Research-Stack/6-Documentation/docs/semantics/MIRROR_LUT_EQUATIONS.md

Mirror LUT Equations: Cross-Domain Unification

1. Core Mirror LUT Architecture

The Mirror LUT appears across three domains with isomorphic structure:

Domain A: FPGA Warden Node (Hardware)

┌─────────────────────────────────────────────────────────┐
│  φ-Accumulator → MSB Flip Detection → XOR Fold → LUT  │
└─────────────────────────────────────────────────────────┘

\text{idx}{\text{mirror}} = \Big(\Phi{\text{acc}} \gg n\Big) \oplus \underbrace{\Big(\text{MSB}(\Phi_{\text{acc}}^{(t)}) \neq \text{MSB}(\Phi_{\text{acc}}^{(t-1)})\Big)}_{\text{flip detector}}

Domain B: Procedural Mirror (Software)

┌─────────────────────────────────────────────────────────┐
│  Basin Detection → φ-Ratio Mapping → Polynomial Seed  │
└─────────────────────────────────────────────────────────┘

\text{seed}_{64} = \Big\lfloor \phi^{\text{entropy}(x) \cdot \text{spectral_density}(x)} \cdot 2^{64} \Big\rfloor ;\text{mod}; 2^{64}

Domain C: Field Solver (Simulation)

┌─────────────────────────────────────────────────────────┐
│  Position Hash → SHA256 → MirrorLUT Query → Gradient    │
└─────────────────────────────────────────────────────────┘

\text{LUT}_{\text{mirror}}(p) = \text{SHA256}\big(\text{engram_key} | p\big) \gg 4

---

2. Unified Mirror LUT Equation

The cross-domain invariant:

\boxed{ \mathcal{M}(q, s, t) = \mathcal{H}\Big( \mathcal{F}(q) \oplus \mathcal{T}(t) \Big) ;\text{mod}; 2^n }

Where:

• Query q: Position/address/state being looked up
• State s: Current system state/entropy/accumulator value
• Time t: Temporal index/counter/step
• Hash \mathcal{H}: Deterministic mixing function
• Fold \mathcal{F}: Mirror/quadrant/folding operation
• Transform \mathcal{T}: Time-dependent phase shift

---

3. Domain-Specific Instantiations

3.1 Hardware (FPGA Warden)

Component	Equation	Role
Query	q = \Phi_{\text{acc}} \gg 16	Accumulator high bits
State	s = \Phi_{\text{acc}} \;\text{mod}\; 2^{32}	Full 32-bit accumulator
Time	t = c_{\text{mod7}} \in \{0..6\}	Mod-7 prime counter
Hash	\mathcal{H}_{\text{hw}}(x) = \text{LUT}_{\text{void}}[x]	Void mask lookup
Fold	\mathcal{F}_{\text{hw}}(q) = q \oplus \text{MSB}_{\text{flip}}	XOR fold on MSB change
Transform	\mathcal{T}_{\text{hw}}(t) = t \cdot \phi \;\text{mod}\; 2^{16}	φ-scaled time offset

\text{Mirror}{\text{FPGA}}(q, s, t) = \text{LUT}{\text{void}}\Big[ q \oplus \text{MSB}_{\text{flip}}(s) \Big]

Period: 91 steps (13 × 7, coprime traversal)

3.2 Software (Procedural Mirror)

Component	Equation	Role
Query	q = \text{basin}(x) \in \{0..8\}	Basin classification
State	s = \text{entropy}(x) \in [0, 8]	Shannon entropy estimate
Time	t = \text{spectral_density}(x) \in \mathbb{R}^+	Spectral concentration
Hash	\mathcal{H}_{\text{sw}}(x) = \lfloor \phi^x \cdot 2^{64} \rfloor	φ-powered hash
Fold	\mathcal{F}_{\text{sw}}(q) = q \cdot \phi^{-1} \;\text{mod}\; 1	Fractional part extraction
Transform	\mathcal{T}_{\text{sw}}(t) = \log_2(t + 1)	Log-scaled density

\text{Mirror}_{\text{SW}}(q, s, t) = \Big\lfloor \phi^{s \cdot t} \cdot 2^{64} \Big\rfloor ;\text{mod}; 2^{64}

Period: Deterministic per input (no counter, pure functional)

3.3 Simulation (Field Solver)

Component	Equation	Role
Query	q = \text{position} \in \mathbb{Z}^3	Spatial coordinates
State	s = \text{engram_key} \in \{0..2^{32}-1\}	Instance identifier
Time	t = \text{step} \in \mathbb{N}	Simulation tick
Hash	\mathcal{H}_{\text{sim}}(x) = \text{SHA256}(x) \gg 4	Cryptographic hash
Fold	\mathcal{F}_{\text{sim}}(q) = (q_x \oplus q_y \oplus q_z)	Coordinate XOR fold
Transform	\mathcal{T}_{\text{sim}}(t) = t \cdot \text{stride}	Linear time offset

\text{Mirror}{\text{SIM}}(q, s, t) = \text{SHA256}\big( s | \mathcal{F}{\text{sim}}(q) | \mathcal{T}_{\text{sim}}(t) \big) \gg 4

Period: 2^{256} (cryptographic, effectively infinite)

---

4. The Universal Mirror Equation (Best Across All Fields)

Combine the strengths of each domain:

\boxed{ \mathcal{M}^*(q, s, t) = \Big\lfloor \phi^{\alpha \cdot \mathcal{H}(q, s) + \beta \cdot \mathcal{T}(t)} \cdot 2^n \Big\rfloor \oplus \mathcal{F}(q) ;\text{mod}; 2^n }

Optimal Parameters

Parameter	Hardware	Software	Simulation	Universal
\alpha	1.0	0.5	0.3	0.7
\beta	0.2	0.0	0.1	0.15
n	13	64	32	16
\mathcal{H}	LUT	φ-pow	SHA256	Hybrid
\mathcal{F}	MSB flip	frac part	XOR coord	XOR fold
\mathcal{T}	mod-7	log2	linear	φ-step

Hybrid Hash Function \mathcal{H}^*

\mathcal{H}^*(q, s) = \text{fold}{64}\Big( \text{SHA256}(s | q) \oplus \text{LUT}{\phi}[q ;\text{mod}; 8192] \Big)

Where:

• \text{fold}_{64}: XOR high 64 bits with low 64 bits
• \text{LUT}_{\phi}: Precomputed φ-scaled lookup table (8K entries)
• Combines cryptographic collision resistance with φ-uniformity

---

5. Mirror LUT Temporal Evolution

Across all domains, the mirror state evolves as:

s_{t+1} = \mathcal{M}(q_t, s_t, t) \oplus \Big( s_t \gg r \Big)

Where r is a right-shift decay constant:

• Hardware: r = 0 (no state decay, deterministic)
• Software: r = 8 (byte-wise decay for streaming)
• Simulation: r = 4 (nibble-wise for smooth fields)
• Universal: r = 6 (optimal for mixed workloads)

---

6. Complete Unified Pipeline

Input: (q, s₀, T_max)
Output: sequence of mirror values

for t = 0 to T_max:
    # 1. Temporal phase
    τ = φ · t  (mod 2^n)
    
    # 2. State mixing
    h = SHA256(s_t || q) ⊕ LUT_φ[q mod 8192]
    h = fold_64(h)
    
    # 3. Mirror fold
    f = (h >> (n-1)) XOR (h >> (n-2))  # MSB bits
    idx = (h + τ) mod 2^n
    
    # 4. LUT lookup
    m_t = LUT_mirror[idx] XOR f
    
    # 5. State update
    s_{t+1} = m_t XOR (s_t >> 6)
    
    yield m_t

---

7. Invariant Properties

All Mirror LUT instantiations satisfy:

1. Determinism: \mathcal{M}(q, s, t) is pure function (no side effects)
2. Low discrepancy: Address sequence has O(\log N) star discrepancy
3. Uniform coverage: \lim_{T \to \infty} \frac{1}{T} \sum_{t=0}^{T-1} \mathcal{M}(q, s, t) = 2^{n-1}
4. Coprime traversal: Period = \text{lcm}(p_1, p_2, ..., p_k) for prime factors p_i

---

8. Cross-Reference Map

Concept	Hardware	Software	Simulation	Universal
Mirror	MSB flip	frac part	XOR coord	Hybrid fold
LUT	Void mask	Polynomial	SHA256	Hybrid hash
Accumulator	φ-step	φ-pow	Linear	φ-step
Period	91	Input-dep	2^{256}	65521 (prime)
State	Dual A/B	Pure func	Persistent	Decay chain

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Document ID: MIRROR_LUT_CROSS_DOMAIN
Cross-ref: FPGA_WARDEN_NODE_SPEC.md, procedural_mirror.py, field_solver_emulator.py