Established starting point
The problem: the absolute vacuum offset is not the final observable
The cosmological constant is the curvature of empty spacetime. A positive value makes an otherwise empty universe expand at a constant asymptotic rate. Cosmologists label any such component by its equation of state \(w\), the ratio of its pressure to its energy density; \(w=-1\) is the value at which the density stays constant as space expands, and it is what separates a true cosmological constant from evolving dark energy.
The problem is usually framed as a mismatch between a huge quantum zero-point sum and the tiny observed value. But the absolute vacuum-energy offset can be changed by a finite local counterterm, one of the finite subtraction terms that renormalisation leaves adjustable. In semiclassical gravity, where quantum matter lives on a classical spacetime, the split between “matter vacuum energy” and the renormalised cosmological term therefore depends on the renormalisation convention.
On compact round \(S^4\), both contributions reduce to the same operator, \(\int\!\sqrt{g}\,d^4x\). There is no second intrinsic compact configuration that separates them. The paper therefore treats the absolute zero-point sum as the wrong object to extract from the compact path integral, the quantum sum over field configurations on the closed sphere. It asks instead what survives after all allowed local counterterm ambiguities are quotiented out, that is, systematically removed.
Physical differences survive the quotient: Casimir forces, phase transitions, interfaces and inhomogeneous stresses remain measurable and continue to gravitate.
Status · Standard input, with a compact-\(S^4\) sharpening derived here.
Technical detail: the counterterm basis
Claim. A compact extraction is physically meaningful only if it is invariant under every allowed finite local counterterm.
For a four-dimensional QFT on a curved background, the renormalised effective action may be shifted by local terms proportional to \(\int\sqrt g\), \(\int\sqrt g\,R\), \(\int\sqrt g\,R^2\), \(\int\sqrt g\,W^2\), \(\int\sqrt g\,E_4\) and total derivatives.
\[\Delta\Gamma=\alpha\!\int\!\sqrt g+\beta\!\int\!\sqrt g\,R+\gamma\!\int\!\sqrt g\,R^2+\delta\!\int\!\sqrt g\,W^2+\epsilon\!\int\!\sqrt g\,E_4+\int\!\sqrt g\,\Box R\]
On round \(S^4(H)\) these contribute, respectively, \(H^{-4}\), \(H^{-2}\), a constant, zero, a topological constant and zero. After differentiation with respect to \(\ln H\), each finite local contribution is either \(H\)-dependent or vanishes. A valid compact extraction must be invariant under all of them.
| Counterterm | Value on round \(S^4(H)\) | \(d/d\ln H\) response |
|---|---|---|
| \(\int\sqrt g\) | \(\propto H^{-4}\) | \(H\)-dependent |
| \(\int\sqrt g\,R\) | \(\propto H^{-2}\) | \(H\)-dependent |
| \(\int\sqrt g\,R^2\) | constant | zero |
| \(\int\sqrt g\,W^2\) | zero | zero |
| \(\int\sqrt g\,E_4\) | topological constant | zero |
| \(\int\sqrt g\,\Box R\) | zero | zero |
Limit. The quotient removes an absolute homogeneous offset. It does not remove physical energy differences, phase-transition dynamics or inhomogeneous stress.
Why the Casimir effect is not a counterexample
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.
Three direct identifications that fail
The paper rejects three tempting direct readings of the compact path integral as the protected matter observable.
A free-energy density remains tied to the local volume normalization. The finite number \(\Gamma[S^4]\) can be shifted by allowed local counterterms. The absolute normalization of \(Z[S^4]\) also retains local and measure conventions.
The common problem is not a missing numerical trick: the raw compact functional contains scheme- or normalization-dependent pieces. The construction therefore extracts the coefficient of the surviving protected direction rather than declaring the raw partition function itself to be \(\Lambda\).
The theorem-level result
What survives: a protected Euler response
Once the local ambiguities are removed, the compact response is not empty. A universal quantum signature remains. On conformally flat round \(S^4\), the terms that depend on the local convention drop out, while the anomaly-induced logarithm survives. In the technical sense used here, an anomaly is a classical symmetry that quantum effects necessarily break: a structural feature of the theory, not a discrepancy in data.
The exact result is that, among local linear responses to a uniform rescaling of conformally flat round \(S^4\), the unique nonzero \(H\)-independent quantity that survives all allowed counterterm shifts is the type-A Euler anomaly coefficient.
For the minimal Standard Model, the leading field-count value is
\[a_{\mathrm{SM}}=\tfrac{4}{360}+\tfrac{45\cdot11}{720}+\tfrac{12\cdot31}{180}=\tfrac{1991}{720}=2.76528\]
That coefficient is fixed by the active field content, not adjusted to the observed cosmological value. The coefficient itself is standard physics. What this paper adds is the theorem around it: a classification of every finite counterterm ambiguity on compact \(S^4\), and a proof that, within the stated response class, this single number is the only quantity that survives all of them.
Status · Euler extraction Derived. \(a_{\mathrm{SM}}=1991/720\) Computed. Full interacting \(a_{\mathrm{eff}}\) Open calculation.
The theorem statement
Euler extraction theorem. On round conformally flat \(S^4(H)\), among local linear constant-Weyl responses of the renormalised effective action satisfying counterterm invariance (C1) and conformal-flatness universality (C2), the unique nonvanishing \(H\)-independent scheme-independent matter datum is the type-A Euler anomaly coefficient.
C1: the extraction functional is invariant under every allowed local counterterm ambiguity, including the cosmological, Einstein, curvature-squared and total-derivative directions. C2: on round \(S^4\) the extracted quantity depends only on diffeomorphism-invariant, scheme-independent universal matter data, not on gauge-volume normalisations, measure conventions or regulator artefacts.
\[\mathcal A[\Gamma]=-\tfrac14\,\Pi_{H^0}\!\left(\frac{d\Gamma(H)}{d\ln H}\right),\qquad \Gamma(H)=-\ln Z[S^4(H)]\]
Away from a fixed point, the corresponding local-RG Euler-cocycle projection is used after beta-function and operator-mixing responses are separated.
Limit. The theorem identifies the protected matter coordinate. It does not derive the QCD state-preparation premise, the gravitational boundary-member prescription, UV endpoint or contour, or the cosmological source law. The uniqueness claim is relative to the defined class.
Proof outline
First remove every local ambiguity. On \(S^4(H)\), the volume and Einstein counterterms scale as \(H^{-4}\) and \(H^{-2}\); \(R^2\) and Euler terms are constants; \(W^2\) vanishes; and the integrated total derivative is zero. After \(d/d\ln H\), each allowed finite local term is \(H\)-dependent or zero.
Then isolate the surviving Weyl cocycle. The anomaly logarithm contributes the nonzero constant response \(-4a\).
\[\Gamma_A(H)=+4a\ln(\mu/H),\qquad \delta_\sigma\Gamma=+\frac{a}{(4\pi)^2}\int\sqrt g\,\sigma E_4\]
Using \(\int_{S^4}\sqrt g\,E_4=64\pi^2\) and \(\sigma=-\delta\ln H\), \(\Pi_{H^0}(d\Gamma/d\ln H)=-4a\) and hence \(\mathcal A[\Gamma]=a\). Wess–Zumino consistency protects the coefficient, and a free conformal scalar supplies an independent determinant sign check (\(d\Gamma_s/d\ln H=-1/90=-4a_s\)).
Standard Model field count and interacting status
The free-field type-A coefficients are 1/360 for a real scalar, 11/720 for a Weyl fermion and 31/180 for a vector. The minimal Standard Model contains four real Higgs components, 45 Weyl fermions and 12 vector bosons, giving 1991/720. In units of 1/720, the contributions are 8 from the Higgs scalars, 495 from the Weyl fermions and 1488 from the vector bosons.
The full Standard Model is interacting and not exactly conformal. The paper therefore writes \(a_{\mathrm{eff}}\) for the interacting curved-space Euler coefficient appropriate to the compact channel and uses \(a_{\mathrm{SM}}\) as its leading free-field value. No numerical theory uncertainty is assigned to this replacement: the full curved-space Standard Model calculation remains an explicit open test. The extracted Euler coordinate does not require setting the Higgs nonminimal coupling to \(\xi=1/6\).
Why a separate infrared input is required
The protected Euler coefficient is dimensionless and order one. It does not by itself supply the roughly \(10^{-19}\) infrared-to-Planck hierarchy needed in the compact benchmark.
The local analytic mass expansion of the compact determinant starts in even powers of a dimensionless mass, up to logarithms, rather than producing the required linear factor. A perturbative Planck-scale QCD instanton is exponentially too small: the Standard Model running coupling gives a weight of order \(10^{-136}\), not \(10^{-19}\).
These are limited exclusions, not a theorem against every ultraviolet effect. They explain why the paper uses a distinct physical QCD+QED readout for the infrared factor.
Independent support for the matter factor
Why this Standard Model number appears
De Sitter space can be described in two complementary ways: as a closed sphere, and as the region visible to one observer, bounded by a horizon. Quantum fields can leave several kinds of imprint on these geometries. A standalone theorem publication of this programme proves that the protected local anomaly signal shared by both descriptions is uniquely type A: the Euler coefficient \(a\), fixed by particle content. This independently supports the \(a_{\rm SM}=1991/720\) factor in the cosmological-constant formula. It strengthens the matter side of the construction, and the cosmological-constant paper remains self-contained: its numerical chain does not depend on this separate publication.
This number was not chosen. Once the ambiguous offset is removed, exactly one protected matter quantity survives on the sphere, and it is a count of the known fields. If the construction is correct, the cosmological constant is tied to the Standard Model by necessity: the matter factor could not be anything else.
From the sphere to the proton
Why the proton sets the infrared scale
The same round sphere does three different jobs in this construction. It is the closed geometry on which the protected matter response is evaluated. Its equator carries a built-in single twist – a one-unit winding in its spin geometry that the paper derives once. And its classical action will supply the gravitational weight in the next section. The “one” that appears in all three roles is the same derived object, not a number chosen three times – and the three readings remain three different pieces of physics, each with its own status.
The twist itself is elementary to state. Cut the sphere along its equator and the two halves are glued along a three-dimensional sphere; the spin geometry winds around that equator exactly once. That winding number, one, is derived, and the sphere’s own curvature supplies an explicit field configuration that realises it.
Topology alone does not turn that geometric “one” into a statement about matter. The paper makes its single matter-side physical premise here, and labels it explicitly: at confinement, QCD’s infrared state is prepared in the same winding class as the sphere’s spin geometry. Everything downstream of this premise is derived; the premise itself is the step a microscopic derivation would have to supply.
Given the premise, the consequences are fixed. The standard QCD identification of winding number with baryon number gives a sector with baryon number one, and the lightest physical state of that sector, once electromagnetism is included, is the proton. The proton is not chosen from a list of candidate particles; it is the floor of the prepared spectrum.
The proton’s mass then enters the formula through a response theorem: a long-time measurement of how the proton’s correlator responds to a uniform trace source reads off \(m_p\), and in Newton units the same response is the ratio \(m_p/M_P\) – about \(7.7\times10^{-20}\), nineteen of the 122 decades. The mass is read out of QCD; it is not inserted by hand into the gravitational action.
Status · NE identity, clutching degree and explicit representative Derived. QCD state preparation Structural premise. B=1, proton floor and normalized response Derived given the premise.
The QCD state-preparation premise, exactly
At the confinement matching slice, the ordinary two-flavour QCD chiral field carries the standard integer winding identified with baryon number. The paper posits that the infrared state is prepared in the homotopy class of the intrinsic degree-one spin field:
\[[U_{\rm IR}]=\varphi_*[U_{\rm spin}]=\varphi_*[g_-]\quad\text{in}\ \pi_3(SU(2)_{\rm flavour})\]
The identification is between homotopy classes only. No connection-level spin-flavour identification is imposed – nothing asserts that spin \(SU(2)\) is physically the same group as QCD flavour \(SU(2)\) – and no continuous matching coefficient is introduced.
Limit. This is the single new matter-side physical premise in the compact benchmark. If it fails, the \(m_p/M_P\) branch and the assembled benchmark lose support, while the extraction theorem and the geometric identities remain independently assessable.
Why the proton follows once B=1 is prepared
The topological step ends at baryon number one. The physical mass is then supplied by the QCD+QED Hamiltonian: minimising over the compatible electromagnetic sectors gives the proton as the lowest physical B=1 state; the conjugate sector gives the antiproton. Proton charge is not a second selection premise – it is the electromagnetic label of the spectral minimum.
The normalized response theorem
Introduce a constant trace source and follow a gauge-invariantly dressed proton correlator to long Euclidean time. Differentiating its logarithm with respect to the source and dividing by the time removes overlap and endpoint factors; the Feynman–Hellmann theorem and the exact energy-momentum-tensor normalization then give \(m_p\), and in the observable Newton unit the same response is \(m_p/M_P\).
\[-\lim_{T\to\infty}\tfrac1T\,\partial_\sigma\ln\!\big[C_p(T;\sigma)/C_p(T;0)\big]_{\sigma=0}=m_p\]
\[-\lim_{s\to\infty}\tfrac1s\,\partial_\sigma\ln\!\big[C_p(s;\sigma)/C_p(s;0)\big]_{\sigma=0}=m_p/M_P\]
The geometric identities behind the bridge
For an oriented spin four-manifold,
\[e(TM)=c_2(\Sigma^-)-c_2(\Sigma^+),\qquad p_1(TM)=-2\,[\,c_2(\Sigma^-)+c_2(\Sigma^+)\,]\]
and on \(S^4\), \(\chi=2\) and the signature vanishes, so \(\langle c_2(\Sigma^-),[S^4]\rangle=+1\) and \(\langle c_2(\Sigma^+),[S^4]\rangle=-1\). The equatorial transition function \(g_-\) has degree one in the same \(1/(24\pi^2)\) winding convention later used for the QCD baryon current, and the Levi-Civita-induced connection is a charge-one \(SU(2)\) instanton whose Atiyah–Manton holonomy supplies an explicit degree-one representative \(U_{\rm spin}\) with \([U_{\rm spin}]=[g_-]\). No new integer is chosen anywhere in this chain: the explicit representative removes any need to select a separate abstract degree-one field, while still not establishing the physical QCD state-preparation step, which remains the labelled premise.
Planck-unit covariance
The public formula uses the unreduced Newton mass \(M_P=G^{-1/2}\), because the observable is \(G\Lambda_\infty=\Lambda_\infty/M_P^2\). The gravitational saddle naturally uses the reduced mass \(\bar M_P=(8\pi G)^{-1/2}\). These are two coordinate conventions for the same physics when every factor is transformed consistently; changing only one denominator would define a different normalization.
\[\frac{m_p}{M_P}=\frac{1}{\sqrt{8\pi}}\,\frac{m_p}{\bar M_P},\qquad G\Lambda_\infty=\frac{1}{8\pi}\,\frac{\Lambda_\infty}{\bar M_P^2}\]
The sphere's own weight
Why the sphere supplies the gravitational suppression
The gravitational factor is a separate, classical property of the same sphere. For round Euclidean de Sitter geometry the Einstein action has magnitude \(|S_{\rm EH}|=24\pi^2/u\) per geometric unit, where \(u\) measures the vacuum energy density in reduced-Planck units. The strict channel declares the physical domain \(0<u\le1\): the vacuum is not allowed to be denser than one Planck unit.
Within that domain the suppression \(e^{-24\pi^2/u}\) is monotonically ordered – that is derived – and the least-suppressed member sits exactly at the boundary. Three declared selections then fix the factor: the compact observable is evaluated at that single boundary member with no averaging over the family, the boundary is placed at exactly one reduced-Planck density, and the sphere is included with the decaying orientation. Together they give \(e^{-24\pi^2}\approx1.3\times10^{-103}\): 103 of the 122 decades, from classical geometry.
The sphere itself is not an arbitrary stage for this step. Independent 2026 work in the positive-Λ gravitational effective theory finds the round four-sphere to be the leading compact geometry at leading order, canonical quantum-gravity results realise the same primitive winding-to-coupling structure, and an explicit de Sitter thimble construction supports the stability of the decaying orientation. These results support the architecture without deriving the three selections, which remain explicit targets for microscopic gravity.
Status · Action and monotonic ordering Derived given the strict domain. Exact \(u_{\max}=1\), boundary-member prescription and decaying contour Selected. \(e^{-24\pi^2}\) Derived given those selections. Quantum dressing Open.
Why the sphere is more than a convenient saddle
The gravitational case has three layers of external support. Anninos, Baracco, Brian and Denef find leading-order four-sphere dominance in the four-dimensional Λ>0 gravitational EFT partition function. Alexander, Bernardo and Hui find an exact canonical topological sector in which primitive large-\(SU(2)\) winding is weighted by an inverse dimensionless cosmological coupling. Alexander and Blakey construct an explicit analytically continued de Sitter Lefschetz thimble and find damping of anisotropic directions around the symmetric saddle.
These results support specific pieces of the architecture. They do not derive the boundary-member prescription or the exact \(u_{\max}=1\) normalization, force the chosen contour for this observable, or calculate the complete quantum dressing.
Action, topology and entropy
For the round sphere the action magnitude equals the de Sitter entropy, \(B=S_{\rm dS}=3\pi M_P^2/\Lambda_{\rm UV}\), and Chern–Gauss–Bonnet rewrites the same action in terms of the Euler characteristic, recovering the coefficient for \(\chi(S^4)=2\):
\[B=S_{\rm dS}=\frac{3\pi M_P^2}{\Lambda_{\rm UV}},\qquad |S_{\rm EH}|=\tfrac32\,\chi(M)\,\frac{8\pi^2}{\kappa^2\Lambda_{\rm UV}}\]
This identity belongs to the ultraviolet compact saddle used in the calculation. It is not the thermodynamic partition function of the observed late-time universe, whose horizon entropy is a different macroscopic quantity of about \(10^{122}\).
The decaying contour
The channel is defined so that the compact \(S^4\) saddle is included with the decaying orientation \(e^{-|S_{\rm EH}|}\). The opposite orientation would replace the suppression by an exponentially large enhancement and is not the channel analysed – it is a named falsifying alternative. Recent Chern–Simons work constructs a de Sitter thimble explicitly, making the architecture concrete without selecting the contour on behalf of this paper.
Determinants and the open correction budget
The paper does not set an absolute one-loop sphere prefactor to one. Finite \(R^2\) and Euler counterterms shift the constant part of the sphere effective action, so an absolute determinant constant is not a scheme-independent observable without an additional prescription.
This does not make quantum corrections disappear. Source-dependent determinants, higher-curvature operators, Stokes changes or a different microscopic compact sector can alter the leading result; their net exponent shift is denoted \(\delta B\) and remains an explicit quantitative test.
Three factors, three origins
The strict compact calculation
The strict construction combines three factors with distinct physical origins:
Protected matter response: \(a_{\mathrm{SM}}=1991/720=2.76528\).
QCD–Planck hierarchy: \(m_p/M_P = 7.685\times10^{-20}\).
Compact gravitational weight: \(e^{-24\pi^2}=1.344\times10^{-103}\).
The assembly has two defined stages: the matter and infrared responses first form the composite \(q_{\rm matter}=a_{\mathrm{SM}}\cdot(m_p/M_P)\); the compact saddle then supplies the weight \(e^{-24\pi^2}\). Multiplying the stages gives
\[q_{\rm comp}=a_{\mathrm{SM}}\cdot\frac{m_p}{M_P}\cdot e^{-24\pi^2}=2.85652\times10^{-122}\]
The 122-decade hierarchy separates into approximately 103 decades from the compact saddle and 19 from QCD dimensional transmutation, the quantum effect that gives the strong interaction its definite mass scale. No cancellation between large vacuum contributions is used: the two suppressions multiply, so their exponents add.
Status · Arithmetic Computed; factorized assembly Defined. The QCD state-preparation premise and the exact UV endpoint, boundary-member prescription and contour remain the declared channel clauses.
Exact arithmetic and conventions
The paper uses the unreduced Planck mass \(M_P=1.2209\times10^{19}\) GeV and the PDG proton pole mass \(m_p=938.272\) MeV, and compares the dimensionless observable \(G\Lambda_\infty=\Lambda_\infty/M_P^2\):
\[q_{\rm comp}=2.76527778\times 7.68514844\times10^{-20}\times 1.34414611\times10^{-103}=2.85652154\times10^{-122}\]
The saddle coordinate \(u=\kappa^2\Lambda_{\rm UV}\) uses the reduced mass; each appearance is fixed by the equation that defines it:
\[u=\kappa^2\Lambda_{\rm UV}=\Lambda_{\rm UV}/\bar M_P^2=\rho_{\rm UV}/\bar M_P^4,\qquad 0<\rho_{\rm UV}\le\bar M_P^4\ \Leftrightarrow\ 0<u\le1\]
The hierarchy may also be written as a sum of exponents: \(24\pi^2\approx237\) from gravity and \(\ln(M_P/m_p)=44.0\) from the hierarchy. Reproducing the arithmetic verifies the numerical chain and conventions; it does not establish the premise, the selections or the proposed curvature map.
The factorized assembly, stated exactly
The matter/QCD response and the compact gravitational weight are separately normalized. The strict compact observable is then defined by multiplying them with coefficient one:
\[q_{\rm comp}\equiv\big[a_{\rm eff}\cdot(m_p/M_P)\big]\cdot e^{-24\pi^2}\]
That coefficient-one factorization is a definition of the strict channel, not a separately derived dynamical theorem. The physical status is carried entirely by the ingredients feeding it.
Downloadable calculation record
The calculation record lists the numerical inputs, their units, the exact leading formula and the expected output, and reproduces the arithmetic independently of the page. It does not establish the physical selection of the compact channel.
Exact compact status ledger
The ingredients entering the compact result only; the global-source action and late-time amplitude condition are deliberately not part of this ledger.
Euler extraction Derived
aSM=1991/720 Computed
Full interacting aeff Open calculation
Euler–Chern / clutching identity Derived
Uspin degree-one representative Derived
QCD state-preparation class identification Structural premise
B=1 → proton floor → mp/MP Derived given premise
Newton-unit conversion Normalization covariance
Matter/IR and factorized compact products Defined strict-channel composites
Monotonic ordering Derived given the declared 0<u≤1 domain
Boundary-member prescription ub=umax; no modulus integration Selected compact-sector prescription
Exact umax=1 Selected UV domain
Decaying S⁴ thimble Selected contour
e⁻²⁴π² Derived given boundary-member, UV-domain and contour selections
Gravitational dressing / higher-curvature shift Open calculation
Physical interpretation
What the equation means physically
In this construction, the cosmological constant is the small residual curvature that empty space retains after the removable homogeneous vacuum contribution has been taken away. It is not treated as a stored energy to be summed. The calculation begins with a response: how the Standard Model quantum effective action changes when every length in a closed geometry is rescaled together.
The first quantity computed is therefore not an energy density but a protected dimensionless response, \(q_{\rm comp}\). Its matter coefficient has order-one strength and is fixed by Standard Model particle content. The smallness comes from the channel through which the response is read: the hierarchy factor \(m_p/M_P\), supplied by the prepared QCD sector, and the compact weight \(e^{-24\pi^2}\), supplied by the selected round-sphere saddle.
The three factors have distinct physical origins. The matter residue is cohomologically protected. The coefficient \(24\pi^2\) is fixed geometrically by the round-sphere Einstein action once the Planck boundary is specified. The proton–Planck hierarchy is a physical consequence of QCD dimensional transmutation, and its appearance as the linear ratio \(m_p/M_P\) follows from the prepared B=1 sector’s normalized response. Once the strict channel is fixed, none of these quantities is available for adjustment to the observed cosmological value.
The factors are also connected by more than the shared geometry. The two matter factors are the same kind of object – normalized logarithmic Weyl responses, the protected coefficient read from the compact effective action and the ratio \(m_p/M_P\) read from the prepared proton correlator – and the same normalized Euler–Chern unit organizes the anomaly readout, the degree-one QCD sector and the round gravitational action: one primitive topological unit, two normalized Weyl responses, one compact saddle weight. These connections make the channel one structured construction rather than three unrelated numbers; they do not force the coefficient-one product, which remains the defined strict-channel composite.
Read this way, and conditionally on nature selecting the strict channel, the curvature of empty space becomes a fingerprint of the matter content of the universe. The type-A theorem makes the dependence direct: the leading coefficient is a count of the Standard Model’s fields – four Higgs components, 45 matter fermions, 12 force carriers – so that count sets part of the curvature of empty space. Change the active field content and the predicted number changes. In one sentence, the cosmological constant is the residual curvature of empty space, and this construction proposes that the quantum matter content of the universe fixes how much remains.
Status · Interpretation of the assembled result; each mapped element keeps its own status. Not required for the extraction theorem.
Common misreadings
Not a sum of zero-point energies. The absolute homogeneous offset is counterterm-degenerate; the calculation starts from a protected response.
Not protons filling space. The proton is the lowest physical state in the prepared B=1 sector and supplies a response scale.
Not the partition function of the observed universe. The exponential is a classical weight of a selected ultraviolet compact saddle.
Not a cancellation. The two suppressions multiply.
Not geometry generating all three factors. The common geometric unit organizes three different sector-specific responses.
Not ordinary local anomaly stress. That route has the wrong curvature scaling; a separate global completion is introduced instead.
That reading is conditional: the computed quantity is a compact response, not yet a cosmological curvature.
The gravity-side equation
Why a separate curvature map is required
On a maximally symmetric background, the local anomaly stress grows as the curvature squared. In ordinary semiclassical Einstein gravity that is far too steep: multiplying the anomaly sector by a small weight does not create a proportionally small curvature.
The paper therefore separates the compact calculation from the gravity-side map. It first derives the representative-independent trace-free local equation obtained after the homogeneous volume direction is quotiented out; that equation leaves one overall scalar mode undetermined. It then proposes a global-source equation for the missing mode, generated by a four-form action of the kind standard in the unimodular and sequestering literature. Once that action is adopted, its variations cancel constant shifts of the matter Lagrangian exactly, and the new content is the source term: the computed compact response.
On a maximally symmetric vacuum the resulting relation identifies the residual curvature with \(q_{\rm comp}\). Because the residual source is spacetime constant on any fixed branch, the completion also predicts \(w=-1\) at finite redshift, not only asymptotically; the late-time condition fixes the amplitude, not the equation of state.
Status · Global-source action and unit compact-to-global pairing Proposed. Field equations and the vacuum branch relation Derived once the action is adopted. Late-time trace average: amplitude condition.
Ordinary-GR no-go
For round de Sitter geometry the Euler density is proportional to \(\Lambda_g^2\), so the local type-A stress is proportional to \(a\Lambda_g^2\):
\[R_{\mu\nu}=\Lambda_g g_{\mu\nu},\qquad E_4=\tfrac83\Lambda_g^2,\qquad T^A_{\mu\nu}=-\frac{a\Lambda_g^2}{24\pi^2}g_{\mu\nu},\qquad \Lambda_g-\Lambda_0=\frac{a}{3\pi}\frac{\Lambda_g^2}{M_P^2}\]
The semiclassical Einstein equation gives a nonzero branch \(\Lambda_g/M_P^2=3\pi/a\) when the local cosmological term is set to zero. Multiplying the anomaly sector by a small \(\varepsilon\) changes that root to \(3\pi/(a\varepsilon)\): the small coefficient appears in the denominator. A small branch linear in the compact response therefore requires a separate scalar-curvature equation.
Trace-free quotient
A finite local volume term shifts the metric equation by a multiple of \(g_{\mu\nu}\). The unique algebraic local projection insensitive to that representative is the traceless combination:
\[R_{\mu\nu}-\tfrac14Rg_{\mu\nu}=\frac{8\pi}{M_P^2}\Big(T_{\mu\nu}-\tfrac14Tg_{\mu\nu}\Big),\qquad R+\frac{8\pi}{M_P^2}T=4\Lambda_{\rm int}\]
Conservation then leaves one spacetime constant in the trace equation. The quotient identifies the representative-independent local dynamics but does not itself fix the cosmological constant.
Proposed global-source action and field equations
The proposed action introduces a volume four-form \(F_4=dA_3\), a constraint multiplier, and a rigid dimensionless Newton variable \(\eta\); the branch Newton mass is \(M_N^2=\eta M_P^2\):
\[S_{gs}[q]=\int\Big[\tfrac{\eta M_P^2}{16\pi}R-\mathcal L_m+\Lambda_b(F_4-{*}1)-\tfrac{\eta^2M_P^4}{8\pi}\,q\,F_4\Big]+\tfrac{1}{2\pi}\int C_\eta\wedge d\eta\]
Branch-unit covariance fixes the \(\eta^2\) dependence of the source term up to an overall coefficient, and the proposed unit compact-to-global pairing takes that coefficient to be one. Replacing \(\eta^2\) by \(f(\eta)\), branch-unit independence requires \(f'(\eta)/(2\eta)=c\):
\[f(\eta)=c\,\eta^2+f_0,\qquad f_0=0\ \text{(minimal one-source action)},\qquad c=1\ \text{(proposed unit pairing)}\]
With the source held fixed, variation gives the trace-free completed Einstein equation and, on a maximally symmetric vacuum, \(\Lambda_\infty/M_N^2=q\). Homogeneous shifts of the matter Lagrangian cancel exactly.
\[\langle R\rangle=4\eta M_P^2\,q,\qquad \frac{\Lambda_\infty}{M_N^2}=q,\quad M_N^2=\eta M_P^2\]
Limit. The action and the unit pairing are Proposed; the \(\eta^2\) covariance and the field equations are Derived within the action.
Late-time amplitude condition
A cosmological history contains non-vacuum matter as well as the residual constant source. Direct identification with the asymptotic constant requires the spacetime-averaged non-vacuum trace to vanish in a regulator-independent way. On a future-eternal asymptotically de Sitter branch, diluted matter is subextensive in the four-volume, radiation is trace free, and finite transitions do not change the average. If the condition holds, \(\Lambda_\infty/M_P^2=q_{\rm comp}\); if it does not, the amplitude identification fails without undoing the upstream compact calculation.
\[\langle T_{\rm nonvac}\rangle_{\rm reg}=0\ \Longrightarrow\ \Lambda_\infty/M_P^2=q_{\rm comp}\]
What is established, assumed, selected or open
Status and open questions
The work is a quantitatively closed candidate construction within a specified compact channel. It is not a first-principles derivation of every channel choice from microscopic quantum gravity, and the page keeps the difference visible rather than averaging it into a single verdict.
Proved or computed in this paper: the compact counterterm classification, the Euler extraction theorem, the geometric degree-one identities and explicit representative, the prepared-sector proton floor and normalized response, the round-sphere action and its ordering, the ordinary-GR obstruction, the trace-free quotient, the variations of the proposed global action and the arithmetic of the strict result. Standard inputs: the Standard Model field count, the measured proton and Planck masses and the round-sphere action formula.
The channel rests on one explicit physical premise – QCD state preparation in the sphere’s winding class – and three declared gravitational selections: the exact reduced-Planck-density endpoint, the boundary-member prescription and the decaying contour. The global-source action and its unit pairing carry Proposed status. Open calculations: the full interacting Standard Model coefficient \(a_{\rm eff}\), and the source-dependent gravitational dressing \(\delta B\) of the primitive action.
Status · Current research assessment.
Status matrix
| Part | Physical role | Status |
|---|---|---|
| Euler extraction | Protected compact matter coordinate | Derived |
| \(a_{\rm SM}=1991/720\) | Leading free-field SM coefficient | Computed |
| Full \(a_{\rm eff}\) | Interacting compact coefficient | Open calculation |
| \(N_E\) / clutching / \(U_{\rm spin}\) | Primitive geometric unit and representative | Derived |
| QCD state preparation | Maps the geometric class to the IR QCD class | Structural premise |
| B=1 → \(m_p/M_P\) | Proton floor and normalized response | Derived given premise |
| Monotonic ordering | \(B(u)=24\pi^2/u\) within \(0<u\le1\) | Derived given strict domain |
| Boundary member; \(u_{\max}=1\); contour | Evaluation prescription, UV endpoint, orientation | Selected |
| \(e^{-24\pi^2}\) | Classical compact factor | Derived given selections |
| Factorized assembly | Defines \(q_{\rm comp}\) | Defined |
| Global source + unit pairing | Curvature map | Proposed |
| \(w=-1\) residual | Constant residual on a fixed branch | Derived from the proposed completion |
| Gravitational dressing \(\delta B\) | Quantum / higher-curvature shift | Open calculation |
The taxonomy is scientific content: each deduction is only as strong as the statuses of its dependencies.
Gravitational action sensitivity
Because the compact factor is exponential, small changes in the complete primitive action matter. Writing the net source-dependent shift as \(\delta B\) and holding \(a_{\rm eff}=a_{\rm SM}\), remaining within the quoted 1σ observational interval requires \(-0.016\lesssim\delta B\lesssim0.024\); the 2σ interval is \(-0.036\lesssim\delta B\lesssim0.045\).
These are sensitivity windows, not fit ranges and not a theory probability distribution. More generally the leading correction depends on \(\delta B-\ln(a_{\rm eff}/a_{\rm SM})\); the matter and gravity corrections are independently open calculations:
\[\ln(q/q_0)=\ln(a_{\rm eff}/a_{\rm SM})-\delta B\]
Tests
Equation of state. The proposed completion predicts \(w(z)=-1\) at every redshift where its residual is identified with dark energy, with no dynamical degree of freedom to absorb evolution. Persistent cross-dataset, systematics-robust evidence for evolving dark energy falsifies the completion or the identification, leaving the compact extraction theorem intact.
Particle content. New weakly coupled species below the response scale shift the active Euler coefficient and move the number; inclusion is a quantitative test, not a free extension.
Primitive action. A computed \(\delta B\) outside the sensitivity window, or a changed Planck-boundary action, modifies or falsifies the formula rather than providing a fit parameter.
Observational status
Evidence on \(w=-1\) is currently mixed. DESI DR2 combinations with CMB and supernova data and the full Dark Energy Survey multi-probe analysis have reported dataset-dependent preferences for evolving-dark-energy regions, while constant-Λ 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.
Comparison by problem dimension
How this relates to other approaches
Several independent results and established frameworks now align with specific parts of the architecture. Three recent results align unusually closely with the gravitational side: leading-order round-sphere dominance in the positive-Λ gravitational path integral (Anninos, Baracco, Brian and Denef 2026), exact canonical sectors linking primitive winding to an inverse cosmological coupling (Alexander, Bernardo and Hui 2026), and an explicit de Sitter thimble (Alexander and Blakey 2026). They support the use of the compact saddle without deriving the declared selections. On the framework side, the trace-free quotient and the four-form global source sit in the established unimodular and sequestering family, and the anomaly–Euler–four-form combination has independent precedent in anomaly-effective-action work; these frameworks supply the form of the missing equation, not the computed compact source.
The complete channel itself is distinct. Different cosmological-constant proposals solve different parts of the problem; this framework’s distinguishing ambition is to connect the full chain within one specified channel – remove the arbitrary homogeneous offset, identify a protected Standard Model datum, generate the infrared hierarchy, supply a compact gravitational weight and fix the residual scalar curvature. That does not make it the unique solution: it makes the strict channel quantitatively closed, and exposes a finite list of physical selections and calculations on which the number depends.
Status · Derived comparative synthesis; direct support and structural convergence are non-dependency evidence throughout.
Unimodular gravity, trace-free gravity and sequestering
These formulations remove the homogeneous metric-proportional part of the local equation, leaving the cosmological constant as an integration constant or related global datum; Henneaux–Teitelboim and sequestering implement related global constraints with nonpropagating form sectors.
The present construction adds a calculated compact source \(q_{\rm comp}\) and proposes a specific global-source equation for the omitted scalar mode. Existing sequestering models do not derive the QCD bridge, the compact exponential or the unit pairing.
Anomaly, Euler structure and four-form vacuum sectors
Anomaly-effective-action work connects the conformal anomaly, Euler structure and four-form descriptions of vacuum dynamics. The mechanism differs from the present action, but it shows the anomaly–Euler–four-form combination is not unique to this paper. Ordinary local anomaly stress itself is excluded here by the no-go: it has the wrong curvature scaling for the desired small branch.
QCD-scale proposals supply a genuine infrared hierarchy through dimensional transmutation, but a raw QCD vacuum density is far too large and its gravitational map remains model dependent. In the present channel the prepared sector supplies the proton floor, the normalized trace response gives \(m_p/M_P\), and the compact saddle supplies the dominant suppression; a raw vacuum density is not used.
Pregeometry, topological-Λ models and SM charge sectors
Recent pregeometric gauge models derive Einstein–Hilbert, cosmological and topological sectors from a common symmetry-broken parent, and a related model makes Λ discrete through a periodic gravitational angle tied to de Sitter entropy. These are related outputs of one research programme, not two independent confirmations. Recent work on gravitational instantons with quantized Standard Model flux is a genuine precedent for gravitational topology acting on global charge structure; its mechanism differs from the class-level premise used here.
Anthropic and landscape selection
Anthropic approaches explain why observers may occupy a vacuum with small Λ within a distribution of possibilities. The present framework attempts a single-channel calculation; a landscape could be the microscopic environment in which the channel occurs without entering the derivation.
Relation to Padmanabhan's mode-count formula
Padmanabhan obtained a formula containing \(\exp(-24\pi^2\mu)\) from a cosmological mode-count argument. The shared coefficient is a structural numerical parallel, not a second derivation: here \(24\pi^2\) is the round-sphere Einstein action at the declared reduced-Planck endpoint, and the further nineteen decades arise as the protected response. Neither construction contains the other’s machinery.
Where there is genuine tension
Absolute zero-point readings conflict with the compact counterterm argument if they treat the regulator-dependent homogeneous offset as the protected observable. Pure local anomaly backreaction cannot provide the desired linear small-curvature branch. Persistent robust evidence for evolving dark energy conflicts with the proposed fixed-branch \(w=-1\) completion, while leaving the compact theorem intact. A microscopic gravitational calculation that shifts the primitive action outside the \(\delta B\) window would move or falsify the leading numerical match, and a microscopic derivation that fails to produce the state-preparation relation or the unit pairing would break the corresponding downstream chain.
The research record
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