JHEP 12, 213 (2021), arXiv:2012.06345, doi:10.1007/JHEP12(2021)213.",
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        "bibliographic_record": "I. Antoniadis, P. O. Mazur, and E. Mottola, Conformal invariance, dark energy, and CMB non-Gaussianity, J. Cosmol. Astropart. Phys. 09, 024 (2012) [arXiv:1103.4164].",
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      "scientific_summary": "the v2 architecture preserves the counterterm quotient, Euler theorem, leading Standard Model coefficient, round-S^4 action, ordinary-GR obstruction, global-source concept and w=-1 test, while replacing the superseded v1 midpoint/paired-cap mechanism with an intrinsic geometric spine, an explicit QCD state-preparation premise, a proton spectral response and a more sharply decomposed compact-gravity sector.",
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        "The v2 scientific architecture preserves the vacuum quotient, Euler extraction theorem, leading Standard Model coefficient, ordinary-GR obstruction, trace-free quotient, round-S^4 action, global-source concept and w=-1 test. It supersedes the v1 midpoint/paired-cap construction and replaces it with an intrinsic Euler-Chern/spin-clutching unit, an explicit QCD state-preparation premise, a B=1 QCD+QED proton spectral readout, a more sharply decomposed compact gravitational sector, explicit determinant non-canonicity and a Defined coefficient-one compact assembly.",
        "The numerical headline formula is unchanged. The logical provenance is not: v2 removes several old selections, makes the surviving matter premise explicit, separates the derived monotonic ordering from the selected exact UV boundary and selected single-member prescription, and exposes the open quantum-gravity action correction as δB."
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        "curvature-charge source",
        "R0-R3 architecture",
        "paired-cap determinant cancellation",
        "legacy 0.5% matter-theory uncertainty",
        "hadron candidate-scoring"
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        "intrinsic Euler-Chern/spin-clutching unit",
        "QCD state-preparation premise",
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        "decomposed compact gravitational sector",
        "determinant non-canonicity",
        "Defined coefficient-one compact assembly"
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        "title": "Midpoint/fixed-plane scale h_*²=m_pM_P",
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      "FOLD-02": {
        "title": "Two-part paired-cap matching hypothesis",
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        "archive_source_id": "ARCH-V1.6-CLAIMS"
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        "canonical_path": "/downloads/cc-claim-register-v1.6.json",
        "canonical_url": "https://cosmological-constant.org/downloads/cc-claim-register-v1.6.json",
        "status": "immutable_archival_source",
        "authority_for": [
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          "v1 claim qualifiers and dependencies",
          "v1 claim version histories",
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}