Skip to content

self-optimizing-memory.md

Latest commit

 

History

History
296 lines (112 loc) · 5.29 KB

self-optimizing-memory.md

File metadata and controls

296 lines (112 loc) · 5.29 KB
Raw
title Self-Optimizing Memory via Cohomological Renormalization
author unknown
tags
monograph
identity
constraint
type essay

Self-Optimizing Memory via Cohomological Renormalization

1. The Foundation: The Binding Invariant and Constraint Persistence

Every system undergoing irreversible transformation faces a fundamental topological necessity: determining what persists across regimes of possibility. In contemporary AI, memory is often treated as a static repository of state. This framework replaces that view with the Binding Invariant — the structured property that persists under constraint closure as a system evolves.

Identity is not stored in the substrate. It is preserved in constraints that bind a trajectory, ensuring that change constitutes development rather than rupture.

Formalizing the Constraint Site

We define the site of constraint histories JS over a partially ordered time set (T, ≤) with Alexandrov topology τ.

  • Constraint sheaf 𝒞 assigns constraints to each time t

  • Stalk 𝒞ₜ represents active constraints at time t

  • Restriction maps ρₜ′ₜ are monotone for t ≤ t′, enforcing irreversibility

  • Constraint history:

    Hₜ(S) = lim← 𝒞ₜ′ (over all t′ ≤ t)

This history acts as structural sediment, embedding past closures into present reasoning.

The Binding Invariant Theorem

A system possesses a binding invariant if and only if its history admits a coherent section in the topos of feasible trajectories.

Equivalent conditions:

  • Non-empty global section: Bₜ(S) ≠ ∅

    → At least one viable future exists

  • Global trajectory compatibility

    → No contradiction with historical constraints

  • Existence of a binding section

    → Identity thread persists across time

  • Coherent limit in Pro(Traj(S))

    → Trajectory has a well-defined boundary

This establishes the formal boundary between continuation and rupture.

2. Geometric Mechanics: Synthetic Grounding and Cohomological Flow

Ambiguity is not noise but structure. Memory optimization becomes a geometric descent process.

Modeling Derived Ambiguity

Candidate interpretations form a derived stack:

𝓜 : Ωᵒᵖ → ∞-Grpd

Mapping space interpretation:

  • Empty → Contradiction

  • Nontrivial → Ambiguity

  • Contractible → Equivalence

Higher cohomology Hᵏ captures multi-scale inconsistencies.

The Cohomological Energy Functional

E(x) = Σₖ≥1 λₖ‖Obsₖ(x)‖² + μ·S(x) + ν·C(x) + η·Σₐ∈A(x) r(a, x)

Components:

  • Obstruction classes → gluing failures

  • Semantic dispersion S(x) → entropy

  • Substrate cost C(x) → resource burden

  • Residual constraints → unsatisfied logic

Minimization yields descent on inconsistency.

Fixed Points as Identity

Identity is a homotopy-stable minimum where:

∇E(x) = 0

The self is the equivalence class of trajectories stable under descent.

3. Operational Architecture: CLIO and LCC

CLIO acts as a monoidal endofunctor over semantic states, enabling compositional reasoning.

Process-Algebraic Mapping

  • tell(c) → add local section

  • ask(c) → reduce obstruction

  • P ∥ Q → attempt gluing of structures

Lyapunov Stability

Energy E acts as a Lyapunov function:

dE/dt < 0 along valid trajectories

Identity is preserved at the level of equivalence classes, not raw states.

Bisimulation as Homotopy

Two systems are equivalent if their trajectories are homotopy equivalent.

Operational equivalence = geometric equivalence.

4. Empirical Realization: Neural Garbage Collection (NGC)

NGC implements learned forgetting as constraint-driven descent.

From Eviction to Renormalization

State is modeled as an RSVP field:

X = (Φ, v, S)

  • Φ → semantic density

  • v → inferential flow

  • S → entropy

NGC performs entropic decimation, removing components that do not support future constraint closure.

Uncertainty as Lyapunov Signal

  • Successful trajectories: dE/dt < 0

  • Failed trajectories: dE/dt > 0

Uncertainty tracks descent quality.

Graph Aggregation

NGC approximates a homotopy colimit:

  1. Expand candidate structures

  2. Identify equivalence classes

  3. Collapse to invariant core

Result: approximation of B∞(S)

5. Strategic Roadmap: Toward Resource-Aware Architectures

Transition from storage-based memory to renormalized semantic structure.

Core Pillars

  1. Field-derived stability

    → derive Lyapunov behavior from system dynamics

  2. Information-geometric optimization

    → apply Fisher metric and natural gradients

  3. Realizability closure

    → unify derived semantics with LCC execution

Design Principles

  • Path-faithful admissibility

    → preserve historical constraint visibility

  • Budget-aware interoception

    → treat compute and energy as constraints

  • Invariant-preserving compression

    → retain binding invariant, compress redundancy

Conclusion: Identity as Coherence Under Descent

The Binding Invariant defines continuity in evolving systems.

Identity is not a stored object.

It is a coherence condition maintained under irreversible transformation.

A system persists if it remains within a stable homotopy class of trajectories.

To remember is to compute.

To exist is to remain coherent under descent.