[ { "Claim ID": "VAC-01", "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.", "Approved public wording": "Absolute zero-point energy is not by itself a scheme-independent observable.", "Controlled status": "Standard input", "Qualifier": "", "Dependencies": "", "Primary internal source": "CC Paper I v1.0, Sec. 3, lines/section “compact-S4 vacuum-energy ambiguity”", "External foundations": "BirrellDavies1982; Wald1994; BianchiRovelli2010; HollandsWald2004", "Open effects / objections": "Do not broaden this to “zero-point energy does not exist” or “never gravitates”.", "Version history": "v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.", "Relations": "" }, { "Claim ID": "VAC-02", "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.", "Approved public wording": "On compact round S4, the ambiguity is especially sharp: the zero-point term and cosmological counterterm are literally the same local operator.", "Controlled status": "Derived here", "Qualifier": "", "Dependencies": "VAC-01", "Primary internal source": "CC Paper I v1.0, Sec. 3", "External foundations": "BirrellDavies1982; Wald1994", "Open effects / objections": "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.", "Relations": "" }, { "Claim ID": "VAC-03", "Technical statement": "Scheme-independent energy differences, phase-transition dynamics, interfaces and inhomogeneous stresses are not removed by quotienting a strictly homogeneous constant shift.", "Approved public wording": "The argument does not erase Casimir effects, phase transitions or inhomogeneous stresses.", "Controlled status": "Standard input", "Qualifier": "", "Dependencies": "VAC-01", "Primary internal source": "CC Paper I v1.0, Sec. 3, “Casimir objection”", "External foundations": "Casimir1948; Jaffe2005", "Open effects / objections": "Keep visible beside VAC-01/VAC-02.", "Version history": "v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.", "Relations": "" }, { "Claim ID": "EXT-01", "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.", "Approved public wording": "Within the stated compact response class, one nonzero universal matter datum survives: the type-A Euler anomaly coefficient.", "Controlled status": "Derived here", "Qualifier": "", "Dependencies": "VAC-01; VAC-02; conditions C1 and C2", "Primary internal source": "CC Paper I v1.0, Theorem “Euler extraction on conformally flat S4”", "External foundations": "Duff1994; Deser1993; WessZumino1971", "Open effects / objections": "Class-relative uniqueness; do not state universal uniqueness across all observables or backgrounds.", "Version history": "v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.", "Relations": "" }, { "Claim ID": "EXT-02", "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.", "Approved public wording": "A constant-Weyl projector removes local counterterm contamination and isolates the Euler response.", "Controlled status": "Derived here", "Qualifier": "", "Dependencies": "EXT-01", "Primary internal source": "CC Paper I v1.0, Sec. 4, Eq. (10) and proof", "External foundations": "WessZumino1971; Riegert1984; Osborn1991; Aminov2026; Frob2025", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "EXT-03", "Technical statement": "Let V_loc = span{E_4, C^2, 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 C^2 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.", "Approved public wording": "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.", "Controlled status": "Derived here", "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.", "Dependencies": "", "Relations": "extends EXT-01 (non-dependency): cross-realisation extension and observer-centred boundary-sector stress test; not required for the downstream numerical derivation.", "Primary internal source": "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", "Open effects / objections": "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" }, { "Claim ID": "SM-01", "Technical statement": "For the minimal Standard Model field census, a_SM = 4/360 + 45(11/720) + 12(31/180) = 1991/720 = 2.76528.", "Approved public wording": "For the minimal Standard Model free-field census, the protected type-A coefficient is a_SM = 1991/720.", "Controlled status": "Derived here", "Qualifier": "", "Dependencies": "EXT-01", "Primary internal source": "CC Paper I v1.0, Sec. 5, Eq. (13) and Appendix B", "External foundations": "Brown1977; DowkerCritchley1977; Christensen1980; Fradkin1984; Ferrero2024; Larue2023; PDG2026", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "IR-01", "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.", "Approved public wording": "A separate infrared input is required; the direct Planck-scale mechanisms tested do not generate the needed linear QCD hierarchy.", "Controlled status": "Derived here", "Qualifier": "", "Dependencies": "EXT-01", "Primary internal source": "CC Paper I v1.0, Sec. 6", "External foundations": "Vassilevich2003; Buttazzo2013; tHooft1976", "Open effects / objections": "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.", "Relations": "" }, { "Claim ID": "QCD-01", "Technical statement": "QCD dimensional transmutation generates a nonperturbative infrared scale; using the current proton and unreduced Planck masses gives m_p/M_P = 7.685e-20.", "Approved public wording": "QCD dimensional transmutation produces the infrared scale hierarchy; the measured proton-to-Planck ratio is m_p/M_P = 7.685e-20.", "Controlled status": "Standard input", "Qualifier": "", "Dependencies": "", "Primary internal source": "CC Paper I v1.0, Sec. 8, Eqs. (19)-(21)", "External foundations": "GrossWilczek1973; Politzer1973; PDG2026; Ji1995; Yang2018; Wang2024; Tanaka2025; Bollweg2026", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "QCD-02", "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 =2 m_p^2; the neutron and eta-prime lie numerically close but fail the stability criterion, and the quoted pure-Yang-Mills glueball scale is not a full-Standard-Model asymptotic pole.", "Approved public wording": "The strict QCD endpoint is the proton: the lightest stable hadronic colour-singlet pole satisfying the stated zero-temperature source criteria.", "Controlled status": "Selected", "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.", "Dependencies": "QCD-01; endpoint criteria P/S/C", "Primary internal source": "CC Paper I v1.0, Secs. 8, 10, 13 and Appendix D", "External foundations": "PDG2026; Ji1995; Yang2018; Bhattacharya2014; MorningstarPeardon1999; Wang2024; Tanaka2025; Bollweg2026", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "GR-01", "Technical statement": "Ordinary local type-A anomaly backreaction on a maximally symmetric de Sitter branch scales as Lambda_g^2/M_P^2 and produces no small branch linear in a suppressed compact weight.", "Approved public wording": "Ordinary local anomaly backreaction has the wrong scaling: it produces an H^4 stress, not the required small linear-curvature branch.", "Controlled status": "Derived here", "Qualifier": "", "Dependencies": "EXT-01", "Primary internal source": "CC Paper I v1.0, Proposition “Ordinary-GR no-go”", "External foundations": "Duff1994", "Open effects / objections": "Applies to ordinary local semiclassical Einstein variation.", "Version history": "v1.2; v1.3 (2026-08-01): source citation updated to the paired-cap revision.", "Relations": "" }, { "Claim ID": "GR-02", "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.", "Approved public wording": "Removing the volume-counterterm representative leaves a unique trace-free local equation and one undetermined scalar-curvature mode.", "Controlled status": "Derived here", "Qualifier": "", "Dependencies": "VAC-01", "Primary internal source": "CC Paper I v1.0, Lemma “Trace-free quotient projector”", "External foundations": "Percacci2018; Jirousek2023", "Open effects / objections": "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.", "Relations": "" }, { "Claim ID": "SAD-01", "Technical statement": "For the round S4 Einstein saddle, |S_EH| = 24 pi^2/(kappa^2 Lambda); at the declared Planck-curvature representative u=kappa^2 Lambda=1, the tree-level exponent is B=24 pi^2.", "Approved public wording": "At the specified Planck-curvature representative, the round-S4 saddle has tree-level action magnitude B = 24 pi^2.", "Controlled status": "Derived here", "Qualifier": "", "Dependencies": "", "Primary internal source": "CC Paper I v1.0, Sec. 9, Eqs. (22)-(25)", "External foundations": "GH1977; GHP1978; Chern1945", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "SAD-02", "Technical statement": "The defining Feynman-Lefschetz contour includes the compact round-S4 thimble with the decaying orientation e^{-|S_EH|}.", "Approved public wording": "The strict channel selects the suppressed compact S4 thimble: the damped weight e^(-B) = e^(-24 pi^2).", "Controlled status": "Selected", "Qualifier": "", "Dependencies": "SAD-01", "Primary internal source": "CC Paper I v1.0, Sec. 9, “Sourced Feynman/Lefschetz contour”", "External foundations": "Witten2011; FLT2017; DDHHHT2017; Lehners2023", "Open effects / objections": "Contour inclusion, orientation and Stokes data are not independently derived; the opposite Hartle-Hawking orientation would give exp(+24 pi^2), about 10^206 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", "Relations": "" }, { "Claim ID": "SAD-03", "Technical statement": "Within the declared sub-Planckian saddle family x=M_P^2/H^2 >=1 with B(x)=24 pi^2 x, the leading sector is localized at the boundary x=1.", "Approved public wording": "Within the declared saddle family, the least-suppressed sector lies at the Planck boundary.", "Controlled status": "Derived here", "Qualifier": "The smallest geometry within the specified family; not a proof that it is the smallest possible in every ultraviolet completion.", "Dependencies": "SAD-01; SAD-02; SAD-04", "Primary internal source": "CC Paper I v1.0, Sec. 9 and Appendix D.4", "External foundations": "", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "SAD-04", "Technical statement": "The ultraviolet boundary is assigned the dimensionless unit u_b=(kappa^2 Lambda)_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.", "Approved public wording": "The strict channel fixes the boundary unit at kappa^2 Lambda = 1.", "Controlled status": "Selected", "Qualifier": "", "Dependencies": "", "Primary internal source": "CC Paper I v1.0, Sec. 9 and Appendix D.5", "External foundations": "", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "SAD-05", "Technical statement": "On the declared sub-Planckian domain z=e^{2 sigma_0}>=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.", "Approved public wording": "The declared Planck boundary, not the sign of the anomaly coefficient, makes the reduced thimble integral finite.", "Controlled status": "Derived here", "Qualifier": "Domain-regulated finiteness conditional on the selected contour and measure.", "Dependencies": "SAD-02; SAD-04", "Primary internal source": "CC Paper I v1.0, Sec. 9, Eq. (26) and Appendix C", "External foundations": "GHP1978; MazurMottola1990; AntoniadisMazurMottola1997", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "FOLD-01", "Technical statement": "Under the two-part matching hypothesis, the reflection-even local one-point coefficient lies on the unique fixed plane of Theta: s_*=L/2, equivalently h_*^2=m_p M_P; in multiplicative variables I(h)=m_p M_P/h.", "Approved public wording": "Given the two-part matching hypothesis, the matched response scale is fixed at h_*^2 = m_p M_P.", "Controlled status": "Derived here", "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.", "Dependencies": "QCD-02; SAD-04; FOLD-02", "Primary internal source": "CC Paper I v1.0, Sec. 10 and Appendix D", "External foundations": "", "Open effects / objections": "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)", "Relations": "" }, { "Claim ID": "FOLD-02", "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 Theta: 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.", "Approved public wording": "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.", "Controlled status": "Proposed", "Qualifier": "Physical matching hypothesis in two parts, each exposed to microscopic derivation; the algebraic involution is its coordinate representation, not an independent rule.", "Dependencies": "QCD-02; SAD-04", "Primary internal source": "CC Paper I v1.0, Sec. 10", "External foundations": "", "Open effects / objections": "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)", "Relations": "" }, { "Claim ID": "SRC-00", "Technical statement": "The strict compact channel declares the one-cap Einstein curvature charge as its source coordinate: C_+(h)=(1/(4 kappa^2)) Int_cap sqrt(g-hat) R[g-hat] = 12 pi^2 x_h with x_h=M_P^2/h^2, reference value C_+0=C_+(M_P)=12 pi^2, and conjugate derivative D_+=C_+0 d/dC_+. A constant rescaling of C_+ cancels between C_+0 and d/dC_+.", "Approved public wording": "The strict channel reads the connected response through the one-cap Einstein curvature charge, whose overall normalisation cancels by construction.", "Controlled status": "Selected", "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.", "Dependencies": "SAD-01", "Primary internal source": "CC Paper I v1.0, Sec. 11, Proposition 4", "External foundations": "", "Open effects / objections": "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)", "Relations": "" }, { "Claim ID": "SRC-01", "Technical statement": "Given the curvature-charge coordinate, the one-cap Euler response DeltaGamma_A+ = A_m(h) ln x_h gives Xi_WZ(h)=D_+ DeltaGamma_A+ = A_m(h) h^2/M_P^2; 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.", "Approved public wording": "In the declared curvature-charge coordinate, the anomaly response yields the factor h^2/M_P^2.", "Controlled status": "Derived here", "Qualifier": "The coordinate is selected; the evaluation within it is derived.", "Dependencies": "EXT-02; SRC-00; FOLD-01", "Primary internal source": "CC Paper I v1.0, Sec. 11, Proposition 4, Eqs. (35) and (36)", "External foundations": "Riegert1984", "Open effects / objections": "The h^2/M_P^2 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_*", "Relations": "" }, { "Claim ID": "NORM-01", "Technical statement": "In the sector-normalised paired-cap ratio DeltaGamma_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.", "Approved public wording": "Source-independent determinants, normalisations and common phases cancel identically in the paired-cap response.", "Controlled status": "Derived here", "Qualifier": "", "Dependencies": "SRC-01", "Primary internal source": "CC Paper I v1.0, Sec. 11 and Appendix A", "External foundations": "", "Open effects / objections": "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.", "Relations": "" }, { "Claim ID": "NORM-02", "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(exp(-2 B_s)).", "Approved public wording": "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.", "Controlled status": "Open calculation", "Qualifier": "Replaces the former unit relative-sector clause, which the paired-cap construction dissolves into a derived cancellation.", "Dependencies": "SAD-02; NORM-01", "Primary internal source": "CC Paper I v1.0, Sec. 11 and Appendix A", "External foundations": "Anninos2022; Law2021; Maldacena2024; Chen2025; GiombiSun2026", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "NUM-01", "Technical statement": "The leading protected compact response is q_WZ = a_SM (m_p/M_P) exp(-24 pi^2) = 2.856e-122.", "Approved public wording": "Under the fixed strict-channel hypotheses, the calculation predicts 2.856 x 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).", "Controlled status": "Derived here", "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.", "Dependencies": "SM-01; QCD-02; SAD-02; SAD-04; FOLD-01; FOLD-02; SRC-00; SRC-01; NORM-01", "Primary internal source": "CC Paper I v1.0, Proposition 5 and Eqs. (40), (48)-(49)", "External foundations": "", "Open effects / objections": "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)", "Relations": "" }, { "Claim ID": "MAP-01", "Technical statement": "Given the proposed global four-form action and its derived field equation = 4 M_P^2 q_WZ, the compact charge is identified with the residual curvature on a maximally symmetric vacuum: Lambda/M_P^2 = q_WZ.", "Approved public wording": "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.", "Controlled status": "Proposed", "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).", "Dependencies": "NUM-01; GS-02", "Primary internal source": "CC Paper I v1.0, Theorem 6 and Appendix E", "External foundations": "HenneauxTeitelboim1989; KaloperPadilla2014; KaloperPadillaStefanyszynZahariade2016", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "GS-01", "Technical statement": "A compact four-form global-source action with rigid P^2 and the fixed dimensionless compact charge q_WZ is proposed to supply the scalar-curvature equation while adding no local propagating degree of freedom.", "Approved public wording": "A proposed compact global-source action supplies the scalar-curvature mode omitted by the trace-free local equation.", "Controlled status": "Proposed", "Qualifier": "", "Dependencies": "GR-02", "Primary internal source": "CC Paper I v1.0, Sec. 12 and Appendix E", "External foundations": "HenneauxTeitelboim1989; KaloperPadilla2014; KaloperPadillaStefanyszynZahariade2016", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "GS-04", "Technical statement": "The action is written directly in the dimensionless charge with coefficient 8 pi (P^2)^2; replacing 8 pi by c gives Lambda/M_P^2=(c/(8 pi)) q_WZ, and the observable convention q_WZ = G Lambda with G=(8 pi P^2)^{-1} fixes c=8 pi 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 Lambda; 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.", "Approved public wording": "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.", "Controlled status": "Proposed", "Qualifier": "The coefficient audit is derived; the convention is part of the proposed gravity-side action.", "Dependencies": "GS-01", "Primary internal source": "CC Paper I v1.0, Sec. 12 and Appendix E", "External foundations": "HenneauxTeitelboim1989; KaloperPadilla2014; KaloperPadillaStefanyszynZahariade2016", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "GS-02", "Technical statement": "Variation of the proposed action yields =4 M_P^2 q_WZ and the vacuum-shift-invariant completed Einstein equation; a maximally symmetric vacuum satisfies Lambda/M_P^2=q_WZ.", "Approved public wording": "Once the global-source action is adopted, its variations fix the residual curvature and cancel homogeneous vacuum shifts.", "Controlled status": "Derived here", "Qualifier": "", "Dependencies": "GS-01; GS-04", "Primary internal source": "CC Paper I v1.0, Theorem 6 and Appendix E", "External foundations": "", "Open effects / objections": "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.", "Relations": "" }, { "Claim ID": "GS-03", "Technical statement": "The late-time observational identification requires the regulated non-vacuum trace average to vanish on the asymptotic branch.", "Approved public wording": "The observed late-time value is identified with the compact response on a future asymptotic branch whose regulated non-vacuum trace average vanishes.", "Controlled status": "Selected", "Qualifier": "Fixes the late-time amplitude; the w=-1 equation of state follows separately from the spacetime-constant residual source.", "Dependencies": "GS-02", "Primary internal source": "CC Paper I v1.0, Sec. 12 and Appendix E", "External foundations": "", "Open effects / objections": "Explicit cosmological branch condition.", "Version history": "v1.2; v1.3 (2026-08-01): qualifier added: amplitude, not equation of state.", "Relations": "" }, { "Claim ID": "OBS-01", "Technical statement": "The Planck 2018 flat-LambdaCDM-inferred dimensionless cosmological constant used in the paper is (Lambda/M_P^2)_obs=(2.846 +/- 0.057)e-122.", "Approved public wording": "Planck 2018 flat-LambdaCDM-inferred value: 2.85 x 10^-122.", "Controlled status": "Standard input", "Qualifier": "", "Dependencies": "", "Primary internal source": "CC Paper I v1.0, abstract, Eqs. (14), (48)-(49) and Ref. [3]", "External foundations": "Planck2018", "Open effects / objections": "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.", "Relations": "" }, { "Claim ID": "OBS-02", "Technical statement": "The predicted central value differs from the Planck flat-LambdaCDM 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.", "Approved public wording": "The strict-channel prediction and the Planck 2018 flat-LambdaCDM-inferred central value differ by about 0.4%.", "Controlled status": "Derived here", "Qualifier": "Calculated within selected channel", "Dependencies": "MAP-01; GS-03; OBS-01", "Primary internal source": "CC Paper I v1.0, abstract, introduction and conclusion", "External foundations": "Planck2018", "Open effects / objections": "Never express this as 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", "Relations": "" }, { "Claim ID": "CORR-01", "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 3e-15 and higher source powers are negligible at the matched scale. A complete calculation must include interacting Standard Model effects on S^4, running couplings, operator mixing and any source-dependent corrections to the extraction map.", "Approved public wording": "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.", "Controlled status": "Open calculation", "Qualifier": "", "Dependencies": "SM-01; SRC-01", "Primary internal source": "CC Paper I v1.0, Secs. 5 and 13", "External foundations": "Frob2025; JackPoole2015; Buttazzo2013; PDG2026", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "CORR-02", "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.", "Approved public wording": "The principal open gravitational corrections are the source-dependent gravitational dressing and ultraviolet corrections to the primitive action; source-independent factors cancel by construction.", "Controlled status": "Open calculation", "Qualifier": "", "Dependencies": "SAD-01; NORM-02", "Primary internal source": "CC Paper I v1.0, Secs. 9, 11, 13 and Appendix A", "External foundations": "AnninosDenefLawSun2022; Law2021; Maldacena2024", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "TEST-01", "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.", "Approved public wording": "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.", "Controlled status": "Test", "Qualifier": "", "Dependencies": "GS-02", "Primary internal source": "CC Paper I v1.0, Secs. 12 and 13.2", "External foundations": "DESIDR2; PantheonPlus2022; DES2026; CortesLiddle2025; ZhongJain2025; ACT2025", "Open effects / objections": "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)", "Relations": "" }, { "Claim ID": "TEST-02", "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.", "Approved public wording": "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.", "Controlled status": "Test", "Qualifier": "", "Dependencies": "SM-01; CORR-02", "Primary internal source": "CC Paper I v1.0, Sec. 13.1 and Tables 5 and 7", "External foundations": "PDG2026; AppelquistCarazzone1975; GiacchiniShapiro2022; Hiller2024", "Open effects / objections": "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.", "Relations": "" }, { "Claim ID": "TEST-03", "Technical statement": "Because B=24 pi^2/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 2e-4.", "Approved public wording": "The strict result is exponentially sensitive to the exact Planck-boundary definition.", "Controlled status": "Test", "Qualifier": "Tests the selected boundary model; it is not a fitted uncertainty.", "Dependencies": "SAD-04; NUM-01", "Primary internal source": "CC Paper I v1.0, Sec. 13.2, “Boundary-unit sensitivity” and Appendix D.5", "External foundations": "", "Open effects / objections": "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.", "Relations": "" }, { "Claim ID": "CMP-01", "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.", "Approved public wording": "The proposal closes the full numerical chain within a specified compact channel while exposing the physical selections that remain.", "Controlled status": "Derived here", "Qualifier": "Comparative synthesis; not a theorem or claim of unique microscopic selection.", "Dependencies": "VAC-02; EXT-01; QCD-02; SAD-01; GS-02; NUM-01", "Primary internal source": "CC Paper I v1.0, Secs. 1, 14 and 15", "External foundations": "Weinberg1989; Percacci2018; KaloperPadilla2014; Weinberg1987; Bousso2000; Schutzhold2002; GH1977", "Open effects / objections": "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.", "Relations": "" }, { "Claim ID": "METH-01", "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.", "Approved public wording": "The research was developed through human-AI collaboration; the author remains responsible for every published claim.", "Controlled status": "Author disclosure", "Qualifier": "", "Dependencies": "", "Primary internal source": "Website Strategy v11, Research Method", "External foundations": "", "Open effects / objections": "Do not disclose operational workflow as a credibility substitute or call the work 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", "Relations": "" }, { "Claim ID": "INT-01", "Technical statement": "In the completed model, Lambda is the residual vacuum curvature fixed by the global-source equation, not a separately observable absolute zero-point-energy sum.", "Approved public wording": "In this construction the cosmological constant is the residual curvature of the vacuum, fixed by the global-source equation.", "Controlled status": "Derived here", "Qualifier": "Given the proposed global-source action.", "Dependencies": "GS-01; GS-02", "Primary internal source": "CC Paper I v1.0, Secs. 12 and 14", "External foundations": "", "Open effects / objections": "Interpretation of the assembled result; not required for the extraction theorem.", "Version history": "v1.3 (2026-08-01): added.", "Relations": "" }, { "Claim ID": "INT-02", "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.", "Approved public wording": "The 122-decade hierarchy separates into about 103 decades from compact gravity and 19 from the QCD-to-Planck ratio.", "Controlled status": "Derived here", "Qualifier": "Arithmetic decomposition of the strict result.", "Dependencies": "NUM-01", "Primary internal source": "CC Paper I v1.0, Sec. 1", "External foundations": "", "Open effects / objections": "None beyond those of NUM-01.", "Version history": "v1.3 (2026-08-01): added.", "Relations": "" }, { "Claim ID": "INT-03", "Technical statement": "The product exp(-24 pi^2) (m_p/M_P) may be written as one combined non-perturbative exponential, exp(-(S_grav+S_trans)) with S_grav=24 pi^2 and S_trans ~= 44.", "Approved public wording": "The two suppressions can be written as a single combined exponential.", "Controlled status": "Derived here", "Qualifier": "Does not imply a single underlying thermal ensemble; the exponents have distinct origins.", "Dependencies": "NUM-01", "Primary internal source": "CC Paper I v1.0, Secs. 1-2", "External foundations": "", "Open effects / objections": "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", "Relations": "" }, { "Claim ID": "INT-04", "Technical statement": "For the selected round-S4 saddle, |S_E| = 3 pi M_P^2/Lambda = S_dS, with the selected damped orientation (SAD-02) the sector weight is exp(-S_dS); at the selected boundary S_dS=24 pi^2. An exact identity for the classical on-shell saddle in the stated conventions; not asserted as an all-orders quantum-gravity identity.", "Approved public wording": "With the selected damped orientation, the compact weight is the entropy suppression exp(-S_dS), where S_dS is the de Sitter entropy of the compact saddle.", "Controlled status": "Standard input", "Qualifier": "The entropy belongs to the ultraviolet compact saddle, not the observed late-time universe.", "Dependencies": "SAD-01; SAD-02", "Primary internal source": "CC Paper I v1.0, Sec. 9, Eq. (24)", "External foundations": "", "Open effects / objections": "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)", "Relations": "" }, { "Claim ID": "CMP-02", "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, =4 M_P^2 q_WZ.", "Approved public wording": "The compact response acts as a candidate selector for the residual curvature that trace-free and sequestering formulations leave undetermined.", "Controlled status": "Derived here", "Qualifier": "Given the proposed global-source action.", "Dependencies": "GR-02; GS-02", "Primary internal source": "CC Paper I v1.0, Secs. 1 and 14", "External foundations": "", "Open effects / objections": "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.", "Relations": "" }, { "Claim ID": "CMP-03", "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.", "Approved public wording": "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.", "Controlled status": "Test", "Qualifier": "Falsification-or-derivation test for microscopic completions.", "Dependencies": "SAD-02; GS-01; FOLD-02; NORM-02", "Primary internal source": "CC Paper I v1.0, Secs. 14-15", "External foundations": "", "Open effects / objections": "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", "Relations": "" } ]