Phase-Coupled Dimensionality Collapse (PCDC)

Part of SCBE-AETHERMOORE v3.0.0
Patent: USPTO #63/961,403
Date: January 18, 2026
Status: Novel Mathematical Construction

0. Motivation

The SCBE system uses 6-dimensional context vectors, but not all dimensions are equally “active” at all times. The “demi/quasi/polly” breathing concept suggests dimensionality should adapt based on system state. PCDC formalizes this as a continuous, phase-derived dimensionality reduction that makes authorization decisions tighten automatically when dimensions drift.

1. Domain

Input: Complex-valued context vector c(t) ∈ ℂ^D where:

  • D = 6 (KO, AV, RU, CA, UM, DR dimensions)
  • Each c_i(t) has magnitude and phase: c_i(t) = |c_i(t)| · e^(i·φ_i(t))
  • Time series available: {c(t-Δ), c(t-2Δ), ..., c(t-NΔ)}

Parameters:

  • Δ: Time window for phase coherence measurement (default: 1 second)
  • N: Number of historical samples (default: 10)
  • τ: Coherence threshold for dimension activation (default: 0.7)

2. Operator

2.1 Phase Coherence (Per-Dimension)

For each dimension i ∈ {1,...,D}, compute phase coherence over time window:

ρ_i(t) = |𝔼_Δ[e^(i·Δφ_i(t,Δ))]|

where:

Δφ_i(t,Δ) = arg(c_i(t)) - arg(c_i(t-Δ))

In English:

  • If phase is stable (small Δφ), then e^(i·Δφ) ≈ 1 → ρ_i ≈ 1
  • If phase drifts randomly, then e^(i·Δφ) averages to 0 → ρ_i ≈ 0

Discrete Implementation:

ρ_i(t) = (1/N) · |Σ_{k=1}^N e^(i·Δφ_i(t-kΔ,Δ))|

2.2 Collapse Operator

Define the collapse operator that attenuates incoherent dimensions:

Π_ρ(c)_i = ρ_i · c_i

Result: c' = Π_ρ(c) where incoherent dimensions are suppressed.

2.3 Effective Dimensionality

Define two measures of effective dimensionality:

Hard threshold:

D_eff^hard(t) = Σ_{i=1}^D 𝟙[ρ_i(t) > τ]

Soft (continuous):

D_eff^soft(t) = Σ_{i=1}^D ρ_i(t)

Interpretation:

  • D_eff = 6: All dimensions coherent (polly mode)
  • D_eff ≈ 3: Half dimensions drifting (demi mode)
  • D_eff < 2: Severe drift (quasi mode, high alert)

3. Invariant

Phase Coherence Preservation:

For a stable system with no external perturbations:

d/dt ρ_i(t) ≈ 0  (for all i)

Proof Sketch: If φ_i(t) = ω_i·t + φ_0 (constant angular velocity), then:

Δφ_i(t,Δ) = ω_i·Δ  (constant)
e^(i·Δφ_i) = e^(i·ω_i·Δ)  (constant)
ρ_i = |e^(i·ω_i·Δ)| = 1

Corollary: Coherence drops only when phase velocity changes (drift/attack).

4. Metric

Define the dimensionality collapse metric:

M_collapse(t) = D - D_eff^soft(t) = Σ_{i=1}^D (1 - ρ_i(t))

Interpretation:

  • M_collapse = 0: No collapse (all dimensions active)
  • M_collapse = D: Complete collapse (all dimensions incoherent)

Optimization Goal: Minimize M_collapse for trusted nodes, maximize for attackers.

5. Theorem

Theorem 1 (Drift Detection):

Let c(t) be a context vector with phase drift σ_φ (standard deviation of phase velocity). Then:

𝔼[ρ_i] ≤ e^(-σ_φ²·Δ²/2)

Proof:

Assume phase increments are Gaussian: Δφ ~ 𝒩(0, σ_φ²·Δ²).

Then:

𝔼[e^(i·Δφ)] = ∫ e^(i·x) · (1/√(2πσ²)) · e^(-x²/(2σ²)) dx
            = e^(-σ²/2)  (characteristic function of Gaussian)

where σ² = σ_φ²·Δ².

Therefore:

ρ_i = |𝔼[e^(i·Δφ)]| ≤ e^(-σ_φ²·Δ²/2)

Corollary: High drift (large σ_φ) → low coherence (small ρ_i) → dimension collapses.

6. Integration with SCBE

6.1 Layer 3 (Langues Metric Tensor)

Current formula:

L(x,t) = Σ_{l=1}^6 w_l · exp[β_l · (d_l + sin(ω_l·t + φ_l))]

PCDC Enhancement:

L_PCDC(x,t) = Σ_{l=1}^6 ρ_l(t) · w_l · exp[β_l · (d_l + sin(ω_l·t + φ_l))]

Effect: Incoherent dimensions contribute less to trust score.

6.2 Dimensional Breathing Modes

Map D_eff to breathing modes:

mode(t) = {
  'polly'  if D_eff(t) > 5.5
  'demi'   if 3.0 ≤ D_eff(t) ≤ 5.5
  'quasi'  if D_eff(t) < 3.0
}

Flux Coefficients:

flux_polly  = 1.0  (all dimensions active)
flux_demi   = 0.5  (half dimensions active)
flux_quasi  = 0.1  (minimal dimensions active)

6.3 Authorization Tightening

Decision Rule:

threshold(t) = base_threshold · (D / D_eff(t))

Effect: As D_eff drops, threshold increases → harder to pass authorization.

7. Implementation

7.1 TypeScript Types 

interface PhaseCoherence {
  rho: number[]; // Per-dimension coherence [0,1]
  D_eff_hard: number; // Hard threshold count
  D_eff_soft: number; // Soft continuous sum
  M_collapse: number; // Collapse metric
  mode: 'polly' | 'demi' | 'quasi';
}

interface PCDCConfig {
  D: number; // Number of dimensions (6)
  Delta: number; // Time window (seconds)
  N: number; // Historical samples
  tau: number; // Coherence threshold
}

7.2 Core Algorithm 

class PCDC {
  private config: PCDCConfig;
  private history: Complex[][]; // [time][dimension]

  constructor(config: PCDCConfig) {
    this.config = config;
    this.history = [];
  }

  /**
   * Update with new context vector
   */
  update(c: Complex[]): void {
    this.history.push([...c]);
    if (this.history.length > this.config.N) {
      this.history.shift();
    }
  }

  /**
   * Compute phase coherence for dimension i
   */
  computeCoherence(i: number): number {
    if (this.history.length < 2) return 1.0;

    let sum_real = 0;
    let sum_imag = 0;
    let count = 0;

    for (let k = 1; k < this.history.length; k++) {
      const phi_curr = this.history[k][i].arg();
      const phi_prev = this.history[k - 1][i].arg();
      const delta_phi = phi_curr - phi_prev;

      sum_real += Math.cos(delta_phi);
      sum_imag += Math.sin(delta_phi);
      count++;
    }

    const avg_real = sum_real / count;
    const avg_imag = sum_imag / count;

    return Math.sqrt(avg_real * avg_real + avg_imag * avg_imag);
  }

  /**
   * Compute full phase coherence state
   */
  computeState(): PhaseCoherence {
    const rho: number[] = [];
    let D_eff_hard = 0;
    let D_eff_soft = 0;

    for (let i = 0; i < this.config.D; i++) {
      const rho_i = this.computeCoherence(i);
      rho.push(rho_i);

      if (rho_i > this.config.tau) {
        D_eff_hard++;
      }
      D_eff_soft += rho_i;
    }

    const M_collapse = this.config.D - D_eff_soft;

    let mode: 'polly' | 'demi' | 'quasi';
    if (D_eff_soft > 5.5) {
      mode = 'polly';
    } else if (D_eff_soft >= 3.0) {
      mode = 'demi';
    } else {
      mode = 'quasi';
    }

    return {
      rho,
      D_eff_hard,
      D_eff_soft,
      M_collapse,
      mode,
    };
  }

  /**
   * Apply collapse operator
   */
  collapse(c: Complex[]): Complex[] {
    const state = this.computeState();
    return c.map((c_i, i) => c_i.multiply(state.rho[i]));
  }
}

8. Test Suite

8.1 Invariant Tests 

describe('PCDC Invariants', () => {
  it('coherence is in [0,1]', () => {
    const pcdc = new PCDC({ D: 6, Delta: 1, N: 10, tau: 0.7 });
    // Add stable phases
    for (let t = 0; t < 20; t++) {
      const c = Array.from({ length: 6 }, (_, i) => Complex.fromPolar(1, i * 0.1 * t));
      pcdc.update(c);
    }

    const state = pcdc.computeState();
    state.rho.forEach((rho_i) => {
      expect(rho_i).toBeGreaterThanOrEqual(0);
      expect(rho_i).toBeLessThanOrEqual(1);
    });
  });

  it('stable phases → high coherence', () => {
    const pcdc = new PCDC({ D: 6, Delta: 1, N: 10, tau: 0.7 });
    // Constant phase velocity
    for (let t = 0; t < 20; t++) {
      const c = Array.from(
        { length: 6 },
        (_, i) => Complex.fromPolar(1, 0.5 * t) // Same velocity for all
      );
      pcdc.update(c);
    }

    const state = pcdc.computeState();
    expect(state.D_eff_soft).toBeGreaterThan(5.5);
    expect(state.mode).toBe('polly');
  });

  it('drifting phases → low coherence', () => {
    const pcdc = new PCDC({ D: 6, Delta: 1, N: 10, tau: 0.7 });
    // Random phase drift
    for (let t = 0; t < 20; t++) {
      const c = Array.from({ length: 6 }, () => Complex.fromPolar(1, Math.random() * 2 * Math.PI));
      pcdc.update(c);
    }

    const state = pcdc.computeState();
    expect(state.D_eff_soft).toBeLessThan(3.0);
    expect(state.mode).toBe('quasi');
  });

  it('collapse operator preserves magnitude scaling', () => {
    const pcdc = new PCDC({ D: 6, Delta: 1, N: 10, tau: 0.7 });
    // Add some history
    for (let t = 0; t < 10; t++) {
      pcdc.update(
        Array.from({ length: 6 }, () => Complex.fromPolar(1, Math.random() * 2 * Math.PI))
      );
    }

    const c = Array.from({ length: 6 }, () => Complex.fromPolar(2, 0));
    const c_collapsed = pcdc.collapse(c);

    // Collapsed magnitudes should be ≤ original
    c_collapsed.forEach((c_i, i) => {
      expect(c_i.abs()).toBeLessThanOrEqual(c[i].abs());
    });
  });
});

8.2 Drift Detection Test 

it('detects drift according to theorem', () => {
  const pcdc = new PCDC({ D: 6, Delta: 1, N: 10, tau: 0.7 });

  // Simulate Gaussian drift with σ_φ = 0.5
  const sigma_phi = 0.5;
  const Delta = 1;

  for (let t = 0; t < 20; t++) {
    const c = Array.from({ length: 6 }, () => {
      const drift = sigma_phi * Math.sqrt(Delta) * randn();
      return Complex.fromPolar(1, drift);
    });
    pcdc.update(c);
  }

  const state = pcdc.computeState();
  const expected_rho = Math.exp((-sigma_phi * sigma_phi * Delta * Delta) / 2);

  // Average coherence should be close to theoretical bound
  const avg_rho = state.D_eff_soft / 6;
  expect(avg_rho).toBeLessThan(expected_rho * 1.5); // Allow some slack
});

9. Patent Claims

Claim 1: Phase-Coupled Dimensionality Collapse Method

“A method for adaptive dimensionality reduction comprising: (a) receiving complex-valued context vector c(t) ∈ ℂ^D; (b) computing per-dimension phase coherence ρi(t) = |𝔼[e^(i·Δφ_i)]|; (c) defining collapse operator Πρ(c)_i = ρ_i · c_i; (d) computing effective dimensionality D_eff = Σ ρ_i; (e) adjusting authorization threshold based on D_eff; wherein incoherent dimensions are automatically suppressed.”

Claim 2: Breathing Mode Classification

“The method of claim 1, wherein dimensional modes are classified as:

  • polly mode if D_eff > 5.5 (all dimensions active);
  • demi mode if 3.0 ≤ D_eff ≤ 5.5 (partial activation);
  • quasi mode if D_eff < 3.0 (minimal activation).”

Claim 3: Drift Detection Bound

“The method of claim 1, wherein phase coherence satisfies: ρi ≤ e^(-σφ²·Δ²/2) where σ_φ is phase drift standard deviation.”

10. Comparison to Prior Art

What’s NOT New

  • Phase coherence measurement (signal processing)
  • Dimensionality reduction (PCA, autoencoders)
  • Adaptive thresholds (anomaly detection)

What IS New

  • Phase-derived continuous dimensionality for security contexts
  • Automatic dimension suppression based on temporal coherence
  • Breathing modes (polly/demi/quasi) as formal mathematical states
  • Integration with hyperbolic geometry (Langues Metric Tensor)

11. Future Work

  1. Multi-Scale Coherence: Measure ρ_i at multiple Δ values
  2. Cross-Dimension Coupling: Detect when dimensions drift together (coordinated attack)
  3. Adaptive τ: Learn optimal coherence threshold from data
  4. Quantum Extension: Apply to quantum state coherence

Status: ✅ MATHEMATICALLY SPECIFIED | ⏳ IMPLEMENTATION PENDING | 🔐 PATENT-READY
Generated: January 18, 2026 21:20 PST
Patent Deadline: 13 days remaining

© 2026 Aethermoore - Issac Davis, Founder | Patent Pending (63/961,403) | Products | Demo

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