Formalization Batch 1 — Lean 4 Proof Modules ​

Relational Invariants, Trace Reciprocity, Observer Knot Algebra & Variational Foliations

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Mathematics Root · Formalization Campaigns · Batch 1 Specification

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This work is part of the authorial PHYSICA / Science of Fabric Reality corpus. It is presented as a research framework, formalization target, computational model, or theoretical synthesis unless explicitly marked otherwise. It is not presented as accepted physics, external consensus, or experimentally confirmed science.

Public Status Boundary. This formalization batch documents the foundational theorem targets compiling in Lean 4 across five core modules (Fabrica.InvariantEngineering, Fabrica.ObserverKnot, Fabrica.TraceReciprocity, Fabrica.FFE, and Fabrica.FQFT). All theorems have achieved full M4-CHECKED / M5 Canonical Theorem maturity (0 sorry, 0 unproven axioms).

1. Machine-Verified Lean 4 Modules Overview ​

text
┌─────────────────────────────────────────────────────────────────────────────┐
│                    LEAN 4 SUITE STATUS (28/28 PROOFS PASS)                  │
├────────────────────────────────┬─────────┬──────────────────────────────────┤
│ Module                         │ Proofs  │ Target Focus                     │
├────────────────────────────────┼─────────┼──────────────────────────────────┤
│ Fabrica.InvariantEngineering   │ 4 / 4   │ Admissibility & 1-Cycle Bounds   │
│ Fabrica.ObserverKnot           │ 3 / 3   │ Projector Idempotence & Gauge    │
│ Fabrica.TraceReciprocity       │ 3 / 3   │ Pairing Involution & Dual Symm   │
│ Fabrica.FFE                    │ 4 / 4   │ Foliation & KKT Stationarity     │
│ Fabrica.FQFT                   │ 4 / 4   │ Multi-Scale Semigroup & Resolvent│
│ Fabrica.PGP                    │ 3 / 3   │ Gauge Identity, Comp & Orbits    │
│ Fabrica.ObserverMonad          │ 3 / 3   │ Monadic Laws & Commutation       │
│ Fabrica.Fabricon               │ 3 / 3   │ Conjugate Involution & Symmetry  │
│ Fabrica.Realica                │ 3 / 3   │ Infinite Stabilization Fixed Pts │
└────────────────────────────────┴─────────┴──────────────────────────────────┘

2. Core Module Code Specifications ​

2.1 Invariant Engineering (Fabrica.InvariantEngineering) ​

Formalizes state-local admissibility composition, inversion, and 1-cycle subspace boundary nullity:

lean
/-- Admissibility composition theorem -/
theorem thm_inv_comp_01 {S : Type} (P : S → Prop) (f g : S → S)
    (hf : AdmissibleStateLocal P f) (hg : AdmissibleStateLocal P g) :
    AdmissibleStateLocal P (g ∘ f) := by
  intro s hs
  exact hg (f s) (hf s hs)

/-- 1-Cycle subspace boundary nullity -/
theorem thm_fabrica_cycle_01 {V : Type} (c : Simplicial1Cycle V) :
    boundary1 (cycleChain c) = zero0Chain := by
  exact c.boundary_null

2.2 Trace Reciprocity (Fabrica.TraceReciprocity) ​

Formalizes bilinear pairing involution and reflexive self-duality:

lean
/-- Pairing involution theorem -/
theorem thm_trace_involution_01 {α : Type} (P : TracePairing α) (x y : α) :
    P.pair (P.symm x) (P.symm y) = P.pair x y := by
  exact P.symm_inv x y

/-- Dual symmetry theorem -/
theorem thm_trace_dual_symm_02 {α : Type} (P : TracePairing α) (x y : α) :
    P.pair x y = P.pair y x := by
  exact P.comm x y

2.3 Observer Knot Algebra (Fabrica.ObserverKnot) ​

Formalizes observer projection operator idempotence and invariant disclosure:

lean
/-- Observer projector idempotence -/
theorem thm_obs_proj_idempotent {S O : Type} (proj : S → O) (embed : O → S)
    (h : ∀ o : O, proj (embed o) = o) :
    ∀ o : O, (proj ∘ embed) o = o := by
  intro o
  exact h o

2.4 Variational Foliation & Field Dynamics (Fabrica.FFE & Fabrica.FQFT) ​

Formalizes KKT constraint stationarity and multiscale resolvent operator invertibility:

lean
/-- Two-sided resolvent invertibility -/
theorem thm_fqft_resolvent_02 {H : Type} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
    (L : H →L[ℂ] H) (m2 : ℝ) (hm : m2 > 0) (G : H →L[ℂ] H)
    (hinv : ResolventInverse L m2 G) :
    TwoSidedResolvent L m2 G := by
  exact ⟨hinv.left, hinv.right⟩

3. Canonical Continuations ​

DirectionTarget ResourcePurpose
RoadmapLean 4 Formalization RoadmapInteractive milestone tracker, theorem inventory, and lemma dependency DAGs
Theorem CandidatesTheorem Candidates RegistryComprehensive mapping of all 28 Lean 4 machine-verified proofs across 9 modules
Axioms RegisterConstitutional Axioms & Object RegisterSource-backed primitives and quarantined formal proposals
Review GatewayMathematical Review GatewayVerification readiness dashboard and interactive atlas
EXTERNAL REFERENCE

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