The spectral action for a spectral triple $(\mathcal{A},\mathcal{H},D)$ with cutoff scale $\Lambda$ is [CC:1996, CC:1997]

where $f\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a positive, smooth, rapidly decreasing function. For the almost-commutative spectral triple encoding the Standard Model [CCM:2006], the asymptotic expansion for $\Lambda\to\infty$ gives [Vassilevich:2003]

where $f_{2k}=\int_{0}^{\infty}f(u)\,u^{k-1}\,\mathrm{d}u$ are the moments and $a_{2k}$ are the Seeley–DeWitt coefficients. The first three terms produce:

• •

  $a_{0}$: cosmological constant ($\sim\Lambda^{4}$),

• •

  $a_{2}$: Einstein–Hilbert action $\sim\Lambda^{2}\int R\sqrt{g}\,\mathrm{d}^{4}x$,

• •

  $a_{4}$: higher-derivative terms proportional to $C^{2}$ and $R^{2}$.

The coefficients of $C^{2}$ and $R^{2}$ in $a_{4}$ depend on the SM particle content but not on the cutoff function $f$.

The one-loop effective action contains nonlocal form factors $F_{1}(\Box/\Lambda^{2})$ and $F_{2}(\Box/\Lambda^{2},\xi)$ multiplying $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$ and $R^{2}$ respectively [BV:1987, BV:1990]. For the choice $f(u)=e^{-u}$, these are constructed from the master function

satisfying $\varphi(0)=1$ and $\varphi^{\prime}(0)=-1/6$. Figure 1 compares $\varphi_{n}(x)$ for $n=1,2,3,5$; the functions share $\varphi_{n}(0)=1$ but differ at all $x>0$.

The per-spin Weyl form factors, expressed as functions of $x=-\Box/\Lambda^{2}$ (Euclidean momentum squared in units of $\Lambda^{2}$), are [CZ:2012, Alfyorov:2026paper1]:

Equations (4)–(6) are derived for the choice $f(u)=e^{-u}$. The algebraic structure (rational functions of $x$ with $\varphi$-dependent coefficients) relies on the fact that $\varphi^{\prime}(0)=-1/6$ cancels the $1/x$ divergence at $x=0$. For alternative cutoffs $f(u)=e^{-u^{n}}$ with $n\geq 2$, the master function satisfies $\varphi_{n}^{\prime}(0)=0$, so this cancellation does not occur and the form factors require a separate heat kernel derivation. In Section 4.2 we use the formal substitution $\varphi\to\varphi_{n}$ in (4)–(6) to estimate the propagator zeros; this is valid at finite $z$ (where no divergence arises) but does not yield a consistent $z\to 0$ limit for $n\geq 2$.

The Dirac form factor (5) evaluates to $h_{C}^{(1/2)}(0)=-1/20$ (negative): the factor $(3\varphi-1)\to 2$ and $(\varphi-1)\to 0$ cancel the $1/x$ terms leaving a negative finite limit. This sign encodes the fermionic functional trace [ParkerToms:2009].

The total SM Weyl coefficient is

with SM multiplicities $N_{s}=4$ (real Higgs scalars), $N_{D}=N_{f}/2=22.5$ (Dirac fermions), $N_{v}=12$ (gauge vectors) [CPR:2008]. At $x=0$, using $h_{C}^{(0)}(0)=+1/120$,  $h_{C}^{(1/2)}(0)=-1/20$,  $h_{C}^{(1)}(0)=+1/10$:

This value depends only on the SM particle content and is independent of the cutoff function $f$. The count $N_{f}=45$ Weyl fermions is the standard SM without right-handed neutrinos, following [CPR:2008]. The Chamseddine–Connes spectral triple [CCM:2006] includes right-handed neutrinos with Majorana mass; their inclusion would modify $\alpha_{C}$. A detailed treatment of the Majorana contribution is beyond the scope of this paper.

The one-loop effective action for quantum fields on a curved background has the universal form [BV:1987, BV:1990]

where $F_{1}$, $F_{2}$ are nonlocal form factors. Linearization around flat space $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ and the Barnes–Rivers decomposition into projectors $P^{(2)}$ (tensor, spin-2, 5 components) and $P^{(0-s)}$ (scalar, 1 component) give the inverse propagator [BV:1987]

where $z=k^{2}/\Lambda^{2}$ and

The dressed spin-2 propagator denominator is [BV:1987, Alfyorov:2026paper1]

where $c_{2}=2\alpha_{C}=13/60$, $z=k^{2}/\Lambda^{2}$, and $\hat{F}_{1}(z)=F_{1}(z)/F_{1}(0)$ is the normalized shape function ($\hat{F}_{1}(0)=1$), with $F_{1}(z)=\alpha_{C}(z)/(16\pi^{2})$ from (7). For $f=e^{-u}$, $\Pi_{\mathrm{TT}}$ has its first positive real zero at

This zero corresponds to a massive mode treated via the fakeon prescription [Anselmi:2017, Anselmi:2018] (building on the Lee–Wick framework [LeeWick:1969]): the mode does not appear in asymptotic states and contributes only through virtual exchange. The fakeon prescription is formulated for propagators with finitely many poles; its extension to the SCT propagator, which as an entire function of order 1 has countably many complex zeros, requires a convergence proof not provided here. Unitarity under this prescription has been verified at one loop [Alfyorov:2026paper1]; the all-orders proof requires extending Anselmi’s finite-threshold argument to infinitely many poles.

The Stelle Lagrangian mass [Stelle:1977] $m_{\mathrm{Stelle}}=\Lambda/\sqrt{c_{2}}=\Lambda\sqrt{60/13}\approx 2.148\,\Lambda$ differs from the propagator pole mass (12) because the nonlocal form factor $\hat{F}_{1}(z)$ modifies the zero position relative to the local approximation $\Pi_{\mathrm{TT}}\approx 1+c_{2}z$.

The scalar mode propagator is

At $\xi=1/6$: $\Pi_{s}=1$ identically, and the scalar mode decouples.

The function $f$ in (1) is constrained but not uniquely determined by the spectral action principle. Entire-function form factors (required for the absence of propagator branch cuts) demand that $f(u)$ extend to an entire function of the complex variable $u$. The family $f(u)=e^{-u^{n}}$ for $n\in\mathbb{N}$ satisfies this constraint; non-integer powers (e.g., $e^{-u^{3/2}}$) produce branch cuts at $u=0$ and are excluded.

Different choices of $n$ give different master functions

different form factors, and different propagator zeros. The $a_{4}$ coefficient $\alpha_{C}(0)=13/120$ is independent of $n$.

The value $\alpha_{C}=13/120$ follows from (8) and depends only on the SM particle content counted in the $a_{4}$ heat kernel coefficient. It is independent of the cutoff function $f$, since $a_{4}$ is a universal coefficient in the heat kernel expansion (2). The value has been formally verified in Lean 4 and cross-checked against three independent literature sources [CZ:2012, CPR:2008, ParkerToms:2009] (source code in [Alfyorov:2026repo]). 354 numerical checks pass at 100-digit precision.

The local part of the one-loop effective action (9) contains two curvature-squared invariants: $\Gamma^{(1)}_{\mathrm{loc}}=\int(c_{1}\,C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}+c_{2}\,R^{2})\sqrt{g}\,\mathrm{d}^{4}x$, where $c_{1}$ and $c_{2}$ are the coefficients of $C^{2}$ and $R^{2}$. In the Shapiro normalization [Shapiro:2004] (without the $1/(16\pi^{2})$ prefactor): $c_{1}=-2\alpha_{C}/3$, $c_{2}=2\alpha_{C}$. In the (9) normalization (with the $1/(16\pi^{2})$ prefactor): $c_{1}=F_{1}(0)=\alpha_{C}/(16\pi^{2})$. The ratio $c_{1}/c_{2}$ is normalization-independent.

At conformal coupling $\xi=1/6$ (the value predicted by the spectral triple; see Section 3.4):

The one-loop matter beta functions are

Because $\beta_{\alpha_{R}}=0$ at $\xi=1/6$ (this coupling is a fixed point of the one-loop RG), the ratio

Consequently, $c_{1}(t)/c_{2}(t)=-1/3$ for all $t=\ln(\mu/\Lambda)$ along the matter-only one-loop trajectory.

The SCT ratio $c_{1}/c_{2}=-0.333$ differs from the perturbative asymptotic freedom branch ($-0.326$; [Shapiro:2004]) by $2\%$ and from the Reuter non-Gaussian fixed point ($-1.09$, truncation-dependent; [BMS:2009]) by a factor of $3.3$.

The logarithmic correction to the Bekenstein–Hawking entropy is

where the coefficient $c_{\log}=37/24\approx 1.54$ (SM field content; with ensemble zero-mode uncertainty $\delta c_{\log}\in[-3/4,\,0]$; sign robust) follows from the Sen formula [Sen:2012], Section 4.2 (Appendix B). The formula uses zeta-function regularisation on the Euclidean Schwarzschild instanton with Dirichlet boundary conditions at the horizon. The result depends on the fermion count $N_{F}=22.5$ Dirac (without right-handed neutrinos; see Section 2.2) and on the graviton contribution $+424$ (gauge-invariant, determined by the one-loop determinant on the Euclidean Schwarzschild instanton).

The sign of $c_{\log}$ discriminates between programs: SCT gives $c_{\log}>0$ (value $37/24$ for SM content, $8/5$ for SM$+\nu_{R}$; both positive). LQG robustly predicts $c_{\log}<0$: $-1/2$ [Meissner:2004] or $-3/2$ [KaulMajumdar:2000] depending on the method. The discriminator is the sign, not the precise numerical value, since the latter depends on the ensemble choice and field content. In asymptotic safety, the result is definition-dependent: $0$ for thermodynamic entropy or $\pi/g_{*}$ for Clausius entropy [FallsLitim:2012]. IDG produces no logarithmic correction (power-law only) [Myung:2017].

The non-minimal Higgs–gravity coupling $\xi$ is determined by the spectral triple. In the standard Chamseddine–Connes spectral action, the $a_{4}$ coefficient contains a common Yukawa-dependent factor multiplying both the curvature coupling $R|H|^{2}$ and the kinetic term $|\nabla H|^{2}$. After canonical normalization, this gives $\xi=1/6$ (conformal coupling). This has been confirmed in five independent works [CCM:2006, CC:2010, vS:2015, CCS:2013, DLM:2014]. At $\xi=1/6$, the RG beta function $\beta_{\xi}\propto(\xi-1/6)$ [ParkerToms:2009] vanishes, so $\xi=1/6$ is an exact one-loop fixed point.

At $\xi=1/6$: $\alpha_{R}=2(\xi-1/6)^{2}=0$ and $\Pi_{s}=1$ identically (13). This is a consequence of the conformal invariance of massless fermions and gauge bosons in $d=4$: their one-loop $R^{2}$ beta functions vanish ($\beta_{R}^{(1/2)}=\beta_{R}^{(1)}=0$), and only scalars ($\beta_{R}^{(0)}=\frac{1}{2}(\xi-\frac{1}{6})^{2}$) contribute to $\alpha_{R}$. At $\xi=1/6$ the scalar contribution also vanishes. The scalar gravitational mode does not propagate. Consequence: exactly two gravitational wave polarizations (tensor modes only).

All known BSM scalars arising from NCG spectral triples also acquire $\xi^{\prime}=1/6$, because the same Yukawa-factor mechanism applies to any scalar from the finite Dirac operator [BFS:2010, CCS:2013ps, vdD:2018].

Detection of a scalar gravitational wave polarization would imply $\xi\neq 1/6$, requiring modification of the standard spectral triple.

Since $\alpha_{R}=0$ at $\xi=1/6$, the $R^{2}$ term in the effective action is absent. There is no scalaron. Starobinsky inflation, which requires a propagating $R^{2}$ scalar with mass $M_{\mathrm{inf}}\approx 1.28\times 10^{-5}\,M_{P}$, is excluded in the standard spectral action.

The only path not definitively excluded is reinterpretation of $\Lambda$ as a sub-Planckian intermediate scale, or framework extension (zeta spectral action [KLV:2015], dilatonized action [CC:2006dilaton]). These require departing from the standard Chamseddine–Connes spectral action. No symmetry argument is known that would protect $\alpha_{R}=0$ beyond one loop.

The requirement that the form factors $F_{1}$, $F_{2}$ be entire functions of $z=\Box/\Lambda^{2}$ restricts the class of admissible cutoff functions. For the family $f(u)=e^{-u^{\alpha}}$, entireness holds if and only if $\alpha\in\mathbb{N}$ (positive integer). At non-integer $\alpha$ (e.g., $\alpha=3/2$), the function $u^{\alpha}$ has a branch point at $u=0$, and the resulting form factors inherit branch cuts. Power-law cutoffs $f(u)=(1+u)^{-N}$ are likewise excluded.

The conventional choice $f(u)=e^{-u}$ ($n=1$) is computationally convenient but not uniquely determined. The choices $e^{-u^{2}}$, $e^{-u^{3}}$, etc., are equally admissible.

To illustrate the sensitivity of cutoff-dependent predictions to the choice of $f$, we compute the propagator zeros by formally substituting $\varphi\to\varphi_{n}$ in the form factors (4)–(6). This substitution is evaluated at finite $z$ (the propagator zeros lie at $z_{0}>2$, far from the $z\to 0$ divergence that arises for $n\geq 2$; see Section 2.2). Table 4 reports the results for $n\in\{1,\ldots,5\}$.

For $n\geq 2$, the fakeon mass lies in the range $m_{2}/\Lambda\in[2.274,2.291]$ (spread $0.7\%$). The $n=1$ (exponential) cutoff is an outlier at $m_{2}/\Lambda=1.554$; the exponential form factor changes sign at a lower value of $z$ than the sharper ($n\geq 2$) cutoffs, pulling the propagator zero closer to the origin.

The Stelle Lagrangian mass $m_{\mathrm{Stelle}}=\Lambda\sqrt{60/13}\approx 2.148\,\Lambda$ (from the local approximation $\Pi_{\mathrm{TT}}\approx 1+c_{2}z$) differs from all five pole masses in Table 4 because the nonlocal form factor modifies the zero position.

At $\xi=1/6$ (scalar mode absent), the modified Newtonian potential takes the single-Yukawa form. In the static limit, the Fourier transform of the propagator (10) gives $V(r)\propto\int\mathrm{d}^{3}k\,e^{i\mathbf{k}\cdot\mathbf{r}}/[k^{2}\Pi_{\mathrm{TT}}(k^{2}/\Lambda^{2})]$. The residue of the spin-2 pole at $\Pi_{\mathrm{TT}}(z_{0})=0$ is $R_{0}=1/[z_{0}\Pi^{\prime}_{\mathrm{TT}}(z_{0})]<0$ (negative, since $\Pi_{\mathrm{TT}}$ crosses zero from above). The coefficient $-4/3$ arises from the weight of the $P^{(2)}$ projector in the tensor structure of the inverse propagator (10):

where $m_{2}=\sqrt{z_{0}}\,\Lambda$ is the fakeon mass from Table 4. In the Yukawa approximation, $V(0)/V_{N}=-1/3$ (repulsive at the origin). This is an artifact of the local approximation: the full nonlocal potential, defined by the Fourier integral of $1/\Pi_{\mathrm{TT}}-1$, diverges as $r\to 0$, since $1/\Pi_{\mathrm{TT}}-1\to-1$ as $z\to\infty$. The Yukawa approximation is valid for $r\gg 1/\Lambda$.

At solar system distances ($r\sim 1\;\mathrm{AU}$), with $\Lambda>3.53\;\mathrm{meV}$. This bound is obtained as follows: at $\xi=1/6$ the scalar mode is absent (Section 3.4), and the potential (19) has a single Yukawa term with the nonlocal mass $m_{2}=1.554\,\Lambda$ (12). The Eöt-Wash experiment [Kapner:2007] constrains the Yukawa correction $|\alpha|\,e^{-r/\lambda}$ at $\lambda=1/m_{2}=1/(1.554\,\Lambda)$ and $|\alpha|=4/3$, giving $\Lambda>3.53\;\mathrm{meV}$ (update of the bound [Alfyorov:2026paper2], previously obtained in the two-component Stelle parametrization): $m_{2}r\sim 10^{8}$ and $V/V_{N}=1-(4/3)\,e^{-10^{8}}=1$ to exponential precision.

The linearized equation of motion for tensor perturbations is $\Pi_{\mathrm{TT}}(\Box/\Lambda^{2})\,\Box\,h_{\mu\nu}=0$, which in momentum space becomes $\Pi_{\mathrm{TT}}(z)\cdot z=0$ with $z=(-\omega^{2}+k^{2})/\Lambda^{2}$. This factors into two branches:

1. (i)

   $z=0$, i.e., $\omega^{2}=k^{2}$: the massless graviton with $v_{\mathrm{ph}}=v_{g}=c$. This mode is unmodified because $\Pi_{\mathrm{TT}}$ is a scalar multiplicative factor acting on the Lorentz-invariant combination $k^{2}$.

2. (ii)

   $\Pi_{\mathrm{TT}}(z_{0})=0$: massive fakeon modes. These do not propagate as physical particles under the fakeon prescription.

The massless graviton dispersion relation ($z=0$ branch) is $\omega^{2}=k^{2}$ exactly at one loop: there is no birefringence, and the signal velocity equals $c$. This is a result about free propagation on flat background; on curved backgrounds (e.g., Schwarzschild), the perturbation equation acquires corrections $\sim c_{2}(\omega/\Lambda)^{2}$ from the nonlocal form factor — see Section 5.2. Both statements (Euclidean derivation) are conditional on the correctness of the Wick rotation for form factors $F_{1}$ that are entire functions of the complex argument.

This is compatible with the GW170817 bound [GW170817:2017] $|c_{T}-c|/c<10^{-15}$, which SCT satisfies exactly (not approximately) at one loop.

The cosmological constant $\Lambda_{\mathrm{cc}}$ enters the spectral action through the $a_{0}$ coefficient and the moment $f_{4}=\int_{0}^{\infty}f(u)\,u\,\mathrm{d}u$ (in the Chamseddine–Connes convention where $a_{0}\propto f_{4}\,\Lambda^{4}$). The physical value of $\Lambda_{\mathrm{cc}}$ is not predicted; it is a free parameter set by the choice of $f_{4}$ and the renormalization-group trajectory. The spectral action generically produces $a_{0}\propto f_{4}\Lambda^{4}$, which exceeds the observed $\Lambda_{\mathrm{cc}}\sim 10^{-122}\,M_{P}^{4}$ by ${\sim}\,120$ orders of magnitude; SCT inherits the cosmological constant problem from quantum field theory. This gap is shared by LQG, AS, and IDG.

QNM frequency shifts in SCT receive two independent contributions:

1. 1.

   Metric modification (Level 2, computed): $\delta\omega/\omega\sim\exp(-m_{2}r_{\mathrm{peak}})$. For GW150914 ($M=62\,M_{\odot}$): $m_{2}r_{\mathrm{peak}}=7.78\times 10^{9}$, giving $\log_{10}(\delta\omega/\omega)\approx-3.4\times 10^{9}$.

2. 2.

   Perturbation-equation correction (parametric estimate): $\delta\omega/\omega\sim\mathcal{O}(1)\cdot c_{2}(\omega/\Lambda)^{2}$, where $c_{2}=13/60$ and $\mathcal{O}(1)$ is an unknown dimensionless coefficient depending on the open problem Gap G1 (computation of $\delta\Theta^{(C)}_{\mu\nu}$ on the Schwarzschild background). For GW150914: $\omega/\Lambda\approx 3.4\times 10^{-10}$, giving $\delta\omega/\omega\sim 10^{-20}$ up to the $\mathcal{O}(1)$ factor.

Contribution (2) dominates by $\sim 10^{3\times 10^{9}}$ orders of magnitude. Both are $\geq 15$ orders below LIGO sensitivity ($\sim 10^{-1}$). Results for observed black holes are given in Table 5. Numerical values are computed in the Stelle approximation ($m_{2}^{\mathrm{Stelle}}=\Lambda\sqrt{60/13}\approx 2.148\,\Lambda$); for the exact propagator zero (12) ($m_{2}=1.554\,\Lambda$ at $n=1$) the values of $m_{2}r_{\mathrm{peak}}$ decrease by $1.554/2.148\approx 0.72$, shifting $\log_{10}(\delta\omega/\omega)$ by $\sim 0.14$ — within the order-of-magnitude accuracy of the estimate. At $\xi=1/6$ the scalar mode decouples and $h(r)=1-(4/3)\,e^{-m_{2}r}$; the table is given for $\xi=0$ (both Yukawa modes) for generality.

Beyond QNMs, the following observables are indistinguishable from GR:

• •

  Neutron star structure (unmodified TOV equation),

• •

  Late-time cosmology ($\delta H^{2}/H^{2}\sim 10^{-64}$; [Alfyorov:2026paper4]).

The SCT scalaron potential (if the scalaron existed) would violate the refined de Sitter Swampland conjecture [Obied:2018] at $\mathcal{O}(1)$ parameter values. The curvature condition has a hard ceiling $\eta_{\min}=-1/3$, which cannot satisfy the Swampland parameter $\tilde{c}\sim 1$.

Since $\xi=1/6$ eliminates the scalaron entirely, this tension is moot for the standard spectral action: there is no scalar potential to test against the conjecture. Confirmation of pure Starobinsky inflation by CMB-S4 or LiteBIRD [LiteBIRD:2022] would increase the tension between the Swampland programme and $R^{2}$-type models in general.

We compare SCT with five quantum gravity programs: Loop Quantum Gravity (LQG) [Rovelli:2004, Thiemann:2007], based on non-perturbative canonical quantization with spin-network states; Asymptotic Safety (AS) [ReuterSaueressig:2012], based on the existence of a non-Gaussian UV fixed point of the gravitational RG flow; Causal Dynamical Triangulations (CDT) [AJL:2012], a lattice approach with a causal constraint on the path integral; string theory [Polchinski:1998], based on one-dimensional extended objects replacing point particles; and Infinite Derivative Gravity (IDG) [BMS:2006, Modesto:2012], which modifies the graviton propagator with an entire-function form factor to achieve ghost-freedom. We do not include Causal Set Theory [BLMS:1987] as a separate entry: it does not by itself produce the quantitative predictions listed in the comparison axes, though the CJ bridge formula [Alfyorov:2026paper7] connects SCT to causal-set observables.

Table 7 presents the comparison across nine quantitative axes. Each cell contains the best available numerical value (or status assessment) from the primary literature.

Three axes yield predictions that are mutually incompatible across programs:

1. (1)

   $c_{\log}$. SCT: $+37/24$. LQG: $-1/2$. Opposite sign. AS: definition-dependent. IDG: no logarithmic correction. A measurement of $c_{\log}$ (e.g., through black hole area quantization signatures in LISA data [LISA:2023]) would sharply discriminate between programs.

2. (2)

   UV propagator. SCT: entire function of order 1. IDG: Gaussian $e^{-k^{2}/M^{2}}/k^{2}$. AS: power-law $G(k)\to g_{*}/k^{2}$. LQG: discrete spinfoam amplitude. These are four qualitatively distinct analytic structures.

3. (3)

   Matter coupling. SCT: $\alpha_{C}=13/120$, determined by the SM particle content alone. AS: the non-Gaussian fixed point constrains the number of matter fields but does not fix the coupling coefficient. LQG, string, IDG: matter coupling is a free input. SCT is the only program in this comparison where the zero-momentum gravitational coupling coefficient $\alpha_{C}(0)=13/120$ is determined by the SM particle content alone, without additional free parameters.

SCT and AS share identical matter one-loop form factors: the Codello–Zanusso basis function $f_{C}(x)=1/(12x)+(\varphi-1)/(2x^{2})$ [CZ:2012] coincides with $h_{C}^{(0)}(x)$ (4). This identity holds for any cutoff function (it is a property of the heat kernel, not of the RG scheme).