Dual-Channel Consensus Gate: Mathematical Specification

Part of SCBE-AETHERMOORE v3.0.0
Patent: USPTO #63/961,403
Layer Integration: Layer 11 (Triadic Consensus) + Audio Axis
Date: January 18, 2026

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0. Goal

Given a request event at time t, output:

decision_t ∈ {ALLOW, QUARANTINE, DENY}

by requiring agreement between two independent channels:

1. Crypto channel: transcript authenticity + freshness + nonce uniqueness
2. Voice/acoustic channel: challenge-bound acoustic evidence (liveness / response binding)

This is “dual lattice” in the operational sense: a cryptographic transcript lattice plus a frequency-bin lattice (discrete spectral coordinates).

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1. Notation

Symbol	Meaning
K	Master key (or session root)
P_t	Request payload (bytes)
AAD_t	Canonical metadata (bytes)
τ_t	Timestamp
n_t	Nonce (unique within defined scope)
c_t ∈ {0,1}^b	Acoustic challenge bitstring
y_t[n]	Audio samples (PCM), n=0,…,N-1
SR	Sample rate (Hz)
N	Segment length (samples)
T_s = N/SR	Segment duration
k ∈ {0,...,N-1}	DFT bin index
f_k = k·SR/N	Bin frequency

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2. Crypto Channel

2.1 Transcript Construction (Authenticated Envelope)

Define a transcript:

C_t := "scbe.v1" | AAD_t | τ_t | n_t | P_t

Compute a MAC tag (HMAC shown; signatures can be substituted later):

tag_t := HMAC_K(C_t)

2.2 Verification Predicate

Let the verifier maintain a nonce set (or database) N_seen for a TTL window.

Define:

MAC validity:

V_mac(t) = 1 if tag_t = HMAC_K(C_t), else 0

Freshness window:

V_time(t) = 1 if |τ_recv - τ_t| ≤ W, else 0

Nonce uniqueness:

V_nonce(t) = 1 if n_t ∉ N_seen, else 0

Crypto score:

S_crypto(t) := V_mac(t) · V_time(t) · V_nonce(t) ∈ {0,1}

State update on accept: if S_crypto(t) = 1, insert n_t into N_seen atomically.

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3. Voice/Acoustic Channel (Challenge-Bound Evidence)

3.1 Rationale

Audio alone is replayable. To make it meaningful, the audio must depend on a fresh challenge c_t. The cleanest mechanism is a spectral watermark bound to c_t and verified by correlation.

3.2 Challenge Generation

Generate:

c_t ← ${0,1}^b

Optionally include protocol metadata:

chal_t := (τ_t, n_t, c_t, mode)

3.3 Deterministic Bin Selection (The “Frequency Lattice”)

Define an allowed bin range [k_min, k_max] and spacing constraint Δk_min to reduce leakage/collisions.

Derive a seed:

s_t := HMAC_K("bins" | τ_t | n_t | c_t)

Use s_t as a PRNG seed to deterministically choose b distinct bins:

{k_1,...,k_b} ⊆ [k_min, k_max]

with |k_i - k_j| ≥ Δk_min for i ≠ j.

Also derive per-bin phases (optional but improves correlation under some pipelines):

φ_j := 2π · u_j,  u_j ∈ [0,1) derived from s_t

3.4 Watermark Waveform (Challenge Encoding)

Choose amplitudes a_j (normalized):

a_j := 1/√b

Define the watermark:

s_c_t[n] := Σ(j=1 to b) a_j · (-1)^(c_t,j) · sin(2π k_j · n/N + φ_j)

for n = 0,...,N-1

Client emits audio:

y_t[n] := v_t[n] + γ · s_c_t[n]

where:

• v_t[n] is the user’s voice (or any acoustic carrier)
• γ > 0 is a small mixing gain

This construction makes the watermark mathematically checkable even if the voice content varies.

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4. Acoustic Verification Statistic

4.1 Matched-Filter Projections (Bin Probes)

Define per-bin projection (a matched filter):

p_j(t) := (2/N) · Σ(n=0 to N-1) y_t[n] · sin(2π k_j · n/N + φ_j)

Under the ideal model (bin-aligned, no clipping), this behaves like:

p_j(t) ≈ γ · a_j · (-1)^(c_t,j) + η_j

where η_j is noise/interference (voice energy leakage, channel noise, mic filtering).

4.2 Correlation Score (Challenge Binding)

Define the correlation:

corr(t) := Σ(j=1 to b) w_j · (-1)^(c_t,j) · p_j(t)

with weights w_j ≥ 0 (often w_j = 1, or inverse-variance weights).

Decision rule:

V_audio(t) := 1 if corr(t) ≥ β, else 0

Audio score:

S_audio(t) := V_audio(t) ∈ {0,1}

4.3 Optional Robustness Gates (Recommended)

To reduce false accepts from random audio energy:

Minimum watermark-band energy:

Σ(j=1 to b) p_j(t)² ≥ E_min

No heavy clipping detected:

max_n |y_t[n]| < 1 - ε

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5. Final Decision Logic (ALLOW / QUARANTINE / DENY)

Use a conservative 3-outcome rule:

DENY if crypto fails:

S_crypto(t) = 0 ⇒ DENY

ALLOW if both pass:

S_crypto(t) = 1 ∧ S_audio(t) = 1 ⇒ ALLOW

QUARANTINE if crypto passes but audio fails:

S_crypto(t) = 1 ∧ S_audio(t) = 0 ⇒ QUARANTINE

(“Quarantine” means step-up verification, rate limit, restricted capability set, or human confirmation.)

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6. Parameter Selection Guidelines

These are engineering constraints that make the math behave:

6.1 Nyquist and Harmonic Safety

Ensure watermark frequencies are below Nyquist:

k_j < N/2  ⟺  f_k_j < SR/2

6.2 Bin Alignment (Important)

The whole matched-filter / orthogonality story works best when bins are DFT-aligned:

• Choose a fixed N and verify over exactly N samples (or window consistently)
• Derive bins k_j directly (not arbitrary Hz values)

6.3 Choose a Practical Band

Typical mics/speakers roll off in high frequencies. A pragmatic band is often mid-high (example only):

• f_min ~ 1200–2000 Hz
• f_max ~ 6000–8000 Hz

Convert to bins:

k_min = ⌈f_min · N/SR⌉
k_max = ⌊f_max · N/SR⌋

6.4 Bit-Length (b) vs Detectability

• Larger b improves challenge binding and reduces chance acceptance
• But requires more bins and increases detectability demands

A practical starting point: b ∈ [16, 64].

6.5 Recommended Defaults

Profile 1: High-Quality Audio (44.1 kHz)

SR = 44100 Hz
N = 22050 samples (0.5 seconds)
f_min = 2000 Hz → k_min = 1000
f_max = 8000 Hz → k_max = 4000
Δk_min = 50 bins (~100 Hz spacing)
b = 32 bits
β = 0.6 (correlation threshold)
γ = 0.05 (5% watermark mixing)

Profile 2: Telephony/VoIP (16 kHz)

SR = 16000 Hz
N = 16000 samples (1.0 second)
f_min = 1200 Hz → k_min = 1200
f_max = 6000 Hz → k_max = 6000
Δk_min = 30 bins (~30 Hz spacing)
b = 24 bits
β = 0.5 (correlation threshold)
γ = 0.08 (8% watermark mixing)

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7. What This Does and Does Not Claim

What You Can Defend

✅ Envelope authenticity reduces to MAC unforgeability (standard cryptographic assumption)

✅ Replay resistance requires and reduces to:

• Nonce uniqueness enforcement + timestamp window enforcement

✅ Challenge binding: The verifier checks for a deterministic watermark tied to c_t; a stale replay will not correlate for new c_t

What You Should NOT Claim Without Empirical Work

❌ “Deepfake-proof”
❌ “Guaranteed liveness”
❌ “Biometric identity”
❌ Any fixed “accuracy %” unless you publish protocol + dataset + operating point

This scheme is best framed as step-up liveness / response binding plus anomaly gating, not “voice biometric authentication.”

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8. Reference Pseudocode

"""
Dual-Channel Consensus Gate
Inputs: AAD_t, P_t, tau_t, n_t, tag_t, audio y[0..N-1]
Secret: K
State: N_seen
"""

def verify_request(AAD_t, P_t, tau_t, n_t, tag_t, y, c_t, K, N_seen, W, beta):
    # --- Crypto channel ---
    C = "scbe.v1" || AAD_t || tau_t || n_t || P_t

    S_crypto = (
        tag_t == HMAC(K, C) and
        abs(tau_recv - tau_t) <= W and
        n_t not in N_seen
    )

    if not S_crypto:
        return "DENY"

    # --- Audio channel (challenge-bound) ---
    # Deterministically re-derive bins/phases from (tau_t, n_t, c_t)
    seed = HMAC(K, "bins" || tau_t || n_t || c_t)
    bins_and_phases = select_bins_and_phases(seed, k_min, k_max, delta_k_min, b)

    # Matched-filter projections
    projections = []
    for j, (k_j, phi_j) in enumerate(bins_and_phases):
        p_j = (2/N) * sum(
            y[n] * sin(2*pi*k_j*n/N + phi_j)
            for n in range(N)
        )
        projections.append(p_j)

    # Correlation score
    corr = sum(
        w_j * (-1)**c_t[j] * p_j
        for j, (w_j, p_j) in enumerate(zip(weights, projections))
    )

    S_audio = (corr >= beta)

    # Decision logic
    if S_audio:
        N_seen.add(n_t)  # atomic
        return "ALLOW"
    else:
        N_seen.add(n_t)  # atomic (still prevent replay)
        return "QUARANTINE"

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9. Integration with SCBE-AETHERMOORE

Layer Mapping

Component	SCBE Layer	Integration
Crypto Channel	Layer 11 (Triadic Consensus)	Crypto + Temporal + Spatial alignment
Audio Channel	Audio Axis (FFT Telemetry)	Frequency-domain pattern detection
Challenge Binding	Layer 1 (Context Commitment)	SHA-256(d + id) binding
Nonce Management	Layer 10 (Lyapunov Stability)	State evolution with uniqueness

Implementation Files

src/
├── symphonic_cipher/
│   ├── audio/
│   │   ├── dual_channel_consensus.py    # Main implementation
│   │   ├── watermark_generator.py       # Challenge encoding
│   │   ├── matched_filter.py            # Bin projections
│   │   └── correlation_verifier.py      # Challenge binding
│   │
│   └── connectors/
│       └── triadic_bridge.py            # L10→L11 dynamics→consensus

tests/
└── symphonic_cipher/
    └── test_dual_channel_consensus.py   # Verification suite

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10. Mathematical Properties

Theorem 1: Replay Resistance

Statement: Given nonce uniqueness enforcement and timestamp window W, a replayed transcript C_t will be rejected with probability 1.

Proof:

1. If n_t ∈ N_seen, then V_nonce(t) = 0 ⇒ S_crypto(t) = 0 ⇒ DENY
2. If |τ_recv - τ_t| > W, then V_time(t) = 0 ⇒ S_crypto(t) = 0 ⇒ DENY ∎

Theorem 2: Challenge Binding

Statement: Given a fresh challenge c_t, a stale audio recording y_old will fail correlation with probability ≥ 1 - 2^(-b).

Proof:

1. Old recording contains watermark for c_old ≠ c_t
2. Correlation corr(t) = Σ w_j · (-1)^(c_t,j) · p_j
3. For random c_old, expected correlation ≈ 0 (orthogonal)
4. Probability of accidental match ≤ 2^(-b) (birthday bound) ∎

Theorem 3: MAC Unforgeability

Statement: Under HMAC security assumptions, forging tag_t without knowledge of K is computationally infeasible.

Proof: Reduces to HMAC-SHA256 PRF security (standard cryptographic assumption). ∎

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11. Performance Characteristics

Computational Complexity

Operation	Complexity	Notes
HMAC computation	O(	C_t	)	Linear in transcript size
Bin selection	O(b log b)	PRNG + sorting
Watermark generation	O(N · b)	N samples, b bins
Matched filtering	O(N · b)	N samples, b projections
Correlation	O(b)	b bins
Total	O(N · b)	Dominated by audio processing

Latency Estimates

Profile 1 (44.1 kHz, N=22050, b=32):

• Watermark generation: ~5 ms
• Matched filtering: ~10 ms
• Total: ~15 ms

Profile 2 (16 kHz, N=16000, b=24):

• Watermark generation: ~8 ms
• Matched filtering: ~15 ms
• Total: ~23 ms

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12. Security Analysis

Attack Vectors

Attack	Mitigation	Effectiveness
Replay	Nonce uniqueness + timestamp	✅ Provably secure
Forgery	HMAC unforgeability	✅ Cryptographically secure
Challenge prediction	HMAC-derived bins	✅ Computationally infeasible
Watermark removal	Spread-spectrum embedding	⚠️ Requires empirical validation
Deepfake synthesis	Challenge binding	⚠️ Not claimed as defense

Threat Model

In Scope:

• Replay attacks (stale audio/transcript)
• Forgery attacks (fake transcripts)
• Challenge prediction (guessing bins)

Out of Scope:

• Deepfake synthesis (not claimed)
• Side-channel attacks (timing, power)
• Physical attacks (mic tampering)

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13. Patent Claims

Claim 1: Dual-Channel Consensus Method

“A method for authenticating requests comprising: (a) verifying cryptographic transcript authenticity via HMAC; (b) enforcing nonce uniqueness and timestamp freshness; (c) generating a fresh acoustic challenge bitstring; (d) deterministically deriving frequency bins from challenge; (e) embedding challenge-bound watermark in audio; (f) verifying watermark correlation at receiver; (g) outputting ALLOW/QUARANTINE/DENY based on dual-channel consensus.”

Claim 2: Challenge-Bound Watermark

“The method of claim 1, wherein the acoustic watermark is generated as:

s[n] = Σ(j=1 to b) a_j · (-1)^(c_j) · sin(2π k_j · n/N + φ_j)

where bins {k_j} and phases {φ_j} are deterministically derived from challenge c_t.”

Claim 3: Matched-Filter Verification

“The method of claim 1, wherein verification comprises: (a) computing per-bin projections via matched filtering; (b) computing correlation score with challenge-dependent signs; (c) accepting if correlation exceeds threshold β.”

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14. References

1. HMAC Security: Bellare, M., Canetti, R., & Krawczyk, H. (1996). “Keying Hash Functions for Message Authentication.”
2. Spread-Spectrum Watermarking: Cox, I. J., et al. (2007). “Digital Watermarking and Steganography.”
3. Matched Filtering: Turin, G. L. (1960). “An Introduction to Matched Filters.”
4. Acoustic Holography: Maynard, J. D., et al. (1985). “Nearfield Acoustic Holography.”

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Status: ✅ MATHEMATICALLY SPECIFIED | ⏳ IMPLEMENTATION PENDING | 🔐 PATENT-READY
Generated: January 18, 2026 21:15 PST
Patent Deadline: 13 days remaining