Mathematical Foundations of Invariants & Admissibility

Invariant Families, Functorial Transport Morphisms & Machine-Verified Composition

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Mathematics Root · Algebraic Invariant Theory & Admissibility Calculus

Public Status Boundary. This document establishes the category-theoretic and algebraic foundation for invariant preservation, state-local admissibility, and functorial transport maps in the Science of Fabric Reality. Core algebraic composition theorems are formally machine-verified in Lean 4 without ungrounded axioms.

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1. Formal Invariant Spaces

Let $\mathbf{Sys}$ be a category of relational systems $\mathcal{S}=(X,R,\partial ,\mathcal{I},\mathcal{T},\mathcal{O},\mathcal{M})$ with morphisms given by system transformations $T:{\mathcal{S}}_1\to {\mathcal{S}}_2$.

Definition 1.1 (Invariant Family)

An Invariant Family $\mathcal{I}$ is an indexed family of functors or evaluation functionals:

$$\mathcal{I}=\{I_k:\mathbf{Sys}⟶{\mathbf{Val}}_k{\}}_{k\in K}$$

where each ${\mathbf{Val}}_k$ is a value space equipped with an equivalence relation ${\sim }_k$.

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2. Invariant-Admissibility Calculus

Definition 2.1 (Transport Morphism)

For each morphism $T\in \mathrm{Hom}({\mathcal{S}}_1,{\mathcal{S}}_2)$ and each index $k\in K$, an Invariant Transport Morphism is a map:

$${\tau }_{T,k}:{\mathbf{Val}}_k⟶{\mathbf{Val}}_k$$

Definition 2.2 (State-Local Admissibility)

The admissibility indicator of transformation $T$ on state $\mathcal{S}$ under invariant family $\mathcal{I}$ is:

$${\mathrm{Adm}}_{\mathcal{I}}(T,\mathcal{S})=\prod _{k\in K}\mathbf{1}[I_k(T(\mathcal{S})){\sim }_k{\tau }_{T,k}(I_k(\mathcal{S}))]$$

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3. Machine-Verified Invariant Theorems

Theorem 3.1: Admissibility Composition (THM-INV-COMP-01)

• Epistemic Status: MACHINE_VERIFIED_LEAN4 (Fabrica.InvariantEngineering).
• Formal Lean 4 Target: thm_state_local_admissibility_composition.
• Hypotheses: Let $T_1:{\mathcal{S}}_0\to {\mathcal{S}}_1$ and $T_2:{\mathcal{S}}_1\to {\mathcal{S}}_2$ satisfy ${\mathrm{Adm}}_{\mathcal{I}}(T_1,{\mathcal{S}}_0)=1$ and ${\mathrm{Adm}}_{\mathcal{I}}(T_2,{\mathcal{S}}_1)=1$.
• Coherence Conditions: Functorial transport ${\tau }_{T_2\circ T_1,k}={\tau }_{T_2,k}\circ {\tau }_{T_1,k}$ and transitivity of ${\sim }_k$.
• Deductive Consequence:$${\mathrm{Adm}}_{\mathcal{I}}(T_2\circ T_1,{\mathcal{S}}_0)=1$$

Theorem 3.2: Inversion Admissibility (THM-INV-INV-01)

• Epistemic Status: MACHINE_VERIFIED_LEAN4 (Fabrica.InvariantEngineering).
• Formal Lean 4 Target: thm_state_local_admissibility_inversion.
• Statement: For any invertible morphism $T$, state-local admissibility is preserved under morphism inversion $T^{-1}$ with inverse transport ${\tau }_{T,k}^{-1}$.

Theorem 3.3: 1-Cycle Linear Subspace Closure (THM-FABRICA-CYCLE-01)

• Epistemic Status: MACHINE_VERIFIED_LEAN4 (Fabrica.InvariantEngineering).
• Formal Lean 4 Target: thm_one_cycle_linear_combination.
• Statement: Linear combinations of discrete simplicial 1-cycles on directed graphs strictly preserve 1-cycle membership.

Theorem 3.4: 1-Cycle Boundary Nullity (THM-FABRICA-CYCLE-02)

• Epistemic Status: MACHINE_VERIFIED_LEAN4 (Fabrica.InvariantEngineering).
• Formal Lean 4 Target: thm_one_cycle_zero_boundary.
• Statement: Any linear combination of 1-cycles evaluates to an exact zero topological boundary (${\partial }_1\sum c_i{\gamma }_i=0$).

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4. Canonical Continuations

Direction	Target Resource	Purpose
Axioms Register	Constitutional Axioms & Object Register →	Formal primitive definitions and quarantined axiom records
Engineering Discipline	Invariant Engineering & IRP →	Operational pre/post-conditions and protocol verification
Proof Tracker	Lean 4 Formalization Roadmap →	26 machine-verified theorem records and lemma dependency DAGs
Trace Reciprocity	Trace Reciprocity Principle →	Bilinear trace pairing symmetry and dual self-evaluation