Paper Stub: The Mirror LUT — Cross-Domain Unification

Equation ID: MIRROR-LUT-004
Status: Theoretical framework (3 domains unified)
Lineage: Abstraction layer above LUT-DSP-003
Date: 2026-04-19
Ancestry: Discovers isomorphism between hardware, software, and simulation domains

The Equation

𝒰(q, s, t) = ℋ(ℱ(q) ⊕ 𝒯(t)) mod 2^n

where:
  q = Query (position/address/state)
  s = State (entropy/accumulator)
  t = Time (counter/step)
  ℋ = Hash function (deterministic mixing)
  ℱ = Fold operation (mirror/quadrant)
  𝒯 = Transform (phase shift)

Domain Instantiations

Domain	Query	State	Time	Hash	Fold	Transform
Hardware	Φ_acc ≫ 16	Φ_acc mod 2^32	c_mod7 ∈ {0..6}	LUT_void	MSB_flip	φ·t mod 2^16
Software	basin(x) ∈ {0..8}	entropy(x) ∈ [0,8]	spectral_density(x)	φ^pow	frac part	log₂(t+1)
Simulation	position ∈ ℤ³	engram_key ∈ 2^32	simulation tick	SHA256	XOR coords	linear stride

Mutation: The Universal Form

𝒰*(q, s, t) = ⌊φ^(α·ℋ(q,s) + β·𝒯(t)) · 2^n⌋ ⊕ ℱ(q) mod 2^n

Optimal parameters:
  α = 0.7 (hybrid weight)
  β = 0.15 (temporal weight)
  n = 16 (hardware-native)

Half-Thought: The Hybrid Hash

"ℋ*(q,s) = fold_64(SHA256(s ‖ q) ⊕ LUT_φ[q mod 8192])"

This "hybrid hash" concept is documented but never implemented:

Combines cryptographic collision resistance with φ-uniformity
Requires 8K-entry LUT_φ (precomputed, but content undefined)
The fold_64 operation is described but no specification given

The gap: The universal form depends on LUT_φ, but the generation of this table is unspecified. It's the software analog of the void mask hardware secret.

Ancestral Tree

LUT-DSP-003 (hardware-specific)
    ↓ abstraction
MIRROR-LUT-004 (cross-domain)
    ├─→ Hardware (MSB flip, 91-step)
    ├─→ Software (φ^entropy, input-dependent)
    └─→ Simulation (SHA256, 2^256 period)
    ↓ universal synthesis
UNIVERSAL-003c (hybrid parameters)
    ↓ DNA encoding bridge
QUATERNION-SLUG-005 (ternary via quaternion)

The Period Convergence Problem

Each domain has different natural period:

Domain	Period	Source
Hardware	91 = 13×7	Coprime traversal
Software	Input-dependent	Deterministic per input
Simulation	2^256	Cryptographic
Universal	65521 (prime)	Proposed compromise

The universal form suggests period 65521 (a Mersenne-like prime), but there's no derivation for why this specific number.

Unfinished: Cross-Domain Verification

Can a computation verified in one domain be trusted in another?

Hardware → Software: Timing attacks on φ-accumulator
Software → Simulation: Floating-point vs fixed-point divergence
Simulation → Hardware: Cryptographic security vs physical probing

The isomorphism is structural, not operational.

Part of the Equation Ancestry Project

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