AetherBrain / PHDM – Mathematical Framework

Extracted from: AetherBrain Architecture: The Case for Geometric Governance and Stateful Control Layers in Large Language Models (Executive Summary, February 2026)

This document isolates the mathematical and systems-theoretic content from the AetherBrain/PHDM executive summary. No new primitives are introduced; the innovation is in composition.

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1. System Decomposition

1.1 Two-Layer Intelligence Model

The system defines a two-layer computational architecture:

Layer 1 – Probabilistic Generator (Cortex)

A stochastic function:

f_theta : (x, h_t) -> y

Layer 2 – Deterministic Control Shell (Skull / Brain Case)

A stateful, rule-driven operator:

G : (s_t, y, c_t) -> {ALLOW, BLOCK, TRANSFORM}

Key property: one layer is non-deterministic (generation), one is deterministic and auditable (control). This is control theory, not ML.

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2. Geometric Governance

2.1 Hyperbolic Permission Space

The system operates on the Poincare disk / ball model:

B^n = { u in R^n : ||u|| < 1 }

Hyperbolic distance d_H(u, v) serves as:

• A partial order on permissions
• A monotone risk scalar
• A hierarchical embedding (tree-like structure)

2.2 Hierarchical Permissioning

Hyperbolic space is chosen because:

• Tree depth is proportional to radial distance
• Boundary = infinite cost
• Near-origin = dense, low-cost region

Permission function:

Permission(s) = phi(d_H(s, s_0))

where phi is monotone. This is order theory embedded in geometry.

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3. Distributed Concealment Storage

3.1 Erasure Coding

Reed-Solomon codes over finite fields:

Encode: D -> {S_1, ..., S_n}
Decode: {S_i1, ..., S_ik} -> D    (any k of n shards reconstruct)

Loss tolerance: n - k shards can be lost without data loss.

3.2 “Holographic” Property

Mathematically, this is:

• Redundant linear reconstruction
• Information not localized to a single address
• Distributed linear algebra with fault tolerance

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4. Reflexive Gating (Formal Control)

4.1 Deterministic Gating

Pre-execution constraint checks with hard decision boundaries:

Gate(a_t, s_t) = {
    ALLOW   if g(s_t, a_t) <= tau
    DENY    otherwise
}

This is static analysis + runtime enforcement, comparable to hardware interrupt handling.

4.2 Tokenized Authority

Capability-based access using cryptographic tokenization:

Capability = <id, scope, constraints>

Aligns with object-capability models and access control logic.

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5. Control Physics

5.1 Energy Landscape

An implicit potential function:

V(s) = f(d_H(s, s_0))

Where:

• Low energy = allowed behavior
• High energy = forbidden behavior

Equivalent to Lyapunov-style stability analysis and barrier functions in control theory.

5.2 Timing Separation (Reflex Arcs)

Two distinct timing domains:

• Fast deterministic layer (control) – reflex arcs
• Slow stochastic layer (generation) – deliberation

This matches real-time systems and safety-critical control loop architecture.

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6. Structural Novelty

Not new math, but new composition:

Innovation	Description
A	Geometry used as authorization lattice, not embedding trick
B	Erasure coding used as concealment, not durability only
C	Deterministic gating placed outside the probabilistic model
D	Stateful control treated as physics, not policy

This is systems mathematics, not ML novelty.

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7. Scope Boundaries

What this framework does not contain:

• No new cryptographic primitives
• No new hardness assumptions
• No new learning algorithm
• No new geometry theorems

The contribution is architectural composition of known mathematical structures.

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8. Minimal Mathematical Identity

Compressed to one statement:

AetherBrain = deterministic control system over a hyperbolic state space, regulating a stochastic generator via geometric constraints and algebraic storage redundancy

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9. Mapping to SCBE-AETHERMOORE Implementation

Mathematical Object	Implementation
Poincare ball B^n	src/harmonic/hyperbolic.ts
Hyperbolic distance d_H	src/harmonic/hyperbolic.ts, src/harmonic/adaptiveNavigator.ts
Potential function V(s)	src/harmonic/harmonicScaling.ts (Harmonic Wall, L12)
Deterministic gate G	src/harmonic/pipeline14.ts (L13 risk decision)
Erasure coding	src/crypto/envelope.ts, src/crypto/pqc.ts
Capability tokens	src/crypto/nonceManager.ts, src/crypto/replayGuard.ts
Permission ordering	src/harmonic/languesMetric.ts (Sacred Tongues 6D metric)
Energy landscape	H(d,R) = phi^d / (1 + e^-R) in harmonicScaling.ts
Timing separation	14-layer pipeline (fast L1-L8 control, slow L9-L14 analysis)
PHDM drift monitor	src/harmonic/phdm.ts
Quantum Axiom Mesh	src/symphonic_cipher/scbe_aethermoore/axiom_grouped/

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References

• SCBE-AETHERMOORE 14-Layer Architecture: LAYER_INDEX.md
• System Architecture: SYSTEM_ARCHITECTURE.md
• Security Model: SECURITY.md
• Hyperbolic geometry implementation: src/harmonic/hyperbolic.ts
• Harmonic wall formula: src/harmonic/harmonicScaling.ts