Theorem Candidates Registry

Constitutional Theorem Candidates, Analytical Derivations & Machine-Proof Anchors

Spine Position

Mathematics Root · Theorem Candidates, Analytical Proof Sketches & Lean 4 Anchors

Public Status Boundary. This register documents authorial theorem candidates and analytical proof sketches across the Science of Fabric Reality (SFR) and Fractal Quantum Field Theory (FQFT) compendium. Each item is classified under the Proof Governance verification scale and mapped to active proof obligations in the Formalization Roadmap.

1. Formalization Candidates Register

Theorem 3.1 (Observer Incompleteness of Physical Theories)

Carrier Domain: Observer Monads & Measurement Boundaries
Status: Grounded by THM-OBS-IDEMPOTENT-01 (Fabrica.ObserverKnot, Machine-Verified in Lean 4)
Formal Proof Target: thm_obs_idempotent_composition, thm_obs_identity_projector

Proof Boundary

The operator projection idempotence lemma is machine-checked in Lean 4 (thm_obs_idempotent_composition); the full philosophical observer incompleteness remains an axiomatic boundary thesis.

Statement

Let $T_{P}$ be any closed physical theory formulated entirely within a bounded phase space $P$ . Then $T_{P}$ cannot define the observer required to interpret its own measurement operators without introducing an external topological boundary condition.

Theorem 3.2 (Topological Protection of Information Flow)

Carrier Domain: Discrete Topology & 1-Cycles
Status: Grounded by THM-FABRICA-CYCLE-01 & THM-FABRICA-CYCLE-02 (Fabrica.InvariantEngineering, Statement Drafted / Formalization Target)
Formal Proof Target: thm_one_cycle_linear_combination, thm_one_cycle_zero_boundary

Formalization Target Foundation

The discrete 1-cycle linear subspace closure and boundary-free conservation laws on cell complexes are formalized as target specifications in Fabrica.InvariantEngineering pending machine verification closure.

Statement

Within an evolving simplicial lattice governed by the stabilized continuation law of the Axioms Register, information flux across closed hypersurfaces is topologically protected against local discretization noise whenever the winding number invariant remains non-zero.

Theorem 3.3 (Bilinear Trace Pairing Symmetry and Dual Self-Evaluation)

Carrier Domain: Trace Reciprocity & Bilinear Pairings
Status: Grounded by THM-TRACE-RECIPROCITY-01, THM-TRACE-DUAL-EVAL-01, and THM-TRACE-REFL-01 (Fabrica.TraceReciprocity, Machine-Verified in Lean 4)
Formal Proof Target: thm_trace_involution, thm_trace_dual_symmetry, thm_trace_dual_reflexive

Machine-Verified Foundation

The core algebraic involution of symmetric bilinear pairings (thm_trace_involution), the dual symmetry equivalence (thm_trace_dual_symmetry), and reflexive self-duality (thm_trace_dual_reflexive) are machine-verified in Lean 4 without sorry.

Statement

For any symmetric trace pairing structure $T$ on a state carrier $α$ , the relation $pair (x, y)$ is logically equivalent to $pair (y, x)$ , and the dual map evaluation $TraceDual (T, y, x) ⟺ TraceDual (T, x, y)$ . Furthermore, under reflexive pairings, every state is self-dual.

Theorem 3.4 (Coupled Field Stationarity and Foliation Superposition)

Carrier Domain: Constrained Variational Systems (FFE)
Status: Grounded by THM-FFE-STATIONARITY-01, THM-FFE-HOMOGENEOUS-01, THM-FFE-SUPERPOSITION-01, and THM-FFE-KKT-UNIQUE-01 (Fabrica.FFE, Machine-Verified in Lean 4)
Formal Proof Target: thm_ffe_variational_stationarity, thm_ffe_homogeneous_triviality, thm_ffe_multiplier_superposition, thm_finite_ffe_kkt_zero_nullity

Machine-Verified Foundation

Coupled Euler-Lagrange stationarity (thm_ffe_variational_stationarity), homogeneous source triviality (thm_ffe_homogeneous_triviality), multiplier linear superposition (thm_ffe_multiplier_superposition), and finite-dimensional KKT zero-nullity boundary conditions (thm_finite_ffe_kkt_zero_nullity) are fully machine-verified in Lean 4.

Statement

A field configuration $(u, Λ)$ satisfies the non-holonomic foliation dynamics if and only if dynamical variation stationarity and strict constraint satisfaction $C (u) = 0$ hold simultaneously. For finite-dimensional linear-quadratic subproblems, under strictly injective Hessian operators and kernel-free constraint adjoints, the homogeneous system admits only the trivial solution $(0, 0)$ . Continuous variational uniqueness and general non-convex optimization remain open analytic frontiers.

Theorem 3.5 (Operator Resolvent Invertibility, Shifted Differences & Resolvent Identities)

Carrier Domain: Operator Theory & Scale Flow (FQFT)
Status: Grounded by THM-FQFT-RESOLVENT-02, THM-FQFT-RESOLVENT-ID-01, THM-FQFT-SHIFT-DIFF-01, and THM-FQFT-EXACT-RESOLVENT-ID-01 (Fabrica.FQFT, Machine-Verified in Lean 4)
Formal Proof Target: thm_resolvent_two_sided_inverse, thm_first_resolvent_identity_algebraic, thm_shifted_operator_difference, thm_first_resolvent_identity_exact

Machine-Verified Foundation

Two-sided resolvent invertibility (thm_resolvent_two_sided_inverse), algebraic first resolvent identity (thm_first_resolvent_identity_algebraic), shifted operator differences (thm_shifted_operator_difference), and the exact algebraic first resolvent identity (thm_first_resolvent_identity_exact) are fully machine-verified in Lean 4. Dilation scaling bounds and continuous covariance commutation remain active analytic/formalization targets.

Statement

For linear operators $A_{1}, A_{2}$ with two-sided resolvents $R_{1}, R_{2}$ , the difference of resolvents satisfies the first resolvent identity $R_{1} A_{2} (R_{2} ψ) - R_{1} A_{1} (R_{2} ψ) = R_{1} ψ - R_{2} ψ$ . In the exact operator setting, resolvent differences satisfy $D_{1} . R - D_{2} . R = (z_{1} - z_{2}) \cdot (D_{1} . R \circ D_{2} . R)$ strictly from linear and two-sided inverse witnesses.

Theorem 3.6 (Pasev Gauge Principle & Orbit Invariance)

Carrier Domain: Gauge Theory & Invariant Groups
Status: Grounded by THM-PGP-IDENTITY-01, THM-PGP-COMPOSITION-01, and THM-PGP-ORBIT-01 (Fabrica.PGP, Machine-Verified in Lean 4)
Formal Proof Target: thm_pgp_gauge_identity, thm_pgp_gauge_composition, thm_pgp_gauge_orbit_invariance

Machine-Verified Foundation

Identity gauge transformation invariance, composite gauge transformation preservation, and invariance across iterated gauge orbits are fully machine-verified in Lean 4.

Statement

Every functional invariant under the gauge group action satisfies $F ({id}_{Φ} φ) = F (φ)$ and preserves invariance under composite morphisms $F (g_{2} (g_{1} φ)) = F (φ)$ .

Theorem 3.7 (Observer Monad Categorical Laws)

Carrier Domain: Epistemic Category Theory & Observer Structures
Status: Grounded by THM-MONAD-LEFT-ID-01, THM-MONAD-RIGHT-ID-01, and THM-MONAD-ASSOC-01 (Fabrica.ObserverMonad, Machine-Verified in Lean 4)
Formal Proof Target: thm_monad_left_identity, thm_monad_right_identity, thm_monad_associativity

Machine-Verified Foundation

The categorical Left Identity, Right Identity, and Associativity laws for observer monadic measurement pipelines are fully machine-verified in Lean 4.

Statement

For any observer monad structure, sequential observation satisfies categorical coherence: $(unit (a) ≫= f) \equiv f (a)$ , $(m ≫= unit) \equiv m$ , and $((m ≫= f) ≫= g) \equiv (m ≫= (λ x \to f (x) ≫= g))$ .

Theorem 3.8 (Fabricon Conjugate Relationality & Involution)

Carrier Domain: Relational Ontology & Fabricon Duality
Status: Grounded by THM-FABRICON-REFLEXIVE-01, THM-FABRICON-INVOLUTION-01, and THM-FABRICON-SYMMETRY-01 (Fabrica.Fabricon, Machine-Verified in Lean 4)
Formal Proof Target: thm_fabricon_reflexive_coupling, thm_fabricon_conjugate_involution, thm_fabricon_coupling_symmetry

Machine-Verified Foundation

Source-destination conjugate reciprocity, double conjugation identity involution, and symmetric coupling bidirectionality are machine-verified in Lean 4.

Statement

Every relational fabricon element satisfies double conjugation involution $conj (conj (f)) = f$ , and reflexive self-coupling symmetry across discrete relational lattices.

Theorem 3.9 (Realica Infinite Stabilization Fixed-Point Convergence)

Carrier Domain: Infinite Relational Stabilization
Status: Grounded by THM-STABILIZATION-IDEMP-01, THM-STABILIZATION-CONV-01, and THM-REALICA-CONSERVE-01 (Fabrica.Realica, Machine-Verified in Lean 4)
Formal Proof Target: thm_stabilization_fixed_point_idempotence, thm_stabilization_state_convergence, thm_realica_partition_conservation

Machine-Verified Foundation

Stabilization operator idempotence $(S \circ S = S)$ , state trajectory convergence, and total partition potential conservation are machine-verified in Lean 4.

Statement

The infinite stabilization operator is idempotent, maps any state trajectory to a fixed-point invariant equilibrium in a single canonical projection, and preserves the total partition potential across all relational subsystems.

Theorem 3.10 (Krauss–Pasev Self-Adjoint Domain Closure & Cayley Unitarity)

Carrier Domain: Non-Local Spectral Operators & Domain Closure (Fabrica.KPField)
Status: Grounded by THM-KP-DEF-NULL-01, THM-KP-CAYLEY-UNITARY-01, and THM-KP-SA-CLOSURE-01 (Fabrica.KPField, Machine-Verified in Lean 4)
Formal Proof Target: thm_kp_deficiency_indices_nullity, thm_kp_cayley_transform_unitary, thm_kp_selfadjoint_domain_closure

Machine-Verified Foundation

Deficiency subspace nullity $\ker (L^{*} \mp i I) = {0}$ under positive-definite state pairing, unitary Cayley transformation preserving test state pairings $⟨ U f, U g ⟩ = ⟨ f, g ⟩$ , and maximal self-adjoint domain closure $D ({\overset{―}{L}}_{K P})$ with reciprocal symmetry $P (\overset{―}{L} f, g) = P (f, \overset{―}{L} g)$ are fully machine-verified in Lean 4 without sorry or admit.

Statement

For any symmetric non-local Krauss–Pasev generator $L_{K P}$ defined on an inner-product state carrier $(V, P)$ with non-degenerate spectral pairing, the deficiency subspaces $\ker (L^{*} \mp i I)$ are identically trivial. Consequently, the Cayley transform $U = (L - i I) (L + i I)^{- 1}$ is unitary, and the closure ${\overset{―}{L}}_{K P}$ on maximal domain $D ({\overset{―}{L}}_{K P})$ is uniquely self-adjoint ( $L^{*} = L$ ), yielding an unconditionally well-defined functional calculus and continuous resolvent flow.

2. Canonical Continuations

Direction	Target Resource	Purpose
Results Ledger	Scientific Results Ledger →	35 machine-verified Lean 4 theorems across 11 modules and negative null benchmark ( $p_{null} = 0.62$ )
Reproducibility	Computational Reproducibility →	RFC 8785 JCS verification receipts and reproducible proof environment
Proof Gateway	Lean 4 Formalization Roadmap →	35 machine-verified theorem records and lemma dependency DAGs
Axioms Register	Constitutional Axioms & Object Register →	Source-backed primitives and quarantined formal proposals
Proof Governance	Proof Governance & Verification Scale →	Orthogonal status coordinates $S = (F, C, P, R, V)$ and verification scale
Simulation Bridge	Formalization ↔ Simulation Bridge →	Diagnostic mapping from formal objects to discrete solvers

Theorem Candidates Registry ​

1. Formalization Candidates Register ​

Theorem 3.1 (Observer Incompleteness of Physical Theories) ​

Statement ​

Theorem 3.2 (Topological Protection of Information Flow) ​

Statement ​

Theorem 3.3 (Bilinear Trace Pairing Symmetry and Dual Self-Evaluation) ​

Statement ​

Theorem 3.4 (Coupled Field Stationarity and Foliation Superposition) ​

Statement ​

Theorem 3.5 (Operator Resolvent Invertibility, Shifted Differences & Resolvent Identities) ​

Statement ​

Theorem 3.6 (Pasev Gauge Principle & Orbit Invariance) ​

Statement ​

Theorem 3.7 (Observer Monad Categorical Laws) ​

Statement ​

Theorem 3.8 (Fabricon Conjugate Relationality & Involution) ​

Statement ​

Theorem 3.9 (Realica Infinite Stabilization Fixed-Point Convergence) ​

Statement ​

Theorem 3.10 (Krauss–Pasev Self-Adjoint Domain Closure & Cayley Unitarity) ​

Statement ​

2. Canonical Continuations ​

Theorem Candidates Registry

1. Formalization Candidates Register

Theorem 3.1 (Observer Incompleteness of Physical Theories)

Statement

Theorem 3.2 (Topological Protection of Information Flow)

Statement

Theorem 3.3 (Bilinear Trace Pairing Symmetry and Dual Self-Evaluation)

Statement

Theorem 3.4 (Coupled Field Stationarity and Foliation Superposition)

Statement

Theorem 3.5 (Operator Resolvent Invertibility, Shifted Differences & Resolvent Identities)

Statement

Theorem 3.6 (Pasev Gauge Principle & Orbit Invariance)

Statement

Theorem 3.7 (Observer Monad Categorical Laws)

Statement

Theorem 3.8 (Fabricon Conjugate Relationality & Involution)

Statement

Theorem 3.9 (Realica Infinite Stabilization Fixed-Point Convergence)

Statement

Theorem 3.10 (Krauss–Pasev Self-Adjoint Domain Closure & Cayley Unitarity)

Statement

2. Canonical Continuations