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, 35 Lean theorem declarations across 11 formal modules have achieved machine-verified status under Lean 4.16.0 with 0 sorry, 0 admit, and 0 quarantined axioms (the former source-declared axiom FQFT_closure has been formally de-quarantined and discharged). 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 (35/35 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 proposition-valued symmetric pairing and conditional reflexive self-duality:

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

/-- Trace dual predicate symmetry theorem -/
theorem thm_trace_dual_symmetry
    (T : TracePairing α) (x y : α) :
    TraceDual α T y x ↔ TraceDual α T x y

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

Note on Fabrica.TraceReciprocity: Formalizes proposition-valued symmetric relations on abstract carrier types. It does not claim machine verification of operator trace cyclicity, categorical traced monoidal functors, or continuous boundary Green identities.

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 separate-kernel zero nullity strictly for finite-dimensional subproblems under injective Hessian $H$ and constraint adjoint $C^*$. It does not assert coupled KKT saddle-point uniqueness, 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 ​

Results Ledger35 Proofs

Scientific Results Ledger

35 machine-verified Lean 4 theorems across 11 modules and canonical negative null benchmark (p_null = 0.62).

ReproducibilityRFC 8785

Computational Reproducibility

RFC 8785 JCS verification receipts and reproducible Lean 4 proof environment.

Proof GatewayLemma DAGs

Lean 4 Formalization Roadmap

Interactive milestone tracker, theorem inventory, and transitive lemma dependency DAGs across all 11 modules.

Frontier CandidatesCandidates

Theorem Candidates Registry

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

Axiomatic LatticeAxioms

Constitutional Axioms Register

Source-backed primitives, declared baseline postulates, and quarantined formal proposals.

Verification LifecycleM-Lifecycle

Proof Governance Framework

Orthogonal status coordinate standards $S = (F, C, P, R, V)$ governing theorem candidate promotion.

Referee GatewayM-Review

Mathematical Review Gateway

Verification readiness dashboard, referee intake dossier, and interactive mathematical atlas.

Numerical SolversNumerical Atlas

Simulation Laboratory

Discretized numerical finite-difference solvers, null tests (p_null = 0.62), and verification receipts.

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.