10 KiB
PBACS: Policy-Based Adaptive Constraint System
Canonical Signal Transport Architecture (NOT Compression)
Version: 2026-04-16
Status: Specification Complete, Implementation In Progress
Core Reframe: This is NOT compression. This is deterministic signal transport + constraint-based reconstruction.
The Canonical 5-Layer Stack
┌─────────────────────────────────────────────────────────────┐
│ LAYER 5: RECONSTRUCTION [BracketedDIAT] │
│ Constraint-preserving interval arithmetic │
│ ├─ Gap conservation: g_l + g_u = u - l │
│ ├─ Conservative operations: min/max bounds │
│ └─ Truth-preserving: value always ∈ [l, u] │
├─────────────────────────────────────────────────────────────┤
│ LAYER 4: VALIDATION [PBACS Core / CMYK-SLUQ] │
│ Adaptive stress routing with policy-based state machine │
│ ├─ K (Black/00): Fast path → no extra checks │
│ ├─ C (Cyan/01): Monitor → widen observation window │
│ ├─ M (Magenta/10): Verify → secondary attestation │
│ └─ Y (Yellow/11): Prune/Reset → swap shadow table │
├─────────────────────────────────────────────────────────────┤
│ LAYER 3: CORRECTION [Void Mask LUT] │
│ Blue-noise structured thresholding replaces arithmetic │
│ ├─ LUT_void: 8Kbit pre-baked blue noise mask │
│ ├─ θ_t = LUT[(Φ_t >> n) ⊕ MSB_flip] │
│ └─ Popcount validation: AND(LUT, deviation) │
├─────────────────────────────────────────────────────────────┤
│ LAYER 2: SCHEDULING [φ-Traversal Engine] │
│ Deterministic coprime traversal (no PRNG) │
│ ├─ Φ_{t+1} = Φ_t + 106070 (mod 2^32) │
│ ├─ 91-step full coverage (13 × 7 coprime) │
│ └─ MSB mirror: quadrant fold on bit flip │
├─────────────────────────────────────────────────────────────┤
│ LAYER 1: TRANSPORT [1-Bit Noise-Shaped Encoder] │
│ Structure-preserving encoding over time │
│ ├─ b_t = 1[v_t + e_{t-1} > θ_t] │
│ ├─ e_t = v_t + e_{t-1} - b_t (error never discarded) │
│ └─ Information in: time + structure + constraints │
└─────────────────────────────────────────────────────────────┘
PBACS Core Equations
The Unified State Vector
\boxed{
X_t = (\Phi_t, e_t, a_t, s_t, b_t, \mathcal{B}_t)
}
Where:
\Phi_t: φ-accumulator (scheduling state)e_t: Error accumulator (transport state)a_t: SLUQ stress accumulator (validation state)s_t: CMYK policy state (2-bit routing)b_t: 1-bit output symbol\mathcal{B}_t: BracketedDIAT constraint structure
The 8-Step Canonical Update
\begin{align}
\text{Step 1: } & \Phi_{t+1} = \Phi_t + \Delta_\phi \pmod{2^{32}} \
\text{Step 2: } & \theta_t = \text{LUT}{\text{void}}\big[(\Phi_t \gg n) \oplus \text{MSB}{\text{flip}}\big] \
\text{Step 3: } & b_t = \mathbb{1}[v_t + e_{t-1} > \theta_t] \
\text{Step 4: } & e_t = v_t + e_{t-1} - b_t \
\text{Step 5: } & \text{stress}_t = \alpha|e_t| + \beta|\hat{\mu}t - \mu^*t| + \gamma \cdot \text{popcount}(\text{LUT}{\text{void}}[i] \land \text{deviation}) \
\text{Step 6: } & a{t+1} = a_t - (a_t \gg 6) + \text{stress}t \
\text{Step 7: } & s_t = a_t \gg 14 \in {00, 01, 10, 11} \
\text{Step 8: } & \mathcal{B}{t+1} = \text{BracketOp}(\mathcal{B}_t, b_t, \pi(s_t))
\end{align}
Layer Specifications
Layer 1: TRANSPORT — 1-Bit Noise-Shaped Encoder
Purpose: Move signal through time with minimal precision
Core Equation:
\boxed{
b_t = \begin{cases} 1 & \text{if } v_t + e_{t-1} > \theta_t \ 0 & \text{otherwise} \end{cases}, \quad e_t = v_t + e_{t-1} - b_t
}
Key Properties:
- Error never discarded — feeds forward
- Precision → moved into time domain
- Single comparator, one register
Hardware Cost: 1 LUT (comparator) + 16 FF (error reg)
Layer 2: SCHEDULING — φ-Traversal Engine
Purpose: Deterministic uniform coverage without PRNG
Core Equation:
\boxed{
\Phi_{t+1} = \Phi_t + \lfloor \phi \cdot 2^{16} \rfloor \pmod{2^{32}} = \Phi_t + 106070 \pmod{4294967296}
}
Coverage Guarantee:
\text{Period} = \frac{13 \times 7}{\gcd(13,7)} = 91 \text{ steps (full traversal)}
Key Properties:
- Most irrational step (φ) = lowest discrepancy
- No multiplication at runtime (iterative add)
- Coprime ensures no harmonic aliasing
Hardware Cost: 32 FF + adder
Layer 3: CORRECTION — Void Mask LUT
Purpose: Structured thresholding replaces arithmetic transforms
Core Equation:
\boxed{
\theta_t = \text{LUT}{\text{void}}\big[(\Phi_t \gg n) \oplus \underbrace{(\text{MSB}(\Phi_t) \neq \text{MSB}(\Phi{t-1}))}_{\text{mirror fold}}\big]
}
Attestation Validation:
\text{attest}t = \text{popcount}(\text{LUT}{\text{void}}[i_t] \land \text{Deviation}_{\text{seg}})
Key Properties:
- Blue noise mask baked at synthesis
- Self-attesting via JTAG readback
- F2b immunity by construction (not calibration)
Hardware Cost: 8Kbit LUTRAM
Layer 4: VALIDATION — PBACS / CMYK-SLUQ
Purpose: Adaptive routing based on stream stability
Stress Accumulator:
\boxed{
\text{stress}_t = \alpha|e_t| + \beta|\hat{\mu}_t - \mu^*_t| + \gamma \cdot \text{attest}_t
}
SLUQ Update:
\boxed{
a_{t+1} = a_t - (a_t \gg 6) + \text{stress}_t
}
Policy State Decode:
\boxed{
s_t = a_t \gg 14 \mapsto \begin{cases} 00 & \rightarrow \text{K (Fast)} \ 01 & \rightarrow \text{C (Monitor)} \ 10 & \rightarrow \text{M (Verify)} \ 11 & \rightarrow \text{Y (Prune)} \end{cases}
}
Policy Actions:
| State | Binary | Stress Range | Action | Effort |
|---|---|---|---|---|
| K | 00 | [0, 16384) | Fast path, no extra checks | Low |
| C | 01 | [16384, 32768) | Widen observation window | Medium |
| M | 10 | [32768, 49152) | Secondary attestation check | High |
| Y | 11 | [49152, 65536) | Prune branch, swap shadow table | Terminal |
Hardware Cost: 16 FF (accumulator) + 2 LUT (decode)
Layer 5: RECONSTRUCTION — BracketedDIAT
Purpose: Constraint-preserving interval arithmetic
Structure:
\boxed{
\mathcal{B} = \langle l, u, v, g_l, g_u, s, p, d \rangle
}
Invariant:
\boxed{
g_l + g_u = u - l \quad \text{(Gap Conservation Law)}
}
Conservative Multiplication:
\boxed{
\mathcal{B}_1 \otimes \mathcal{B}2 = \langle \min(v_i v_j), \max(v_i v_j), v_1 v_2, s{\max} \rangle
}
Policy-Driven Update:
\mathcal{B}_{t+1} = \begin{cases}
\text{bracketAdd}(\mathcal{B}_t, b_t) & s_t = K \
\text{bracketWiden}(\mathcal{B}_t) & s_t = C \
\text{bracketVerify}(\mathcal{B}_t, \text{attest}_t) & s_t = M \
\text{bracketReset}() & s_t = Y
\end{cases}
Hardware Cost: 6 × 16 FF (Fix16 fields) + comparison logic
The "Not Compression" Manifesto
| Traditional Compression | PBACS Canonical Signal Transport |
|---|---|
| Minimize entropy | Preserve structure under quantization |
| Store fewer bits | Represent value as trajectory |
| Information in bits | Information in time + structure + constraints |
| Perfect reconstruction | Constraint-based recovery |
| Global optimization | Local constraint satisfaction |
| Floating-point elegance | Add/shift/LUT/compare only |
Implementation Status
| Component | Spec | Lean | Verilog | Testbench |
|---|---|---|---|---|
| 1-bit encoder | ✅ | ❌ | ❌ | ❌ |
| φ-accumulator | ✅ | ❌ | ❌ | ❌ |
| Void mask LUT | ✅ | ❌ | ❌ | ❌ |
| CMYK-SLUQ | ✅ | ❌ | ❌ | ❌ |
| BracketedDIAT | ✅ | ✅ | ❌ | ⚠️ |
| LocalDerivative | ✅ | ✅ | ❌ | ⚠️ |
| Full 8-step loop | ✅ | ❌ | ❌ | ❌ |
Critical Path: Void mask generator → 1-bit encoder Verilog → SLUQ→Bracket wiring
Single-Sentence Definition
PBACS is a Policy-Based Adaptive Constraint System that uses deterministic φ-traversal, blue-noise LUT thresholding, and 2-bit stress routing to transport signals through 1-bit time-domain encoding while preserving truth via bracketed interval constraints.
Resource Budget (Lattice iCE40 HX8K)
| Layer | LUTs | FFs | BRAM | Notes |
|---|---|---|---|---|
| Transport | 1 | 16 | 0 | 1-bit encoder |
| Scheduling | 2 | 32 | 0 | φ-accumulator |
| Correction | 4 | 0 | 8K | Void mask LUT |
| Validation | 2 | 16 | 0 | SLUQ + CMYK decode |
| Reconstruction | 10 | 96 | 0 | BracketedDIAT |
| TOTAL | ~200 | ~160 | 8K | Zero DSP blocks |
Canonical Test Suite
Test 1: Stability Loop
Input: constant v_t
Verify: |e_t| < ε, s_t ∈ {K, C}
Test 2: Stress Response
Input: threshold distortion
Verify: M triggers, recovery completes, no runaway
Test 3: Gap Conservation
Input: bracket operations sequence
Verify: ∀t: g_l(t) + g_u(t) = u(t) - l(t)
Test 4: Adversarial Resilience
Input: corrupted LUT region
Verify: Y triggers early, system self-prunes
Document ID: PBACS_CANONICAL_ARCHITECTURE
Cross-ref: LUT_AS_DSP_EQUATION.md, FPGA_WARDEN_NODE_SPEC.md, ADAPTIVE_1BIT_CMYK_MERGED.md, BracketedCalculus.lean