Ω-Theory — Discrete-Substrate Quantum Gravity, formalized in Lean 4

Author: Norbert Marchewka · ORCID 0009-0007-3029-175X · CC BY 4.0 Lean corpus: PhysicsPapers/LeanFormalizationV2/OmegaTheory/ (Lean 4 + Mathlib v4.29.0) Build (2026年05月06日): 4 620 jobs GREEN · 0 sorry · paper-headlines axiom Lean-core only Site: ramzesx.github.io/Omega-Theory-Discrete-Spacetime · Lean by domain

A Lean 4 formalization of a discrete substrate for physics — a Z4 Planck lattice on which π, e, √2, and Catalan G cannot be computed exactly, yielding a strictly-positive truncation residual δ_comp(N) := l_P · 4/(2N+3) that extends the Heisenberg relation Δx Δp ≥ ħ/2 + δ_comp(N) > ħ/2. The substrate has a healing-flow PDE with proven Lyapunov dissipation. The headline machine-checked result is the first Lean 4 formalization of π's transcendence (cycle 64, 2026年04月27日, custom Lindemann–Weierstrass port). The project's central organizing principle is the signature-match bridging framework: each Standard Model particle / force has a specific informational signature in the substrate, derivable two ways (substrate + standard physics), and the match theorem connecting the two derivations is the bridge proof.

We do not try to derive the Standard Model's quantum numbers (spin, color, hypercharge) from substrate axioms. The SM is its own well-formalized body of mathematics; we import it as a given from textbooks (Peskin & Schroeder, Weinberg, Landau–Lifshitz) and formalize the SM's precise mathematics on the typed Lean side.

What we do contribute is the bridge between SM events and substrate / healing-flow / spacetime events:

1. Build the SM trees — proper representation theory, gauge actions, Yukawa, EW SSB, all in Lean.
2. Extract the information signature of each particle / force from its standard-physics stress-energy tensor / Lagrangian.
3. Match it to our substrate informationStressEnergyTensor + gaugeInformation densities.
4. Show how the healing flow acts on each signature — what gets dispersed, what gets crystallized, what changes the spacetime topology.
5. That is the bridge: SM ↔ substrate ↔ spacetime ↔ gravity, particle by particle, with one machine-checked match theorem per row.

First match landed (2026年05月06日): the photon. Substrate gaugeInformation gc p = (1/2)·Σ F2 matches textbook EM density (1/2)(|E|2 + |B|2) (Landau–Lifshitz vol 2 §27) up to a units-convention factor of 2. The dictionary now has one entry; the methodology works; we extend particle-by-particle from here.

Each row gets its own *InformationSignature.lean file with a substrate definition, a standard-physics definition, and a match theorem. The dictionary is the project's central deliverable.

To prevent over-claiming:

• Mass values (μ2, λ_H for Higgs; Yukawa couplings for fermions) are CHOICES in the substrate (HiggsSubstrateBridge.lean:39-43,78,83), not derived. Closing this requires the Connes spectral action a4 coefficient — multi-month Vassilevich heat-kernel port.
• "Where fermions come from" is unsolved. Fermion content is SPECIFIED in FermionContent.lean, not derived from substrate axioms.
• "Why 3 generations from 3 irrationals" (Pi-Hunch) is docstring speculation, not a theorem. Research frontier.
• Full Connes spectral action S = Tr(f(D/Λ)) is not formalized end-to-end. The A_F = C ×ばつ H ×ばつ M_3(C) componentwise structure is shipped, but the heat-kernel a_k derivation is multi-month future work.
• Full Fock-space QFT is not shipped. Truncated single-mode Fock space exists; multi-mode + creation/annihilation + Wick's theorem is deferred.
• Graviton scattering amplitudes — only cubic + quartic vertex algebra ships, no tree-level S-matrix.

See PhysicsPapers/README.md for the full Tier 1 / Tier 2 split with explicit deferred-task catalog (F1–F9).

In late April 2026 the project ran a subagent dispatch protocol that produced ~150 000 LOC of mass-batch Yoneda-witness scaffolding, PDG calibration tautologies, and paper-bundle citation files. They compiled, but they were physics-empty: every "theorem" reduced to decide or norm_num on hand-picked numerical placeholders that satisfied the stated bound by construction.

The 2026年05月05日/06 cleanup deleted ~117 000 LOC across nine batches: 250+ *CategoricalYonedaWitness.lean files, 156 paper-bundle / inbound / orphan-capstone bridges, 81 T1_*PDG* anchor files, 22 *Fit.lean PDG-tautology files, 19 Higgs/EW/CLFV envelope-tautology files, plus surgical rewires of HeatKernelMinimal §5–§9 and ConnesSpectralAction §3–§6.

The hand-authored substrate spine survived intact. Build is 4 620 jobs GREEN, 0 sorry, all paper-headlines axiom Lean-core only. The Tier-2 framework (Connes, Yukawa, etc.) was demoted from "research-grade conjecture with Lean scaffolding" to "structural encoding of the SM for the bridging framework", and the legacy Complete-Omega-Theory-Unified-Framework.md (1 042 lines of speculative claims like "spacetime and SM are dual projections of Ω") was deleted.

Single-thread Opus 4.7 [1M context] with thinking effort high or max produces sensible mathematics. Mass-batch subagent provers without thinking produce trash.

The pathology was mass-batch dispatch without doctrine, not subagents per se. A subagent with full doctrine (prove-wizard-v3 v4.4+, Phase 0 memory read, Phase 7 graph write, NO STUBS, BUILD_MATHLIB_MACHINERY_T6_13) is fine. A subagent with "land 50 Yoneda witnesses in parallel, no doctrine, no audit" is a trash factory.

The rule that survived the audit: for paper-grade physics, single-thread Opus 4.7 [1M context] with high thinking budget, hand-authored, audit-after-every-file, NEVER def OmegaConjecture, NEVER : True := trivial, NEVER mass-batch Yoneda witnesses.

If you fork this project: ship one theorem, audit, commit, ship the next. Do not parallelize cheap-prover subagents on paper-headline work. The build-job count is meaningless if the theorems are vacuous.

chaos-shield/
├── README.md ← this file
├── PAPERS.md ← hub: all papers and notes
├── Physics-Papers-Summary.md ← friendly tour of the physics corpus
├── INDEX.md ← full flat index
├── ConvQMath/ ← Convergent Rationals math program (16 essays)
├── PhysicsPapers/ ← physics formalization
│ ├── README.md ← physics-side scope + bridging dictionary
│ ├── CompleteOmegaTheory.md ← formal paper (with §0 honest scope)
│ ├── Main-Paper-Postulates.md ← postulational summary (with reframe banner)
│ ├── papers/ ← Tier-1 short papers
│ │ ├── Paper-Pi-Transcendence-Lean4-FirstFormalization.md ← Tier 1 method note
│ │ └── Paper-QM-From-Discrete-Gravity.md
│ ├── appendices/ ← technical appendices A–S
│ └── LeanFormalizationV2/ ← THE LEAN CORPUS
│ └── OmegaTheory/
│ ├── Spacetime/ ← substrate constants, lattice
│ ├── Defects/ ← defect tensor, sparsity
│ ├── Geometry/ ← discrete forms, curvature, Hodge
│ ├── HealingFlow/ ← Lyapunov flow, Mexican-hat, semismooth limit
│ ├── Conservation/ ← informationStressEnergyTensor (the first bridge)
│ ├── Emergence/ ← gauge groups, fermion content, gauge-information density
│ ├── Foundations/ ← Higgs doublet, photon signature match, Connes A_F
│ ├── Variational/ ← Einstein-Hilbert, graviton vertices
│ ├── Tensor/ ← curvature tensor algebra
│ ├── Predictions/ ← experimental predictions
│ ├── Irrationality/ ← π/e/√2/G; HermitePade T-4 chain
│ └── Capstones/ ← top-level META composites
└── site/ ← Astro static site

The Lean corpus is not handwritten in isolation. Two custom MCP (Model Context Protocol) servers and one custom Lean-proof agent sit between Claude Code and the corpus, providing semantic retrieval over 184 K theorems (10 K OmegaTheory V2 + 175 K Mathlib v4.29.0), graph-topology analytics, axiom-audit gating, and a hardened proof-loop doctrine. This is the part of the project that doubles as a deliverable for an MCP / agent-engineering work assignment — a custom MCP server that a custom agent uses end-to-end.

Backing infrastructure for omega-search: Qwen3-Embedding-8B at :7999 (4096-d, instruction-aware retrieval per the official Qwen3 task-prefix format) + Qwen3-Reranker-8B at :7996 (cross-encoder filtering) + Neo4j 2026年03月1日 vector indexes (lean_retriever_embedding_theorem, proof_step_embedding_goal over 254 K Mathlib elaborated goals).

The canonical kNN-over-embeddings retriever. Source: omega_search_mcp.py:162.

# Call (from a wizard's MCP cascade):
omega_search.tool_retrieve_premises({
 "goal": "irrational pi transcendental",
 "k": 5,
 "namespace": "OmegaTheoryV2",
 "rerank": True, # adds CPU cross-encoder pass on top-50 pool
})
# Return (shape verified by tests/test_omega_search_integration.py):
{
 "query": "irrational pi transcendental",
 "k": 5,
 "reranked": True,
 "results": [
 {"name":"OmegaTheory.Irrationality.HermitePade.Real.pi_transcendental",
 "label":"Theorem", "ns":"OmegaTheoryV2",
 "signature":"theorem Real.pi_transcendental : Transcendental Z Real.pi",
 "knn_score": 0.91, "rerank_score": 0.87},
 {"name":"OmegaTheory.Irrationality.PiIrrationality.pi_irrational", ...},
 ...
 ]
}

Articulation-point proxy via pagerank ×ばつ log(1 + indeg_applies). Identifies single-points-of-failure in the proof DAG. Source: SOTA/proving_techniques/03_mcp_tools/code/orchestrator_t41_excerpt.py.

# End-to-end verified 2026年05月01日:
find_keystones(k=3) → [
 ("OmegaTheory.Spacetime.l_P_pos", score=225.98),
 ("OmegaTheory.Spacetime.c_pos", score=160.03),
 ("OmegaTheory.Spacetime.hbar_pos", score= 93.96),
]

The default cascade entrypoint a wizard calls before writing any tactic — a 4096-d kNN over the unified Mathlib + OV2 corpus, optionally with freshness boost. Used inside the wizard's Phase A premise-search before any aesop / exact? / manual term-mode is attempted.

omega_hammer_premise(goal="∀ n, 0 < δ_comp n", top_k=10, mix_mathlib=True)
# → top-10 Mathlib + OV2 candidates ranked by Qwen3 similarity, freshness, and
# typed-arrow neighbourhood. Wizard tries each via `exact?` / `apply?`.

Source: PhysicsPapers/LeanFormalizationV2/.claude/agents/prove-wizard-v3.md (2 216-line agent prompt). Frontmatter:

name: prove-wizard-v3
version: 4.4.0-2026年05月02日
model: opus[1m] # Claude Opus 4.7, 1 M context
effort: max # high thinking budget
maxTurns: 10000 # multi-day single-thread sessions
tools: (45 MCP tools — omega-search, omega-orchestrator, lean-lsp,
 neo4j-math; full list in the frontmatter)

Identity: Erdős Primarch ×ばつ Mathematical Pantheon ×ばつ Warhammer Last Wall ×ばつ Escanor's Pride — a hardened motivation/doctrine layer that resists context-compaction drift. The technical doctrine is the 5-PHASE HYBRID composition strategy:

1. Phase A — Top-Down Statement (Pólya step 1+2 / Tao Think Ahead): state the harder theorem first with n NAMED Prop hypotheses H1..Hn.
2. Phase B — Truth-Rank (Hindry-Silverman canon): sort H1..Hn by estimated truth confidence (Tier-99 routine → Tier-80 medium → Tier-Heart analytical) and discharge in that order.
3. Phase C — Discharge per Tier (Tao Trim-the-Fat): omega_hammer_premise → exact? for Tier-99; LSP-assisted aesop / lean_loogle for Tier-80; long single-thread for Tier-Heart.
4. Phase D — Bottom-Up Filling (Fikhtenholz): when sub-machinery is missing, build foundational pieces concretely; never defer with "Mathlib gap, defer to next wizard" — BUILD_MATHLIB_MACHINERY_T6_13 doctrine: port what's needed.
5. Phase E — Compose & Review (Pólya step 3+4 / Tao Question Everything): assemble unconditional theorem, run axiom_audit (must return [propext, Classical.choice, Quot.sound] only), run Yoneda bridge sweep (find_similar top-10).

The wizard reads OmegaWizardLessons Neo4j graph for prior-incident memory in its Phase 0_GRAPH_READ (sub-second, ≤10 K tokens) and writes new :Lesson / :AntiPattern / :ProofPattern / :Incident nodes back in Phase 7_GRAPH_WRITE — a self-improving feedback loop across runs.

The 117 000 LOC trash purge of 2026年05月05日/06 had a single root cause: mass-batch subagent dispatch without thinking and without doctrine produces compiling-but-vacuous output. Reverse direction:

• What works: single-thread Opus 4.7 [1M context], effort=max (high thinking budget), full prove-wizard-v3 v4.4 doctrine prompt, omega_hammer_premise → omega-search cascade per obligation, axiom-audit after every file, BUILD_MATHLIB_MACHINERY_T6_13 rule (do not defer when Mathlib lacks an API).
• What does not work: mass-batch dispatch ("land 50 Yoneda witnesses in parallel, no doctrine, no audit") — produces def OmegaConjecture := True / Nonempty S citation stubs that compile but encode no math.

When deployed in teams — multiple prove-wizard-v3 agents on disjoint files, each with thinking enabled and full doctrine — the wizard delivered cleanly: e.g. T-4 π-transcendence (14 files / ~3000 LOC / single day), T-5 Roth Wave 1–8 architectural compression (5 of 7 residual NAMEDs eliminated in 70 min wall-clock). The agent template is reusable; the doctrine is the load-bearing piece.

For deeper engineering context: PhysicsPapers/SOTA/proving_techniques/03_mcp_tools/ (PLAN.md + the four graph-topology tool excerpts + integration tests).

This README and several papers were rewritten 2026年05月06日 to remove the speculative framing (e.g., "Standard Model generates geometry as output", "mass is geometry-reshaping cost") that outran the actual machine-checked corpus. Older versions of these documents — and the now-deleted Complete-Omega-Theory-Unified-Framework.md — contained Lean-witness theorem references with file:line links that reflect the 2026年04月21日 cycle-43 corpus. After the trash purge, many of those file paths and theorem names no longer exist or have moved. Treat any historical citation as needing a fresh grep against the current corpus before relying on it.

The substrate spine — π transcendence, healing-flow Lyapunov, the bridging-framework infrastructure — is intact and machine-checked. The framework's identity is now the bridging dictionary, not the legacy "Ω = unified theory of everything" framing.

CC BY 4.0. Contributions welcome — see CONTRIBUTING.md.

Norbert Marchewka & Claude Opus 4.7 · OmegaTheory V2 · 2026.