SCT Theory

Spectral Causal Theory is a research program investigating whether gravity and its quantum corrections can be derived from the spectral data of the Dirac operator.

This repository is an active research workspace rather than a single frozen release snapshot. The badges above reflect a local repository audit on 2026-04-11 (Release 4.0) rather than a harmonized semantic version tag.

Quick Start

git clone https://github.com/davidichalfyorov-wq/sct-theory.git
cd sct-theory
python -m pip install -r requirements.txt

python -m pytest analysis/ -x -q              # run 5047+ verification tests
python analysis/run_ci.py                     # full CI pipeline
python papers/build.py                        # compile all TeX targets

---

The Problem

Modern physics rests on two pillars:

Framework	Describes	Works at
General Relativity	Gravity, spacetime, black holes, cosmology	Large scales
Quantum Field Theory	Particles, forces, the Standard Model	Small scales

Both work extraordinarily well in their domains. The open problem is what happens in regimes where both matter: the early universe, black hole interiors, and quantum corrections to gravity itself.

The Idea

Instead of writing down an arbitrary higher-derivative gravitational action, SCT starts from a different premise:

Geometry leaves a fingerprint in the spectrum of the Dirac operator. We read physics from that fingerprint.

Concretely, the spectral action principle uses the spectrum of a generalized Dirac operator to construct an action functional. Expanding it in curvature invariants recovers the Einstein-Hilbert action at low energy, but also produces specific, calculable quantum corrections governed by nonlocal form factors.

These form factors are not free parameters. They are fixed by the spectrum and the particle content of the Standard Model.

Current Repository Focus

The repository currently has two active layers:

• the main spectral-action / form-factor / phenomenology program
• an exploratory finite-nerve / FND-1 formalization under theory/lean/SCTLean/FND1/

The section below summarizes the current audited state of the main spectral-action line, not a tagged release note.

Chirality proof and perturbative UV finiteness

One central current result is a four-line algebraic proof that all perturbative counterterms in D²-quantization of the spectral action are block-diagonal in the chiral basis. The chirality of the counterterms is proven. Whether chirality implies absorbability by spectral function deformation is a conjecture (see Erratum in Paper 5).

The proof rests on the identity: since the Dirac operator D anticommutes with the chirality operator γ₅, the perturbation δ(D²) automatically commutes with γ₅. This forces the kinetic operator, propagator, vertices, and all multi-loop diagrams into the chirality-preserving subalgebra.

• UV finiteness holds through two loops without additional assumptions
• At all perturbative orders under five BV axioms (two proven, one natural, two verified to one loop)
• The algebraic identity is formally verified in Lean 4 (13 theorems, zero sorry)

Black hole entropy

The Wald entropy for Schwarzschild black holes in SCT is computed with full Standard Model content:

S = A/(4G) + 13/(120π) + (37/24) ln(A/ℓ_P²) + O(1)

The logarithmic coefficient c_log = 37/24 is determined by the SM particle content via the Sen formula (2012) and has opposite sign to the Loop Quantum Gravity prediction (c_log = −3/2), providing a potential observational discriminant. For the SM+3ν_R extension, c_log = 8/5.

Black hole singularity: softened, not resolved

In the local Yukawa approximation, the modified Newtonian potential has V(0)/V_N = −1/3 (post-no-scalaron, spin-2 only). In the full nonlocal treatment, V(0) is not well-defined at one loop (the Fourier integral diverges for r → 0). The Kretschner scalar is softened from K ~ r⁻⁶ (Schwarzschild) to K ~ r⁻⁴. However, singularity resolution requires exponential UV suppression of the propagator (entire order ≥ 2, as in infinite derivative gravity). SCT’s propagator denominator Π_TT(k²) approaches a finite constant at high momenta, giving the same 1/k² UV behavior as GR. This is a structural limitation: the master function φ(z) is entire of order 1, producing algebraic rather than exponential modification. The Giacchini–de Paula Netto threshold (2018) requires ≥ 6 derivatives for curvature regularity; SCT effectively has 4.

Upgraded consistency results

Five previously open questions about the theory's internal consistency are now resolved or narrowed:

• Two-loop finiteness is now unconditional (previously required a specific absorption scheme)
• Unitarity in D²-quantization: the massive spin-2 ghost (z₀ = 2.41, m₂/Λ = 1.55) is treated via the Anselmi-Piva fakeon prescription, which projects it out of the asymptotic spectrum
• Optical theorem verified at one loop
• All-orders finiteness is conjectured under five BV axioms (two proven, one natural, two verified to one loop only; extending to all orders is an open problem)

Literature cross-check

An equation-by-equation comparison document verifies every SCT form factor result against published literature:

• Codello-Zanusso (2013, J. Math. Phys. 54, 013513)
• Codello-Percacci-Rahmede (2009, Annals Phys. 324, 414)
• Tseytlin-Shapiro-Ribeiro (2020, Phys. Lett. B 808, 135645)

All results match to 15+ significant digits. Zero discrepancies found. One convention difference (sign of endomorphism, notational).

Honest limitations identified

• Inflation: The standard spectral action predicts conformal coupling ξ = 1/6, at which the R² sector and the associated scalaron are entirely absent. Even at minimal coupling (ξ = 0), the scalaron mass is M = 15.4 M_Pl — six orders of magnitude too heavy for Starobinsky inflation. SCT does not explain inflation without BSM extensions to the spectral triple.
• Singularity resolution: Softened (K: r⁻⁶ → r⁻⁴) but not resolved. The propagator denominator Π_TT → const at UV, giving 1/k² behavior. The Giacchini–de Paula Netto threshold requires ≥ 6 derivatives for curvature regularity; the spectral action effectively has 4. See “Known Problems” below.
• D²-quantization vs metric quantization: Physical equivalence established through one loop. All-orders equivalence conditional on two BV axioms (Jacobian well-definedness and anomaly freedom), verified to one loop.

What This Repository Does

The project derives, computes, and verifies predictions step by step:

Spectral geometry  ->  Spectral action  ->  Heat kernel expansion
     ->  Form factors (per spin)  ->  Combined SM coefficients
     ->  Field equations  ->  Modified gravity  ->  Testable predictions

Key results established

One-loop form factors for all Standard Model sectors (scalar, Dirac, vector) are computed and cross-verified. All form factors are governed by a single master function:

This function is entire (no poles in the complex plane), which guarantees that the form factors themselves do not introduce additional ghost poles beyond those already present in the dressed propagator Π_TT.

The plot uses x = -k^2/\Lambda^2, so the left branch corresponds to x < 0 and the right branch to x > 0. The left branch is the Euclidean continuation and grows as x -> -infinity because |x| increases toward the left; the right branch decays as \varphi(x) ~ 2/x for x -> +infinity. In particular, the explicit factor e^{-x/4} in the closed form does not imply exponential decay on the positive branch, because \operatorname{erfi}(\sqrt{x}/2) contributes the compensating e^{x/4} asymptotic.

Modified Newtonian potential. At distances comparable to the spectral scale 1/Λ, the gravitational potential departs from Newton's law. In the Yukawa approximation (valid for r ≫ 1/Λ):

Standard Model contributions. Each particle sector (scalars, fermions, gauge bosons) contributes different heat kernel coefficients and form factor profiles:

Standard Model coefficients. The combined Weyl-squared coefficient α_C = 13/120 and the ratio c₁/c₂ = −1/3 at conformal coupling follow from standard heat kernel theory (Gilkey 1975, Vassilevich 2003) applied to the SM particle content. They are not free parameters of the spectral action — they are fixed by the spectrum.

Formal Verification (Lean 4)

The repository currently contains two distinct Lean layers:

• a legacy theorem layer in theory/lean/proofs/ for spectral-action and form-factor identities
• an active finite-nerve / FND-1 layer in theory/lean/SCTLean/FND1/ for boundary, chain-complex, and H1 constructions

The legacy core algebraic identities remain machine-verified in Lean 4 with Mathlib:

Theorem	Statement	File
chiral_q_identity	(AB + BA + B²)C = C(AB + BA + B²) given AC = −CA, BC = −CB	theory/lean/proofs/chiral_q_identity.lean
bv_canonical_transformation	BV canonical transformations preserve the antibracket	same
centralizer_inv_closed	If [K, γ5] = 0 then [K−1, γ5] = 0	same
+ 10 more	Even Clifford comm, spin connection, diffeo generator, CME, ...	same

These 13 legacy theorem files are still present, but they are no longer the whole formal picture. The active SCTLean/FND1/ stack adds a much larger exploratory formalization layer, including finite-nerve support, boundary operators, d1 ∘ d2 = 0, and a first-homology interface.

Literature Cross-Check

Every key equation is traced to published sources:

SCT result	Published source	Status
Master function φ(x)	Codello-Zanusso (2013) eq. (2.3)	Exact match
Five CZ form factors	Codello-Zanusso (2013) eq. (2.21)	Exact match
βW per spin	Codello-Percacci-Rahmede (2009) eq. (III.9)	Exact match
αC = 13/120	CPR counting with SM content	Exact match
All local limits	CZ (2013) eq. (2.22)	Exact match

Full comparison: theory/derivations/SCT_literature_comparison.tex

Repository Structure

theory/           Formal theory content
  axioms/           Foundational postulates
  derivations/      Step-by-step mathematical derivations
  predictions/      Testable predictions with observables and precision targets
  consistency-checks/  Internal consistency proofs and audit notes
  lean/proofs/      Legacy Lean 4 theorem files for spectral-action identities
  lean/SCTLean/FND1/  Active finite-nerve boundary/homology formalization

analysis/         Computational backbone
  sct_tools/        Python package (18 top-level modules)
  scripts/          Verification and computation scripts
  figures/          Publication-quality figures

papers/           Publication drafts and build tools
docs/             Roadmap, overview, presentations

Verification Philosophy

Hard derivations fail for boring reasons: wrong signs, mismatched conventions, silent transcription errors. This project uses an 8-layer verification pipeline instead of trusting any single calculation:

Layer	Method	Purpose
1	Analytic checks	Dimensions, limits, symmetries, pole cancellation
2	Numerical (100+ digits)	High-precision evaluation at multiple test points
2.5	Property fuzzing	1000+ randomized hypothesis tests
3	Literature comparison	Cross-check against 13+ independent references
4	Dual derivation	Independent method, different approach
4.5	Triple CAS	SymPy, GiNaC, and mpmath must agree to 12+ digits
5	Lean 4 formal proofs	Machine-verified rational identities
6	Multi-backend	Multiple Lean backends must independently pass

Some pipeline stages use automated and AI-assisted tooling. All mathematical content, physical interpretation, and claims are verified and approved by the human author.

Papers

#	Paper	DOI
1	Nonlocal one-loop form factors of the spectral action with Standard Model content	10.5281/zenodo.19039242
2	Solar system and laboratory tests of the spectral action scale	10.5281/zenodo.19045796
3	Chirality of the Seeley-DeWitt coefficients and quartic Weyl structure in the spectral action	10.5281/zenodo.19056204
4	Nonlinear field equations and FLRW cosmology of the spectral action with Standard Model content	10.5281/zenodo.19056349
5	Perturbative UV finiteness of the spectral action in D²-quantization: a chirality proof	preprint in repository
6	Auxiliary boundary data and the failure of intrinsic coherence in a finite-nerve route for spectral causal theory	preprint in repository
7	Weyl curvature from the Hasse diagram: a parameter-free bridge formula for causal sets	10.5281/zenodo.19364212
8	Non-perturbative spectral gravity measure: pro-torsor structure and the obstruction to canonical expectations	Submitted to Annales Henri Poincaré

Research Status

Selected research highlights:

Topic	Key result	Status
One-loop form factors	All SM sectors computed, master function entire	Complete
Nonlinear field equations	Full variational equations + FLRW reduction	Complete
Lorentzian formulation	Wick rotation of the spectral action	Complete (perturbative)
Unitarity	Spin-2 ghost handled via Anselmi-Piva fakeon prescription	Conditional (fakeon for infinite poles open)
Causality	Signal speed = c (macroscopic); micro-violation at ℓ ~ 1/Λ	Conditional
Two-loop finiteness	Counterterm uniquely absorbed	Unconditional
All-orders finiteness	Conjectured via chirality + two unproven BV axioms	Open conjecture
Graviton scattering	Tree-level SCT = GR; one-loop finite	Verified (internal)
No-scalaron theorem	Πs(z,ξ) > 1 for all z > 0 and ξ ≠ 1/6; scalar graviton absent; m₂ = 1.554Λ	Complete
Solar system tests	Spectral scale Λ > 8.50 meV from GWTC-3 (post-no-scalaron, parameter-free)	Complete
SM+3νR extension	Full one-loop sector: ODE, zeros, Hadamard (aH=1/15), magic number 43/12, dS comparison	Complete
Black hole entropy	clog = 37/24 (SM) / 8/5 (SM+3νR); opposite sign to LQG	Verified (internal)
Black hole singularity	Kretschner softened r−6 → r−4; not resolved (ΠTT → const, same 1/k² UV as GR)	Negative
Late-time cosmology	Corrections 60+ orders below observability	Consistent
Inflation	Scalaron mass too heavy; requires BSM extension. At NCG-predicted ξ = 1/6 no scalaron exists.	Negative
De Sitter conjecture	Refined Swampland dS conjecture violated; ηmin = −1/3	Resolved
Non-minimal coupling	ξ = 1/6 follows from the standard Chamseddine-Connes spectral triple normalization; depends on the choice of spectral triple	Resolved (convention-dependent)
CJ bridge formula	Parameter-free relation CJ = C₀ N8/9 E² T⁴ linking Hasse-diagram observable to electric Weyl tensor; R = 1.016 ± 0.015; factor 4 = 2alg × 2dyn (Mss → 2 verified N=1k–10k); 105 Lean theorems; Paper 7	Conditional (two conditions unproven)
FND-1 finite-nerve route	Auxiliary chain complex + H₁ formalized (63 Lean modules); intrinsic coherence obstruction proved; Paper 6	Negative (support-only) / Open (causal order)

Open Problems Collection

The repository includes a structured collection of 50 open research problems in open-problems/, organized by domain and ranked by impact. Each problem file is self-contained with statement, known results, failed approaches, success criteria, and references.

Domain	Count	Status
Foundations	6	1 resolved (OP-04), 5 open
Unitarity	6	1 partial (OP-07), 5 open
UV finiteness	4	2 resolved (OP-13, OP-14), 2 open
Cosmology	4	2 resolved (OP-17, OP-20), 2 open
Black holes	3	all open
Spectral dimension	2	all open
Predictions	8	3 resolved (OP-30, OP-31, OP-33), 5 open
Causal sets	10	1 resolved (OP-40), 1 closed (OP-41), 8 open
Scalar sector	1	resolved (OP-44)
Numerical	5	all open
Formal verification	1	open

Resolved problems:

• OP-20 (de Sitter conjecture): The refined Swampland dS conjecture (Ooguri-Vafa 2018) is violated by the SCT scalaron potential for c₁, c₂ ~ O(1). The minimum value of the normalized Hessian is η_min = −1/3, imposing a hard ceiling on c₂. The gradient condition fails above φ ≈ 1.19 M_Pl. Both ratios |V′|/V and V″/V are independent of the scalaron mass M₀.

• OP-44 (critical coupling ξ): Under the standard Chamseddine-Connes spectral triple normalization, the Higgs non-minimal coupling takes the conformal value ξ = 1/6. The a₄ Seeley-DeWitt coefficient structure gives this value after canonical Higgs normalization. This is confirmed by five independent groups (2006-2015) and is an exact one-loop RG fixed point (β_ξ vanishes identically at ξ = 1/6). At conformal coupling the scalar graviton mode decouples entirely, and the Starobinsky scalaron is absent.

• OP-04 (parameter counting): The cutoff function f in the spectral action S = Tr(f(D²/Λ²)) is not uniquely determined by physical requirements (entireness + unitarity + causality). The robust prediction core of SCT consists only of a₄-level quantities: α_C = 13/120, c₁/c₂, PPN parameters, and c_T = c on FLRW. All finite-momentum observables (form factors, effective masses, modified potential, spectral dimension) depend on the choice of f. However, the variation across admissible entire cutoffs is small (~5% for effective masses).

• OP-17 (scalaron mass): No known mechanism within the standard NCG spectral action can produce a scalaron mass compatible with Starobinsky inflation while preserving α_C and the geometric scalar couplings. Combined with OP-44 (ξ = 1/6): the scalaron is entirely absent at the NCG-predicted conformal coupling. All known BSM scalars from NCG spectral triples also have conformal coupling, contributing zero to α_R. The only surviving path is reinterpretation of Λ as a sub-Planckian intermediate scale.

• OP-33 (cross-program comparison): A systematic 6×9 quantitative comparison table across SCT, LQG, Asymptotic Safety, CDT, String Theory, and IDG, with equation-level citations. Three discriminating axes identified (c_log, UV propagator, matter coupling); one quasi-universal axis (d_S → 2 in UV).

Details, methodology, and full problem files: open-problems/README.md

Known Problems and Open Questions

The spectral action framework has well-known theoretical vulnerabilities. This project does not claim to have solved them.

The spin-2 ghost. The Weyl-squared term in the spectral action produces a massive spin-2 mode with wrong-sign residue (Ostrogradsky ghost). This is not an artifact of our computation — it is inherent to all quadratic gravity theories (Stelle 1977). The nonlocal form factors prevent new ghost poles from appearing, but they do not eliminate the fundamental one. Our approach uses the fakeon prescription (Anselmi-Piva 2017), which projects the ghost out of the physical spectrum. This prescription is mathematically consistent but not universally accepted.

All-orders finiteness is a conjecture. The chirality theorem proves that the counterterm space is one-dimensional at any loop order. Combined with two BV axioms (well-defined Jacobian and anomaly freedom), this implies all-orders finiteness. However, these axioms are verified only through one loop. Extending the proof to all orders is an open mathematical problem. The claim is a conjecture supported by one-loop evidence, not a theorem.

No singularity resolution. The propagator denominator Π_TT approaches a finite constant at UV, giving 1/k² behavior — the same as GR. Singularity resolution requires exponential UV suppression (as in infinite derivative gravity), which the spectral action with Schwartz-class test function f does not produce. The Kretschner scalar is softened (r⁻⁶ → r⁻⁴) but still diverges at r = 0. The sometimes-quoted V(0) = 0 is an artifact of the local Yukawa approximation; the full nonlocal Fourier integral diverges at the origin in one-loop EFT.

No infrared predictions. The spectral action is a UV modification of gravity. All corrections are exponentially suppressed at distances above 1/Λ ≈ 0.08 mm. The theory cannot address the Hubble tension, dark energy, or large-scale structure anomalies.

The spectral action is Euclidean. The Chamseddine-Connes spectral action is defined on Riemannian manifolds. Its extension to Lorentzian signature relies on Wick rotation, which is well-defined perturbatively but has no established non-perturbative Lorentzian formulation. At the non-perturbative level, the Euclidean functional integral over Dirac operators does not admit a canonical expectation value: background-dependent Gaussian sectors are mutually singular (Feldman-Hajek), and measure classes form a pro-torsor rather than a single measure (Paper 8). The Lorentzian extension of this pro-torsor structure is an open problem.

Interpretation. The spectral action can be viewed as a fundamental principle (the spectrum of the Dirac operator determines all of physics) or as an effective field theory (a convenient generating functional for higher-derivative corrections). This project adopts the former as a working hypothesis but does not claim it is the only valid interpretation.

What This Project Is Not

• It is not a claim that all of fundamental physics is finished.
• It is not a replacement for peer review.
• It is not a promise that every research direction will survive future checks.

It is a serious research workspace built to make derivations reproducible, inspectable, and falsifiable.

Author

Formal theory documents and papers are authored by David Alfyorov (ORCID).

Credits

Research-assistance and workflow support: Igor Shnyukov.

Licensing

• Source code: Apache-2.0
• Research text, derivations, and documentation: CC BY 4.0

Star History