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

Status Boundary

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. Within the current canonical formal corpus, 26 Lean theorem declarations across 10 formal modules have achieved machine-verified status under Lean 4.16.0 with 0 sorry and 0 admit. One source-declared axiom, FQFT_closure, remains unverified and quarantined outside the canonical theorem authority (earlier project records cited 28 theorems across 9 modules; the current compiler-closed canonical corpus establishes 26 across 10). Verification applies strictly to the encoded formal statements and hypotheses; it does not imply machine verification of the complete scientific theory or empirical physical validation.

1. Machine-Verified Lean 4 Modules Overview ​

text
┌─────────────────────────────────────────────────────────────────────────────┐
│                    LEAN 4 SUITE STATUS (26/26 PROOFS PASS)                  │
├────────────────────────────────┬─────────┬──────────────────────────────────┤
│ Module                         │ Proofs  │ Target Focus                     │
├────────────────────────────────┼─────────┼──────────────────────────────────┤
│ 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.InvariantEngineering   │ 0 / 0   │ Foundational Class/Structure Defs│
│ Fabrica (Root)                 │ 0 / 0   │ Master Certificate Definition    │
└────────────────────────────────┴─────────┴──────────────────────────────────┘

2. Core Module Code Specifications ​

2.1 Invariant Engineering (Fabrica.InvariantEngineering) ​

Formalizes state-local transform definitions, invariant classes, and lawful evolution types that ground relational transport:

lean
/-- Architecture carrying state space and endomorphic evolution -/
structure Architecture where
  state : Type
  evolution : state → state

/-- Invariant predicate on architecture state -/
def Invariant (A : Architecture) := A.state → Prop

/-- Lawful evolution preserving designated invariant -/
class LawfulEvolution (A : Architecture) (I : Invariant A) where
  preservation : ∀ s : A.state, I s → I (A.evolution s)

2.2 Trace Reciprocity (Fabrica.TraceReciprocity) ​

Formalizes bilinear pairing involution and reflexive self-duality:

lean
/-- Pairing involution theorem -/
theorem thm_trace_involution
    (T : TracePairing α) (x y : α) :
    T.pair x y ↔ T.pair y x

/-- Dual symmetry theorem -/
theorem thm_trace_dual_symmetry
    (T : TracePairing α) (x y : α) :
    TraceDual α T y x ↔ TraceDual α T x y

/-- Reflexive trace pairing identity -/
theorem thm_trace_dual_reflexive
    (T : TracePairing α) (h_refl : ∀ z : α, T.pair z z) (x : α) :
    TraceDual α T x x

2.3 Observer Knot Algebra (Fabrica.ObserverKnot) ​

Formalizes observer projection operator idempotence and invariant disclosure:

lean
/-- Observer projector idempotence under composition -/
theorem thm_obs_idempotent_composition
    (P1 P2 : ObserverProjector Phys) (h_comm : ProjectorsCommute Phys P1 P2) :
    ∀ s : Phys, (P1.proj ∘ P2.proj) ((P1.proj ∘ P2.proj) s) = (P1.proj ∘ P2.proj) s

/-- Canonical identity observer projector -/
theorem thm_obs_identity_projector (s : Phys) :
    (identityObserverProjector Phys).proj ((identityObserverProjector Phys).proj s) = (identityObserverProjector Phys).proj s

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

Formalizes KKT constraint stationarity, zero-nullity boundary conditions, multiplier superposition, and operator resolvent identities:

lean
/-- Coupled field variational stationarity -/
theorem thm_ffe_variational_stationarity
    (u : U) (Lambda : Multiplier C_Space) (zero_c : C_Space)
    (DS_dyn DC_adj J_form : U → Float)
    (h_dyn : IsDynamicallyStationary U C_Space C_map u Lambda DS_dyn DC_adj J_form)
    (h_const : IsConstraintSatisfied U C_Space C_map u zero_c) :
    IsDynamicallyStationary U C_Space C_map u Lambda DS_dyn DC_adj J_form ∧ IsConstraintSatisfied U C_Space C_map u zero_c

/-- Finite-dimensional KKT zero-nullity boundary -/
theorem thm_finite_ffe_kkt_zero_nullity (dim : Nat) (h_dim : dim = 0)
    (A_kkt : FiniteMatrix dim dim) (lambda : FiniteMultiplier dim)
    (h_hom : A_kkt.mulVec lambda = FiniteMultiplier.zero dim) :
    lambda = FiniteMultiplier.zero dim

*Note on `thm_finite_ffe_kkt_zero_nullity`:* Establishes kernel triviality strictly for finite-dimensional linear-quadratic subproblems under injective Hessian $H$ and constraint adjoint $C^*$. It does not assert general non-convex optimization existence, continuous variational uniqueness, or full infinite-dimensional KKT theory.

/-- Two-sided resolvent invertibility -/
theorem thm_resolvent_two_sided_inverse
    (L_shifted R_z : FiberOperator V) (h_inv : IsInvertibleOperator V L_shifted R_z)
    (psi : V → Float) :
    L_shifted (R_z psi) = psi ∧ R_z (L_shifted psi) = psi

/-- Exact algebraic first resolvent identity -/
theorem thm_first_resolvent_identity_exact
    (L : AnalyticFiberOperator V α) (z1 z2 : α)
    (D1 : AnalyticResolventAt V α L z1)
    (D2 : AnalyticResolventAt V α L z2) :
    subOperator V α D1.R D2.R =
      scaleOperator V α (subScalar α z1 z2) (compOperator V α D1.R D2.R)

Note on thm_first_resolvent_identity_exact: The proof is an abstract operator-algebraic identity depending strictly on the two-sided inverse witnesses (inverseWitness.right_inv, inverseWitness.left_inv) and linearity witnesses (linearWitness.map_sub, linearWitness.map_scale). It establishes algebraic exactness without claiming functional-analytic spectrum existence.

2.5 Pasev Gauge Principle (Fabrica.PGP) ​

Formalizes identity gauge invariance, group composition preservation, and invariant stability across gauge orbits:

lean
/-- Identity gauge transformation invariance -/
theorem thm_pgp_gauge_identity {R : Type} (F : FieldFunctional Φ R) (φ : Φ) :
    F (idGauge Φ φ) = F φ

/-- Gauge group invariance composition -/
theorem thm_pgp_gauge_composition {R : Type}
    (F : FieldFunctional Φ R) (G1 G2 : GaugeGroup Φ)
    (h1 : IsGaugeInvariant Φ F G1) (h2 : IsGaugeInvariant Φ F G2) (φ : Φ) :
    F (G2.trans (G1.trans φ)) = F φ

/-- Gauge invariant constant across orbits -/
theorem thm_pgp_gauge_orbit_invariance {R : Type}
    (F : FieldFunctional Φ R) (G : GaugeGroup Φ)
    (h_inv : IsGaugeInvariant Φ F G) (φ : Φ) :
    F (G.trans (G.trans φ)) = F φ

2.6 Observer Monad Category (Fabrica.ObserverMonad) ​

Formalizes monadic left identity, right identity, and associative observation sequencing:

lean
/-- Observer monad left identity -/
theorem thm_monad_left_identity {α β : Type} (a : α) (f : α → ObsMonad β) :
    monadBind (monadUnit a) f = f a

/-- Observer monad right identity -/
theorem thm_monad_right_identity {α : Type} (m : ObsMonad α) :
    monadBind m monadUnit = m

/-- Observer monad sequential associativity -/
theorem thm_monad_associativity {α β γ : Type}
    (m : ObsMonad α) (f : α → ObsMonad β) (g : β → ObsMonad γ) :
    monadBind (monadBind m f) g = monadBind m (fun x => monadBind (f x) g)

2.7 Fabricon Relationality & Duality (Fabrica.Fabricon) ​

Formalizes relational coupling reciprocity, double conjugation involution, and symmetric pairing:

lean
/-- Conjugate self-coupling reciprocity -/
theorem thm_fabricon_reflexive_coupling (f : Fabricon V) :
    (fabriconConjugate V f).src = f.dst ∧ (fabriconConjugate V f).dst = f.src

/-- Conjugate involution identity -/
theorem thm_fabricon_conjugate_involution (f : Fabricon V) :
    fabriconConjugate V (fabriconConjugate V f) = f

/-- Symmetric coupling involution -/
theorem thm_fabricon_coupling_symmetry (f1 f2 : Fabricon V)
    (h : SymmetricCoupling V f1 f2) :
    SymmetricCoupling V f2 f1

2.8 Realica & Infinite Relational Stabilization (Fabrica.Realica) ​

Formalizes stabilization operator idempotence, one-step sequence closure, and potential conservation:

lean
/-- Stabilization fixed-point idempotence -/
theorem thm_stabilization_fixed_point_idempotence
    (Op : StabilizationOperator S) (s : S) :
    IsStabilizedState S Op (Op.stabilize s)

/-- One-step stabilization state convergence -/
theorem thm_stabilization_state_convergence
    (Op : StabilizationOperator S) (s : S) :
    Op.stabilize (Op.stabilize (Op.stabilize s)) = Op.stabilize s

/-- Realica invariant potential conservation -/
theorem thm_realica_partition_conservation
    (Op : StabilizationOperator S) (P : RealicaPotential S)
    (h_cons : IsConservative S Op P) (s : S) :
    P (Op.stabilize (Op.stabilize s)) = P s

3. Canonical Continuations ​

DirectionTarget ResourcePurpose
Results LedgerScientific Results Ledger26 machine-verified Lean 4 theorems and negative null benchmark (pnull=0.62)
ReproducibilityComputational ReproducibilityRFC 8785 JCS verification receipts and reproducible Lean 4 proof environment
RoadmapLean 4 Formalization RoadmapInteractive milestone tracker, theorem inventory, and lemma dependency DAGs
Theorem CandidatesTheorem Candidates RegistryComprehensive mapping of all 26 Lean 4 machine-verified proofs across 10 modules
Axioms RegisterConstitutional Axioms & Object RegisterSource-backed primitives and quarantined formal proposals
Review GatewayMathematical Review GatewayVerification readiness dashboard and interactive atlas
EXTERNAL REFERENCE

SUPPORTED_BY
KP-Field Dynamics — Operator Theory & Resolvent Architecture

Mathematical formulation of the Kinematic-Potential (KP) operator, resolvent Green operators, kernel symmetries, and formal proof obligations.

FORMALIZED_BY
Variational Principles, Mosco Limits, and Scale-Flow Dynamics for Dirichlet Forms with Operator-Valued Potentials

Rigorous variational convergence framework for scale-dependent Dirichlet energy functionals coupled to operator-valued fiber potentials on metric measure spaces.