Research-Stack/6-Documentation/docs/semantics/O_AMMR_CRC_PATTERN_MEMORY.md

Orthogonal AMMR and CRC Pattern Memory

This document specifies Orthogonal AMMR (O-AMMR) as a projection-based generalization of AMMR and defines its integration into AMMR-backed CRC pattern memory.

The central move is:

• replace additive or scalar summaries with orthogonal projection summaries
• preserve integrity via cryptographic commitment
• preserve informational structure via retained orthogonal subspaces

Section 1: Orthogonal AMMR - Projection-Based Merge Laws

1. Overview

Orthogonal AMMR (O-AMMR) extends the standard Algebraic Merkle Mountain Range by replacing scalar or associative summaries with projection-based geometric summaries.

Each node encodes:

1. a cryptographic commitment
2. a structured orthogonal representation of its underlying data

The structure functions as a verifiable dimensionality-reduction engine, preserving both:

1. integrity, through hashing
2. informational structure, through orthogonal bases

2. Node Structure

Each node N is defined as:

N = {
  hash: H(N_left.hash || N_right.hash || Q || R),
  summary: {
    Q: orthonormal basis matrix,
    R: projection coefficients,
    shape: (rows, cols),
    energy: ||R||_F
  }
}

Where:

• Q spans the retained subspace of inputs
• R encodes projections of inputs into that basis
• energy captures total signal magnitude

3. Canonical Merge Law

To ensure determinism across distributed systems, merge is defined as:

merge(A, B) = Orthogonalize(Canonicalize(A ∪ B))

3.1 Canonicalization

Inputs are sorted deterministically:

1. by hash, or
2. by structural index

This ensures:

identical input sets -> identical ordering -> identical summaries

3.2 Orthogonalization

Instead of naive Gram-Schmidt, the implementation should use one of:

1. Modified Gram-Schmidt (MGS)
2. Householder QR
3. SVD, preferred for numerical stability

The factorization target is:

X -> Q R

with:

• Qᵀ Q = I
• R upper triangular in QR form
• or diagonal / singular-value aligned when using SVD-style retention

4. Order-Independent Variant

The recommended order-independent form is covariance accumulation:

C = Σ_i x_i x_iᵀ

Then:

C = U Σ Uᵀ

Where:

• U becomes the orthogonal basis
• Σ encodes signal strength

This form is:

1. associative
2. commutative
3. appropriate for distributed AMMR aggregation

5. Hash Stability

Floating-point summaries must be stabilized before commitment.

Canonical rule:

Q_quant = round(Q, ε)
R_quant = round(R, ε)
hash = H(... || Q_quant || R_quant || ...)

Without quantization:

identical math -> different hashes -> consensus failure

6. Basis Truncation

To prevent unbounded basis growth:

retain i iff Σ_i >= λ

Only dominant components are retained.

This yields:

1. compressed summaries
2. bounded memory
3. suppression of low-energy directions

7. Interpretation

O-AMMR transforms the structure into:

a committed projection space of computation history

Each node encodes:

1. what directions of information exist
2. what signal strength is retained in those directions
3. what redundancy has been removed through projection

Section 2: Integration into AMMR-Backed CRC Pattern Memory

1. Redefinition of CRC Squares

Previously:

• CRC squares cached discrete pattern signatures

Now:

• CRC squares store local orthogonal summaries

Canonical regional form:

CRC_region:
  Q_local
  Σ_local
  trust
  stability

2. CRC Signature Function

Instead of:

CRC = hash(pattern bits)

use:

CRC = H(Q_quant, Σ_quant)

This encodes:

1. pattern structure
2. pattern strength
3. dimensional composition

3. Local n-map Schema

Each region maps:

n-map:
  index  -> basis vector direction
  weight -> singular value

So each cell contributes:

1. to a direction in feature space
2. not merely to a binary local state

4. Pattern Matching

Pattern matching becomes projection-based:

similarity(A, B) = ||Q_Aᵀ Q_B||

Meaning:

• similarity is the degree of alignment between retained subspaces

This replaces:

1. Hamming distance
2. bitwise comparison

5. Nutrient Dynamics

The nutrient layer becomes geometric rather than symbolic.

Nutrient Gain Law

gain = ||x - Q Qᵀ x||

This is the residual after projection.

Interpretation:

• high residual -> new information -> high nutrient
• low residual -> redundant information -> low nutrient

Nutrient Decay Law

Σ_i down -> branch pruned

Interpretation:

• weak directions lose energy
• low-energy directions are eventually removed

Duty Threshold Law

A region activates when:

||Σ|| > θ

Routing Export Law

Mycorrhizal routing weight is:

w_(A->B) = ||Q_Aᵀ Q_B|| * trust_A

Meaning:

• regions share signals when they span similar subspaces

AMMR Lock-In Rule

A pattern is committed when all of the following hold:

1. basis is stable across steps
2. singular values are stable across steps
3. projection error is below threshold

6. Solver Update Order

Each orthogonal update step proceeds as:

1. Observe Convert cell states into vectors.

2. Project Apply:

   x -> Q Qᵀ x
   

3. Compute residual

   r = x - Q Qᵀ x
   

4. Update proposals If residual is large, propose a new retained direction.

5. Update CRC bases Re-orthogonalize the local region.

6. Update AMMR Merge orthogonal summaries upward through the tree.

7. Nutrient update Reward residual energy and decay weak singular directions.

7. Mirror Algorithm

AMMR provides:

(Q, Σ)

The mirror layer computes:

coefficients = Qᵀ x

Then:

output = LUT(coefficients)

This provides:

1. constant-time inference after projection
2. no repeated basis recomputation at execution time

8. Resulting System Behavior

The resulting system is:

a self-compressing, projection-driven field solver

Where:

1. patterns are stored as subspaces
2. learning means discovering new orthogonal directions
3. memory means persistent basis vectors
4. pruning means removing low-energy directions

Final Synthesis

The integrated system is:

1. AMMR Cryptographic commitment and hierarchical structure

2. Orthogonalization Redundancy removal through basis extraction

3. Nutrients Residual-driven learning and decay

4. Mycorrhizal routing Connectivity driven by subspace similarity

5. Mirror LUT Constant-time execution over projected coordinates

In plain terms, the system becomes:

a distributed, verifiable, self-pruning basis-learning machine

This is more than:

1. pattern matching
2. storage
3. ordinary cached execution

It is:

structure-preserving computation memory

Completion Metrics

This specification is implementation-ready only when all of the following are mapped to code or theorem targets:

1. node structure with Q, R, and committed hash
2. deterministic merge law
3. quantized hash commitment
4. truncation rule with explicit threshold
5. projection-based CRC signature
6. projection similarity routing law
7. residual-based nutrient update law
8. mirror LUT execution keyed by projected coefficients

Immediate Lean Targets

The smallest truthful Lean-facing targets suggested by this spec are:

1. AMMRSummary Fields:

   • basis witness
   • coefficient witness
   • shape
   • energy

2. CRCRegionSummary Fields:

   • local basis witness
   • local strength witness
   • trust
   • stability

3. projectionSimilarity Purpose:

   • typed similarity witness between retained subspaces

4. residualEnergy Purpose:

   • typed nutrient gain witness

5. MirrorLUTIndex Purpose:

   • projected-coordinate key for execution

The first theorem targets should be:

1. deterministic merge produces equal committed parent for equal canonical inputs
2. residual energy is zero for perfectly represented inputs
3. equal projected coefficients yield equal mirror LUT index