Research-Stack/6-Documentation/papers/OTOM/08_Invariant_Theory.md

Invariant Theory: Conservation Laws and Constraints

Authors: Research Stack Team
Date: April 2026
Domain: TTM Layer D (Invariants)
OTOM Version: 2.2

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Abstract

Invariant Theory establishes the conservation laws and constraints governing all OTOM computations. It formalizes the Crystallization Front Invariant, Epistemic Inhibitory Control, and other critical system constraints.

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1. Core Invariants

1.1 Crystallization Front Invariant (Sisyphus Inverse)

\Phi_{\text{si}} = \frac{\text{formation rate}}{\text{dissolution rate}} \geq \tau_{\text{critical}}

Prevents premature convergence to suboptimal states.

1.2 Epistemic Inhibitory Controller (The Warden)

\mathcal{W}(s) = \mathbb{1}[\text{prohibited}(s)] \cdot \infty

Enforces prohibited state avoidance.

1.3 Trajectory Quality Invariant (Metatyping)

\Sigma = \int_0^T \sqrt{g_{ij} \dot{x}^i \dot{x}^j}\, dt

Measures path quality through manifold geometry.

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2. Conservation Laws

2.1 Information Conservation

H(X) + I(X;Y) = H(X|Y) + H(Y)

2.2 Energy Conservation

\Delta E_{\text{system}} + \Delta E_{\text{environment}} = 0

2.3 Action Conservation

\oint_C p\, dq = 2\pi n \hbar

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3. ACI: Automatic Convergence Inhibition

3.1 Inhibition Condition

\text{inhibit}(s_t) = \text{entropy}(s_t) < \theta_{\text{entropy}} \land \text{variance}(s_{t-k:t}) < \theta_{\text{variance}}

3.2 Golden Stratum Gate (Jupiter Regime)

G_{\text{gate}} = \{s \in \mathcal{M} \mid \Phi_{\text{si}}(s) \geq \Phi_{\text{critical}}\}

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4. Prohibited States

4.1 Definition

\mathcal{P} = \{s \in \mathcal{S} \mid \exists p \in \text{Predicates}, \neg p(s)\}

4.2 Warden Enforcement

\forall s \in \mathcal{P}, \mathcal{W}(s) = \text{active}

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5. Implementation

Lean 4 Modules:

• Prohibited.lean — Prohibited state framework
• Witness.lean — Witness verification
• StructuralAttestation.lean — Attestation logic

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6. Theorems

6.1 Invariant Preservation

\forall \text{bind } b, \text{lawful}(b) \implies \text{invariant}(\text{source}(b)) = \text{invariant}(\text{target}(b))

6.2 Convergence Guarantee

\text{ACI}(s_0) \land \text{wellFormed}(s_0) \implies \exists! s_\infty, \lim_{t \to \infty} s_t = s_\infty

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7. References

• Noether, E. (1918). Invariante Variationsprobleme.
• Research Stack, AGENTS.md §1.9
• Research Stack, OTOM Ontology v2.2.