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95 lines
3 KiB
Markdown
95 lines
3 KiB
Markdown
# Paper Stub: The Mirror LUT — Cross-Domain Unification
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**Equation ID:** MIRROR-LUT-004
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**Status:** Theoretical framework (3 domains unified)
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**Lineage:** Abstraction layer above LUT-DSP-003
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**Date:** 2026-04-19
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**Ancestry:** Discovers isomorphism between hardware, software, and simulation domains
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---
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## The Equation
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```
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𝒰(q, s, t) = ℋ(ℱ(q) ⊕ 𝒯(t)) mod 2^n
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where:
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q = Query (position/address/state)
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s = State (entropy/accumulator)
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t = Time (counter/step)
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ℋ = Hash function (deterministic mixing)
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ℱ = Fold operation (mirror/quadrant)
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𝒯 = Transform (phase shift)
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```
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## Domain Instantiations
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| Domain | Query | State | Time | Hash | Fold | Transform |
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|--------|-------|-------|------|------|------|-----------|
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| **Hardware** | Φ_acc ≫ 16 | Φ_acc mod 2^32 | c_mod7 ∈ {0..6} | LUT_void | MSB_flip | φ·t mod 2^16 |
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| **Software** | basin(x) ∈ {0..8} | entropy(x) ∈ [0,8] | spectral_density(x) | φ^pow | frac part | log₂(t+1) |
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| **Simulation** | position ∈ ℤ³ | engram_key ∈ 2^32 | simulation tick | SHA256 | XOR coords | linear stride |
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## Mutation: The Universal Form
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```
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𝒰*(q, s, t) = ⌊φ^(α·ℋ(q,s) + β·𝒯(t)) · 2^n⌋ ⊕ ℱ(q) mod 2^n
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Optimal parameters:
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α = 0.7 (hybrid weight)
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β = 0.15 (temporal weight)
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n = 16 (hardware-native)
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```
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## Half-Thought: The Hybrid Hash
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> "ℋ*(q,s) = fold_64(SHA256(s ‖ q) ⊕ LUT_φ[q mod 8192])"
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This "hybrid hash" concept is documented but never implemented:
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- Combines cryptographic collision resistance with φ-uniformity
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- Requires 8K-entry LUT_φ (precomputed, but content undefined)
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- The fold_64 operation is described but no specification given
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**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.
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## Ancestral Tree
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```
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LUT-DSP-003 (hardware-specific)
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↓ abstraction
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MIRROR-LUT-004 (cross-domain)
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├─→ Hardware (MSB flip, 91-step)
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├─→ Software (φ^entropy, input-dependent)
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└─→ Simulation (SHA256, 2^256 period)
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↓ universal synthesis
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UNIVERSAL-003c (hybrid parameters)
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↓ DNA encoding bridge
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QUATERNION-SLUG-005 (ternary via quaternion)
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```
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## The Period Convergence Problem
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Each domain has different natural period:
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| Domain | Period | Source |
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|--------|--------|--------|
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| Hardware | 91 = 13×7 | Coprime traversal |
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| Software | Input-dependent | Deterministic per input |
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| Simulation | 2^256 | Cryptographic |
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| **Universal** | **65521 (prime)** | Proposed compromise |
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The universal form suggests period 65521 (a Mersenne-like prime), but there's no derivation for why this specific number.
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## Unfinished: Cross-Domain Verification
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Can a computation verified in one domain be trusted in another?
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- Hardware → Software: Timing attacks on φ-accumulator
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- Software → Simulation: Floating-point vs fixed-point divergence
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- Simulation → Hardware: Cryptographic security vs physical probing
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The isomorphism is structural, not operational.
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---
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*Part of the Equation Ancestry Project*
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