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- the exact technical claim;
- its public statement;
- its status;
- its assumptions and dependencies;
- the source location in the paper;
- external foundations where relevant;
- open objections or calculations.
The register prevents the public explanation from silently becoming broader than the paper. A public sentence may be simpler than the technical statement, but it may not change its scope or status. A claim may be Derived while depending on Selected or Proposed premises; its dependency chain therefore remains part of its scientific status.
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VAC-01 Standard input
Technical statement In renormalised semiclassical gravity, the absolute vacuum-energy offset is scheme dependent because a constant shift of the matter Lagrangian renormalises the cosmological counterterm.
Public statement Absolute zero-point energy is not by itself a scheme-independent observable.
Source in the paper CC Paper I v1.0,
Sec. 3, lines/section “compact-S⁴ vacuum-energy ambiguity”
Limits and open issues The claim does not extend to “zero-point energy does not exist” or “zero-point energy never gravitates”.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.
VAC-02 Derived
Technical statement On compact round S⁴ without boundary, the vacuum-energy functional and cosmological counterterm are the same local volume operator, proportional to H⁻⁴, with no intrinsic geometrical label separating them.
Public statement On compact round S⁴, the ambiguity is especially sharp: the zero-point term and cosmological counterterm are literally the same local operator.
Source in the paper CC Paper I v1.0,
Sec. 3 Limits and open issues This is a compact-background sharpening of standard renormalisation lore.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.
VAC-03 Standard input
Technical statement Scheme-independent energy differences, phase-transition dynamics, interfaces and inhomogeneous stresses are not removed by quotienting a strictly homogeneous constant shift.
Public statement The argument does not erase Casimir effects, phase transitions or inhomogeneous stresses.
Downstream claims none
Source in the paper CC Paper I v1.0,
Sec. 3, “Casimir objection”
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.
EXT-01 Derived
Technical statement Among local linear constant-Weyl responses of the renormalised effective action on round conformally flat S⁴ satisfying counterterm invariance and conformal-flatness universality, the unique nonvanishing H-independent scheme-independent matter datum is the type-A Euler anomaly coefficient.
Public statement Within the stated compact response class, one nonzero universal matter datum survives: the type-A Euler anomaly coefficient.
Source in the paper CC Paper I v1.0, Theorem “Euler extraction on conformally flat S⁴”
Limits and open issues Class-relative uniqueness; not a universal uniqueness across all observables or backgrounds.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.
EXT-02 Derived
Technical statement At a conformal fixed point the Euler coefficient is isolated by A[Gamma] = -(1/4) Pi_H0(d Gamma/d ln H), because d Gamma_Euler/d ln H = -4a; away from a fixed point the corresponding local-RG Euler-cocycle projection is used after beta-function and operator-mixing terms are separated.
Public statement A constant-Weyl projector removes local counterterm contamination and isolates the Euler response.
Limits and open issues Running-theory statement requires beta-function and operator-mixing responses to be separated by the local-RG Euler projector.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.; v1.4 (2026-08-03): projector sign corrected to -(1/4), following d Gamma_Euler/d ln H = -4a; fixed-point and local-RG cases as before
EXT-03 Derived
Technical statement Let V_loc = span{E_4, C², j_1, j_2, D} be the standard parity-even local four-derivative anomaly basis, and let rho_dS = rho_S4 (+) rho_hor be the paired evaluation into the closed-sphere and spherical-horizon response spaces. Then Im(Pi_dS^univ o rho_dS |_V_loc) = span{e_A^dS}, where e_A^dS is the unit-normalised generator of the Euler/Wess-Zumino quotient line, and the transmitted coordinate is a_dS(Q) = a(Q); a QFT Q has projected local response 4 a(Q) e_A^dS. The C² and j_1 sectors vanish by conformal flatness; the trivial sector is removed by finite-counterterm invariance; and the nonzero j_2 sector belongs to the full auxiliary codimension-one boundary problem rather than to the common no-boundary response.
Public statement Within the protected local anomaly-response sector studied in Paper II, the closed-sphere and static-patch descriptions of de Sitter share one convention-independent matter coefficient: the type-A Euler coefficient a. For the minimal Standard Model free-field census, a_SM = 1991/720.
Downstream claims none
Qualifier The public wording is a gloss; the exact theorem is restricted to the parity-even fixed-point local four-derivative logarithmic anomaly sector of exact round de Sitter. Established in the standalone theorem paper, not in the cosmological-constant manuscript.
Relations extends EXT-01 (non-dependency): cross-realisation extension and observer-centred boundary-sector stress test; not required for the downstream numerical derivation.
Source in the paper The Universal Local de Sitter Anomaly Channel Is Uniquely Type A, version 2.0: Sec. 2 (channel and theorem), Sec. 3 (closed-S⁴ extraction), Sec. 4 (static-patch realisation and j_2 test), Sec. 5 (Standard Model coordinate), Sec. 6 (application). doi:10.5281/zenodo.21809544
External foundations Deser & Schwimmer 1993 Duff 1994 Wess & Zumino 1971 Herzog Huang & Jensen 2016 Law 2021 Anninos Denef Law & Sun 2022
Limits and open issues The theorem does not establish the proton endpoint rule; the two-part paired-cap matching hypothesis or its fixed plane; the one-cap Einstein curvature-charge coordinate; the suppressed compact thimble; the exact Planck-boundary unit; the global-source action or source-charge convention; the asymptotic trace-average branch; the full static-patch edge theory; or the numerical cosmological result.
Version history v1.5 (2026-08-05): added at the Zenodo deposit of the standalone theorem paper, version 2.0, doi:10.5281/zenodo.21809544; public wording follows the plain-language gloss with the exact theorem in the technical statement.; v1.6 (2026-08-08): public statement shortened; the exact theorem remains in the technical statement and qualifier
SM-01 Derived
Technical statement For the minimal Standard Model field census, a_SM = 4/360 + 45(11/720) + 12(31/180) = 1991/720 = 2.76528.
Public statement For the minimal Standard Model free-field census, the protected type-A coefficient is a_SM = 1991/720.
Limits and open issues Leading free-field value; independent of the Higgs nonminimal coupling in the projected response; interaction and running effects are separately budgeted. A free-field type-A census of the minimal Standard Model; not a theorem that no additional ultraviolet degrees of freedom exist below the matching scale. Additional light fields shift the result by their anomaly contribution.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.; v1.4 (2026-08-03): census caveat added: minimal-SM free-field count, not a no-new-fields theorem.; v1.6 (2026-08-08): public wording names the free-field census, keeping CORR-01's interacting budget distinct
IR-01 Derived
Technical statement Direct zero-mode sampling, the local analytic heat-kernel mass expansion (including logarithms multiplying even powers), and Planck-scale QCD instantons do not produce the required linear infrared factor m_p/M_P in the stated one-saddle treatment.
Public statement A separate infrared input is required; the direct Planck-scale mechanisms tested do not generate the needed linear QCD hierarchy.
Downstream claims none
Source in the paper CC Paper I v1.0,
Sec. 6 Limits and open issues Limited to the tested local/direct mechanisms; not a no-go theorem for every nonlocal or UV-complete compact effect.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.
QCD-01 Standard input
Technical statement QCD dimensional transmutation generates a nonperturbative infrared scale; using the current proton and unreduced Planck masses gives m_p/M_P = 7.685 × 10⁻²⁰.
Public statement QCD dimensional transmutation produces the infrared scale hierarchy; the measured proton-to-Planck ratio is m_p/M_P = 7.685 × 10⁻²⁰.
Limits and open issues The proton pole is a scale representative, not a claim that the vacuum contains protons or that one mass-decomposition term is the endpoint.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.; v1.4c (2026-08-04): IR-01 dependency removed: dimensional transmutation and the measured mass ratio are standard independently of the IR-necessity argument.; v1.6 (2026-08-08): public wording carries no selection work; endpoint selection lives entirely in QCD-02
QCD-02 Selected
Technical statement Within the declared QCD-endpoint candidate class, the proton is selected as the infrared endpoint because it is the lightest colour-singlet hadronic state with a scheme-independent physical pole mass that is stable in the full zero-temperature Standard Model, and it directly carries the normalised trace charge ⟨p|T^μ_μ|p⟩=2 m_p²; the neutron and η′ lie numerically close but fail the stability criterion, and the quoted pure-Yang-Mills glueball scale is not a full-Standard-Model asymptotic pole.
Public statement The strict QCD endpoint is the proton: the lightest stable hadronic colour-singlet pole satisfying the stated zero-temperature source criteria.
Dependencies QCD-01; endpoint criteria P/S/C
Qualifier A selected matching criterion, physically motivated by confinement, stability and the QCD-generated trace scale; not a theorem that quantum gravity must choose the proton pole.
Limits and open issues The endpoint criteria are selected channel criteria, not a theorem of every gravitational trace source; the choice does not assert proton dominance of the full trace spectral function.
Version history v1.2; v1.3 (2026-08-01): trace-charge criterion made explicit; near-miss exclusions noted.; v1.4 (2026-08-03): qualifier states the endpoint rule as a selected matching criterion, not a theorem of quantum gravity.; v1.6 (2026-08-08): hadronic/QCD-endpoint candidate class made explicit in both statements; the previous unrestricted wording ranged beyond the intended QCD endpoint class
GR-01 Derived
Technical statement Ordinary local type-A anomaly backreaction on a maximally symmetric de Sitter branch scales as Λ_g²/M_P² and produces no small branch linear in a suppressed compact weight.
Public statement Ordinary local anomaly backreaction has the wrong scaling: it produces an H⁴ stress, not the required small linear-curvature branch.
Downstream claims none
Source in the paper CC Paper I v1.0, Proposition “Ordinary-GR no-go”
Limits and open issues Applies to ordinary local semiclassical Einstein variation.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.
GR-02 Derived
Technical statement The unique local algebraic projection invariant under adding c g_mn to the metric equation is the trace-free Einstein equation, leaving one spacetime-constant scalar mode.
Public statement Removing the volume-counterterm representative leaves a unique trace-free local equation and one undetermined scalar-curvature mode.
Source in the paper CC Paper I v1.0, Lemma “Trace-free quotient projector”
Limits and open issues The quotient identifies equation shape, not a complete physical law.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.
SAD-01 Derived
Technical statement For the round S⁴ Einstein saddle, |S_EH| = 24π²/(κ²Λ); at the declared Planck-curvature representative u=κ²Λ=1, the tree-level exponent is B=24π².
Public statement At the specified Planck-curvature representative, the round-S⁴ saddle has tree-level action magnitude B = 24π².
Limits and open issues Geometry fixes the action and action-entropy identity; the exact boundary unit and contour are selected.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.; v1.4 (2026-08-03): dependency direction corrected: the action formula is primitive and no longer depends on SAD-03/SAD-04.; v1.4c (2026-08-04): public wording restated as the derived magnitude; the suppressing orientation belongs to SAD-02
SAD-02 Selected
Technical statement The defining Feynman-Lefschetz contour includes the compact round-S⁴ thimble with the decaying orientation e^(−|S_EH|).
Public statement The strict channel selects the suppressed compact S⁴ thimble: the damped weight e^(-B) = e⁻²⁴π².
Source in the paper CC Paper I v1.0,
Sec. 9, “Sourced Feynman/Lefschetz contour”
Limits and open issues Contour inclusion, orientation and Stokes data are not independently derived; the opposite Hartle-Hawking orientation would give e⁺²⁴π², about 10²⁰⁶ too large, and is a named falsifying alternative.
Version history v1.2; v1.3 (2026-08-01): open effects gain the named falsifying orientation.; v1.4c (2026-08-04): public wording states the selected damped weight explicitly
SAD-03 Derived
Technical statement Within the declared sub-Planckian saddle family x=M_P²/H² ≥ 1 with B(x)=24π² x, the leading sector is localized at the boundary x=1.
Public statement Within the declared saddle family, the least-suppressed sector lies at the Planck boundary.
Downstream claims none
Qualifier The smallest geometry within the specified family; not a proof that it is the smallest possible in every ultraviolet completion.
Limits and open issues Boundary dominance does not determine the unit assigned to the boundary.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.; v1.4 (2026-08-03): now derived from SAD-01 with the SAD-04 boundary; smallest-in-family scope stated.; v1.4b (2026-08-04): SAD-02 added: boundary localisation uses the damped orientation
SAD-04 Selected
Technical statement The ultraviolet boundary is assigned the dimensionless unit u_b=(κ²Λ)_boundary=1. The boundary value is not derived from the semiclassical action; the final result is exponentially sensitive to it, and its microscopic justification is a central open problem.
Public statement The strict channel fixes the boundary unit at κ²Λ = 1.
Limits and open issues Alternative O(1) boundary definitions are quantitatively different models, not a statistical uncertainty within the strict channel.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.; v1.4 (2026-08-03): stated as an independent selected boundary; sensitivity and open-problem status made explicit
SAD-05 Derived
Technical statement On the declared sub-Planckian domain z=e^(2σ₀) ≥ 1, the reduced damped zero-mode integral is finite: the Planck boundary removes the small-radius endpoint and the damped thimble controls infinity. Extension to z=0 reproduces the divergent R1 reading; there is no separate anomaly-integrability test passed by a_SM>0.
Public statement The declared Planck boundary, not the sign of the anomaly coefficient, makes the reduced thimble integral finite.
Downstream claims none
Qualifier Domain-regulated finiteness conditional on the selected contour and measure.
Limits and open issues Does not determine contour inclusion, orientation, Stokes data, boundary unit or full metric/ghost stability.
Version history v1.2; v1.3 (2026-08-01): corrected in the paired-cap revision: finiteness is regulated by the declared domain, not by the sign of a_SM.; v1.4b (2026-08-04): SM-01 removed (finiteness is regulated by the declared domain, not by the sign of a_SM); SAD-04 added for the boundary
FOLD-01 Derived
Technical statement Under the two-part matching hypothesis, the reflection-even local one-point coefficient lies on the unique fixed plane of Θ: s∗=L/2, equivalently h∗^2=m_p M_P; in multiplicative variables I(h)=m_p M_P/h.
Public statement Given the two-part matching hypothesis, the matched response scale is fixed at h∗^2 = m_p M_P.
Qualifier Derived conditional on both parts of the matching hypothesis; equality of the leading endpoint matching errors at the same point is a consistency check, not an additional selection rule.
Limits and open issues The fixed point inherits the hypothesis status of FOLD-02.
Version history v1.2; v1.3 (2026-08-01): restated for the paired-cap revision: fixed point conditional on endpoint conjugacy.; v1.3a (2026-08-02): aligned with the release manuscript (two-part matching hypothesis; citations updated)
FOLD-02 Proposed
Technical statement The two-part paired-cap matching hypothesis. Part one, endpoint conjugacy: the Planck cap state and the stable-proton trace state are Osterwalder–Schrader-conjugate ends of one finite dilation-transfer amplitude, supplying the reflection Θ: s ↦ L−s on s=ln(h/m_p), 0 ≤ s ≤ L=ln(M_P/m_p). Part two, reflection-even readout: the compact charge is represented by the reflection-even local one-point coefficient of the matched cap amplitude.
Public statement The strict channel adopts a two-part matching hypothesis: endpoint conjugacy supplies the reflected interval, and a reflection-even readout places the insertion on its fixed plane.
Qualifier Physical matching hypothesis in two parts, each exposed to microscopic derivation; the algebraic involution is its coordinate representation, not an independent rule.
Source in the paper CC Paper I v1.0,
Sec. 10 Limits and open issues Not a theorem; a microscopic derivation or refutation of the conjugacy would strengthen or falsify the channel.
Version history v1.2; v1.3 (2026-08-01): restated for the paired-cap revision: the folded readout becomes the endpoint-conjugacy hypothesis.; v1.3a (2026-08-02): aligned with the release manuscript (two-part matching hypothesis; citations updated)
SRC-00 Selected
Technical statement The strict compact channel declares the one-cap Einstein curvature charge as its source coordinate: C_+(h)=(1/4κ²) ∫_cap √ĝ R[ĝ] = 12π² x_h with x_h=M_P²/h², reference value C_+0=C_+(M_P)=12π², and conjugate derivative D_+=C_+0 d/dC_+. A constant rescaling of C_+ cancels between C_+0 and d/dC_+.
Public statement The strict channel reads the connected response through the one-cap Einstein curvature charge, whose overall normalisation cancels by construction.
Qualifier Within the operator basis retained through four derivatives, the curvature charge is the only nonconstant local geometric coordinate; six- and higher-derivative directions belong to the correction budget. A truncation clause, not a coordinate-invariant consequence of the anomaly.
Source in the paper CC Paper I v1.0,
Sec. 11, Proposition 4
Limits and open issues A different source coordinate would define a quantitatively different channel; the swap audit tabulates the alternatives.
Version history v1.2; v1.3 (2026-08-01): restated for the paired-cap revision: the inverse-curvature source derivative is replaced by the one-cap Einstein curvature charge.; v1.3a (2026-08-02): aligned with the release manuscript (two-part matching hypothesis; citations updated)
SRC-01 Derived
Technical statement Given the curvature-charge coordinate, the one-cap Euler response ΔΓ_A+ = A_m(h) ln x_h gives Ξ_WZ(h)=D_+ ΔΓ_A+ = A_m(h) h²/M_P²; at the reflection point h∗^2=m_p M_P this is A_m m_p/M_P. The equality applies to the projected Euler-cocycle term after beta-function, threshold and interaction responses have been separated; the curvature-charge derivative acts on the explicit ln x_h dependence.
Public statement In the declared curvature-charge coordinate, the anomaly response yields the factor h²/M_P².
Qualifier The coordinate is selected; the evaluation within it is derived.
Source in the paper CC Paper I v1.0,
Sec. 11, Proposition 4, Eqs.
(35) and
(36) Limits and open issues The h²/M_P² factor belongs to the declared charge coordinate; it is not a coordinate-invariant consequence of the Euler anomaly.
Version history v1.2; v1.3 (2026-08-01): restated for the paired-cap revision: evaluation through the curvature charge.; v1.4 (2026-08-03): projected-derivative reading made explicit: the derivative acts on the explicit ln x_h dependence of the separated Euler-cocycle term.; v1.4c (2026-08-04): FOLD-01 added: the technical statement evaluates at the reflection point h_*
NORM-01 Derived
Technical statement In the sector-normalised paired-cap ratio ΔΓ_n+ = -ln[Z_n+(h)/Z_n+(M_P)], every factor independent of the external source cancels identically: the reflected bra's normalisation and phase, metric and ghost determinant constants, collective-coordinate Jacobians, zero-mode volumes and finite local constants.
Public statement Source-independent determinants, normalisations and common phases cancel identically in the paired-cap response.
Limits and open issues The cancellation covers source-independent factors only; source-dependent terms remain physical corrections (NORM-02).
Version history v1.2; v1.3 (2026-08-01): restated for the paired-cap revision: cancellation is a property of the paired-cap ratio.; v1.4a (2026-08-03): defensive clause removed from the public wording per the positive-not-defensive prose doctrine.
NORM-02 Open calculation
Technical statement Source-dependent gravitational dressing of the cap response and higher-curvature corrections to the primitive action B_s are not fixed by the construction; higher primitive sectors enter at O(e^(−2B_s)).
Public statement The open gravitational corrections are the source-dependent dressing of the cap response and ultraviolet corrections to the primitive action; both are computable tests of the formula.
Qualifier Replaces the former unit relative-sector clause, which the paired-cap construction dissolves into a derived cancellation.
Limits and open issues A computed dressing away from unity or a shifted boundary action modifies or falsifies the formula; the 0.4% proximity is not evidence that these corrections are small.
Version history v1.2; v1.3 (2026-08-01): restated for the paired-cap revision: the unit relative-sector condition nu C_w=1 is dissolved by the paired-cap cancellation; this record now carries the residual open corrections.; v1.3a (2026-08-02): wording aligned with the release manuscript
NUM-01 Derived
Technical statement The leading protected compact response is q_WZ = a_SM (m_p/M_P) e⁻²⁴π² = 2.856 × 10⁻¹²².
Public statement Under the fixed strict-channel hypotheses, the calculation predicts 2.856 × 10⁻¹²² without fitting a parameter to the cosmological value. The identification of this number with the cosmological constant is a separate proposed step (MAP-01).
Qualifier Calculated prediction within selected channel; the observed value predates the model. Computes the compact charge q_WZ only; the identification with the residual curvature is the separate proposed step MAP-01.
Source in the paper CC Paper I v1.0, Proposition 5 and Eqs.
(40),
(48)-
(49) Limits and open issues Fixed hypotheses are testable but not retunable after comparison; the source-dependent gravitational dressing and boundary-action corrections remain uncomputed.
Version history v1.2; v1.3 (2026-08-01): public wording aligned to the card's 3-decimal value; dependencies updated to the paired-cap structure.; v1.4 (2026-08-03): global-source dependencies removed on internal review; the compact calculation stands before the curvature map (see MAP-01)
MAP-01 Proposed
Technical statement Given the proposed global four-form action and its derived field equation ⟨R⟩ = 4 M_P² q_WZ, the compact charge is identified with the residual curvature on a maximally symmetric vacuum: Λ/M_P² = q_WZ.
Public statement The proposed gravity-side completion identifies the calculated compact response with the residual curvature of empty space. This identification is a separate step from the compact calculation and carries the completion’s Proposed status.
Qualifier Distinct from NUM-01, which computes q_WZ within the compact channel; the identification inherits the global-source action and its source-charge normalisation. The late-time branch (GS-03) enters at the observational comparison (OBS-02).
Source in the paper CC Paper I v1.0, Theorem 6 and
Appendix E Limits and open issues The identification stands or falls with the proposed completion; a completion returning a different action-level normalisation changes the predicted curvature.
Version history v1.4 (2026-08-03): added on internal review; separates the global identification from the compact calculation NUM-01.; v1.6 (2026-08-08): GS-03 moved downstream to OBS-02: the proposed action alone gives the maximally symmetric vacuum relation; the trace-average branch enters at the late-time observational identification
GS-01 Proposed
Technical statement A compact four-form global-source action with rigid P² and the fixed dimensionless compact charge q_WZ is proposed to supply the scalar-curvature equation while adding no local propagating degree of freedom.
Public statement A proposed compact global-source action supplies the scalar-curvature mode omitted by the trace-free local equation.
Limits and open issues The action is a gravity-side postulate and is not derived from the extraction theorem.
Version history v1.2; v1.3 (2026-08-01): light restatement: the source is the fixed dimensionless charge.; v1.4 (2026-08-03): back-dependency on NUM-01 removed; the proposed action is stated independently of the computed value
GS-04 Proposed
Technical statement The action is written directly in the dimensionless charge with coefficient 8π (P²)²; replacing 8π by c gives Λ/M_P²=(c/(8π)) q_WZ, and the observable convention q_WZ = G Λ with G=(8π P²)^{-1} fixes c=8π at the action level. The sign is the Legendre pairing with the positive cap charge. The coefficient is fixed within the proposed source-charge convention q_WZ = G Λ; that convention is part of the gravity-side completion and is not derived by the compact anomaly calculation. It is not fitted after comparison with observation.
Public statement The coefficient and sign of the global-source term are specified at the action level by the stated source-charge convention, rather than imposed after variation.
Qualifier The coefficient audit is derived; the convention is part of the proposed gravity-side action.
Limits and open issues A microscopic parent must derive the action itself; the convention stands or falls with it.
Version history v1.2; v1.3 (2026-08-01): restated for the paired-cap revision: the Z_g unit matching is absorbed into the action-level coefficient.; v1.4 (2026-08-03): normalisation framing clarified: fixed within the proposed convention, part of the completion, not derived by the compact calculation.; v1.6 (2026-08-08): 'fixed' -> 'specified': the convention is part of the proposed completion
GS-02 Derived
Technical statement Variation of the proposed action yields ⟨R⟩=4 M_P² q_WZ and the vacuum-shift-invariant completed Einstein equation; a maximally symmetric vacuum satisfies Λ/M_P²=q_WZ.
Public statement Once the global-source action is adopted, its variations fix the residual curvature and cancel homogeneous vacuum shifts.
Source in the paper CC Paper I v1.0, Theorem 6 and
Appendix E Limits and open issues Derived from the proposed action; does not establish its microscopic origin.
Version history v1.2; v1.3 (2026-08-01): restated for the paired-cap revision: action-level form; the separate unit dictionary no longer appears.
GS-03 Selected
Technical statement The late-time observational identification requires the regulated non-vacuum trace average to vanish on the asymptotic branch.
Public statement The observed late-time value is identified with the compact response on a future asymptotic branch whose regulated non-vacuum trace average vanishes.
Qualifier Fixes the late-time amplitude; the w=-1 equation of state follows separately from the spacetime-constant residual source.
Limits and open issues Explicit cosmological branch condition.
Version history v1.2; v1.3 (2026-08-01): qualifier added: amplitude, not equation of state.
OBS-01 Standard input
Technical statement The Planck 2018 flat-ΛCDM-inferred dimensionless cosmological constant used in the paper is (Λ/M_P²)_obs=(2.846 ± 0.057) × 10⁻¹²².
Public statement Planck 2018 flat-ΛCDM-inferred value: 2.85 × 10⁻¹²².
Source in the paper CC Paper I v1.0, abstract, Eqs.
(14),
(48)-
(49) and Ref. [3]
Limits and open issues Observational error is not the theoretical uncertainty of the mechanism.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.
OBS-02 Derived
Technical statement The predicted central value differs from the Planck flat-ΛCDM inference by approximately 0.4%; numerically this equals about 0.18 times the quoted observational standard deviation. The 0.18 figure is a comparison with the observational error bar only, not a combined theoretical significance, likelihood ratio or goodness-of-fit statistic, because the theoretical uncertainty of the construction has not been calculated.
Public statement The strict-channel prediction and the Planck 2018 flat-ΛCDM-inferred central value differ by about 0.4%.
Downstream claims none
Qualifier Calculated within selected channel
Source in the paper CC Paper I v1.0, abstract, introduction and conclusion
Limits and open issues Not a sigma significance; structural theory effects are unquantified.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.; v1.4 (2026-08-03): restated per internal review: 0.18 defined as a multiple of the observational standard deviation, with the non-significance boundary explicit.; v1.4b (2026-08-04): comparison now inherits the curvature identification MAP-01 (which carries NUM-01, GS-02 and the late-time branch).; v1.6 (2026-08-08): GS-03 dependency received from MAP-01 (late-time branch belongs to the comparison); public wording names the flat-LambdaCDM inference, mirroring OBS-01
CORR-01 Open calculation
Technical statement Standard Model interaction/source-response corrections are provisionally estimated to be of order 0.5%; this is an order-of-magnitude estimate, not a calculated bound. Mass thresholds are at most about 3 × 10⁻¹⁵ and higher source powers are negligible at the matched scale. A complete calculation must include interacting Standard Model effects on S⁴, running couplings, operator mixing and any source-dependent corrections to the extraction map.
Public statement Matter-channel corrections are provisionally estimated at about half a percent; this is an order-of-magnitude estimate, not a calculated bound, and the full curved-background Standard Model calculation remains open.
Downstream claims none
Source in the paper CC Paper I v1.0, Secs.
5 and
13 Limits and open issues Estimate, not a complete interacting Standard Model calculation on S⁴.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.; v1.3a (2026-08-02): wording aligned with the release manuscript.; v1.4 (2026-08-03): estimate/bound distinction aligned with the paper: order-of-magnitude estimate, not a calculated bound
CORR-02 Open calculation
Technical statement Source-dependent gravitational dressing of the cap response and higher-curvature corrections to the primitive Planck-boundary action are not quantified in the leading matter-QFT truncation; source-independent determinants and phases cancel in the paired-cap ratio and are not open items.
Public statement The principal open gravitational corrections are the source-dependent gravitational dressing and ultraviolet corrections to the primitive action; source-independent factors cancel by construction.
Limits and open issues Must remain visible beside the number card; the 0.4% match cannot be used as evidence that the remaining source-dependent corrections are small.
Version history v1.2; v1.3 (2026-08-01): restated for the paired-cap revision: determinants cancel; the open items are the dressing and the boundary action.; v1.6 (2026-08-08): 'dominant structural unknowns' -> 'principal open gravitational corrections': their size is expressly unquantified
TEST-01 Test
Technical statement The proposed completion's residual source is spacetime constant on a fixed branch, so it predicts w(z)=-1 at every redshift where it is identified with dark energy (w_0=-1, w_a=0), with no dynamical dark-energy degree of freedom; the trace-average condition fixes the amplitude, not the equation of state.
Public statement The completion predicts w = −1 exactly, at all redshifts; persistent cross-dataset, systematics-robust evidence for evolving dark energy would falsify the proposed completion or its identification with the observed acceleration, leaving the extraction theorem intact.
Downstream claims none
Source in the paper CC Paper I v1.0, Secs.
12 and
13.2 Limits and open issues DESI DR2 combinations and the six-year DES analysis show dataset- and parameterisation-dependent 2.3-4.2 sigma preferences for evolution: a substantive tension, tracked on the site's observational panel.
Version history v1.2; v1.3 (2026-08-01): falsifier criterion restated across successive pre-release revisions: the former >5 sigma non-parametric threshold is replaced by persistent cross-dataset, systematics-robust evidence across flexible parametric and non-parametric reconstructions, because quoted sigma values are dataset- and parameterisation-dependent. w=-1 extended to finite redshift per the residual-source derivation.; v1.4b (2026-08-04): w=-1 rests on the proposed action and its derived field equation; GS-03 fixes the amplitude, not the equation of state.; v1.6 (2026-08-08): observational note 'material tension' -> 'substantive tension' (register idiom sweep, queued 2026-08-06)
TEST-02 Test
Technical statement Active weakly coupled particle content below h∗ shifts a_eff and the strict result; representative shifts include +1.7% for three light right-handed neutrinos, +8.3% for a fourth generation and about +18% for low-energy MSSM matter.
Public statement Additional active weakly coupled fields below the response scale shift the anomaly residue and the strict result once the source-dependent gravitational dressing and ultraviolet-action corrections are controlled.
Downstream claims none
Source in the paper CC Paper I v1.0,
Sec. 13.1 and Tables 5 and 7
Limits and open issues Conditional sensitivity, not a present BSM exclusion; heavy Majorana right-handed neutrinos can decouple above the response scale.
Version history v1.2; v1.3 (2026-08-01): control conditions restated per the paired-cap revision; h_* notation.
TEST-03 Test
Technical statement Because B=24π²/u_b, a fractional change of the exact boundary unit changes the strict central value exponentially; a roughly 4% output window corresponds to a boundary change of order 2 × 10⁻⁴.
Public statement The strict result is exponentially sensitive to the exact Planck-boundary definition.
Downstream claims none
Qualifier Tests the selected boundary model; it is not a fitted uncertainty.
Limits and open issues Alternative boundary definitions are different models with very different exponents.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.
CMP-01 Derived
Technical statement The framework is quantitatively closed within its specified channel because it supplies an explicit element for the vacuum-energy quotient, protected matter datum, infrared hierarchy, gravitational weight and residual-curvature equation.
Public statement The proposal closes the full numerical chain within a specified compact channel while exposing the physical selections that remain.
Downstream claims none
Qualifier Comparative synthesis; not a theorem or claim of unique microscopic selection.
Source in the paper CC Paper I v1.0, Secs.
1,
14 and
15 Limits and open issues The paper supplies a sharply specified candidate explanation; microscopic selection remains open.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.
METH-01 Author disclosure
Technical statement The published work was developed through human-AI collaboration under the sole author’s judgement and responsibility; model participation is not scientific evidence for the claims.
Public statement The research was developed through human-AI collaboration; the author remains responsible for every published claim.
Downstream claims none
Source in the paper Website Strategy v11, Research Method
Limits and open issues Operational workflow is not a credibility substitute; the work is not presented as AI-verified.
Version history v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.; v1.6 (2026-08-08): reclassified from Standard input to the non-scientific record type Author disclosure; the six scientific statuses remain a closed set
INT-01 Derived
Technical statement In the completed model, Λ is the residual vacuum curvature fixed by the global-source equation, not a separately observable absolute zero-point-energy sum.
Public statement In this construction the cosmological constant is the residual curvature of the vacuum, fixed by the global-source equation.
Downstream claims none
Qualifier Given the proposed global-source action.
Source in the paper CC Paper I v1.0, Secs.
12 and
14 Limits and open issues Interpretation of the assembled result; not required for the extraction theorem.
Version history v1.3 (2026-08-01): added.
INT-02 Derived
Technical statement Within the strict channel the 122-decade suppression decomposes as approximately 103 decades of compact-saddle suppression plus 19 decades of the QCD-to-Planck ratio, weighted by the rational Standard Model Euler residue.
Public statement The 122-decade hierarchy separates into about 103 decades from compact gravity and 19 from the QCD-to-Planck ratio.
Downstream claims none
Qualifier Arithmetic decomposition of the strict result.
Source in the paper CC Paper I v1.0,
Sec. 1 Limits and open issues None beyond those of NUM-01.
Version history v1.3 (2026-08-01): added.
INT-03 Derived
Technical statement The product e⁻²⁴π² (m_p/M_P) may be written as one combined non-perturbative exponential, e^(−(S_grav+S_trans)) with S_grav=24π² and S_trans ~= 44.
Public statement The two suppressions can be written as a single combined exponential.
Downstream claims none
Qualifier Does not imply a single underlying thermal ensemble; the exponents have distinct origins.
Source in the paper CC Paper I v1.0, Secs.
1-
2 Limits and open issues Rewriting, not an additional derivation.
Version history v1.3 (2026-08-01): added.; v1.6 (2026-08-08): 'combine into a single non-perturbative exponential' -> 'can be written as': an algebraic rewriting, not one underlying process
INT-04 Standard input
Technical statement For the selected round-S⁴ saddle, |S_E| = 3π M_P²/Λ = S_dS, with the selected damped orientation (SAD-02) the sector weight is e^(−S_dS); at the selected boundary S_dS=24π². An exact identity for the classical on-shell saddle in the stated conventions; not asserted as an all-orders quantum-gravity identity.
Public statement With the selected damped orientation, the compact weight is the entropy suppression e^(−S_dS), where S_dS is the de Sitter entropy of the compact saddle.
Downstream claims none
Qualifier The entropy belongs to the ultraviolet compact saddle, not the observed late-time universe.
Limits and open issues Standard geometric identity applied to the selected sector; not a separate derivation.
Version history v1.3 (2026-08-01): added.; v1.4 (2026-08-03): classical on-shell scope stated.; v1.4b (2026-08-04): suppressing exponential made explicitly conditional on the selected orientation; SAD-02 added.; v1.6 (2026-08-08): public wording corrected from 'the exponential of the de Sitter entropy' (which reads as exp(+S)) to the explicit suppression exp(-S_dS)
CMP-02 Derived
Technical statement The construction yields the trace-free local equation of unimodular-family approaches and proposes the compact response as the value of their undetermined scalar datum: with the stated source-charge convention, ⟨R⟩=4 M_P² q_WZ.
Public statement The compact response acts as a candidate selector for the residual curvature that trace-free and sequestering formulations leave undetermined.
Downstream claims none
Qualifier Given the proposed global-source action.
Source in the paper CC Paper I v1.0, Secs.
1 and
14 Limits and open issues Does not establish that the proposed action is the unique completion of trace-free gravity.
Version history v1.3 (2026-08-01): added; v1.3a (2026-08-02): wording aligned with the release manuscript.
CMP-03 Test
Technical statement Any microscopic completion of the strict channel must supply the contour orientation and Stokes data, the exact Planck-boundary action, the source-dependent gravitational dressing, the two-part paired-cap matching (endpoint conjugacy and reflection-even readout) and the four-form source; a completion returning different values falsifies the strict channel.
Public statement The construction exposes definite quantities for a parent theory to compute; a parent theory that computes them can either derive the strict channel or rule it out.
Downstream claims none
Qualifier Falsification-or-derivation test for microscopic completions.
Source in the paper CC Paper I v1.0, Secs.
14-
15 Limits and open issues None; this is the channel's exposure statement.
Version history v1.3 (2026-08-01): added.; v1.3a (2026-08-02): aligned with the release manuscript (two-part matching hypothesis; citations updated).; v1.6 (2026-08-08): scoped to microscopic completions of the strict channel rather than every conceivable UV completion