Mathematics

Formal Architecture, Theorem-Status Map & Proof Governance

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Science Root · Formal Mathematical Spine, Axiomatic Governance & Proof Lifecycle

Public Status Boundary. This section presents formal mathematical models, proof obligations, and theorem candidates under development. It separates source objects from empirical conjectures and does not claim solved physics or machine-verified status in Lean 4 unless an explicit proof file or formal certificate is attached.

The Mathematics sector constitutes the formal, deductive spine of the Science of Fabric Reality. It enforces strict mathematical hygiene across all theoretical proposals, separating axiomatic definitions, machine-checked theorems, conditional derivations, open conjectures, and numerical simulations without asserting unverified physical claims.

$$\boxed{{\displaystyle \mathbf{\text{Proof sketch}}\ne \mathbf{\text{machine proof}}\ne \mathbf{\text{numerical verification}}}}$$

Figure 4.1 — The Mathematics Theorem-Status Hierarchy: Strict epistemic demarcation separating primitive definitions, constitutional axioms, Lean 4 artifacts and formalization targets, open conjectures, and numerical simulations.

Demarcates the nine formalization tiers from primitive mathematical objects to machine-checked Lean 4 proofs and numerical surrogates.

Credit: Ivan Pasev / GILC Research·CC BY-NC-SA 4.0·SCHEMATIC

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1. The Nine-Tier Theorem Status Taxonomy

Every mathematical object and claim in the corpus is classified under a strict nine-tier status hierarchy:

Tier	Status Class	Verification Standard	Example in Corpus
1. Objects	CARRIER_OBJECT	Set-theoretic / categorical well-formedness.	Relational state tuples $\mathcal{S}=(X,R,\partial ,\mathcal{I})$.
2. Axioms	CONSTITUTIONAL_AXIOM	Declared baseline postulates; checked for mutual consistency.	Constitutional Axioms Register.
3. Proven in Corpus	PROVEN_IN_CORPUS	Deductive manuscript proofs with verified lemma trees (not unhedged empirical claims).	Finite-dimensional KKT uniqueness theorem for FFE.
4. Imported Theorems	IMPORTED_ESTABLISHED	Established standard literature theorems (comparator math).	Dunford–Pettis resolvent theorem, Noether's theorem, Mosco variational convergence (Mosco 1969).
5. Conditional Derivations	DERIVED_CONDITIONALLY	Derived under explicitly declared asymptotic or regularity assumptions.	Continuous FFE Euler-Lagrange equations under $a\to 0$.
6. Theorem Targets	FORMALIZATION_TARGET	Active conjectures with formal mathematical problem statements.	General infinite-dimensional KP-Field self-adjointness.
7. Counterexamples	COUNTEREXAMPLE_AUDIT	Active adversarial search for boundary failures or non-uniqueness.	Non-convex constraint manifolds in FFE optimization.
8. Lean 4 Artifacts	LEAN4_ARTIFACT	Machine-checked code and compiled formalization targets in Lean 4.	Formal Proof Registry (28 machine-verified theorems across 9 Lean modules).
9. Simulation Surrogates	SIMULATION_SURROGATE	Discretized numerical lattice models. Simulation is not proof.	Simulation Atlas finite-difference solvers.

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2. Proof Governance Lifecycle (M0–M5)

All mathematical propositions progress through the six-stage M-Lifecycle:

• M0 — Heuristic Conjecture: Conceptual idea with heuristic physical motivation.
• M1 — Formal Problem Statement: Rigorous mathematical formulation with typed domains and boundary conditions.
• M2 — Proof Architecture: Complete proof sketch identifying intermediate lemmas and failure modes.
• M3 — Paper Proof: Exhaustive manuscript proof peer-reviewed within the SFR Review Portal.
• M4 — Interactive Formalization: Lean 4 translation with machine-checked lemma closures.
• M5 — Canonical Theorem: Full machine-checked formalization integrated into the permanent proof library.

Review the active governance standards: Proof Governance Framework $\parallel$ Publication Maturity Scale.

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3. Mathematical Foundations & Active Components

The formal program is organized into five dedicated sub-sectors:

3.1 Axiomatic Object Register

Complete catalog of constitutional axioms, carrier sets, topological boundary operators, and invariant predicates.

3.2 Formalization Roadmap

Interactive milestone tracker for active Lean 4 campaigns, lemma dependency DAGs, and interactive theorem proving targets (28 machine-verified proofs in Lean 4).

3.3 Theorem Candidates Registry

Classified repository of candidate theorems across spectral graph theory, operator algebras, and variational dynamics.

3.4 Mathematical Atlas & Taxonomy

Topological navigation atlas mapping formal state spaces, invariant transport theorems, and proof governance registers.

3.5 Simulation Atlas

Open-source finite-difference and lattice solvers computing spectral bounds, dynamical dispersion, and invariant preservation.

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4. Submission-Grade Research Manuscripts (Epochs Ω284G-R3–R5)

The mathematical results of the Science of Fabric Reality research program are crystallized into four submission-grade core journal manuscripts equipped with standalone LaTeX packages and formal proof mappings:

Manuscript	Focus Domain	Key Mathematical Contributions	Epistemic Status
Manuscript A	Relational Foundations & Invariance (OBJ-TH-SFR)	Constitutional 7-tuple state spaces, State-local admissibility (DEF-ADM-01), Admissibility composition (THM-INV-COMP-01, Lean 4 verified), Inversion admissibility (THM-INV-INV-01, Lean 4 verified), and 1-Cycle conservation (THM-FABRICA-CYCLE-01, Lean 4 verified).	SUBMISSION_GRADE_COMPLETE
Manuscript B	Dirichlet Variational Principles & Mosco Limits (OBJ-TH-FQFT)	Weak Euler-Lagrange Dirichlet stationarity (THM-FQFT-EL-WEAK-01), Common-fiber Mosco convergence (THM-FQFT-C4A-MOSCO-01), Scale-flow dynamics (THM-FQFT-C4-FLOW-CONV-01), Non-linear action $\mathrm{\Gamma }$-convergence (THM-FQFT-NONLINEAR-CONV-01), and Two-sided resolvent invertibility (THM-FQFT-RESOLVENT-02, Lean 4 verified).	SUBMISSION_GRADE_COMPLETE
Manuscript C	Spectral Mediator Models & Metrology (OBJ-PRG-P2)	Operator Helmholtz Green functions (THM-KP-FIBER-GREEN-01), Convex Yukawa mixtures (THM-KP-SPECTRAL-YUKAWA-01), Gauge-complete Stückelberg model V1 (THM-P2-V1-GAUGE-01), and Exact hydrogenic $2S-2P$ zero-mode cancellation theorem.	SUBMISSION_GRADE_COMPLETE
Manuscript D	Negative Identifiability Theorems (OBJ-PRG-NEG)	Degree underdetermination (NEG-KP-R1-SPECTRAL-INCOMPLETE-01), Pullback universality (THM-KP-PULLBACK-UNIV-01), Parameter identifiability rank deficiency ($2<3$), and Epistemic prospective firewall (ACTIVE_P4_SEALS = 0).	SUBMISSION_GRADE_COMPLETE

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5. Canonical Continuations

Direction	Node	Route	Focus / Purpose
Active Proof Tracker	Lean 4 Formalization Roadmap	/04-mathematics/formalization-roadmap	28 machine-verified theorem records and lemma dependency DAGs
Axioms Register	Constitutional Axioms & Object Register	/04-mathematics/axioms	Source-backed primitives and quarantined formal proposals
Simulation Lab	Simulation Atlas & Numerical Testbeds	/04-mathematics/simulations/	Discretized numerical solvers, null tests ($p_{\text{null}}=0.62$), and receipts
Review Portal	SFR Review Portal & Critique Intake	/03-research/sfr-review-portal/	Five-gate critique protocol and formal referee inquiry gateway