Can the cosmological constant be calculated from theory rather than fitted to observation?

Claim. A compact extraction is physically meaningful only if it is invariant under every allowed finite local counterterm.

Status. Standard renormalisation input plus a compact-\(S^4\) classification derived here.

For a four-dimensional QFT on a curved background, \[ \Delta\Gamma =\alpha\!\int\!\sqrt g +\beta\!\int\!\sqrt g\,R +\gamma\!\int\!\sqrt g\,R^2 +\delta\!\int\!\sqrt g\,W^2 +\epsilon\!\int\!\sqrt g\,E_4 +\int\!\sqrt g\,\Box R . \]

A constant matter-Lagrangian shift changes only the coefficient of \(\int\sqrt g\). On compact \(S^4\), the putative absolute vacuum-energy functional and the cosmological counterterm are therefore the same operator. A well-posed extraction must quotient this direction and every other finite local ambiguity before assigning physical significance to a response.

Limit. The quotient removes an absolute homogeneous offset. It does not remove physical energy differences, phase-transition dynamics or inhomogeneous stress.

Claims: VAC-01, EXT-01.
Full paper: Compact vacuum-energy ambiguity · Local counterterms.

Claim. The Casimir effect measures a scheme-independent difference and does not establish that a single absolute compact vacuum-energy value is observable.

The plate force compares two configurations, so a local cosmological counterterm contributes equally to both and cancels in the subtraction. A closed round \(S^4\) partition function is one compact configuration and contains no intrinsic second geometry that performs the subtraction.

The same distinction preserves physical QCD phase differences. A change between confined and deconfined phases, an interface or a spacetime-dependent transition is not a uniform shift of the entire matter Lagrangian. Such effects remain in the local stress tensor and continue to gravitate.

Limit. Vacuum phenomena remain real and physical; the claim is that an absolute homogeneous zero-point offset cannot be separated from the local cosmological counterterm by the compact calculation alone.

Claims: VAC-02.
Full paper: The Casimir objection.

The paper tests three direct readings of the compact partition function.

1. Free energy per spatial volume. Interpreting \(-T\ln Z/V_3\) as a vacuum density restores a Planck-scale result: the zero-mode integral leaves an order-one partition function, while the Gibbons–Hawking temperature and spatial volume carry the dimensional scale.

2. The full effective action at the matched response scale. Substituting \(h_*^2=m_pM_P\) into the classical compact action gives \(24\pi^2M_P/m_p\sim3\times10^{21}\), an over-suppression by about \(10^{21}\) orders of magnitude. This confuses the response-evaluation scale with the curvature label of the selected gravitational sector.

3. The partition function itself. The anomaly contribution behaves as \(Z_{\mathrm{matter}}\propto(H/\mu)^{4a_{\mathrm{SM}}}\). No standard thermodynamic operation turns a quantity raised to \(a_{\mathrm{SM}}\) into a response linear in \(a_{\mathrm{SM}}\).

The common problem is that \(Z\) and \(\Gamma=-\ln Z\) contain local scheme- and measure-dependent pieces. The website therefore follows the protected connected-response construction rather than treating the raw partition function as the observable.

Claims: VAC-01, EXT-01, SRC-01.
Full paper: Failed compact identifications.

Euler extraction theorem. On round conformally flat \(S^4(H)\), among local linear constant-Weyl responses of the renormalised effective action satisfying counterterm invariance and conformal-flatness universality, the unique nonvanishing \(H\)-independent scheme-independent matter datum is the type-A Euler anomaly coefficient.

The two conditions, exactly as the paper defines them.

C1 (counterterm/scheme invariance): the extraction functional \(\mathcal F\) is invariant under all local counterterm ambiguities of the six-term basis shown in the counterterm panel above, including \(\alpha\!\int\!\sqrt g\) (the cosmological counterterm), \(\beta\!\int\!\sqrt g\,R\), higher-curvature terms and total-derivative ambiguities, so that \(\mathcal F\) constitutes a well-posed prediction of the path integral.

C2 (conformally-flat universality): on round \(S^4\) (\(W^2=0\)), the extracted quantity depends only on diffeomorphism-invariant, scheme-independent universal data of the matter sector, not on gauge-volume normalisations, measure conventions or regulator artefacts.

The uniqueness claim is relative to this class; the class definition is part of the theorem.

At a conformal fixed point, \[ \mathcal A[\Gamma]=-\frac14\Pi_{H^0}\!\left(\frac{d\Gamma(H)}{d\ln H}\right),\qquad \Gamma(H)=-\ln Z[S^4(H)]. \] Away from a fixed point, the corresponding local-RG Euler-cocycle projection is used after beta-function and operator-mixing responses are separated.

Status. Derived.

Limit. The theorem identifies a protected matter direction. It does not derive the gravitational contour, endpoint, two-part matching hypothesis, curvature-charge coordinate, source action or late-time branch, and it does not say that local anomaly stress directly produces the observed cosmological constant.

Claims: EXT-01, EXT-02.
Full paper: Section 4, Theorem 1.

The proof has two stages.

1. Remove every local ambiguity. On round \(S^4(H)\), the volume and Einstein counterterms scale as \(H^{-4}\) and \(H^{-2}\). The \(R^2\) and Euler terms are constants, \(W^2\) vanishes and the integrated total derivative is zero. After differentiation with respect to \(\ln H\), each finite local contribution is either explicitly \(H\)-dependent or zero. A finite Euler counterterm changes only the constant part of \(\Gamma\) and cannot remove an anomaly logarithm.

2. Isolate the remaining Weyl cocycle. The anomaly-induced term is \[ \Gamma_A(H)=+4a\ln(\mu/H), \] whose constant-Weyl response is \(-4a\). On conformally flat \(S^4\), the Weyl-squared cocycle is absent, while \[ \delta_\sigma\Gamma=+\frac{a}{(4\pi)^2}\int\sqrt g\,\sigma E_4. \] Using \(\int_{S^4}\sqrt g\,E_4=64\pi^2\) and \(\sigma=-\delta\ln H\) gives \(\Pi_{H^0}(d\Gamma/d\ln H)=-4a\) and hence \(\mathcal A[\Gamma]=a\).

Wess-Zumino consistency protects the coefficient. A free conformal scalar fixes the sign independently of the Ward-identity argument: its determinant gives \(d\Gamma_s/d\ln H=-1/90=-4a_s\). In a running theory the local-RG projector removes beta-function and operator-mixing responses before the Euler component is read.

Limit. The uniqueness is class-relative: local, linear, constant-Weyl responses on round conformally flat \(S^4\) satisfying C1 and C2. It is not a uniqueness theorem over every nonlocal functional or gravitational observable.

Claims: EXT-01.
Full paper: Theorem 1 and proof.
The theorem and proof are results of this paper; the line below lists the established results they build on.
Foundations: Duff 1994, Deser & Schwimmer 1993, Wess & Zumino 1971, Osborn 1991, Aminov et al. 2026.

The free-field type-A coefficients are \[ a_{\rm scalar}=\frac1{360},\qquad a_{\rm Weyl}=\frac{11}{720},\qquad a_{\rm vector}=\frac{31}{180}. \] The minimal Standard Model has four real Higgs components, 45 Weyl fermions and 12 vector bosons: \[ a_{\rm SM}=\frac4{360}+\frac{45\times11}{720}+\frac{12\times31}{180}=\frac{1991}{720}=2.76528. \] In units of \(1/720\), the contributions are 8 from the Higgs scalars, 495 from the Weyl fermions and 1488 from the vector bosons.

At \(h_*=\sqrt{m_pM_P}\simeq3.4\times10^9\,{\rm GeV}\), all known Standard Model masses are negligible relative to the response scale; the largest squared threshold ratio is about \(3\times10^{-15}\). The projected Euler datum is independent of the Higgs nonminimal curvature coupling \(\xi\): nonconformal scalar terms either belong to local \(R^2\) and \(\Box R\) structures removed by the projector or lie outside the protected Euler coefficient.

The paper distinguishes the protected field-count coefficient from broader interaction and determinant corrections. It provisionally estimates the matter-channel source-response envelope at about \(0.5\%\), while a complete interacting Standard Model calculation on \(S^4\) remains open.

Claims: SM-01, CORR-01.
Full paper: Section 5 · Appendix B.
Sources: PDG 2026, Brown 1977, Dowker & Critchley 1977, Ferrero et al. 2024, Larue et al. 2023, Jack & Poole 2015.

The theorem answers one compact matter question: which local response remains universal after the finite counterterm quotient on conformally flat round \(S^4\)?

It does not identify the late-time Friedmann curvature by ordinary metric variation. The paper separately proves that local type-A anomaly stress scales as \(H^4\) and yields no small branch linear in a suppressed compact weight.

It also does not derive the infrared endpoint, compact saddle contour, exact boundary unit, two-part matching hypothesis or global-source law. Those enter later and carry different controlled statuses.

This separation is deliberate. The theorem can be assessed independently of the proposed cosmological mechanism, and disagreement with a selected physical clause does not erase the extraction result.

Claims: EXT-01, GR-01.
Full paper: Scope of the theorem.

The paper uses the unreduced Planck mass \[ M_P=G^{-1/2}=1.221\times10^{19}\,{\rm GeV} \] and the PDG proton pole mass \(m_p=938.272\ \mathrm{MeV}\), and compares the dimensionless observable \(G\Lambda=\Lambda/M_P^2\).

This differs from the compact saddle coordinate \(u=\kappa^2\Lambda=8\pi G\Lambda\). The selected gravitational sector is fixed at \(u=1\), where the tree-level exponent is \(B=24\pi^2\). The independent response scale satisfies \[ h_*^2=m_pM_P,\qquad \frac{h_*^2}{M_P^2}=\frac{m_p}{M_P}=7.685\times10^{-20}. \] With \(a_{\rm SM}=2.76528\) and \(e^{-24\pi^2}=1.344\times10^{-103}\), \[ 2.76528\times7.685\times10^{-20}\times1.344\times10^{-103}=2.856\times10^{-122}. \] The hierarchy may also be written as a sum of exponents, \(S_{\rm grav}=24\pi^2\approx237\) and \(S_{\rm trans}=\ln(M_P/m_p)=44.01\), a total exponent of approximately 281; at one loop, dimensional transmutation gives \(S_{\rm trans}\simeq2\pi/[b_0\alpha_s(M_P)]\) up to threshold and order-one matching factors.

No parameter is fitted to the cosmological value. Reproducing the arithmetic verifies the numerical chain and conventions; it does not verify the declared assumptions defining the contour, boundary, endpoint, two-part matching hypothesis, curvature-charge coordinate or global-source map.

Convention note. The curvature label \(H\) used across these panels is defined by \(|S_{\mathrm{EH}}|=24\pi^2M_P^2/H^2\) (equivalently \(H^2=8\pi\Lambda\)). It is distinct from the Lorentzian de Sitter expansion rate \(H_{\mathrm{dS}}^2=\Lambda/3\) (the two differ by a factor of \(24\pi\)) and from the observed Hubble constant \(H_0\).

Planck-mass conventions. The two Planck masses divide by defining equation. The unreduced \(M_P\) enters through Newton’s constant, \(G=1/M_P^2\), and therefore carries the observable \(G\Lambda=\Lambda/M_P^2\); the reduced \(\bar M_{\rm P}=M_P/\sqrt{8\pi}\) enters through the gravitational coupling \(\kappa^2=8\pi G=1/\bar M_{\rm P}^2\), and therefore carries the saddle coordinate \(u=\kappa^2\Lambda\) and the boundary condition \(V_\Lambda=\bar M_{\rm P}^4\). Each appearance is fixed by the equation that defines it.

Claims: NUM-01.
Full paper: Sections 1-2 · Proposition 5 · Eqs. (39)-(40).
Source: PDG 2026.

Proposition 5 (strict-channel compact response). For the primitive suppressed round-\(S^4\) sector, the normalised one-cap curvature-charge response and the endpoint-conjugate Planck–proton cap imply

The extraction theorem gives \(a_{\mathrm{eff}}(h_*)=a_{\mathrm{SM}}\) up to the stated interaction and mass corrections; the one-cap curvature-charge derivative gives \(h_*^2/M_P^2\); the zero-temperature endpoint together with the two-part matching hypothesis, endpoint conjugacy with a reflection-even readout, gives \(h_*^2=m_pM_P\); and source-independent determinants and common phases have already cancelled in the sector-normalised paired-cap ratio. Therefore

This is the compact-response result; the global-source completion supplies its map to vacuum curvature (Curvature Map).

Status. Derived, conditional on the channel clauses; each clause carries its own status (Status).

Full paper: Proposition 5 · Table 1.

The calculation record lists the three numerical inputs, their units, the formula and the expected output. It reproduces the arithmetic independently of the page. It does not establish the physical selection of the compact channel.

Action: Download the calculation record

Claim. The compact gravitational factor has an exact action–entropy identity.

For round de Sitter geometry, the magnitude of the Euclidean Einstein–Hilbert action equals the de Sitter horizon entropy:

At the selected Planck-curvature boundary, \(B=S_{\mathrm{dS}}=24\pi^2\), so the sector weight can be written \(e^{-24\pi^2}=e^{-S_{\mathrm{dS}}}\): an entropy-suppressed weight for the smallest de Sitter geometry in the declared sub-Planckian family, with a Boltzmann-like mathematical form.

Claims: INT-03, INT-04.

Reading the identity. The entropy in this identity belongs to the ultraviolet compact saddle: this geometry enters the calculation suppressed by its own horizon entropy. The observed late-time universe carries a different entropy, about \(10^{122}\), which enters macroscopic partition-function calculations with the opposite sign; the two are distinct objects. The constant itself is the connected cap response constructed in The Calculation above, a quantity that The Problem shows cannot be reached as \(-T\ln Z/V\) or as the raw sphere partition function. And the full exponential is a compound object, a gravitational saddle action added to a renormalisation-group scale-counting exponent, so its Boltzmann-like form is shared structure rather than evidence of a single thermal ensemble. The suppressing sign comes from the selected damped orientation of the compact thimble; the opposite Hartle–Hawking orientation would weight the same geometry by \(e^{+S_{\mathrm{dS}}}\) and stands as a named falsifying alternative (see the saddle panel).

Status. Standard geometric identity applied to the selected sector; not a separate derivation and not required for the extraction theorem.

Full paper: Saddle weight · Failed identifications.

Not a sum of zero-point energies. On compact \(S^4\) the absolute homogeneous offset is counterterm-degenerate; the computed object is the protected response.

Not protons filling space. The proton marks the selected endpoint of the QCD hierarchy; nothing asserts the vacuum contains protons.

Not the partition function of the observed universe. The exponential weights a selected ultraviolet compact sector, not the late-time horizon.

Not a cancellation. Two independent suppressions multiply; nothing large is subtracted.

Not an ordinary local anomaly stress. The protected response is extracted on the compact sector and enters through the global curvature map; it does not act as a local anomaly-induced stress tensor in the field equations.

Claims: VAC-01, VAC-02, QCD-01, SAD-01, NUM-01.

The order-one anomaly coefficient and compact exponent do not supply the required factor linear in \(m_p/M_P\). The paper tests three direct mechanisms.

Zero-mode localisation. With \(x=M_P^2/H^2\ge1\), the action is \(24\pi^2x\). The integral is exponentially localised at the Planck boundary. The confinement region is about 44 e-folds away in \(\ln H\) and carries an additional action of order \(10^{40}\).

Local analytic mass expansion. In the local analytic part of the compact one-loop heat-kernel expansion, mass dependence enters through \(m^2\). The resulting powers are \((ma)^{2n}\), with logarithms multiplying even powers, rather than a term linear in \(ma\). For \(ma=m_p/M_P\), the first term is of order \(10^{-38}\).

Planck-scale QCD instantons. A perturbative Standard Model RG extrapolation gives \(\alpha_s(M_P)\simeq0.02\). The corresponding instanton action is about 314 and the amplitude about \(10^{-136}\), negligible compared with the required \(10^{-19}\).

These are limited exclusions within the stated treatment. They do not rule out every nonlocal or ultraviolet-complete effect. They establish that the strict compact calculation needs a genuine infrared input separate from the Planck-curvature saddle.

Claims: IR-01.
Full paper: Section 6.
Sources: Vassilevich 2003, Buttazzo et al. 2013, ’t Hooft 1976.

The endpoint is the stable, gauge-invariant, zero-temperature representative of QCD dimensional transmutation in the full Standard Model. For a relativistically normalised proton state, energy-momentum conservation gives \(\langle p|T^\mu{}_{\mu}|p\rangle=2m_p^2\): the proton pole is directly a physical trace-charge standard.

The strict criteria are:

1. Physical pole: an observable mass rather than a scheme-dependent running parameter.
2. Colour singlet: a physical confined state.
3. Stable in the full Standard Model: a state that remains available at late times.

These criteria exclude \(\Lambda_{\rm QCD}\), which is a running parameter; pions and rho mesons, which are unstable; \(2\pi T_c\), which is a thermal crossover diagnostic; and the quoted \(0^{++}\) pure-Yang-Mills scale, which is not a stable full-Standard-Model asymptotic pole. The proton is the lightest hadronic state satisfying all three conditions. The neutron and \(\eta'\) lie numerically close to the proton but fail the stability criterion; their inclusion in the scored table makes explicit that numerical proximity is not the endpoint rule.

The proton mass is predominantly generated by QCD dynamics. Recent lattice and NNLO work clarifies the decomposition of that mass and its scheme dependence, but the endpoint is the measured physical pole rather than any individual decomposition term.

Status. Selected physical endpoint.

Limit. The criteria are physically motivated channel clauses, not a theorem for every gravitational trace source. The nearby \(2\pi T_c\) scale shifts the result by about \(4.2\%\) and is treated as a robustness comparison.

Claims: QCD-01, QCD-02.
Full paper: Section 8 · Section 10 · Table 4 · Appendix D.
Sources: PDG 2026, Ji 1995, Yang et al. 2018, Bhattacharya et al. 2014, Morningstar & Peardon 1999, Wang et al. 2024, Tanaka 2025, Bollweg et al. 2026.

Claim. The round-\(S^4\) Einstein saddle has a fixed classical action formula; the strict model selects a damped thimble and the boundary unit \(u_b=\kappa^2\Lambda=1\).

For the round sphere, \[ S_{\rm EH}=-\frac{24\pi^2}{\kappa^2\Lambda},\qquad |S_{\rm EH}|=\frac{24\pi^2}{u}. \] At the declared Planck boundary \(u=1\), the exponent is \(B=24\pi^2\). Equivalently, the boundary is where the vacuum potential reaches one reduced-Planck energy density, \(V_\Lambda=\bar M_{\rm P}^{4}\). This exponent also equals the de Sitter entropy, \(B=S_{\rm dS}\), and Chern-Gauss-Bonnet gives \[ |S_{\rm EH}|=\frac32\chi(M)\frac{8\pi^2}{\kappa^2\Lambda}, \] which recovers the coefficient for \(\chi(S^4)=2\).

Within the declared sub-Planckian family \(0<u\le1\), the action increases away from the boundary. Thus endpoint dominance selects the boundary member of that declared family, while the exact unit assigned to the boundary remains a model definition. Alternative conventions such as \(G\Lambda=1\) or a unit Planck radius give very different exponents and therefore define different models.

The selected Feynman/Lefschetz contour includes the compact thimble with the damped orientation \(e^{-|S_{\rm EH}|}\). On the declared sub-Planckian domain the reduced integral is finite: the Planck boundary, not the sign of \(a_{\rm SM}\), removes the small-radius endpoint, while the damped thimble controls the behaviour at infinity. This check does not select the contour, its orientation or its Stokes data. The opposite Hartle–Hawking orientation would replace \(e^{-24\pi^2}\) by \(e^{+24\pi^2}\), enlarging the result by about \(10^{206}\): a named falsifying alternative. The selected damping has the same semiclassical sign as the tunnelling (Vilenkin) weighting, whose exponent \(3\pi/G\Lambda\) equals \(24\pi^2\) at the selected boundary.

Finite \(R^2\) and Euler terms do not shift the protected source response: constants are annihilated by the source derivative and local counterterm classes are removed by C1. Source-dependent gravitational dressing and higher-curvature operators remain separate open calculations; source-independent determinant factors cancel in the paired-cap response.

Status. Action formula and endpoint dominance: Derived. Suppressed thimble and exact boundary unit: Selected. Domain finiteness: Derived conditional on the selected contour and measure.

Claims: SAD-01, SAD-02, SAD-03, SAD-04, SAD-05.
Full paper: Section 9 · Appendix C · Appendix D.4-D.5.
Sources: Gibbons & Hawking 1977, Gibbons et al. 1978, Witten 2011, Lehners 2023, Mazur & Mottola 1990, Antoniadis et al. 1997.

Claim. Under the two-part matching hypothesis, the evaluation point is the unique reflection fixed point of the endpoint interval.

The matching hypothesis has two parts, both physical. First, endpoint conjugacy: the Planck cap state and the stable-proton trace state are posited to be conjugate ends of one finite dilation-transfer amplitude (Osterwalder–Schrader conjugation in radial quantisation, where logarithmic scale is Euclidean transfer time). Writing \(s=\ln(h/m_p)\) on the interval \(0\le s\le L=\ln(M_P/m_p)\), this supplies the endpoint-exchange reflection \(\Theta:s\mapsto L-s\). Second, a reflection-even readout: the compact charge is represented by the reflection-even local one-point coefficient of the matched cap amplitude. The first part supplies the reflected interval; the second places the insertion on its fixed plane, which is unique:

In multiplicative variables the same reflection is \(\mathcal I(h)=m_pM_P/h\).

The same point also equalises the two leading endpoint matching errors, \(h^2/M_P^2\) and \(m_p^2/h^2\); that is a consistency check rather than an additional selection rule. The algebraic involution \(\mathcal I\) is the coordinate representation of endpoint-conjugate cap reflection, not an independent matching rule.

Status. The two-part matching hypothesis: Proposed physical hypothesis. The fixed point and the scale \(h_*^2=m_pM_P\): Derived, conditional on both parts.

Claims: FOLD-01, FOLD-02.

Full paper: Section 10 · Appendix D.

Four quantitative readings must be distinguished. The first three treat the matter scale as the curvature modulus itself; R3 is a different observable, the paired-cap coefficient. With \(x=M_P^2/H^2\) and \(B=24\pi^2\), the corrected reduced measure is \(x^{-2a_{\mathrm{SM}}-1}e^{-Bx}\,dx\).

R1: unbounded integral. Extended to \(x=0\) the integral diverges: that endpoint is exactly the excluded super-Planckian region. A value assigned by analytic continuation would be contour- and prescription-dependent, and is not an admissible compact observable.

R2: bounded-window integral. \[I_{\mathrm{R2}}\simeq\frac{e^{-B}}{2B} \left[1-\frac{2a_{\mathrm{SM}}+1}{B}+O(B^{-2})\right],\] localised at the ultraviolet boundary. It gives about \(2.8\times10^{-106}\), some sixteen orders of magnitude above the observed value, and the infrared factor is lost.

R0: on-saddle identification. Setting \(h=M_P\) treats the external source scale as a second name for the saddle curvature and gives \(a_{\mathrm{SM}}e^{-24\pi^2}\simeq3.7\times10^{-103}\), nineteen orders too large; the insertion sits at the ultraviolet endpoint and is not fixed under endpoint exchange.

R3: reflection-matched paired-cap coefficient. The gravitational sector is held fixed with weight \(e^{-B}\), and the normalised cap response is differentiated with respect to the external source at the reflection plane: \[q_{\mathrm{WZ}} =e^{-24\pi^2}\left[\mathcal C_{+,0}\frac{d\Gamma_{A,+}}{d\mathcal C_+}\right]_{h=h_*} =a_{\mathrm{SM}}\,\frac{m_p}{M_P}\,e^{-24\pi^2}.\]

R3 is not selected by dominance among the other readings. It is a different observable, defined by the paired-cap construction: distinct arguments for the gravitational sector and the external source, rather than two values assigned to one modulus.

Claims: FOLD-02, NUM-01.

Full paper: Inequivalent compact-channel readings.

Selected source coordinate. After the volume direction is quotiented as the vacuum-energy ambiguity, the two-derivative Einstein curvature charge is the only nonconstant local geometric coordinate in the operator basis retained through four derivatives; six- and higher-derivative directions belong to the correction budget. The charge is

with the conjugate derivative normalised to one primitive cap, \(\mathcal D_+=\mathcal C_{+,0}\,d/d\mathcal C_+\).

Derived evaluation. The extraction theorem gives the universal one-cap Euler response \(\Delta\Gamma_{A,+}=\mathcal A_m(h)\ln x_h\), hence

which at \(h_*^2=m_pM_P\) becomes \(\mathcal A_m(h_*)\,m_p/M_P\). A constant rescaling of \(\mathcal C_+\) cancels between \(\mathcal C_{+,0}\) and \(d/d\mathcal C_+\): the response depends on the charge’s linear Weyl scaling, not on any displayed overall normalisation.

The charge is a coordinate on the external source family, not an additive term in the extracted matter functional; finite \(\int\sqrt g\,R\) counterterms remain excluded by the Euler projector, and \(\mathcal C_+\) is not the complete off-shell Einstein–Hilbert action, whose volume term scales as \(x_h^2\).

The paper’s swap audit makes the coordinate’s role explicit. With \(r=m_p/M_P\): the cap-volume coordinate would give a response smaller by \(r/2\); the full-sphere logarithmic derivative, or evaluation at \(h=M_P\), larger by \(r^{-1}\); evaluation at the arithmetic mean larger by \((4r)^{-1}\); evaluation at \(h=m_p\) smaller by \(r\). Alternative coordinates or evaluation points miss by eighteen orders of magnitude or more.

Status. The four-derivative operator basis is the stated truncation; within it the curvature-charge coordinate is the unique survivor and the evaluation of the Euler response is Derived.

Claims: SRC-00, SRC-01.

Full paper: Section 11, Proposition 4.

The paired-cap response is a normalised ratio,

The same reflected bra appears in numerator and denominator, so its overall normalisation and phase cancel; the ratio likewise removes every factor independent of the external source, including metric and ghost determinant constants, collective-coordinate Jacobians, zero-mode volumes and finite local constants. This is a normalised response quotient, not a unit-prefactor assumption: the cancellation is derived, and no separate unit-source or sector-normalisation clause enters the channel.

Three effects remain, and they are physical corrections rather than normalisations. A determinant that depends on the source dresses the cap response. Higher-curvature operators can shift the primitive action \(B_s\) itself, the principal outstanding ultraviolet calculation; no numerical error is assigned to it before a completion is specified. Higher primitive sectors enter at \(O(e^{-2B_s})\).

Existing sphere path-integral calculations show that unnormalised partition functions do carry nontrivial spin-dependent prefactors and phases; the quotient is precisely why those source-independent quantities never enter the connected cap response. The 0.4% agreement still cannot be used as evidence that the remaining source-dependent corrections are small: they are computable tests of the formula.

Status. Source-independent cancellation: Derived. Source-dependent gravitational dressing and the boundary-action shift: Open calculations.

Claims: NORM-01, NORM-02, CORR-02.

Full paper: Section 11 · Appendix A.

Sources: Anninos et al. 2022, Law 2021, Maldacena 2024, Chen et al. 2025, Giombi & Sun 2026.

For a maximally symmetric geometry, \[ R_{\mu\nu}=\Lambda_g g_{\mu\nu},\qquad E_4=\frac83\Lambda_g^2. \] The local anomaly stress is \[ T^A_{\mu\nu} =-\frac{a\Lambda_g^2}{24\pi^2}g_{\mu\nu}. \]

The ordinary semiclassical Einstein equation gives \[ \Lambda_g-\Lambda_0 =\frac{a}{3\pi}\frac{\Lambda_g^2}{M_P^2}. \]

If \(\Lambda_0=0\), the non-zero root is \(\Lambda_g/M_P^2=3\pi/a\), which is Planckian. Multiplying the anomaly sector by a small \(\varepsilon\) changes the root to \(3\pi/(a\varepsilon)\): the small coefficient appears in the denominator.

A protected coefficient multiplying a local volume term would reintroduce the counterterm-degenerate operator. A two-derivative term only renormalises Newton’s constant, while curvature-squared terms produce \(H^4\) stress. A small branch linear in the compact response therefore requires a separate scalar-curvature equation.

Claims: GR-01.
Full paper: Ordinary-GR obstruction.

Write \[ \mathcal E_{\mu\nu} =G_{\mu\nu}-\frac{8\pi}{M_P^2}T_{\mu\nu}. \] A finite volume term shifts \(\mathcal E_{\mu\nu}\) by \(c\,g_{\mu\nu}\).

The unique algebraic local projection invariant under \[ X_{\mu\nu}\mapsto X_{\mu\nu}+c\,g_{\mu\nu} \] and equal to the identity on traceless tensors is \[ X_{\mu\nu}\mapsto X_{\mu\nu}-\frac14Xg_{\mu\nu}. \]

This gives \[ R_{\mu\nu}-\frac14Rg_{\mu\nu} =\frac{8\pi}{M_P^2} \left(T_{\mu\nu}-\frac14Tg_{\mu\nu}\right). \] Conservation then implies \[ R+\frac{8\pi}{M_P^2}T=4\Lambda_{\mathrm{int}}. \]

The quotient therefore identifies the representative-independent local equation and leaves one scalar mode undetermined.

Limit. The projection is not a complete physical law and does not fix \(\Lambda_{\mathrm{int}}\). The proposed global-source action supplies that missing equation.

Claims: GR-02.
Full paper: Trace-free quotient projector.

The proposed gravity-side action is written directly in the rigid dimensionless charge:

Here \(\mathcal P^2\) is rigid with physical branch \(\mathcal P^2=M_P^2/(8\pi)\), \(F_4=dA_3\) is an auxiliary closed four-form treated as metric independent off shell, and \(q_{\mathrm{WZ}}\) is held fixed under the rigid variation. The independent variations give

and eliminating \(\Lambda_b\) yields the vacuum-shift-invariant completed equation and, on a maximally symmetric vacuum, \(\Lambda/M_P^2=q_{\mathrm{WZ}}\).

Coefficient and sign audit. Replacing the coefficient \(8\pi\) by a general \(c\) would give \(\Lambda/M_P^2=(c/8\pi)\,q_{\mathrm{WZ}}\). The charge is defined in the observable convention \(q_{\mathrm{WZ}}=G\Lambda\) with \(G=(8\pi\mathcal P^2)^{-1}\), and that source–charge convention sets \(c=8\pi\) at the action level; the sign is the standard Legendre pairing between the positive cap charge and its source. Specifying the coupling directly at the action level leaves no separate matching step: the coefficient is fixed by the stated convention rather than imposed after variation.

Status. Action: Proposed. Consequences of the action: Derived. Coefficient and sign: fixed at the action level by the stated source–charge convention.

Claims: GS-01, GS-02, GS-04, MAP-01.

Full paper: Section 12 · Appendix E.

Sources: Henneaux & Teitelboim 1989, Kaloper & Padilla 2014, Kaloper et al. 2016, Khoury et al. 2026.

The compact theorem fixes a vacuum relation. A cosmology with matter history also carries a spacetime-averaged non-vacuum trace. The direct late-time identification requires \[ \langle T\rangle_{\rm reg}=\lim_{\tau\to\infty}\frac{\int_{M_\tau}\sqrt{-g}\,T_{\rm nonvac}\,d^4x}{\int_{M_\tau}\sqrt{-g}\,d^4x}=0. \] For pressureless matter in an asymptotically de Sitter universe, \(\rho_m\propto a^{-3}\), so the numerator grows only linearly in time while the four-volume grows exponentially. Radiation is trace free. Contributions from a finite number of localised transitions, or from components of finite comoving abundance, likewise grow more slowly than the de Sitter four-volume.

If the average does not vanish, then in the stated source-charge convention \[ \Lambda_\infty=M_P^2q_{\rm WZ}+\frac{2\pi}{M_P^2}\langle T\rangle_{\rm reg} \].

Status. Selected late-time branch condition.

Limit. The sufficient dilution argument does not imply that every cosmological history, recollapsing solution or non-asymptotic branch has a vanishing average.

Claims: GS-03.
Full paper: Section 12 · Appendix E.

The taxonomy is scientific content: each deduction is only as strong as the statuses of its dependencies.

Full record: Claim register.
Full paper: Table 1 · Discussion.

The paper separates matter-response corrections, ultraviolet corrections to the primitive action, higher sectors and endpoint robustness. Source-independent normalisations and common phases cancel by construction in the paired-cap response and are absent from the budget.

The \(\sim0.5\%\) matter figure is a provisional estimate, not a calculated uncertainty or upper bound; a complete curved-space Standard Model cap calculation would replace it. The endpoint comparison is not a Gaussian theory error bar: the strict criteria select the proton, and replacing it by the thermal scale relaxes the criteria and defines a nearby model reading.

The paper’s sensitivity table makes the contrast explicit: the pion gives about \(0.15\) of the observed value, the rho \(0.83\), the proton \(1.004\), the neutron \(1.005\), the \(\eta'\) \(1.025\), \(2\pi T_c\) about \(1.042\), and the quoted pure-Yang-Mills \(0^{++}\) proxy about \(1.82\). The neutron and \(\eta'\) lie numerically close but fail the stability criterion: numerical proximity is not the endpoint rule.

Claims: CORR-01, CORR-02, QCD-02, NORM-02.

Full paper: Section 13 and Tables 3-4 · Appendix A.

1. Equation of state. The residual source of the proposed completion is spacetime constant, 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. Persistent cross-dataset and systematics-robust evidence for evolving dark energy across flexible parametric and non-parametric reconstructions would falsify the proposed completion or its identification with the observed acceleration, leaving the compact extraction theorem intact.

2. Particle content. Weakly coupled species below \(h_*\simeq3.4\times10^9\,{\rm GeV}\) shift the active Euler coefficient. Illustrative v1.0 shifts are \(+1.7\%\) for three light right-handed neutrinos, \(+0.1\%\) for one real scalar, \(+8.3\%\) for an additional fermion generation and about \(+18\%\) for low-energy MSSM matter. Heavy Majorana right-handed neutrinos above the response scale decouple; light Dirac neutrinos would contribute.

3. Boundary unit. The boundary is fixed by the reduced-Planck energy-density condition \(V_\Lambda=\bar M_{\rm P}^4\), independently of the cosmological value. Because \(B=24\pi^2/u_b\), a fractional boundary change alters the result exponentially. A roughly \(4\%\) output window corresponds to \(|\delta|\lesssim2\times10^{-4}\) in the boundary exponent convention.

4. Gravitational dressing and UV action. Source-independent determinants and phases cancel in the paired-cap construction. A computed source-dependent dressing away from unity, or a changed Planck-boundary action, would modify or falsify the formula rather than provide a fit parameter.

The particle-content tests become quantitative exclusions once source-dependent gravitational dressing and the higher-curvature correction to the primitive action are controlled below the corresponding shifts.

Claims: TEST-01, TEST-02, TEST-03.
Full paper: Section 13.1 · Section 13.2 · Table 5.
Sources: PDG 2026, Appelquist & Carazzone 1975, Giacchini et al. 2022, Hiller et al. 2024.

Evidence on \(w=-1\) is currently mixed. The completion predicts \(w(z)=-1\) at all redshifts, so evolving dark energy is a direct falsifier. DESI DR2 alone is well described by flat ΛCDM; combinations with CMB and supernova data show dataset-dependent 2.8–4.2σ preferences for \(w_0>-1\), \(w_a<0\), and the full six-year DES multi-probe analysis finds 2.3–3.2σ departures across its tested combinations.

The working assessment:

The current preference for evolution is a substantive tension, dependent on the datasets and the \(w_0w_a\) parameterisation. Both constant-Λ and evolving readings remain active. The construction’s prediction is unchanged: \(w(z)=-1\).

The 0.4% comparison is conditional on the Planck 2018 flat-ΛCDM inference: if evolution is established, that inference no longer directly measures the asymptotic constant predicted here. This assessment is dated July 2026 and will be revisited when a substantially new combined analysis appears.

Limit. Falsification requires persistent cross-dataset, systematics-robust evidence for evolving dark energy across flexible parametric and non-parametric reconstructions; a preference under one dataset combination and parameterisation does not meet that criterion.

Claims: TEST-01.

These formulations remove the homogeneous metric-proportional part of the local equation. A constant vacuum shift therefore does not change the local traceless dynamics.

The residual cosmological constant usually reappears as an integration constant, flux or boundary datum whose value is not fixed locally. The present proposal adds a compact quantum response and a global source equation intended to fix that omitted scalar datum: with the stated source-charge convention, \(\langle R\rangle=4M_P^2\,q_{\mathrm{WZ}}\). Trace-free gravity removes the arbitrary vacuum offset; the compact response acts as a candidate selector for the finite residual curvature that remains.

Comparison dimension: vacuum-shift handling and residual-value determination.
Not claimed: that the proposed global action is the unique completion of trace-free gravity.
Claims: CMP-02.

Sequestering uses global variables or constraints to cancel homogeneous vacuum contributions. The four-form action proposed here belongs to the same broad structural family.

The distinction is the source assigned to the residual curvature: the strict channel identifies it with a calculated compact Wess–Zumino response rather than leaving it as an arbitrary flux datum. The shared architecture is close: rigid global variables, auxiliary four-forms, exact cancellation of constant matter-Lagrangian shifts, and a global equation for the residual curvature. A possible future reading is that the compact anomaly channel provides the value a sequestering-like mechanism needs to select, conditional on the source-charge convention and the source-dependent dressing being derived rather than imposed.

Comparison dimension: vacuum-shift cancellation and residual sourcing.
Not claimed: that existing sequestering models derive the compact response or its source-charge convention.

Anthropic approaches explain why observers may occupy a vacuum with small \(\Lambda\) within a distribution of possibilities. They do not generally calculate the observed central value from minimal Standard Model field content.

The present framework attempts a single-channel calculation. Its strength is quantitative closure within that channel; the limitation is that several channel choices remain selected rather than microscopically derived. A landscape could be the microscopic environment in which the channel occurs; it is not part of the present derivation.

Comparison dimension: environmental selection versus a single-channel calculation.
Not claimed: that environmental selection is excluded; a landscape could host the channel without entering the derivation.

QCD generates a genuine infrared hierarchy, but a raw QCD vacuum density is far too large and its gravitational map remains model dependent.

Here the proton pole is a stable physical endpoint, the source response supplies the linear ratio \(m_p/M_P\), and the compact saddle supplies the much larger exponential suppression. QCD supplies the hierarchy; compact gravity supplies the dominant suppression.

Comparison dimension: origin and power of the infrared scale.
Not claimed: that the proton is mandatory for every gravitational trace source.

Trace anomalies provide universal quantum information, but local type-A anomaly stress on a maximally symmetric background scales as curvature squared. The paper’s no-go proposition shows that a small coefficient does not generate a proportionally small branch in ordinary Einstein gravity.

The construction therefore retains the protected anomaly datum but maps it through a separate global scalar-mode equation. The compatibility is at the data level: in the maximally symmetric setting relevant here, ordinary local type-A stress alone cannot produce the observed small-curvature branch.

Comparison dimension: anomaly data and curvature scaling.
Not claimed: that anomaly physics plays no role; the protected datum is retained, and only the ordinary local-stress route is excluded.

Compact Euclidean saddles naturally generate exponential weights, but their interpretation depends on contour, boundary conditions, measure and sign.

The strict channel selects a suppressed round-\(S^4\) thimble at a declared Planck-curvature boundary, then combines that weight with an independent QCD hierarchy and protected Standard Model coefficient. Compatibility is conditional: a contour or state that excludes the saddle, assigns it the opposite enhancement, carries a nontrivial relative Stokes multiplier or source-dependent phase, or changes the Planck-boundary action would conflict with the channel. The tradition is both a natural parent framework and one of the strongest available tests.

Comparison dimension: origin of the exponential and contour ambiguity.
Not claimed: a universally selected de Sitter contour; compatibility is conditional on the declared thimble, boundary and multipliers.

The overlaps are structural: the compact action equals the de Sitter entropy of the selected saddle, the gravitational factor has an \(e^{-S}\) form, and the curvature is fixed through a global relation rather than an ordinary local vacuum fluid. Programmes that treat the gravitational field equations as thermodynamic or statistical relations share that shape.

The construction is a saddle-point evaluation, not a derivation from \(-T\ln Z/V\) or a conventional thermal ensemble. A future thermodynamic completion could explain why the curvature-charge coordinate, contour and source-charge convention are selected; until then, thermodynamics is a structural interpretation of the assembled result (see the thermodynamic structure panel), not the derivation.

Sources: Jacobson 1995, Padmanabhan & Padmanabhan 2013.

Comparison dimension: statistical reading of the gravitational equations.
Not claimed: that thermodynamics derives the construction; the overlap is structural and the completion remains open.

In 2012 Padmanabhan obtained \(\Lambda L_P^2=3\exp(-24\pi^2\mu)\) from a phase-space count of modes crossing the Hubble radius through an early inflationary era, radiation–matter domination and late acceleration, with \(\mu\) an order-one factor collecting horizon-crossing and cosmic-transition uncertainties.

The shared numerical form is a structural parallel, not a second derivation and not independent corroboration. That construction contains no compact-\(S^4\) counterterm quotient, Euler extraction, Standard Model anomaly census, proton endpoint, paired-cap readout or four-form completion. This construction contains no Hubble-crossing mode count, cosmic-era matching or Planck-scale inflationary history. Here \(24\pi^2\) is the round-\(S^4\) Einstein action at the declared Euclidean Planck boundary, and the further nineteen decades arise as the protected response \(a_{\mathrm{SM}}\,m_p/M_P\). For numerical comparison only, the present result rewrites in that form with \(\mu_{\rm eff}=1.18615\); that is an algebraic translation, not a determination of the mode-count \(\mu\).

Comparison dimension: shared numerical form versus distinct calculational objects.
Not claimed: precedence in either direction, or that the shared coefficient \(24\pi^2\) constitutes a second derivation.

Sources: Padmanabhan 2012, Padmanabhan & Padmanabhan 2013.

A parent quantum-gravity theory could justify the Planck boundary and compute what the channel currently declares: the contour orientation and Stokes data, the boundary action, the source-dependent gravitational dressing, the two-part paired-cap matching and the four-form source. Whatever it returns either derives the strict channel or falsifies it; the construction exposes definite quantities for the parent theory to calculate.

A holographic completion would be the sharpest version, fixing the compact-sector Hilbert space and the sign and magnitude of the saddle weight. Nothing in the present construction requires holography, and no boundary dual is supplied; that reading is speculative.

Comparison dimension: microscopic completion as derivation or falsification.
Not claimed: that any existing completion currently derives the channel; the declared quantities are exposed for the parent theory to compute.
Claims: CMP-03.

Absolute vacuum-energy calculations. Any approach treating a regulator-dependent zero-point sum as the physical observable conflicts with the compact counterterm argument.

Pure local anomaly backreaction. The ordinary-GR no-go rules out a small branch linear in the compact response within that reading.

Dynamical dark energy. The proposed completion predicts a spacetime-constant residual source: \(w(z)=-1\) at every redshift where it is identified with the observed acceleration, with no dynamical degree of freedom to absorb evolution. Persistent cross-dataset and systematics-robust evidence for evolving dark energy falsifies the completion or the identification, leaving the extraction theorem intact. Current preferences for evolution are a substantive, dataset-dependent tension, tracked under Observational status.

Uncompensated low-scale field content. New weakly coupled particles below the response scale shift the active coefficient and move the number; inclusion is a quantitative test, not a free extension.

Claims: VAC-01, GR-01, TEST-01, TEST-02.

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