Spectral-Invariants-MTC

Spectral-Invariants-MTC is a first-principles research codebase for deriving and auditing spectral invariants of finite-screen modular tensor category data arising from affine Kac--Moody vertex-operator-algebra blocks. The repository is intentionally austere: it contains algebraic category data, exact trace closures, high-precision bridge audits, and rigidity regressions, but no phenomenological fitting layer.

The construction replaces empirical scale setting with a finite-capacity current-algebra computation anchored on the anomaly-free branch

SO(10)_312 ⊃ SU(2)_26 × SU(3)_8.

All admissible structures are encoded through integrable affine weights, Weyl/Kac--Peterson modular S-data, Verlinde fusion coefficients, Frobenius--Perron dimensions, finite current-mode traces, and explicitly audited closure tensors.

Architectural Principles

• Affine current-mode truncation: The finite screen is modeled by truncating Kac--Moody current excitations generated by modes J^a_{-n}. The active basis is the affine current-cylinder basis subject to the finite register capacity N, not a legacy Virasoro oscillator tower. Virasoro data appear only through derived Sugawara consistency channels where required by modular phase closure.
• First-principles scale derivation: Core scales are derived from the static affine branch, its modular tensor category data, and finite-capacity trace identities. No external cosmological observation, likelihood, or posterior is allowed to set the scale.
• Exact-first arithmetic: Branch framing, level ratios, and phase locks are evaluated as exact integer or rational statements before any numerical rendering. Arbitrary-precision mpmath arithmetic is used only where transcendental modular data require it.
• High-precision audit floor: The bridge and parent-subspace solvers run at a 250 decimal digit floor and test residuals against the repository noise wall. Ordinary binary double precision is not acceptable for any threshold-sensitive invariant.
• Separation of proof status: Independent topological inputs, internal self-consistency checks, loop-trace closures, and off-shell rigidity probes are rendered as distinct audit channels. This prevents algebraic self-comparisons from being misrepresented as independent external measurements.
• No fitting pipeline: The repository excludes cosmological parameter fitting scripts, differential-equation solver workflows, empirical tuning utilities, and observation-driven scale-setting logic.

Repository Structure

Spectral-Invariants-MTC/
├── README.md
├── requirements.txt
├── context/
│   └── physics_constants.tex
├── paper/
│   └── mtc_mass_gap.tex
├── results/
│   └── final_audit_report.md
├── src/
│   └── mtc_engine/
│       ├── __init__.py
│       ├── bridge_audit.py
│       ├── core.py
│       ├── exact_traces.py
│       ├── export_report.py
│       ├── parent_subspace.py
│       └── run_audit.py
└── tests/
    ├── test_category_properties.py
    ├── test_exact_traces.py
    ├── test_rigidity.py
    └── test_run_audit.py

Package Layout

src/mtc_engine/core.py

Defines the affine WZW sector data for SU(2)_26, SU(3)_8, and SO(10)_312: integrable weights, Weyl groups, Kac--Peterson modular S entries, Verlinde fusion matrices, and Frobenius--Perron dimension extraction. This module is the algebraic kernel of the repository.

src/mtc_engine/exact_traces.py

Implements finite-screen trace objects, exact capacity constants, modular-T trace checks, visible-branch framing defects, and analytical mass-gap trace closures. It is reserved for exact or arbitrary-precision computations where double precision would erase the invariant being tested.

src/mtc_engine/parent_subspace.py

Solves the targeted SO(10)_312 parent block subspace without enumerating the full parent spectrum. The full parent category has 6,563,729,615 integrable primaries, so this module evaluates only the audited four-block subspace using Weyl denominator formulas, Kac--Peterson modular entries, and finite character generators.

src/mtc_engine/bridge_audit.py

Builds the high-precision bridge report. It combines completion-residue loading, Frobenius--Perron checks, loop-trace residuals, exact phase-space zeros, and continuous-coordinate closure tensor probes into one immutable audit ledger.

src/mtc_engine/run_audit.py

Provides the command-line QA renderer for the bridge audit. Its text output preserves the Category A/Category B taxonomy used by the manuscript and generated Markdown report.

src/mtc_engine/export_report.py

Runs the bridge audit, parent-subspace solver, and active perturbation sweep, then exports results/final_audit_report.md. The exporter formats mpmath values directly and never down-casts high-precision values through binary floats.

Audit Taxonomy & Verification Rigor

The repository enforces a strict firewall between independent topological foundations and internal consistency validation.

Category A — Independent Foundations

Category A contains inputs that are not produced by feeding the target closure identity back into itself:

• Exact framing arithmetic: The visible branch checks 312/(2·26)=6, 312/(3·8)=13, and Δ_fr=0 at the rational/integer level before evaluating any phase residual.
• Small-sector Frobenius--Perron extractions: The tractable SU(2)_26 and SU(3)_8 sectors are validated by direct finite primary sets, modular S data, Verlinde fusion, and high-precision Frobenius--Perron dimension comparisons.
• Category-property regressions: The small-sector test suite verifies modular S unitarity, exact integer commutativity and associativity of fusion matrices, and preservation of fusion multiplication by quantum dimensions at the configured precision floor.

Category B — Internal Consistency Validation

Category B contains self-consistency and closure checks for the encoded finite static block:

• Parent-sector bypass checks: The SO(10)_312 parent channel is audited through a targeted four-block Weyl/Kac--Peterson subspace. This is an algebraic bypass of impossible full fusion-matrix diagonalization, not a brute-force independent spectral measurement.
• Phase-sensitive modular-T intersections: Character intersections and diagonal trace closures are evaluated at high precision to verify internal modular consistency.
• Mass-gap scaling traces: The quartic finite-screen trace identity and its -1/4 exponent are checked as internal algebraic consequences of the chosen static trace normalization.
• Active VOA closure tensor probes: Off-anchor coordinate motions are tested by recomputing the finite closure tensor channels and then lifting nonzero defects against the zero tangent norm of the singleton topological moduli space.

The continuous coordinate-perturbation sweeps are not hardcoded mock bypasses and do not assign divergence solely because a nonzero delta was requested. Each perturbation actively recomputes the breakdown of level integrality, root-embedding hyperplanes, visible embedding ratios, modular framing phase, Sugawara central-charge phase, prefactor closure, and capacity closure. Infinite stiffness is reported only after the recomputed finite VOA defect is nonzero and the topological tangent norm remains zero.

Verification Suites

tests/test_category_properties.py

Performs high-precision category-theoretic regression checks for the tractable sectors: full modular S-matrix unitarity, exact Verlinde ring commutativity and associativity, and Frobenius--Perron multiplicativity of the quantum-dimension vector.

tests/test_exact_traces.py

Checks finite-screen trace values, modular trace closure, exact constants, and current-block trace identities.

tests/test_rigidity.py

Validates rigidity claims, exact phase zeros, bridge residual mapping, and active off-shell coordinate responses.

tests/test_run_audit.py

Verifies the command-line audit renderer, required QA ledger sections, precision-log behavior, and explicit on-shell zero-perturbation handling.

Numerical Requirements

Install the minimal numerical stack with:

python -m pip install -r requirements.txt

The dependency set is intentionally small:

• numpy for deterministic array interfaces where needed.
• scipy for vetted scientific utilities where appropriate.
• mpmath for arbitrary-precision transcendental evaluation.

Run the validation suite with:

pytest

Generate the public Markdown audit ledger with:

python src/mtc_engine/export_report.py

Any implementation that evaluates invariants near the finite-screen audit floor must use exact arithmetic or explicitly configured arbitrary precision. Binary floats are rejected in perturbation paths that would otherwise obscure closure residuals.

Non-Goals

This repository does not include:

• Cosmological parameter fitting scripts.
• Differential equation solver pipelines.
• Empirical scale-setting utilities.
• Observation-driven tuning or posterior inference workflows.
• Smooth surrogate models for the singular topological projector.

These exclusions preserve the boundary between first-principles affine current algebra, finite-screen modular tensor category validation, and any downstream phenomenological interpretation.