Comprehensive Mathematical Foundations of SCBE

Spectral Context-Bound Encryption: A Unified Mathematical Framework

Table of Contents

1. Introduction
2. Standing Axioms A1-A12
3. Fourteen-Layer Pipeline
4. Security Proofs and Guarantees
5. Integration with Cryptographic Envelope

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Introduction

The SCBE (Spectral Context-Bound Encryption) Security Gate is a mathematically rigorous framework combining:

1. Hyperbolic Geometry Pipeline: 14-layer transformation from complex context to risk decisions
2. Cryptographic Envelope: AES-256-GCM authenticated encryption for secure message passing

The One-Line Math Contract

All proofs hinge on this contract:

• Hyperbolic state stays inside compact sub-ball 𝔹ⁿ_{1-ε}
• All ratio features use denominator floor ε > 0
• All extra channels are bounded and enter risk monotonically with nonnegative weights

This makes continuity/Lipschitz-on-compact, monotonicity in deviation features, and boundedness provable.

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Standing Axioms A1-A12

Configuration (Choice Script)

Fix integers D ≥ 1 and K ≥ 1 and set n := 2D.

A configuration (“choice script”) is a tuple:

Θ := (α, ε_ball, ε, G, b(·), a(·), Q(·), {μ_k}_{k=1}^K, λ₁,λ₂,λ₃, w_d,w_c,w_s,w_τ,w_a, R, θ₁,θ₂)

Axiom A1 (Input Domain)

The context state satisfies c(t) ∈ ℂᴰ for all t in the time index set.

For end-to-end stability: ‖c(t)‖_ℂ ≤ M for some M < ∞.

Axiom A2 (Realification Isometry)

Define Φ₁: ℂᴰ → ℝⁿ by:

Φ₁(z₁,...,z_D) := (Re(z₁),...,Re(z_D), Im(z₁),...,Im(z_D))

Then Φ₁ is a real-linear isometry: ‖c‖ℂ = ‖Φ₁(c)‖ℝ

Axiom A3 (SPD Weighting)

The weighting matrix G ∈ ℝⁿˣⁿ is symmetric positive definite (SPD).

Define the weighted transform: x_G := G^{1/2} · x

Axiom A4 (Poincaré Embedding + Clamping)

Let α > 0 and ε_ball ∈ (0,1).

Poincaré embedding Ψ_α: ℝⁿ → 𝔹ⁿ:

Ψ_α(x) := tanh(α‖x‖) · x/‖x‖    for x ≠ 0
Ψ_α(0) := 0

Clamping operator Πε: 𝔹ⁿ → 𝔹ⁿ{1-ε}:

Π_ε(u) := u                      if ‖u‖ ≤ 1-ε
Π_ε(u) := (1-ε) · u/‖u‖          otherwise

All hyperbolic states: u := Πε(Ψα(x_G))

Axiom A5 (Fixed Hyperbolic Metric)

The hyperbolic distance d_H on 𝔹ⁿ is the Poincaré ball metric:

d_H(u,v) = arcosh(1 + 2‖u-v‖² / ((1-‖u‖²)(1-‖v‖²)))

Axiom A6 (Breathing Transform)

For each t, b(t) > 0 and the breathing map T_breath(·;t): 𝔹ⁿ → 𝔹ⁿ:

T_breath(u;t) := tanh(b(t) · artanh(‖u‖)) · u/‖u‖    for u ≠ 0
T_breath(0;t) := 0

Assume b(t) ∈ [b_min, b_max] for 0 < b_min ≤ b_max < ∞.

CRITICAL: Breathing is a smooth ball-preserving diffeomorphism, but NOT an isometry unless b(t) = 1.

Axiom A7 (Phase Transform Isometry)

For each t, let a(t) ∈ 𝔹ⁿ and Q(t) ∈ O(n).

Define the phase map:

T_phase(u;t) := Q(t) · (a(t) ⊕ u)

where ⊕ is Möbius addition.

T_phase(·;t) IS an isometry of (𝔹ⁿ, d_H).

Axiom A8 (Realms)

Realm centers satisfy μk ∈ 𝔹ⁿ{1-ε} for k = 1,…,K.

Define realm distance:

d*(u) := min_{k=1,...,K} d_H(u, μ_k)

A8 FIX: All realm centers must be clamped: μk ← Πε(μ_k) before use.

Axiom A9 (Signal Regularization)

All ratio-based features use denominators bounded below by ε > 0.

Example: Replace Σ

Y[k]

² by Σ

Y[k]

² + ε

Axiom A10 (Coherence Features Bounded)

• Spectral coherence: S_spec(t) ∈ [0,1]
• Audio coherence: S_audio(t) ∈ [0,1]
• Spin coherence: C_spin(t) ∈ [0,1]
• Trust score: τ(t) ∈ [0,1]

Axiom A11 (Triadic Temporal Aggregation)

Windows W₁, W₂, W_G are finite.

Let λᵢ > 0 with λ₁ + λ₂ + λ₃ = 1.

Define d_tri(t) as weighted ℓ² norm of windowed averages.

Normalized:

d̃_tri(t) := min(1, d_tri(t)/d_scale) ∈ [0,1]

Axiom A12 (Risk Functional)

Weights satisfy wd, w_c, w_s, wτ, w_a ≥ 0 and Σw = 1.

Harmonic scaling:

H(d*, R) := R^{(d*)²}    where R > 1

Base risk:

Risk_base(t) := w_d·d̃_tri + w_c·(1-C_spin) + w_s·(1-S_spec) + w_τ·(1-τ) + w_a·(1-S_audio)

Amplified risk:

Risk'(t) := Risk_base(t) · H(d*(t), R)

Decision thresholds θ₁ < θ₂:

• Risk’ < θ₁ → ALLOW
• θ₁ ≤ Risk’ < θ₂ → QUARANTINE
• Risk’ ≥ θ₂ → DENY

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Fourteen-Layer Pipeline

Layer	Name	Input	Output	Key Operation
L1	Complex Context	A, Φ	c ∈ ℂᴰ	c_k = a_k·e^{iφ_k}
L2	Realification	c	x ∈ ℝ²ᴰ	x = [Re(c), Im(c)]
L3	Weighted Transform	x	x_G	x_G = G^{1/2}·x
L4	Poincaré Embedding	x_G	u ∈ 𝔹ⁿ	u = Πε(Ψα(x_G))
L5	Möbius Stabilization	u	u’	u’ = u ⊕ (-μ_k)
L6	Breathing	u’	u_b	Diffeomorphism (NOT isometry)
L7	Phase Transform	u_b	u_f	Isometry: Q·(a ⊕ u_b)
L8	Realm Distance	u_f	d*	min_k d_H(u_f, μ_k)
L9	Spectral Coherence	Telemetry	S_spec	FFT energy ratio
L10	Spin Coherence	Phases	C_spin		Σe^{iθ}	/N
L11	Behavioral Trust	x	τ	Hopfield energy sigmoid
L12	Harmonic Scaling	d*	H	R^{(d*)²}
L13	Composite Risk	All signals	Risk’	Weighted sum × H
L14	Audio Telemetry	Audio	S_audio	Phase stability

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Security Proofs and Guarantees

Theorem 1 (Boundedness)

For any input c(t) with ‖c(t)‖ ≤ M:

Risk'(t) ∈ [0, R^{D_max²}]

where D_max is the maximum possible realm distance.

Proof: By A4 clamping, all states stay in 𝔹ⁿ_{1-ε}. By A10, all coherence signals are in [0,1]. By A12, base risk is convex combination of [0,1] values, hence in [0,1]. Harmonic scaling is bounded for bounded d*. ∎

Theorem 2 (Monotonicity)

For fixed other inputs, Risk’ is:

• Increasing in d̃_tri (higher deviation → higher risk)
• Decreasing in C_spin, S_spec, τ, S_audio (higher coherence → lower risk)

Proof: Direct from A12 formula. Each coherence term enters as (1 - signal), so higher signal → lower contribution. ∎

Theorem 3 (Continuity)

The map c(t) → Risk’(t) is continuous on the bounded domain.

Proof: Each layer is continuous:

• L1-L3: Linear/smooth operations
• L4: tanh is smooth, clamping is Lipschitz
• L5-L7: Möbius addition and rotation are smooth on interior
• L8: min of continuous functions
• L9-L14: Bounded ratios with ε floor

Composition of continuous functions is continuous. ∎

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Integration with Cryptographic Envelope

Risk-Gated Envelope Creation

async function createGatedEnvelope(params: CreateParams, riskResult: RiskResult) {
  if (riskResult.decision === 'DENY') {
    throw new Error('Risk threshold exceeded');
  }

  const envelope = await createEnvelope({
    ...params,
    // Include risk metadata in AAD
    schema_hash: computeSchemaHash(params.body, riskResult),
  });

  if (riskResult.decision === 'QUARANTINE') {
    envelope.aad.audit_flag = true;
  }

  return envelope;
}

Audit Trail

Every envelope creation logs:

• Risk’ value
• Decision (ALLOW/QUARANTINE/DENY)
• Coherence signals snapshot
• request_id for correlation

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References

1. Ungar, A. A. “Hyperbolic Trigonometry and its Application in the Poincaré Ball Model”
2. Nielsen & Chuang, “Quantum Computation and Quantum Information”
3. Cover & Thomas, “Elements of Information Theory”

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Document Version: 2.0
Last Updated: 2026-01-13
Authors: SCBE Development Team