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 definitions are grounded in Lean 4 (Fabrica.InvariantEngineering) and classical predicate transformer semantics.

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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:

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

Definition 2.3 (Computational Invariant Residual)

For continuous or numerical evaluations, invariant preservation is quantified via the normalized residual:

$${\rho }_{j,k}=\frac{|I_j(x_{k+1})-I_j(x_k)|}{max(|I_j(x_k)|,s_{j,min})+\epsilon }\le {\tau }_j$$

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3. Formalization Architecture & Theorem Obligations

3.1 Lean 4 Architectural Core (Fabrica.InvariantEngineering)

The foundational type-theoretic structure of invariant engineering is formalized in module Fabrica.InvariantEngineering:

• Architecture: structure Architecture where state : Type; evolution : state → state
• Invariant: def Invariant (A : Architecture) := A.state → Prop
• LawfulEvolution:

  class LawfulEvolution (A : Architecture) (I : Invariant A) where
    preservation : ∀ s : A.state, I s → I (A.evolution s)

3.2 Formalization Targets (Proof Obligations)

The categorical composition theorems function as formalization targets in the formalization roadmap:

• Target: Admissibility Composition (THM-INV-MORPHISM-COMPOSITION-01):$${\mathrm{Adm}}_{\mathcal{I}}(T_1,{\mathcal{S}}_0)=1\wedge {\mathrm{Adm}}_{\mathcal{I}}(T_2,{\mathcal{S}}_1)=1⟹{\mathrm{Adm}}_{\mathcal{I}}(T_2\circ T_1,{\mathcal{S}}_0)=1$$
• Target: Inversion Admissibility: Preserving admissibility across invertible morphism inversion $T^{-1}$.
• Public Grounding: Grounded in Hoare logic and Dijkstra predicate transformer semantics (Hoare 1969, Dijkstra 1976) as established software verification standards.

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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 →	35 machine-verified theorem records and lemma dependency DAGs
Trace Reciprocity	Trace Reciprocity Principle →	Bilinear trace pairing symmetry and dual self-evaluation