GeoSpectra Lab

A falsification-first numerical harness for finite-lattice spectral toy geometries

One-line: This is not a physics claim. It is an instrument that takes a compact product geometry, puts a spectral operator on a finite lattice, and asks how would I be fooled? before accepting any signal as real.

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Two Research Tracks

This repository contains two independent research projects on compact geometries:

	Track A — Numerical	Track B — Algebraic
Geometry	S³×S¹ finite lattice	S³×S⁶ spectral triple
Method	Eigensolver + IPR / r-stat	Index theory (Atiyah-Singer) + exact arithmetic
Verdict	DISCRETIZATION_SENSITIVE	Exact S⁶ index (=1), internally-certified 1D kernel — but the full S³×S⁶ operator has no zero mode for the round/Levi-Civita S³ construction actually used (KT-8, 2026-07-17); N_gen=3 is not yet established as a physical result
Key result	7.07× signal; geometry-agnostic	ind(D_{S⁶}⊗S⁻)=1 per channel — a mathematical index, not yet shown to be a massless 4D fermion mode
Tests	~500 regression tests	2500+ tests (fractions.Fraction, zero float ops; 2512 passed/4 skipped independently reproduced 2026-07-15)
Directory	cc_toy_lab/, scripts/, tests/	tom_s3_spinor_toy/
Entry point	reports/GATE4B_SPECIFICITY_VERDICT_v0.1.24.md	tom_s3_spinor_toy/RESEARCH_STATUS_REPORT.md

Track A explored whether a lattice product structure produces a robust spectral signal. It does — but the signal is DISCRETIZATION_SENSITIVE, not specific to S³×S¹ physics.

Track B constructs an exact Atiyah-Singer index computation on the S⁶ factor of S³×S⁶ geometry, with an internally-certified one-dimensional local kernel:

• G73: ind(D_{S⁶}⊗S⁻) = 1 per triality channel
• G74A: Lichnerowicz gap + G₂-Schur → dim ker = 1 on every non-trivial sector (certified); trivial-component rank = 1 verified by three independent internal routes incl. a closed-form analytic derivation (Round 59, 2026-07-14); external review outstanding (L4B, see below)
• G74B: sign(ind) = +1 → left-handed excess → SM chirality label

Blocking gap, tool-and-literature-verified (KT-8, 2026-07-17): the physical claim requires a zero mode of the full nine-dimensional internal Dirac operator on S³×S⁶, not the S⁶ factor alone. For the round, untwisted Levi-Civita S³ actually used throughout this project (product metric, product connection, twist pulled back from S⁶ only), the full operator provably has no zero mode: $D_{\mathrm{full}}^2=D_{S^3}^2\otimes1+1\otimes D_{S^6,S^-}^2 \geq(3/2\rho_3)^2>0$ regardless of the S⁶ factor's own index. N_gen=3 is therefore not yet an established physical result — the index above computes a mathematical object (zero modes of the S⁶-factor operator alone), not a demonstrated massless 4D fermion. A torsion-deformed S³ connection provides a mathematical (not physical) candidate escape route: the obstruction is removable at computable parameter values, but no physical principle is known for selecting them over the standard connection used elsewhere in this project. See reports/PROJECT_360_ROUND3_SYNTHESIS.md (KT-8 through KT-11) and tom_s3_spinor_toy/preprint.tex §Open Problems for full derivations.

Dimension correction: the total spacetime dimension of the ansatz actually used is 13 (4 external + 3 + 6), not 10 as earlier phrasing implied — "10D" conflated a spinor representation's dimension with a spacetime dimension count. No consistent 13-dimensional parent theory is claimed (standard supergravity is capped at 11D).

Open dependency (honestly flagged in G67/G68/G73 themselves, independent of the KT-8 gap above): even granting a resolution of KT-8, the "×3 channels ⟹ N_gen=3" step assumes three geometrically distinct octonion-multiplication channels (L_p, R_p, T_p) each appear in the S³×S⁶ Dirac action — gate G67-C3, 2/3 closed. G68 (2026-06-21) proves L and R are genuinely inequivalent Clifford(0,7) representations (pseudoscalar Ω_L=+I≠Ω_R=−I). The third (vector, 8_v) channel remains open, needs G72/Tom input. G44 (2026-06-20) shows G₂ (S⁶'s isotropy group) cannot distinguish the three SO(8) triality reps by G₂-content alone (8_v≅8_s≅8_c as G₂-modules) — the same fact G73 uses, for a different purpose. See TOM_RECONSTRUCTION_ACH_MATRIX.md Case 7 for the full reconciliation.

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Industrial Applications

The spectral-fingerprint method built here (Track A: graph-Laplacian eigenvalue signatures on finite lattices) has been adapted for a commercial use case — see GeoSpectra-Industrial (ScanGuard). Same core signal-detection principle, different question: instead of "is this signal specific to S³×S¹ physics?" it asks "is this 3D scan anomalous?". Pilot-ready MVP (Two-Mode Architecture, 20 ADRs), seeking first real-scan validation partners. Independent codebase, independent benchmarks.

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⚠️ Current Status — Track A (2026-06-03)

Item	Value
Latest verdict (v0.1.24)	DISCRETIZATION_SENSITIVE / GEOMETRY_AGNOSTIC (FINAL)
Aggregate true-IPR contrast (W=20 vs W=0)	7.07× (≈ 7.15× before S³ Dirac operator fix — <1.1% change, signal survived correction)
Specificity cascade	5 levels — see Key Result below
Total cases analyzed	306 (216 Gate 4B + 54 negative controls + 18 wilson scrambled + 18 spectral extended)
Active direction	Per-family divergence audit (ring stable, spectral_circle weakening); planned port to S³×S² per Tom Lawrence redirect (CAMP 2026-05-26)
Active falsification tests	FT-1 (r-stat W=0 baseline anomaly), FT-2 (inter-family IPR divergence), FT-3 (FSS strengthening vs denominator artifact) — see docs/CLAIMS_AND_CAVEATS.md

What "DISCRETIZATION_SENSITIVE / GEOMETRY_AGNOSTIC" means in one sentence: the harness can distinguish a lattice product structure from random / scrambled / broken baselines, but it does not distinguish between Wilson-term details inside the lattice family — i.e. it detects a lattice-discretized product-structure pattern, not S³×S¹ physics.

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Current Claim Boundary

Observed signal             ✅ yes
Survives S³ Dirac fix       ✅ yes (7.15× → 7.07×, <1.1% change)
Rejects random noise        ✅ yes (L1)
Rejects scrambled S¹        ✅ yes (L2)
Distinguishes FFT vs lattice ✅ yes (L3)
Specific to S³×S¹ physics   ❌ no
Physics claim               ❌ no
Best verdict                DISCRETIZATION_SENSITIVE / GEOMETRY_AGNOSTIC

Want this on one page? → docs/PROJECT_SKETCH.md (3-minute read)

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Track A — What the Numerical Harness Does

• Builds discretized Dirac-style spectral operators on compact product toy geometries (current case study: S³×S¹, largest operator dimension N = 13824 at s1_size=128; S³ Hilbert dimension 108 per S¹ site. Note: earlier docs cited N≤896 — a single-shell labeling erratum corrected in reports/DIMENSION_DISCREPANCY_AUDIT_v0.1.25.md).
• Runs pre-registered finite-size-scaling sweeps under Anderson-style on-site disorder.
• Measures true eigenvector-based Inverse Participation Ratio (IPR) and adjacent gap ratio (r-statistic).
• Compares the geometric signal against four classes of falsification controls: random Hermitian, scrambled geometry, broken Wilson term, alternative discretization (FFT vs lattice).
• Logs every claim with an explicit evidence marker and an explicit list of what each result does not mean.

Track A — What the Numerical Harness Does NOT Do

• Does not prove covariant compactification.
• Does not prove chiral fermions, protected zero modes, or SM chirality.
• Does not derive the Standard Model gauge group SU(3) × SU(2) × U(1).
• Does not validate any cosmological or physical extra-dimensional model.
• Does not claim a thermodynamic limit (N → ∞) — all results are finite-lattice.
• Does not prove S³×S¹ is the correct geometry for anything physical.
• Does not carry institutional endorsement from any third party, including Tom Lawrence.

Track B — What the Algebraic Project Establishes

See tom_s3_spinor_toy/ for full details and 2210 tests (pytest tom_s3_spinor_toy/tests/ only; running the full directory including experiments/ gives 2488 collected as of 2026-07-13, see tom_s3_spinor_toy/experiments/20260713-round58-readiness-audit/decision.md — different scope, not a discrepancy).

• ind(D_{S⁶}⊗S⁻) = 1 per triality channel exactly (G73, PROMOTE 29/29) — a mathematical index on the S⁶ factor alone. "N_gen = 3" is NOT yet an established physical result: the full internal Dirac operator on S³×S⁶ has no zero mode for the round Levi-Civita S³ construction actually used (KT-8, 2026-07-17, tool-and-literature-verified three times) — see "Status correction" below
• dim ker = 1, internally certified: Lichnerowicz spectral gap + G₂-Schur certify no extra zero modes on every non-trivial isotypic component (G74A + 20260713-round52→round56 general bound, PROMOTE); the trivial-component rank = 1 is verified by three independent internal routes — a from-scratch reimplementation blind to the original code, a full-fibre completeness + Hermiticity audit, and a closed-form analytic derivation (the amplitude IS the Killing-spinor Dirac eigenvalue −√3; twisting correction vanishes by rep theory) — see tom_s3_spinor_toy/experiments/20260714-round59-trivial-rank-certification/; external review outstanding (L4B in tom_s3_spinor_toy/preprint.tex Open Problems)
• SM chirality: sign(ind) = +1 → net left-handed excess, unconditional; all three modes purely left-handed under the same internally-certified L4B rank result above (G74B, PROMOTE 31/31)
• SM fermion content: 3+3̄+1+1 per generation from three independent routes — SO(8) triality (G67), CSDR on G₂/SU(3) (G69), SO(4)×G₂ rep theory (G24)
• Exact arithmetic: fractions.Fraction throughout — zero floating-point operations in the core index chain

Track B does NOT: claim S³×S⁶ is the physical compactification; derive the SM gauge group; fix coupling constant λ (free parameter, enforced by test suite); establish that the full internal Dirac operator has a zero mode (KT-8: it provably does not, for the round Levi-Civita S³ construction used here); claim the ansatz is 10-dimensional (it is 13-dimensional, 4+3+6).

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Key Result — 5-level specificity cascade (v0.1.24, FINAL)

Level	Test	FSS slope	Verdict	What it tells us
L1	Random Hermitian matrix	−1.14 (WEAKENING)	✅ REJECTS	Pure randomness fails the pattern — harness is not fooled by noise.
L2	Scrambled geometry (S¹ permutation)	−0.90 (WEAKENING)	✅ REJECTS	Broken topology fails — product structure matters.
L3	FFT discretization vs lattice discretization	spectral_circle −0.48 vs ring +0.01	✅ DISTINGUISHES	The harness is sensitive to the discretization method itself.
L4	Within lattice family (ring, wilson_ring)	both STABLE	✅ ACCEPTS	Any lattice product passes — robust within method.
L5	Wilson-term internal details (scrambled wilson term)	−0.07 (STABLE)	❌ DOES NOT DISTINGUISH	Wilson structure is irrelevant — sensitivity ends at L3.

Conclusion: the harness has specificity up to L3, not L5. Any external claim about this repository must respect that ceiling.

Full report: reports/GATE4B_SPECIFICITY_VERDICT_v0.1.24.md Unified audit: reports/UNIFIED_RESULT_RECONCILIATION_AUDIT_v0.1.24.md

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Quickstart (10 minutes)

git clone https://github.com/sergeeey/N-7-GeoSpectra-Lab.git
cd N-7-GeoSpectra-Lab

# Minimal environment (Python 3.11+)
pip install -r requirements.txt

# Full regression suite (494 tests across 44 files, cc_toy_lab track)
pytest -q tests/

# Algebraic spinor suite — Track B (2210 tests, exact arithmetic)
pytest -q tom_s3_spinor_toy/tests/ --tb=no

# Run a smoke version of the radion stabilization study
python scripts/radion_stabilization.py --quick

# Run synthetic r-statistic controls (Poisson / GOE / GUE)
python scripts/r_stat_controls.py --quick

Reproduce the Headline Result

The Gate 4B v0.1.24 grid (216 cases) is heavy — peak ≈ 10 GiB RAM on the largest case (N=128, j_max=3). Use a host with at least 32 GiB RAM.

# Full corrected rerun (≈ 1.8 hours on 16 vCPU / 32 GiB host)
python scripts/run_gate4_batched.py \
  --output-base reports/RUNS/gate4_fss_v0.1.24 \
  --protocol-version v0.1.24 \
  --ipr-metric-version v0.1.24_true_ipr_corrected_s3_dirac

Pre-registered protocol: reports/GATE_4B_RERUN_PROTOCOL_v0.1.24.md

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Architecture

flowchart LR
    A["Compact product geometry<br/>S³ × S¹"] --> B["Spectral operator<br/>Dirac / Anderson"]
    B --> C["Discretization<br/>3 lattice families + FFT control"]
    C --> D["Disorder sweep<br/>W = 0, 12, 20"]
    D --> E["Sparse eigendecomposition"]
    E --> F1["True IPR<br/>sum of psi_i^4"]
    E --> F2["r-statistic<br/>adjacent gap ratio"]
    F1 --> G["Pre-registered<br/>decision rules"]
    F2 --> G
    I["Falsification controls<br/>random / scrambled / broken / FFT"] --> G
    G --> H["Verdict<br/>PASS / WEAK / FAIL / DISCRETIZATION_SENSITIVE"]
    G --> J["Audit trail<br/>raw JSON + reports"]

Loading

Source modules: cc_toy_lab/{geometry, spectral, radion, topology, controls, discovery}/.

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Documentation Map

Track A — Numerical

You want to know...	Read this
Why this project exists and what question it actually asks	docs/RESEARCH_CONTEXT.md
Exactly what can and cannot be claimed externally	docs/CLAIMS_AND_CAVEATS.md
15 main artefacts and their outcome cards	docs/OUTCOMES.md
Hardware needed to reproduce the heavy grid	docs/HARDWARE_REQUIREMENTS.md
5-phase roadmap	docs/ROADMAP.md
The current FINAL verdict reasoning	reports/GATE4B_SPECIFICITY_VERDICT_v0.1.24.md
Cross-script reproduction audit (one loader, one formula)	reports/UNIFIED_RESULT_RECONCILIATION_AUDIT_v0.1.24.md
What was tried and failed (first-class results)	reports/NULL_RESULTS.md
Open scientific issues	reports/ISSUES_SCIENTIFIC.md
Full GitHub showcase audit (engineering hygiene + safety)	docs/GITHUB_SHOWCASE_AUDIT.md

Track B — Algebraic

You want to know...	Read this
Overview, gate chain G6–G74B, all results	tom_s3_spinor_toy/README.md
Full technical status report	tom_s3_spinor_toy/RESEARCH_STATUS_REPORT.md
Preprint abstract (T1/T2 theorems)	tom_s3_spinor_toy/preprint_abstract.md
24 falsified approaches (null results)	tom_s3_spinor_toy/null_results/INDEX.md
N_gen=3 core calculation	tom_s3_spinor_toy/experiments/20260621-g73-three-channel-dirac/

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Inspiration and Attribution

This project was initially inspired by broader questions in compact product geometries, Kaluza–Klein-style reasoning, and covariant compactification, including public work by Tom Lawrence.

Tom Lawrence's work is the analytical / theoretical line of inquiry that initially motivated the geometric choice of S³×S¹ as a test case. GeoSpectra is the independent computational line.

Resource	Link
Tom Lawrence — Website	https://warpedandbroken.com/
Tom Lawrence — LinkedIn	https://www.linkedin.com/in/tomlawrence_45533/
Tom Lawrence — ResearchGate	https://www.researchgate.net/profile/Tom-Lawrence
Tom Lawrence — ORCID	0000-0002-2741-8226
(2021) Tangent space symmetries in GR and teleparallelism — IJGMMP	arXiv:2211.07586 · DOI
(2022) Product manifolds as realisations of general linear symmetries — IJGMMP	arXiv:2203.09473 · DOI
(2023) Covariant Compactification: a Radical Revision of Kaluza–Klein Unification — preprint	Preprints.org 202303.0314
(2025) Symmetries of Field Configurations and No-Go Theorems — preprint	Preprints.org 202510.2222
(2024) General Relativity — its beauty, its curves, its rough edges... — essay (Minkowski Institute Press)	PDF
(2021) Do the symmetries of product spaces hold the key to unification? — Symmetry 2021 MDPI	DOI

Independence statement. GeoSpectra is developed independently by Sergey Boyko. It is not endorsed by Tom Lawrence. All errors, interpretations and claims in this repository are the author's own. The numerical results in this repository are not a test of Tom Lawrence's theory directly — per his own assessment at CAMP 2026-05-26, S³×S¹ is closer to original Kaluza–Klein style than to his covariant compactification framework, which requires at least two extra dimensions (next planned port: S³×S²).

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Author

Sergey Boyko — independent researcher Affiliation: Ronin Institute for Independent Scholarship 2.0 (Research Scholar) ORCID: 0009-0009-2178-5701 Email: sergey.boyko@ronininstitute.org (academic) · sergeikuch80@gmail.com (personal)

GitHub: @sergeeey Project repository: https://github.com/sergeeey/N-7-GeoSpectra-Lab

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How to Cite

If you use this methodology or reference this work, please cite via the metadata in CITATION.cff or:

Boyko, S. (2026). A Falsification-First Validation Harness for Discretized Spectral Operators on Compact Product Manifolds. Zenodo. https://doi.org/10.5281/zenodo.20252650

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License

Released under Creative Commons Attribution 4.0 International (CC BY 4.0). You may share and adapt the material, including for commercial purposes, with attribution.

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A Note on the Repository

This repository is intentionally a methodology project, not a marketing project.

• Negative results are first-class artefacts (see reports/NULL_RESULTS.md).
• Operator bugs are openly documented along with their corrected reruns (see reports/INCIDENT_GATE4B_v0.1.24_OOM_2026-05-25.md, commit 093573b).
• Each released verdict carries an explicit list of what it does not entail.
• A separate file docs/CLAIMS_AND_CAVEATS.md governs what may and may not be said externally.

If you find a wrong number, a misleading sentence, or a missing caveat, please open an issue — that is the most valuable contribution this project can receive.