Claim register v2.1 – source paper v2.1

Claims

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Each claim record states the exact technical claim, its approved public statement, status class and manuscript-facing status label, assumptions and dependencies, downstream consequences, equations, source location in the paper, external evidence, limits/open issues and version history.

The register also carries non-propagating evidence roles for literature. “Direct architectural support” and “structural convergence” change how supporting references are presented; neither becomes a dependency or upgrades a manuscript status.

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Canonical claim count: 53. Public release version: v2.

Foundations

VAC-01 Standard inputFoundations

Absolute vacuum offset is scheme dependent

Public statement Absolute zero-point energy is not by itself a scheme-independent observable.
Technical statement In renormalised semiclassical gravity, a constant shift of the matter Lagrangian renormalises the cosmological counterterm. The split between an absolute homogeneous matter-vacuum offset and the renormalised cosmological term is therefore convention dependent.
Dependencies none
Downstream claims EXT-01, GR-02, VAC-02, VAC-03
Qualifier This concerns the absolute homogeneous offset. It does not say that vacuum fluctuations do not exist or that physical energy differences never gravitate.
Source in the paper Sec. “The compact-S⁴ vacuum-energy ambiguity”.
Limits and open issues Casimir differences, phase transitions, interfaces and inhomogeneous stresses remain physical.
Version history Retained from v1; wording aligned to the cosmological-constant paper v2.0.

VAC-02 DerivedFoundations

Compact S⁴ sharpens the vacuum ambiguity

Public statement On compact round S⁴, the ambiguity is especially sharp: the homogeneous vacuum term and cosmological counterterm are the same local volume operator.
Technical statement On closed round S⁴ without boundary, the homogeneous matter-vacuum term and the cosmological counterterm both reduce to the same functional ∫√g d⁴x ∝ H⁻⁴, with no intrinsic geometrical label separating them.
Dependencies VAC-01
Downstream claims CMP-01, EXT-01
Qualifier A compact-background sharpening of standard renormalisation freedom.
Source in the paper Sec. “The compact-S⁴ vacuum-energy ambiguity”.
Evidence REF-009, REF-013
Limits and open issues The statement is about a single compact homogeneous saddle, not differences between physically distinct configurations.
Version history Retained from v1; wording aligned to the cosmological-constant paper v2.0.

VAC-03 Standard inputFoundations

Physical differences survive the quotient

Public statement Removing the homogeneous offset does not erase Casimir effects, phase transitions, interfaces or inhomogeneous stresses.
Technical statement Quotienting a strictly homogeneous constant shift leaves scheme-independent differences between configurations and nonhomogeneous stress-tensor structure untouched.
Dependencies VAC-01
Downstream claims none
Qualifier Scope guard for the vacuum-energy quotient.
Source in the paper Sec. “Why Casimir observables are different”.
Evidence REF-020, REF-089
Limits and open issues No claim is made that all vacuum phenomena are unphysical.
Version history Retained from v1.

Matter / anomaly

EXT-01 DerivedMatter / anomaly

Euler extraction theorem on conformally flat S⁴

Public statement Within the stated compact response class, one nonzero universal matter datum survives: the type-A Euler anomaly coefficient.
Technical statement Among local linear constant-Weyl responses of the renormalised effective action on round conformally flat S⁴ satisfying counterterm invariance (C1) and conformal-flatness universality (C2), the unique nonvanishing H-independent scheme-independent matter datum is the type-A Euler anomaly coefficient a.
Dependencies VAC-01, VAC-02
Downstream claims CMP-01, CORR-01, EXT-02, GR-01, IR-01, MAT-01, SM-01
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.
Equations Eq. (1)
Source in the paper Theorem “Euler extraction on conformally flat S⁴”.
Limits and open issues The theorem identifies the protected matter coordinate. It does not by itself produce the cosmological curvature.
Version history Retained from v1; current theorem wording follows the cosmological-constant paper v2.0.

EXT-02 DerivedMatter / anomaly

Euler projector isolates the protected coordinate

Public statement A constant-Weyl projector removes local counterterm contamination and isolates the Euler response.
Technical statement At a conformal fixed point, A[Γ]=−(1/4) ΠH0(dΓ/d ln H) and ΠH0(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.
Dependencies EXT-01
Downstream claims none
Qualifier The running-theory statement requires separation of beta-function and operator-mixing terms.
Equations Eq. (2) · Eq. (3)
Source in the paper Euler-extraction theorem, Eq. defining A[Γ] and proof.
Limits and open issues The projector extracts the protected coordinate; it does not assert that the rest of the effective action vanishes.
Version history Retained from v1; sign and running-theory scope aligned to v2.0.

EXT-03 DerivedMatter / anomaly

Standalone Type-A theorem across two de Sitter realisations

Public statement 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 statement 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).
Dependencies none
Downstream claims none
Qualifier Independent cross-realisation theorem. It is not an incoming dependency of the cosmological numerical chain.
Source in the paper Standalone Type-A theorem publication, current public record; exact theorem and scope on its own paper page.
Limits and open issues Does not establish the QCD state-preparation premise, the boundary-member prescription, umax=1 UV-domain normalization, contour choice, quantum dressing, global source pairing or numerical cosmological result.
Version history Retained as an independent v2.0 programme record.

SM-01 ComputedMatter / anomaly

Minimal Standard Model Euler coefficient

Public statement For the minimal Standard Model free-field census, the protected type-A coefficient is aSM = 1991/720.
Technical statement For four real Higgs scalars, 45 Weyl fermions and 12 vector bosons, aSM=4/360+45(11/720)+12(31/180)=1991/720=2.76527778.
Dependencies EXT-01
Downstream claims CORR-01, NUM-01, TEST-02
Qualifier Leading free-field Standard Model value; the full interacting curved-space coefficient is CORR-01.
Equations Eq. (5)
Source in the paper Sec. “Standard Model evaluation of the extracted datum”.
Limits and open issues Not a statement that no additional ultraviolet degrees of freedom exist, and not a calculation of the full interacting aeff.
Version history Retained from v1; old provisional 0.5% uncertainty removed.

IR-01 DerivedMatter / anomaly

Direct Planck-scale mechanisms tested do not supply the linear IR factor

Public statement A distinct infrared readout is required; the direct Planck-scale mechanisms tested do not generate the needed linear QCD hierarchy.
Technical statement 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 mp/MP. The protected Euler coefficient itself is order one.
Dependencies EXT-01
Downstream claims none
Qualifier Limited exclusion of the tested direct/local routes, not a no-go theorem for every nonlocal or UV-complete effect.
Source in the paper End of Sec. “Standard Model evaluation of the extracted datum”.
Limits and open issues The construction obtains the infrared factor through the prepared QCD+QED sector instead.
Version history Retained and simplified from v1; no midpoint-scale machinery remains.

Geometry / topology

TOP-01 DerivedGeometry / topology

Normalized Euler-Chern unit on S⁴

Public statement The oriented round spin sphere carries one intrinsic normalized Euler-Chern unit: NE = χ(S⁴)/2 = c₂(Σ⁻) = 1.
Technical statement For the chosen orientation on S⁴, NE=(1/64π²)∫√g E₄=χ(S⁴)/2=1, while ⟨c₂(Σ⁻),[S⁴]⟩=+1 and ⟨c₂(Σ⁺),[S⁴]⟩=−1.
Dependencies none
Downstream claims CMP-01, TOP-02
Qualifier Purely geometric; it does not identify spin SU(2) with QCD flavour SU(2).
Equations Eq. (6) · Eq. (11) · Eq. (8) · Eq. (7)
Source in the paper Sec. “One intrinsic Euler-Chern unit on the round sphere”.
Limits and open issues Fixes the primitive geometric integer, not its physical QCD state preparation.
Version history Introduced in v2.0.

TOP-02 DerivedGeometry / topology

Degree-one equatorial clutching map

Public statement The chiral spin bundle’s equatorial transition map is a primitive degree-one map S³→SU(2).
Technical statement 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 NE=⟨c₂(Σ⁻),[S⁴]⟩=deg g₋=1.
Dependencies TOP-01
Downstream claims TOP-03
Qualifier Relative Chern class is represented by the boundary clutching/Chern-Simons transgression; the half-sphere curvature integral need not itself be integer.
Equations Eq. (10) · Eq. (9) · Eq. (11)
Source in the paper Sec. “The equatorial clutching map has degree one”.
Evidence REF-022, REF-023
Limits and open issues A geometric spin-bundle statement, not yet a QCD claim.
Version history Introduced in v2.0.

TOP-03 DerivedGeometry / topology

Explicit degree-one spin-holonomy representative

Public statement The same primitive spin class has an explicit group-valued representative Uspin with deg Uspin=1 and [Uspin]=[g₋].
Technical statement The Levi-Civita-induced charge-one SU(2) connection on Σ⁻ has Atiyah-Manton holonomy Uspin:S³→SU(2)spin satisfying deg Uspin=⟨c₂(Σ⁻),[S⁴]⟩=1 and [Uspin]=[g₋].
Dependencies TOP-02
Downstream claims QCD-03
Qualifier Connection-level representative of the geometric spin class. No SU(2)spin=SU(2)flavour identification is asserted.
Equations Eq. (12)
Source in the paper Paragraph “An explicit degree-one spin-holonomy representative”.
Evidence REF-082, REF-085
Limits and open issues The physical class transfer to QCD remains QCD-03.
Version history Introduced in v2.0.

QCD / IR

QCD-01 Standard inputQCD / IR

QCD supplies a physical infrared mass scale

Public statement QCD dimensional transmutation generates a hadronic infrared scale, and the measured proton-to-Planck ratio is mp/MP = 7.685×10⁻²⁰.
Technical statement QCD dimensional transmutation generates the confined hadronic spectrum. Using the physical proton mass and unreduced Newton mass MP=G⁻¹/² gives mp/MP=7.68514844×10⁻²⁰.
Dependencies none
Downstream claims QCD-04
Qualifier This record supplies the standard QCD scale and measured ratio only; it performs no state selection or spin-to-QCD matching.
Equations Eq. (23)
Source in the paper QCD section and numerical audit.
Limits and open issues The role of the B=1 sector is established only after QCD-03.
Version history Retained as a standard foundation; the v1 selected-endpoint claim QCD-02 is superseded.

QCD-03 Structural premiseQCD / IR

QCD state-preparation class identification

Public statement 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 statement At confinement the physical premise is [UIR]=φ*[Uspin]=φ*[g₋] in π₃(SU(2)flavour), with φ orientation preserving at the level of the abstract SU(2) groups.
Dependencies TOP-03
Downstream claims CMP-01, CMP-03, QCD-04, TEST-02
Qualifier The paper’s single new matter-side physical premise. It is class-level only, not a connection-level spin-flavour identification.
Equations QCD state-preparation class identification · Eq. (13)
Source in the paper Sec. “One class-level QCD boundary premise”.
Limits and open issues A microscopic derivation from Standard Model plus gravity remains open.
Version history Introduced in v2.0; supersedes the v1 endpoint/fold matching architecture on the matter side.

QCD-04 Derived given premiseQCD / IR

Prepared B=1 QCD+QED sector has proton spectral floor

Public statement 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 statement 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 mp; the conjugate sector gives the antiproton.
Dependencies QCD-03, QCD-01
Downstream claims QCD-05
Qualifier The proton is not chosen from a scored candidate list; it follows as the spectral floor once B=1 is prepared.
Equations Eq. (14) · Eq. (16) · Eq. (15)
Source in the paper “From the B=1 sector to the proton channel” and “Gauge-consistent proton floor”.
Limits and open issues Depends on QCD-03. Electromagnetic edge flux dresses the charged subsystem but does not change baryon winding.
Version history Introduced in v2.0; replaces the selected proton-endpoint record QCD-02.

TRC-01 Defined source normalizationQCD / IR

Hamiltonian trace-source normalization

Public statement 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 statement Define Θ=∫Σ√h Tμμ 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.
Dependencies none
Downstream claims QCD-05
Qualifier Definition of the source coordinate used by the proton response theorem; not a new dynamical assumption.
Equations Eq. (17)
Source in the paper Eq. defining the trace source.
Evidence REF-069
Limits and open issues The source normalization fixes how the derivative is read; it does not prepare the B=1 sector.
Version history Introduced in v2.0.

QCD-05 Derived given premiseQCD / IR

Proton Weyl-slope theorem

Public statement For the prepared proton channel, the normalized long-time trace-source slope is mp/MP.
Technical statement With TRC-01 and the regulated proton correlator, −lims→∞(1/s)∂σ ln[Cp(s;σ)/Cp(s;0)]|σ=0=mp√G=mp/MP. The result follows from spectral projection, field-theoretic Feynman-Hellmann and the exact forward energy-momentum-tensor normalization.
Dependencies QCD-04, TRC-01
Downstream claims MAT-01
Qualifier Derived once the prepared B=1 proton channel exists; endpoint overlaps and subextensive edge dressings drop out of the normalized long-time slope.
Equations Eq. (20) · Eq. (19)
Source in the paper Theorem “Proton Weyl-slope theorem”.
Limits and open issues Does not derive QCD-03.
Version history Introduced in v2.0.

UNIT-01 Normalization covarianceQCD / IR

Planck-unit normalization covariance

Public statement Reduced and unreduced Planck conventions are algebraically equivalent when every normalization is transformed consistently; changing notation does not add a physical choice.
Technical statement With MP=G⁻¹/² and M̄P=(8πG)⁻¹/², mp/MP=(1/√(8π))(mp/M̄P) and GΛ=(1/8π)(Λ/M̄P²). Consistent conversion transforms both response and curvature observable.
Dependencies none
Downstream claims MAT-01
Qualifier Algebraic covariance of conventions, not a dynamical selection.
Equations Eq. (21)
Source in the paper Eq. “Planck covariance”.
Evidence REF-004
Limits and open issues Changing only one denominator while holding a coefficient-one source law fixed would define a different normalization.
Version history Introduced in v2.0.

MAT-01 Defined strict-channel compositeQCD / IR

Normalized matter/IR composite

Public statement The strict channel defines the matter/IR composite as qmatter = aeff·(mp/MP), with coefficient one between two separately normalized response coordinates.
Technical statement Define AE≡aeff and Qp≡mp/MP, then qmatter^(1)≡AE Qp=aeff(mp/MP). 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.
Dependencies EXT-01, QCD-05, UNIT-01
Downstream claims ASM-01
Qualifier Defined observable, not a separately derived dynamical interaction.
Equations Eq. (22)
Source in the paper Sec. “Normalized matter/IR composite”.
Limits and open issues The leading numerical evaluation substitutes aeff→aSM; the full aeff remains CORR-01.
Version history Introduced in v2.0.

Compact gravity

SAD-01 DerivedCompact gravity

Round-S⁴ Einstein action and Euler-normalized form

Public statement For round Euclidean de Sitter, the Einstein-Hilbert action has magnitude |SEH|=24π²/u, and the same result can be written (24π²/u)NE.
Technical statement For RμνUVgμν on round S⁴, SEH=−24π²/(κ²ΛUV). With u=κ²ΛUV and NE=1, |SEH|=(24π²/u)NE. The magnitude also equals the de Sitter entropy.
Dependencies none
Qualifier Classical on-shell Einstein result. It does not by itself select u=1 or a path-integral contour.
Equations Eq. (24) · Eq. (25) · Eq. (29)
Source in the paper Sec. “Round-sphere action”.
Limits and open issues Quantum dressing and higher-curvature shifts are CORR-02.
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 Selected contourCompact gravity

Decaying compact thimble

Public statement The strict channel selects the decaying orientation of the compact round-S⁴ thimble.
Technical statement The cosmological compact observable is defined so that the round-S⁴ thimble contributes with the damped orientation e−|SEH|. The opposite sign would replace suppression by an enormous enhancement and is not the channel analysed.
Dependencies SAD-01
Downstream claims CMP-03, INT-04, SAD-06
Qualifier Contour inclusion/orientation is selected, not derived by the compact action formula.
Equations Eq. (30)
Source in the paper Paragraph “Suppressed-thimble clause”.
Limits and open issues Broader contour selection depends on Stokes data, boundary conditions and observable.
Version history Retained from v1; v2.0 adds direct thimble support.

SAD-04 Selected UV domainCompact gravity

Exact reduced-Planck-density UV endpoint

Public statement The strict compact channel declares the reduced-Planck-density domain 0<u≤1 and fixes its exact upper endpoint at umax=1.
Technical statement With u=κ²ΛUVUV/M̄P⁴, the declared ultraviolet domain is 0<ρUV≤M̄P⁴, equivalently 0<u≤1, with umax=1 ⇔ ρUV=M̄P⁴.
Dependencies SAD-01
Downstream claims CMP-03, INT-04, SAD-03, SAD-06, TEST-03
Qualifier The exact numerical boundary is a selected UV-domain normalization, not a theorem that every ultraviolet completion breaks down at the same coefficient.
Equations Exact UV-endpoint normalization
Source in the paper Reduced-Planck-density UV-domain paragraph.
Evidence REF-006, REF-008
Limits and open issues Microscopic selection of this exact endpoint remains a central gravity target.
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 Derived given strict domainCompact gravity

Monotonic ordering within the declared domain

Public statement 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 statement 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=umax the least-suppressed member, but the ordering does not itself select evaluation there.
Dependencies SAD-01, SAD-04
Downstream claims SAD-07
Qualifier Derived ordering only. Selection of a single boundary member is priced separately in SAD-07; the exact domain endpoint remains SAD-04.
Equations Eq. (25) · Declared reduced-Planck-density UV domain · Exact UV-endpoint normalization
Source in the paper Compact de Sitter saddle section.
Limits and open issues Does not define a modulus measure or select the member at which the compact observable is evaluated.
Version history Retained ID; v2.0 finalization separates derived monotonic ordering from the selected boundary-member prescription.

SAD-07 Selected compact-sector prescriptionCompact gravity

Boundary-member prescription (no modulus integration)

Public statement The strict compact observable is evaluated at the least-suppressed boundary member ub=umax; no integration over u is part of the defined compact observable.
Technical statement After SAD-03 establishes the monotonic ordering of the declared domain, the compact channel selects single-member evaluation at ub=umax rather than introducing a modulus integral over u. Combined with the selected UV endpoint SAD-04, this gives ub=umax=1.
Dependencies SAD-03
Downstream claims CMP-03, INT-04, SAD-06, TEST-03
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.
Equations Eq. (26)
Source in the paper Compact de Sitter saddle section, boundary-member paragraph.
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.
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 Derived given boundary-member, UV-domain and contour selectionsCompact gravity

Classical strict-channel gravitational factor

Public statement Given the selected umax=1 UV endpoint, selected boundary-member prescription ub=umax and decaying contour, the classical compact gravitational factor is e⁻²⁴π².
Technical statement At the selected member ub=umax=1, B1=|SEH|=24π². With the selected damped orientation, the strict leading Einstein-Hilbert saddle contribution is e−B1=e−24π²=1.34414611×10⁻¹⁰³.
Dependencies SAD-01, SAD-02, SAD-04, SAD-07
Downstream claims ASM-01
Qualifier Classical Einstein-Hilbert factor only; the member prescription, UV endpoint and contour are selected, while quantum/source-dependent dressing is open.
Equations Eq. (27) · Eq. (30)
Source in the paper “Classical strict-channel factor”.
Evidence REF-038
Limits and open issues Does not assert a universal absolute one-loop prefactor.
Version history Introduced in v2.0 to price the factor explicitly.

DET-01 DerivedCompact gravity

No canonical absolute determinant constant on round S⁴

Public statement A raw absolute sphere determinant does not carry a universal scheme-independent constant prefactor that can simply be set to one.
Technical statement 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 Cgrav is not scheme independent without an additional renormalisation and measure prescription.
Dependencies SAD-01
Downstream claims CORR-02
Qualifier No unit-prefactor or determinant-cancellation claim is used; source-dependent quantum-gravity corrections remain open.
Equations Eq. (49) · Eq. (48)
Source in the paper Appendix “Determinant status of the strict compact weight”.
Limits and open issues Source-dependent determinants, Stokes changes and higher-curvature shifts remain CORR-02.
Version history Introduced in v2.0; replaces the old paired-cap determinant-normalisation story.

Assembly

ASM-01 Defined strict-channel compositeAssembly

Factorized compact assembly

Public statement The leading compact observable is defined by multiplying the normalized matter/IR composite by the selected classical compact factor.
Technical statement Define qcomp≡qmatter^(1)e−24π²=aeff(mp/MP)e−24π². The coefficient-one factorization is the strict-channel observable definition, not a separately derived interaction vertex.
Dependencies MAT-01, SAD-06
Downstream claims NUM-01
Qualifier Definition of the compact response. The later map from qcomp to cosmological curvature is separate.
Equations Eq. (31)
Source in the paper Sec. “Leading compact-channel result”.
Limits and open issues Does not upgrade the QCD premise or gravitational selections to derived status.
Version history Introduced in v2.0.

NUM-01 Derived given stated premise/selectionsAssembly

Leading strict-channel compact benchmark

Public statement Within the stated leading strict channel, the compact response is qcomp = 2.85652154×10⁻¹²², with no continuous parameter fitted to the cosmological value.
Technical statement Using aeff→aSM=1991/720 in ASM-01 together with the physical mp/MP and e−24π², qcomp=(1991/720)(7.68514844×10⁻²⁰)(1.34414611×10⁻¹⁰³)=2.85652154×10⁻¹²².
Dependencies ASM-01, SM-01
Downstream claims CMP-01, CORR-03, CORR-04, INT-02, INT-03, MAP-01, OBS-02
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.
Equations Eq. (33) · Eq. (32)
Source in the paper Numerical audit and compact-status ledger.
Evidence REF-004
Limits and open issues Full aeff and gravitational dressing remain open; the formula cannot be retuned to their outcome.
Version history Retained ID but completely rebased onto the v2.0 geometric/QCD/gravity chain.

Global completion

GR-01 DerivedGlobal completion

Ordinary-GR no-go for a linear compact source

Public statement 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 statement On a maximally symmetric branch the type-A anomaly stress scales as Λg². In ordinary semiclassical Einstein gravity, Λg−Λ0=(a/3π)(Λg²/MP²); multiplying the anomaly sector by a small ε places ε in the denominator of the nonzero root rather than producing Λg/MP²∝εa.
Dependencies EXT-01
Downstream claims none
Qualifier Applies to the ordinary local metric-variation route.
Equations Eq. (34)
Source in the paper Proposition “Ordinary-GR no-go for a linear compact source”.
Evidence REF-018
Limits and open issues Motivates a separate scalar zero-mode equation; it does not rule out all nonlocal/global completions.
Version history Retained from v1.

GR-02 DerivedGlobal completion

Trace-free quotient leaves one scalar mode

Public statement Removing the volume-counterterm representative leaves a unique trace-free local Einstein equation and one undetermined spacetime-constant scalar mode.
Technical statement The unique algebraic projection invariant under Eμν→Eμν+cgμν and equal to the identity on traceless tensors is Eμν−Egμν/4, yielding Rμν−Rgμν/4=(8π/MP²)(Tμν−Tgμν/4). Conservation leaves R+(8π/MP²)T=4Λint.
Dependencies VAC-01
Downstream claims CMP-02, GS-01
Qualifier Determines the representative-independent local equation shape, not a complete physical law or the value of Λint.
Equations Eq. (36) · Eq. (35)
Source in the paper Lemma “Trace-free quotient projector”.
Evidence REF-028, REF-029
Limits and open issues The missing scalar equation is supplied only by the proposed completion.
Version history Retained from v1.

GS-01 Proposed completionGlobal completion

Four-form global-source completion

Public statement A Henneaux-Teitelboim/sequestering-type four-form action is proposed to supply the scalar-curvature mode omitted by the trace-free local equation.
Technical statement The proposed action Sgs[q] contains a rigid Newton variable η, a volume four-form constraint and a source term proportional to η²MP⁴ q F₄. The compact calculation supplies q=qcomp; the action is varied at fixed rigid q.
Dependencies GR-02
Downstream claims CMP-03, GS-02, GS-04, GS-05
Qualifier Gravity-side completion proposed independently of the extraction theorem.
Equations Eq. (37)
Source in the paper Sec. “From compact response to cosmological curvature: global-source completion”.
Limits and open issues Microscopic selection of this action remains open.
Version history Retained ID; action rewritten and status sharpened in v2.0.

GS-05 Derived from branch-unit covarianceGlobal completion

Branch-unit covariance fixes η² source dependence

Public statement 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 statement Replacing η² by f(η), branch-unit independence of Λ/MN² 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.
Dependencies GS-01
Downstream claims GS-04
Qualifier Derived within the proposed completion; it does not fix the unit coefficient c=1.
Equations Branch-unit covariance of the source function
Source in the paper Appendix “Off-shell variation of the global-source action”.
Limits and open issues The remaining coefficient is GS-04.
Version history Introduced in v2.0.

GS-04 Proposed unit pairingGlobal completion

Unit compact-to-global source pairing

Public statement The proposed completion uses unit pairing between the normalized compact response and the global four-form source.
Technical statement 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 Sgs[q].
Dependencies GS-01, GS-05
Downstream claims GS-02
Qualifier A proposed physical normalization, not fitted after comparison and not derived by the compact anomaly calculation.
Equations Eq. (37) · Branch-unit covariance of the source function
Source in the paper Global-source action and Appendix on branch-unit covariance.
Limits and open issues A microscopic parent could derive a different coefficient and thereby change the curvature map.
Version history Retained ID but fundamentally revised in v2.0; old v1 source-charge convention is superseded.

GS-02 Derived once action and unit pairing are adoptedGlobal completion

Field equations of the proposed global-source action

Public statement Once the stated global-source action and unit pairing are adopted, its variations cancel homogeneous matter shifts and fix the residual vacuum relation Λ/MN²=q.
Technical statement Variation gives F₄=*1, ⟨R⟩=4ηMP²q and Gμν+ηMP²q gμν=(8π/(ηMP²))(Tμν−⟨T⟩gμν/4). On a maximally symmetric vacuum with MN²=ηMP², Λ/MN²=q. Constant shifts of Lm cancel exactly.
Dependencies GS-01, GS-04
Downstream claims CMP-02, GS-03, INT-01, MAP-01, TEST-01
Qualifier The equations are a theorem of the stated action; the action and unit pairing remain proposed.
Equations Eq. (38) · Eq. (39) · Eq. (40)
Source in the paper Theorem “Global-source field equations”.
Limits and open issues Does not establish microscopic selection of GS-01/GS-04.
Version history Retained ID; equations and normalization updated to v2.0.

GS-03 Amplitude conditionGlobal completion

Regulator-independent late-time trace-average condition

Public statement Direct late-time identification of the compact response with the asymptotic cosmological amplitude additionally requires the regulated non-vacuum trace average to vanish.
Technical statement For connected future-directed exhaustions Mτ with subextensive boundary/edge terms and regulator-independent limit, require ⟨Tnonvacreg=limτ→∞[∫√−g Tnonvac]/[∫√−g]=0. Future-eternal asymptotically de Sitter dilution satisfies this standard sufficient regime.
Dependencies GS-02
Downstream claims CMP-03, OBS-02
Qualifier Amplitude condition. It fixes the late-time amplitude, not the equation of state, which follows from the spacetime-constant residual source.
Equations Eq. (43) · Eq. (42)
Source in the paper Late-time exhaustion/trace-average subsection.
Limits and open issues Not asserted for every recollapsing or non-asymptotic cosmological history.
Version history Retained ID but reclassified from selected branch to explicit amplitude condition in v2.0.

MAP-01 Proposed cosmological identificationGlobal completion

Compact response to cosmological curvature map

Public statement 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 statement With q=qcomp inserted into the proposed action, GS-02 gives Λ/MN²=q on a maximally symmetric vacuum. Matching to the measured branch Newton unit yields the residual curvature map; direct asymptotic identification Λ/MP²=qcomp additionally assumes GS-03.
Dependencies NUM-01, GS-02
Downstream claims CMP-02, INT-01, OBS-02
Qualifier Separate from the compact calculation. The identification inherits the proposed action and unit pairing; the observational late-time amplitude inherits GS-03.
Equations Eq. (44) · Eq. (40)
Source in the paper Global-source completion and asymptotic relation.
Limits and open issues A different microscopic global source law or pairing would change the cosmological reading without undoing NUM-01.
Version history Retained ID; v2.0 cleanly separates compact response from curvature map.

Observations

OBS-01 Observational inputObservations

Planck 2018 flat-ΛCDM inferred benchmark

Public statement Planck 2018 flat-ΛCDM-inferred value: (2.846±0.057)×10⁻¹²² in the paper’s GΛ convention.
Technical statement The comparison benchmark is (Λ/MP²)obs=(2.846±0.057)×10⁻¹²² inferred from Planck 2018 under flat ΛCDM.
Dependencies none
Downstream claims CORR-03, OBS-02
Qualifier Model-dependent cosmological inference, not a direct measurement of an asymptotic constant and not a theory error bar.
Source in the paper Abstract/numerical comparison.
Evidence REF-003
Limits and open issues If evolving dark energy is established, this flat-ΛCDM inference is not the direct asymptotic quantity predicted by the completion.
Version history Retained from v1.

OBS-02 Derived comparisonObservations

Numerical comparison with the Planck benchmark

Public statement 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 statement 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.
Dependencies NUM-01, MAP-01, GS-03, OBS-01
Downstream claims none
Qualifier Plain benchmark comparison, not a likelihood ratio or sigma-level theory confirmation.
Equations Eq. (45)
Source in the paper Global-source comparison and correction-budget section.
Evidence REF-003
Limits and open issues Open aeff and gravitational corrections determine how the benchmark ultimately moves.
Version history Retained ID; dependency chain updated to v2.0.

Corrections

CORR-01 Open calculationCorrections

Full interacting curved-space Standard Model coefficient

Public statement The full interacting curved-space Standard Model coefficient aeff in the precise compact channel remains to be calculated.
Technical statement The leading free-field value aSM is known, and local-RG/interacting-QFT results constrain the structure of corrections, but the paper does not assign a numerical uncertainty to aeff without a complete curved-space Standard Model calculation including running, beta-function/operator mixing and the relevant compact projection.
Dependencies EXT-01, SM-01
Downstream claims CMP-03, CORR-04, TEST-02
Qualifier Open matter calculation; no numerical theory uncertainty is assigned without the complete curved-space Standard Model calculation.
Equations Eq. (5)
Source in the paper Standard Model evaluation and correction budget.
Limits and open issues No theory error bar is inferred from partial perturbative information in the released paper.
Version history Retained ID but completely revised in v2.0; old 0.5% estimate removed.

CORR-02 Open calculationCorrections

Quantum gravitational dressing and higher-curvature action shift

Public statement 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 statement The strict leading benchmark is the classical Einstein-Hilbert truncation δB=0. This does not assert a universal canonical determinant value Cgrav=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.
Dependencies SAD-01, DET-01
Downstream claims CMP-03, CORR-04, TEST-02, TEST-03
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.
Equations Eq. (49) · Eq. (48)
Source in the paper Determinant/higher-curvature status and correction budget.
Limits and open issues Their size is not inferred from the 0.4% central-value agreement.
Version history Retained ID but rebased from the old paired-cap cancellation framework.

CORR-03 Derived sensitivity boundCorrections

Gravity-only δB tolerance

Public statement At fixed aeff=aSM, the Planck benchmark allows only a few×10⁻² shift in the complete primitive action.
Technical statement For q(δB)=q₀e−δB at aeff=aSM, remaining within the quoted Planck 2018 1σ interval requires −0.0161≲δB≲0.0239; the 2σ interval is −0.0356≲δB≲0.0446.
Dependencies NUM-01, OBS-01
Downstream claims TEST-03
Qualifier Gravity-only slice of the joint open correction space, not a prior distribution or fitted error bar.
Equations Eq. (46) · Eq. (47)
Source in the paper Eqs. for δB 1σ and 2σ intervals.
Evidence REF-003
Limits and open issues Matter and gravity corrections can both move the benchmark; the general combination is CORR-04.
Version history Introduced in v2.0.

CORR-04 Derived correction relationCorrections

Joint matter-gravity correction relation

Public statement The leading joint correction depends on the combination δB−ln(aeff/aSM), not on two independently fitted offsets.
Technical statement Relative to the leading benchmark, q/q₀=(aeff/aSM)e−δB; equivalently ln(q/q₀)=ln(aeff/aSM)−δB. The constant-q contours are sensitivity relations between two independently open calculations, not a fit space.
Dependencies NUM-01, CORR-01, CORR-02
Downstream claims none
Qualifier Sensitivity relation only. The open calculations determine the point; observation does not select it.
Equations Joint matter-gravity correction relation
Source in the paper Correction-budget paragraph on joint matter-gravity space.
Limits and open issues No cancellation between open sectors is assumed or encouraged.
Version history Introduced in v2.0.

Tests

TEST-01 TestTests

Exact w=-1 observational falsifier of the completion

Public statement 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 statement On any fixed branch of GS-02 the residual source is proportional to gμν 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.
Dependencies GS-02
Downstream claims none
Qualifier Falsifier of the proposed completion/observational identification, not of the upstream compact theorem.
Equations Eq. (39) · Eq. (41)
Source in the paper Observational-tests paragraph.
Evidence REF-042, REF-043
Limits and open issues A preference under one dataset combination/parameterisation is not by itself the stated falsification threshold.
Version history Retained from v1; status and separation from GS-03 preserved.

TEST-02 TestTests

Microscopic field-content sensitivity

Public statement Changes in microscopic field content can move the benchmark through aeff 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 statement New microscopic degrees of freedom change the protected matter coordinate through the full aeff 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.
Dependencies SM-01, CORR-01, CORR-02, QCD-03
Downstream claims none
Qualifier Conditional field-content test, not a present BSM exclusion.
Source in the paper Final paragraph of correction budget/observational tests.
Limits and open issues Quantitative exclusion requires the full matter and gravity calculations.
Version history Retained ID but revised in v2.0; legacy percentage scenarios removed from canonical copy.

TEST-03 TestTests

Primitive-action/UV-boundary precision test

Public statement The strict gravitational boundary is sharply testable because any microscopic change of the primitive action moves the prediction exponentially.
Technical statement A parent gravitational theory must reproduce, replace or falsify the boundary-member prescription, exact umax=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.
Dependencies SAD-04, SAD-07, CORR-02, CORR-03
Downstream claims none
Qualifier Test of the selected UV model and quantum completion, not a fitted uncertainty.
Equations Eq. (46) · Eq. (47)
Source in the paper “Strict compact gravitational channel” and observational tests.
Limits and open issues Alternative boundary conventions or higher-curvature actions are different microscopic outcomes, not statistical variations of one fixed model.
Version history Retained ID but recast around the v2.0 δB budget.

Interpretation

INT-01 Derived within proposed completionInterpretation

Residual-curvature interpretation

Public statement 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 statement 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.
Dependencies GS-02, MAP-01
Downstream claims none
Qualifier Interpretation of the proposed completion; not required for the extraction theorem.
Equations Eq. (39) · Eq. (41)
Source in the paper Global-source section and Discussion.
Limits and open issues Inherits Proposed status through MAP-01.
Version history Retained from v1, updated to the v2.0 action.

INT-02 Derived arithmeticInterpretation

103+19 decade decomposition

Public statement 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 statement From NUM-01, −log10(e−24π²)≈102.87 and −log10(mp/MP)≈19.11; aSM is O(1). The hierarchy is multiplicative rather than a cancellation among large vacuum-energy terms.
Dependencies NUM-01
Downstream claims none
Qualifier Arithmetic decomposition of the leading strict result.
Equations Eq. (33)
Source in the paper Numerical audit discussion.
Limits and open issues Inherits all upstream conditions of NUM-01.
Version history Retained from v1.

INT-03 Derived rewritingInterpretation

Combined-exponent rewriting

Public statement The two suppressions can be rewritten as a single exponential of the sum of two distinct exponents.
Technical statement e−24π²(mp/MP)=exp[−24π²−ln(MP/mp)]. This is an algebraic rewriting of the gravitational and QCD hierarchies, not evidence that they arise from one thermal or microscopic process.
Dependencies NUM-01
Downstream claims none
Qualifier Rewriting only.
Equations Eq. (31)
Source in the paper Numerical audit/interpretive discussion.
Limits and open issues Does not establish a common dynamical origin of the exponents.
Version history Retained from v1 as a non-load-bearing interpretation.

INT-04 Standard identity applied to selected sectorInterpretation

Compact action-de Sitter entropy identity

Public statement 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^(−SdS).
Technical statement For the round de Sitter saddle, |SEH|=3πMP²/Λ=SdS. Under SAD-02 and the selected SAD-04/SAD-07 endpoint-member prescription, SdS=24π² and the classical weight is e−SdS=e−24π².
Dependencies SAD-01, SAD-02, SAD-04, SAD-07
Downstream claims none
Qualifier Classical on-shell identity applied to the selected sector, not an all-orders quantum-gravity identity.
Equations Eq. (28) · Eq. (30)
Source in the paper Action-entropy identity in compact gravity section.
Evidence REF-038
Limits and open issues The entropy belongs to the ultraviolet compact saddle, not the observed late-time horizon.
Version history Retained from v1.

Comparison

CMP-01 Comparative synthesisComparison

Specified compact channel closes the leading numerical chain

Public statement The construction closes the leading compact numerical chain while isolating, rather than hiding, the premise and gravitational selections on which it depends.
Technical statement 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.
Downstream claims none
Qualifier Synthesis, not a claim of unique microscopic selection or completed first-principles derivation.
Equations Eq. (32)
Source in the paper Introduction, Discussion and Conclusion.
Evidence REF-005
Limits and open issues Five remaining programme targets are exposed explicitly.
Version history Retained ID; v2.0 wording reflects the new architecture.

CMP-02 Comparative synthesisComparison

Relation to trace-free and sequestering approaches

Public statement The compact response acts as a candidate selector for the scalar residual that trace-free and sequestering formulations leave undetermined.
Technical statement 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 qcomp. This is structurally close to global-variable/sequestering approaches but uses a different source value.
Dependencies GR-02, GS-02, MAP-01
Downstream claims none
Qualifier Comparison conditional on the proposed completion.
Equations Eq. (39) · Eq. (35) · Eq. (40)
Source in the paper Discussion “Independent structural convergence”.
Limits and open issues Does not establish that existing sequestering models derive qcomp or the unit pairing.
Version history Retained from v1, updated to v2.0.

CMP-03 TestComparison

Microscopic parent-theory derivation-or-falsification test

Public statement 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 statement A microscopic completion must address the full interacting aeff; derive or replace QCD-03; derive, replace or falsify the boundary-member prescription, exact umax=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.
Downstream claims none
Qualifier Programme-level falsification-or-derivation test.
Source in the paper Discussion and Conclusion five-target programme.
Limits and open issues Does not assume such a parent theory exists.
Version history Retained ID but completely revised to the v2.0 five-target programme.

CMP-04 Comparison - non-dependencyComparison

Independent support and structural convergence

Public statement 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 statement 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.
Dependencies none
Downstream claims none
Qualifier Comparative context only; cannot upgrade Derived/Selected/Premise/Proposed/Open manuscript statuses.
Source in the paper Discussion “Independent structural convergence”; website How It Relates.
Limits and open issues Structural convergence is not validation of the numerical cosmological result.
Version history Introduced in v2.0.

Method

METH-01 Author disclosureMethod

Human-AI research-method disclosure

Public statement The research was developed through human-AI collaboration; the author remains responsible for every published claim.
Technical statement 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.
Dependencies none
Downstream claims none
Qualifier Method disclosure, not a scientific premise or credibility substitute.
Source in the paper Website Method/About records and author disclosure.
Limits and open issues Operational prompts and internal transcripts are not part of the scientific claim graph.
Version history Retained from v1.

Historical v1 claim-ID map (archive only)

These v2 archive stubs are not nodes in the canonical v2 dependency graph. Their full historical statements, dependencies and version histories resolve by archive_source_id to the retained final v1.6 claim register at /downloads/cc-claim-register-v1.6.json, which remains the named archival source rather than being duplicated into the current canonical object.

FOLD-01 Archived · v1

Midpoint/fixed-plane scale h_*²=m_pM_P

Disposition in the current release Superseded. The midpoint/fold is absent from the v2.0 canonical chain.

FOLD-02 Archived · v1

Two-part paired-cap matching hypothesis

Disposition in the current release Superseded. Replaced by the explicit QCD state-preparation premise QCD-03 plus downstream spectral response.

NORM-01 Archived · v1

Paired-cap cancellation of source-independent determinants

Disposition in the current release Superseded. v2.0 instead states DET-01: no canonical absolute determinant constant; physical source-dependent dressing is CORR-02.

NORM-02 Archived · v1

Paired-cap residual determinant/open sector record

Disposition in the current release Superseded by DET-01 and CORR-02.

QCD-02 Archived · v1

Selected proton endpoint by candidate criteria

Disposition in the current release Superseded by QCD-03/QCD-04/QCD-05. In v2.0 the proton is not selected by a scored candidate list; it is the spectral floor once the B=1 sector is prepared.

SAD-05 Archived · v1

Reduced zero-mode finiteness test

Disposition in the current release Archived v1 technical record; not a load-bearing canonical v2.0 claim.

SRC-00 Archived · v1

One-cap Einstein curvature-charge source coordinate

Disposition in the current release Superseded. The v2.0 proton response uses TRC-01/QCD-05; the normalized matter composite is MAT-01.

SRC-01 Archived · v1

Curvature-charge response h²/M_P²

Disposition in the current release Superseded with SRC-00; not part of v2.0.
Archival source ARCH-V1.6-CLAIMS. Final v1.6 claim register · cc-claim-register-v1.6.json (44 records) · immutable archival source. SHA-256 097f774cdd69fa8829d23a957b8a881f720e559e0aa6030600c473e47f13812c (computed from the retained byte-stable export at build). It is the authority for full v1 claim statements; v1 claim qualifiers and dependencies; v1 claim version histories; reconstruction of superseded v1 claim records.