Heller-Dirac

Purpose: co-foundational scaffold beside Heller-Godel, organizing the framework's extended Hopf algebraic, time-theoretic, and field-theoretic content around the spectral-triple substrate.

Where Heller-Godel hosts the proof-theoretic core, PFK substrate, and claim-grade discipline, Heller-Dirac hosts the spectral / modular / Hopf / field-theoretic apparatus that physics-flavored Clay-program content may sit on.

Scope: three pillars

1. Extended Hopf scaffold — Hopf algebras, Hopf algebroids, bicrossed products, Connes-Moscovici-type transverse-geometry carriers, Connes-Kreimer-type renormalization carriers, and Hopf-cyclic cohomology.
2. Time theory — Tomita-Takesaki modular flow, KMS structure, thermal time hypothesis, Bisognano-Wichmann correspondence, and type III factor structure of QFT vacuum representations.
3. Field theory — Algebraic QFT (Haag-Kastler), Wightman, constructive field theory, topological field theory (Atiyah-Segal), and factorization-algebra formulations.

The unifying object is the spectral triple (A, H, D) with optional grading and real structure, equipped with Hopf-algebra actions and modular structure on the GNS representation.

Dependency on Heller-Godel

Heller-Dirac consumes Heller-Godel for:

• PFK substrate: schemas, operator catalog, anti-seed;
• framework objects: HG-FND-*, HG-MTH-005;
• claim-grade discipline.

Heller-Dirac publishes HD-* objects for downstream Clay-program consumption when their content is spectral, modular, Hopf-algebraic, or field-theoretic.

Identifier namespace

Prefix	Domain
HD-FND-*	Foundational vocabulary
HD-HA-*	Hopf algebra scaffold
HD-TM-*	Time theory
HD-FT-*	Field theory
HD-SP-*	Spectral apparatus
HD-EX-*	Fixtures
HD-MTH-*	Methodology declarations
A-HD-*	Anti-seed register

Non-claims

This repository does not:

• prove any Clay problem;
• claim that spectral-triple reformulation constitutes proof (A-HD-NC-001);
• claim modular flow is physical time outside Bisognano-Wichmann scope (A-HD-TM-001);
• claim AQFT axiomatization constructs a theory (A-HD-FT-001);
• claim Hopf-scaffold encoding proves renormalizability (A-HD-HA-001);
• transfer methodology into proof-grade content in downstream Clay programs.

Status

Scaffold declaration. No apparatus implemented yet.

Subsequent PRs should:

1. draft HD-FND-* foundational vocabulary;
2. specify the spectral-triple base structure;
3. add HD-EX-* canonical fixtures such as the circle spectral triple and free Dirac operator examples;
4. specify modular-flow apparatus and Bisognano-Wichmann scope;
5. specify Hopf-scaffold catalog entries with Connes-Moscovici and Connes-Kreimer anchor examples;
6. cross-reference into Heller-Godel HG-MTH-006..010 as those drafts land.