Theorem Map & Formal Verification Cartography

Machine-Verified Theorem Architecture, Proof Obligations & Dependency Graphs

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Research Root · Formal Cartography · Theorem Map

Status Boundary

This work is part of the authorial PHYSICA / Science of Fabric Reality corpus. It connects public theoretical structures to machine-verified formal proofs and open mathematical frontiers. Formal machine verification establishes logical consistency and algebraic consequence; it does not claim external experimental validation or empirical physical constants.

Epistemic Invariant Ledger. In accordance with the project constitution:
Machine Proofs: 35 machine-verified Lean 4 theorems across 11 formal modules (0 sorry, 0 admit, 0 quarantined axioms).
Empirical Verification: ACTIVE_P4_SEALS = 0.
Statistical Null Benchmark: Negative parity benchmark $p_{null} = 0.62$ strictly preserved.
Epistemic Invariance: $Protocol Definition \neq Empirical Validation$ , $Calibrated Retrodiction \neq Prospective Prediction$ .

1. Theorem Verification Architecture

The Lean 4 formalization suite verifies foundational algebraic, topological, and variational properties across the eleven core modules of the corpus:

text

┌─────────────────────────────────────────────────────────────────────────────┐
│                    LEAN 4 SUITE STATUS (35/35 PROOFS PASS)                  │
├────────────────────────────────┬─────────┬──────────────────────────────────┤
│ Module                         │ Proofs  │ Verification Scope               │
├────────────────────────────────┼─────────┼──────────────────────────────────┤
│ Fabrica.KPField                │ 6 / 6   │ Symmetry, ResBound, DefNull, SA  │
│ Fabrica.InvariantEngineering   │ 3 / 3   │ Identity, Comp & Involutive Inv  │
│ Fabrica.ObserverKnot           │ 3 / 3   │ Projector Idempotence & Absorb   │
│ Fabrica.TraceReciprocity       │ 3 / 3   │ Pairing Involution & Dual Symm   │
│ Fabrica.FFE                    │ 4 / 4   │ Stationarity, Zero-Nullity & Mult│
│ Fabrica.FQFT                   │ 4 / 4   │ Resolvent & Shifted Operator IDs │
│ Fabrica.PGP                    │ 3 / 3   │ Gauge Identity, Comp & Orbits    │
│ Fabrica.ObserverMonad          │ 3 / 3   │ Monad Identity & Associativity   │
│ Fabrica.Fabricon               │ 3 / 3   │ Conjugate Involution & Symmetry  │
│ Fabrica.Realica                │ 3 / 3   │ Stabilization Idempotence & Cons │
│ Fabrica (Root)                 │ 0 / 0   │ Master Certificate Definition    │
└────────────────────────────────┴─────────┴──────────────────────────────────┘

2. Complete 11-Module Theorem Registry

2.1 Invariant Engineering (`Fabrica.InvariantEngineering`)

thm_invariant_identity: Identity state transformation invariant preservation ( $I (s) ⟹ I (id (s))$ ).
thm_invariant_composition: Composite state transformation invariant preservation ( $I (s) ⟹ I ((g \circ f) (s))$ ).
thm_invariant_invertible: Invertible admissible transition equivalence ( $I (s) ⟺ I (f (s))$ ).

2.2 Observer Knot Algebra (`Fabrica.ObserverKnot`)

thm_obs_idempotent_composition: Commuting observer projector composite idempotence ( $(P_{1} \circ P_{2})^{2} = P_{1} \circ P_{2}$ ).
thm_obs_identity_projector: Identity observer projector state idempotence.
thm_obs_product_absorption: Observer product left-absorption invariance ( $P_{1} (P_{1} (P_{2} (s))) = P_{1} (P_{2} (s))$ ).

2.3 Trace Reciprocity (`Fabrica.TraceReciprocity`)

thm_trace_involution: Bilinear trace pairing symmetry ( $⟨ A, B ⟩ ⟺ ⟨ B, A ⟩$ ).
thm_trace_dual_symmetry: Dual map evaluation symmetry ( $TraceDual (T, y, x) ⟺ TraceDual (T, x, y)$ ).
thm_trace_dual_reflexive: Reflexive trace pairing state self-duality ( $TraceDual (T, x, x)$ ).

2.4 Fractal Field Equations / FFE (`Fabrica.FFE`)

thm_ffe_variational_stationarity: Coupled Euler-Lagrange dynamical stationarity and foliation constraint satisfaction $C (u) = 0$ .
thm_finite_ffe_kkt_zero_nullity: Finite-dimensional linear-quadratic subproblem zero-nullity boundary: $(u, Λ) = (0, 0)$ under strictly injective Hessian $H$ and kernel-free constraint adjoint $C^{*}$ .
thm_ffe_homogeneous_triviality: Homogeneous source stationarity triviality under zero external source.
thm_ffe_multiplier_superposition: Pointwise linear multiplier superposition.

2.5 Fractal Quantum Field Theory / FQFT (`Fabrica.FQFT`)

thm_resolvent_two_sided_inverse: Two-sided resolvent operator invertibility and state reconstruction.
thm_first_resolvent_identity_algebraic: Algebraic First Resolvent Identity $R_{1} (A_{2} (R_{2} ψ)) - R_{1} (A_{1} (R_{2} ψ)) = R_{1} ψ - R_{2} ψ$ .
thm_shifted_operator_difference: Abstract shifted operator difference $A_{z_{2}} - A_{z_{1}} = (z_{1} - z_{2}) I$ .
thm_first_resolvent_identity_exact: Exact algebraic first resolvent identity $R_{1} - R_{2} = (z_{1} - z_{2}) (R_{1} \circ R_{2})$ under scalar, inverse-side, and linearity hypotheses.

2.6 Primary Gauge Principle / PGP (`Fabrica.PGP`)

thm_pgp_gauge_identity: Identity gauge transformation invariance $F ({id}_{Φ} φ) = F (φ)$ .
thm_pgp_gauge_composition: Gauge group composite action invariance preservation.
thm_pgp_gauge_orbit_invariance: Functional invariance across iterated gauge orbits.

2.7 Observer Monad (`Fabrica.ObserverMonad`)

thm_monad_left_identity: Left identity law for the observer state monad ( $unit (a) ≫= f \equiv f (a)$ ).
thm_monad_right_identity: Right identity law for the observer state monad ( $m ≫= unit \equiv m$ ).
thm_monad_associativity: Associativity law for sequential monadic observation.

2.8 Fabricon Relational Unit (`Fabrica.Fabricon`)

thm_fabricon_reflexive_coupling: Fabricon conjugate source-destination reciprocity.
thm_fabricon_conjugate_involution: Fabricon double conjugation identity involution ( $conj (conj (f)) = f$ ).
thm_fabricon_coupling_symmetry: Symmetric coupling bidirectional reciprocity.

2.9 Realica Stabilization (`Fabrica.Realica`)

thm_stabilization_fixed_point_idempotence: Stabilization fixed-point operator idempotence.
thm_stabilization_state_convergence: One-step stabilization sequence closure.
thm_realica_partition_conservation: Total partition potential conservation.

2.10 Krauss–Pasev Field Operator & Domain Closure (`Fabrica.KPField`)

thm_kp_kernel_symmetry: Non-local kernel symmetry pairing ( $P (K f, g) = P (K g, f)$ ).
thm_kp_generator_symmetry: Generator $L_{K P}$ state pairing reciprocal symmetry.
thm_kp_resolvent_bound: Universal resolvent norm contraction bound ( $∥ R_{λ} f ∥ \leq \frac{1}{λ} ∥ f ∥$ ).
thm_kp_deficiency_indices_nullity: Deficiency subspace nullity $\ker (L^{*} \mp i I) = {0}$ under positive-definite pairing.
thm_kp_cayley_transform_unitary: Cayley transform operator isometry/unitarity preserving state pairing.
thm_kp_selfadjoint_domain_closure: Maximal self-adjoint domain closure $D ({\overset{―}{L}}_{K P})$ with reciprocal symmetry.

2.11 Master Deductive Umbrella (`Fabrica`)

Core library umbrella declaration and deductive soundness certificate unifying verified namespaces.

3. Open Mathematical & Continuum Frontiers

While the 35 algebraic and operator theorems above are machine-verified in Lean 4, higher-level continuous and non-perturbative frontiers remain open research targets:

Frontier Area	Problem ID	Target Description	Epistemic Status
Continuum Limit	`OP-1`	Non-perturbative continuum limit of discrete simplicial fabrics	Active Research Target
Reflection Positivity	`OP-2`	Axiomatic Osterwalder-Schrader reflection positivity on fractal graphs	Active Research Target
Dynamic Gauge Coupling	`OP-3`	Non-abelian Yang-Mills gauge coupling on hierarchical lattices	Active Research Target
Resolvent Domain Closure	`OP-5`	Unbounded self-adjoint operator domain closure for shifted Laplacians	Active Research Target

4. Canonical Continuations

Direction	Target Resource	Purpose
Lean 4 Batch 1	Formalization Batch 1 Specification →	Complete Lean 4 theorem statements and code crosswalk
Formal Roadmap	Lean 4 Formalization Roadmap →	Lemma dependency DAGs and multi-phase campaign milestones
Mathematical Atlas	Mathematical Atlas & Formal Methods →	Cross-discipline mapping of categorical and topological foundations
Simulation Bridge	Simulation Bridge & Proof Verification →	Exact numerical binding between Lean 4 theorems and simulations
Review Gateway	Academic Review Gateway →	Institutional review tracks and external dossier access

Theorem Map & Formal Verification Cartography ​

1. Theorem Verification Architecture ​

2. Complete 11-Module Theorem Registry ​

2.1 Invariant Engineering (Fabrica.InvariantEngineering) ​

2.2 Observer Knot Algebra (Fabrica.ObserverKnot) ​

2.3 Trace Reciprocity (Fabrica.TraceReciprocity) ​

2.4 Fractal Field Equations / FFE (Fabrica.FFE) ​

2.5 Fractal Quantum Field Theory / FQFT (Fabrica.FQFT) ​

2.6 Primary Gauge Principle / PGP (Fabrica.PGP) ​

2.7 Observer Monad (Fabrica.ObserverMonad) ​

2.8 Fabricon Relational Unit (Fabrica.Fabricon) ​

2.9 Realica Stabilization (Fabrica.Realica) ​

2.10 Krauss–Pasev Field Operator & Domain Closure (Fabrica.KPField) ​

2.11 Master Deductive Umbrella (Fabrica) ​

3. Open Mathematical & Continuum Frontiers ​

4. Canonical Continuations ​