Spine Position
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.
The Nine-Tier Theorem Status Taxonomy
Machine-Checked Lean 4 Proofs
The highest standard of formal certitude: fully compiled Lean 4 source files certified reproducible by CI and the kernel.
Machine-verified code verified by the Lean 4 kernel with 0 sorry.
26 Verified Theorems across 10 Lean Modules (Lake build clean)
Proof Governance Lifecycle (M0–M5)
Interactive Formalization
Translation of proof lemmas into Lean 4 source modules without unverified axioms.
Accepted manuscript proof (M3) or formalization target (Tier 6).
Lake clean compilation (`lake build`) with lemma DAG verification.
Corpus Mathematical Pillars
Axiomatic Object Register
Complete formal specification of constitutional axioms (A0–A9), carrier state-spaces, and topological boundary operators.
Lean 4 Formalization Roadmap
Machine-checked formalization campaigns, acyclic lemma dependency DAGs, and Lake build verification across 10 Lean modules.
Analytic Proofs & Continuum Core
Continuous weak Euler-Lagrange Dirichlet stationarity, Mosco variational limits, and scale-flow convergence in infinite dimensions.
Simulation Atlas & Solvers
Open-source lattice finite-difference solvers, spectral eigenvalue testbeds, and verified adversarial null model benchmarks.
Demarcates the nine formalization tiers from primitive mathematical objects to machine-checked Lean 4 proofs and numerical surrogates.
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 |
| 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 (analytical derivations; distinct from Tier 8 machine-verified Lean 4 code). | Continuous weak Euler-Lagrange Dirichlet stationarity (THM-FQFT-EL-WEAK-01). |
| 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 |
| 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 (35 machine-verified theorems across 11 Lean modules). |
| 9. Simulation Surrogates | SIMULATION_SURROGATE | Discretized numerical lattice models. Simulation is not proof. | Simulation Atlas finite-difference solvers. |
2. Orthogonal Status Coordinates & Proof Governance (F, C, P, R, V)
All mathematical propositions and physical models are governed by five independent, orthogonal coordinates:
– (Formal / Mathematical Axis): Degrees of deductive mathematical rigor, from informal heuristic ( ) to machine-checked Lean 4 kernel verification ( , 35 verified declarations across 11 modules) and canonical library integration ( ). – (Computational Axis): Discretized sanity checks ( ) and standardized benchmarks ( ); numerical simulation is stress-testing and never formal proof. – (Prediction / Empirical Axis): Formal identities ( ), calibrated retrodictions ( , ), prospective predictions ( , ACTIVE_P4_SEALS = 0), and independent empirical replications (). – (Review Axis): Internal critique protocol in the SFR Review Portal ( ) versus independent external journal peer review ( ). – (Publication Axis): Public corpus release ( ), open repository DOI archiving ( ), external submission ( ), and external venue publication ( ).
Review the active governance standards: Proof Governance Framework
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 (35 machine-verified proofs in Lean 4 across 11 modules).
Limitations & Open Problems
- The current registry records 35 machine‑checked theorem objects across the compiled Lean corpus. Their validity is scoped to the precise formal statements and assumptions under which they were proved. Negative results, identifiability obstructions, open problems, and empirical questions remain separately tracked.
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.
3.6 Mathematical Inventions
Authorial mathematical physics inventions including Trace Reciprocity on discrete transfer operators, Relational Gauge Principles on fiber bundles, and Infinite-Dimensional Asymptotic Symmetries.
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), Inversion admissibility (THM-INV-INV-01), and 1-Cycle conservation (THM-FABRICA-CYCLE-01) (formalization targets in Fabrica.InvariantEngineering). | 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 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 | 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 (ACTIVE_P4_SEALS = 0). | SUBMISSION_GRADE_COMPLETE |
5. Canonical Continuations
Scientific Results Ledger
35 machine-verified Lean 4 theorems across 11 modules and canonical negative null benchmark (p_null = 0.62).
Computational Reproducibility
RFC 8785 JCS verification receipts, hash chains, and reproducible Lean 4 proof environment.
Lean 4 Formalization Roadmap
35 machine-verified theorem records, compilation receipts, and acyclic lemma dependency graphs.
Formalization Batch 1 Specification
Direct Lean 4 formal module specifications across foundations, gauge principles, and spectral theory.
Constitutional Axioms Register
Source-backed primitives, declared baseline postulates, and quarantined formal proposals.
Mathematical Inventions
Trace reciprocity, relational gauge principles, and infinite symmetries across the operator continuum.
Simulation Laboratory
Discretized numerical finite-difference solvers, null tests (p_null = 0.62), and execution receipts.
SFR Review Portal
Five-gate critique protocol, dispute resolution engine, and formal referee inquiry gateway.
Claim–Evidence Architecture
Interactive knowledge graph projecting machine-verified mathematical claims, empirical witness bindings, and defeat conditions.