# OMEGA NAUGHT — Integration Proof Bundle Prepared for Matthew Alan Newman, 14 October 2026\. ## What is delivered This bundle contains **34 precisely scoped mathematical theorem records**, their assumptions and derivations, a JSON corpus of **2,314 exact algebraic checks in** **19 groups**, and native **Python, Julia, and Rust** certificate interpreters and arithmetic verifiers\. It covers the nine original integration bridges and the subsequent geometric–affine, operator, symplectic, manifold, Boolean\-computing, and quantized\-layer extensions\. **Python was executed\. Julia and Rust were not compiled or executed in this** **environment\. Their source is supplied, not certified as a successful build\.** The authoritative run record is `evidence/execution_status.json`\. No Lean or other proof\-assistant kernel run is claimed\. Three implementations are not three independent mathematical reviews\. The written arguments establish their conclusions under their hypotheses\. A test result does not establish an unrelated analytic theorem\. All\-body convex inequalities, external inverse\-problem uniqueness theorems, a universal operator bound of two, universal fluid computation, and neural\-network performance claims are not imported as proved\. ## Files |File |Role | |----------------------------------|--------------------------------------------------------------------------------------------------------------------------------| |`omega_proofs.json` |The theorem ledger: definitions, assumptions, statements, proof steps, dependencies, evidence mappings, exclusions, and sources.| |`PROOFS.md` |A readable rendering of the same theorem records. The JSON remains the structured source. | |`certificates.json` |Exact symbolic instruction programs and explicitly labeled finite algebraic cases. | |`python/verify.py` |Python standard-library implementation, including exhaustive canonical ledger replay. | |`julia/verify.jl` |Native `BigInt` / `Rational{BigInt}` implementation; does not call Python. | |`julia/Project.toml` |Julia project dependencies. | |`rust/src/main.rs` |Native `BigInt` / `BigRational` implementation; does not call Python or Julia. | |`rust/Cargo.toml`, `rust/build.rs`|Rust build configuration and compiler-version recording. | |`run_all.py` |Runs available engines, compares normalized computed reports, and refuses to label missing engines as passed. | |`tests/adversarial_audit.py` |Python malformed-input, false-certificate, exact-field, and optimization-mode regression checks. | |`tests/sympy_crosscheck.py` |Optional second arithmetic implementation for 149 selected checks; requires SymPy. | |`schemas/` |JSON structure schemas. A schema validates structure, not mathematical truth. | |`evidence/` |Actual execution outputs, inventory, and explicit execution status. | |`provenance/` |Three original project sources, retained without changes. Their historical assertions are not silently upgraded. | |`SHA256SUMS.json` |Byte-integrity manifest. This is not an authenticated signature or proof of correctness. | |`check_integrity.py` |Standard-library checker for the manifest and source hashes. | ## Read the proof and evidence levels correctly **Deductive arguments\.** Each theorem states its assumptions, dependencies, and limits\. Some use classical results such as the transcendence of pi and the cyclotomic\-field degree formula; those dependencies are named, not reproved\. Analytic limits and operator theorems are proved by the written arguments, not by pretending to simulate an infinite\-dimensional space\. **Universal symbolic identities\.** Programs normalize multivariate polynomials over `Q(phi)`, with `phi^2 = phi + 1`\. A zero polynomial proves equality for all specializations of its free variables\. Rational denominators in a theorem may be cleared in its executable identity; the theorem retains the required nonzero\-denominator hypothesis\. Positivity and inequality instructions are restricted to exact constants, not arbitrary variable assumptions\. **Finite checks\.** Other programs verify exact fixed matrices, code minors, icosahedral moments, counterexamples, or declared sample cases\. In addition, each native implementation is designed to recompute the complete canonical 15\-bit ledger audit and the exact finite floor\-prefix certificate\. Only the Python execution of those implementations was observed here\. **Unestablished applications\.** The JSON records 12 excluded, conditional, unsupported, or refuted broader claims\. In particular, exact reconstruction is not a performance benchmark, and a geometric drawing is not a physical model\. ## Geometry and number conventions OMEGA begins with the unit circle, the naught stage\. The canonical quantities are `phi=(1+sqrt(5))/2`, `delta=pi/1296000`, `s=sin(delta)`, and `c=cos(delta)`\. The six ordered marks use coordinates `a*(z_i,c)`, where `z=(-s-phi^2,-s-1,-s,s,s+1,s+phi^2)`\. The earlier integration note uses the orthogonally exchanged coordinates `(c,z_i)`\. The selected readouts `G=a*phi` and `C=a*pi` define `lambda=phi/(phi+pi)`\. The selected `C` is **not** the retained major\-arc length\. The rational slope `q=339949771344778/10^15` is **not** lambda\. The canonical sine\-defined landmarks are algebraic in a larger cyclotomic field\. The exact `Q(phi)` worked examples with auxiliary rational seeds such as `s=1/2` are **not** substitutions for that canonical sine\. Canonical cyclotomic membership is proved in the ledger; the checker verifies its order and ambient\-degree arithmetic, not an expansion in a 345,600\-dimensional basis\. ## Reproduce the main runs Run these commands from the extracted bundle directory\. ### Python — no third\-party packages ```sh python python/verify.py certificates.json python -O python/verify.py certificates.json python tests/adversarial_audit.py python check_integrity.py ``` The primary checker targets Python 3\.10 or newer\. It was executed with Python 3\.13\.5\. Acceptance checks use explicit exceptions and remain active under `-O`\. ### Julia — source supplied, not executed here ```sh julia --project=julia -e 'using Pkg; Pkg.instantiate()' julia --project=julia julia/verify.jl certificates.json ``` The project specifies Julia 1\.10 or newer, JSON3 1\.14\.3, and SHA\. Package installation requires access to the Julia package registries unless dependencies are already cached\. No Julia `Manifest.toml` was generated or tested here\. ### Rust — source supplied, not compiled here ```sh cargo run --release --manifest-path rust/Cargo.toml -- certificates.json cargo test --manifest-path rust/Cargo.toml ``` The source uses Rust edition 2021 and the dependencies listed in `Cargo.toml`\. The first resolution creates `Cargo.lock`; there is no claimed tested lockfile in this bundle\. Preserve the generated lockfile for a reproducible local build\. Release\-profile integer overflow checks are explicitly enabled\. Mathematical scalar arithmetic uses arbitrary\-precision integers and rationals\. ### Compare the native reports ```sh python run_all.py ``` This exits nonzero unless all three engines execute successfully and their computed mathematical reports agree\. Fresh outputs go to `runs/`, not the preserved `evidence/` directory\. `--allow-missing` permits a partial run, but its status remains `PARTIAL_NOT_RUN`; it never manufactures Julia or Rust results\. Runtime version strings are excluded from mathematical report comparisons\. Recorded stderr streams use JSON envelopes, including their exact text, raw byte length, and hash; this preserves empty streams in file stores that reject zero\-byte uploads\. The optional CAS cross\-check is: ```sh python tests/sympy_crosscheck.py ``` It requires SymPy and was run with the version recorded in its evidence JSON\. It re\-evaluates the selected programs with SymPy’s exact polynomial arithmetic, not with the primary checker’s custom coefficient arithmetic\. ## Observed mathematical checks The main Python reports contain 2,314 exact algebraic checks, all 32,768 seeds in the canonical 15\-bit interval, 753,664 forward transitions, 32,768 complete ledger reversals and telescoping identities, and 32,768 first\-residue inversions\. The maximum observed canonical extinction length is ten\. An exact rational enclosure and floor\-sum proof certify equality of the real and rational floor maps for every integer `0 <= d < 42189065`\. At `42189065` the floors are respectively `14342163` and `14342162`\. The common prefix sum is `302541195258184`\. The 42\-million\-state prefix is proved by an exact lattice\-sum certificate, not falsely described as an exhaustive per\-state loop\. The Boolean audit checks 261 floor\-transition cases, all eight full\-adder truth table rows, and all eight Toffoli rows\. Its proof of realizability uses the adder and restoring\-division invariants; this is not a fabricated hardware synthesis or gate\-timing result\. A complete auxiliary example serializes six exact affine\-field readings into 24 bytes, replays each byte’s 23\-step ledger, and reconstructs the original slope and intercept\. Additional evidence includes 65 adversarial audit entries, 149 selected exact checks in the second arithmetic implementation, and a new rerun of the original 13\-group historical Python verifier\. The original normal/optimized outputs differ only in their optimization\-mode header\. New primary and adversarial normal/optimized outputs match byte for byte\. ## Corrections and limits that remain essential The unnormalized canonical ellipse chart is not symplectic\. Its determinant\- normalized replacement is symplectic, but changes overall size\. Positive\-length feet lie outside the derived numerical\-range ellipse\. Six point samples and five interval samples are different observation models with different error budgets\. The full quartic interval decoder has no redundancy for detecting model mismatch\. Finite boundary probes do not identify an arbitrary manifold\. An integer linear layer with its remainders is injective exactly when its integer linear map is injective on the allowed inputs\. Full column rank gives an explicit inverse; a rank\-deficient map requires a restricted\-domain test\. Discarded information must be stored somewhere\. No improvement in model accuracy, training, generalization, energy use, or compression is demonstrated\. A valid witness verifies an arithmetic history, not its provenance\. Another valid history is not detectable as an attack without a separate authentication scheme\. The 23\-step record is not a minimal\-size encoding of a 15\-bit seed\. No peer\-review acceptance, current website audit, external manuscript proof, formal\-kernel execution, physical experiment, or successful Julia/Rust build is certified by this bundle\.