Paper Stub: LUT-as-DSP Hardware Collapse

Equation ID: LUT-DSP-003
Status: Hardware specification (Verilog + Lean formalization)
Lineage: Hardware extraction from ENE cost functions
Date: 2026-04-19
Ancestry: FPGA Warden Node — the first equation designed for silicon

The Equation

φ_corr = LUT_void[(Φ_acc ≫ n) ⊕ MSB_flip]

where:
  Φ_acc^(t+1) = (Φ_acc^(t) + ⌊φ · 2^16⌋) mod 2^32
  
  c_mod7^(t+1) = { 0            if c_mod7^(t) = 6
                 { c_mod7^(t) + 1 otherwise
  
  idx_mirror = (Φ_acc ≫ n) ⊕ (MSB(Φ_acc^(t)) ≠ MSB(Φ_acc^(t-1)))

Mutation History

Variant	Domain	Key Difference
LUT-DSP-003	Hardware	φ-accumulator + mod-7 counter
LUT-SW-003a	Software	φ^entropy · 2^64 polynomial seed
LUT-SIM-003b	Simulation	SHA256(engram_key ‖ position) ≫ 4
UNIVERSAL-003c	Cross-domain	Hybrid hash + XOR fold (see MIRROR_LUT_EQUATIONS.md)

Half-Thought: The Void Mask

"LUT_void[i] = void-and-cluster(i) ∈ {0,1}^2^13"

The void mask is described as "baked at synthesis time — no runtime PRNG, no seed, no non-determinism." But the generation algorithm (void-and-cluster) is never specified in the codebase. This is intentional opacity:

The mask generation is a manufacturing secret
Different Warden nodes may have different void masks
The security property relies on mask diversity, not mask algorithm

Hardware-to-Lean Correspondence

Hardware	Lean Formalization	File
φ-accumulator	`DynamicCanal.phiAccumulator`	`DynamicCanal.lean`
mod-7 counter	`BraidBracket.mod7Counter`	`BraidBracket.lean`
Void mask LUT	`SLUG3.landauerCostBits` (placeholder)	`SLUG3.lean`
XOR fold	`UnitQuaternion.conjugate` (partial)	`QuaternionGenomic.lean`

The Resource Collapse

The equation achieves zero DSP blocks by replacing all arithmetic with LUT lookups:

Operation	Traditional	LUT-as-DSP
φ^(i mod 7)	`pow(PHI, i % 7)` floating-point	`phi_acc + 106070`
Multiplication	`*` operator	LUT lookup
Entropy	`os.urandom()` / PRNG	Void mask (baked)

Ancestral Connection

ENE-BIND-001 (cost: UInt32 Q16.16)
    ↓ hardware extraction
LUT-DSP-003 (cost: LUT lookup cycles)
    ↓ cross-domain unification
UNIVERSAL-003c (hybrid hash function)
    ↓ DNA encoding
QUATERNION-SLUG-005 (ternary gate)

The LUT-DSP-003 equation is the hardware anchor — it proves the cost function can be evaluated without floating-point, making all upstream equations hardware-extractable.

Open Hardware Questions

What is the actual void-and-cluster algorithm? (Trade secret vs. open specification)
Can the 91-step period (13×7) be extended for higher security?
Dual-port A/B swap: race-free by construction, but what about power-glide attacks?

Part of the Equation Ancestry Project

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