From b4c73e596573448b0fc3cd43000a7309a1be8c1c Mon Sep 17 00:00:00 2001 From: Allaun Silverfox <28494262+allaunthefox@users.noreply.github.com> Date: Thu, 2 Jul 2026 03:29:51 +0200 Subject: [PATCH] Remove --- docs/REDERIVATION.md | 289 ------------------------------------------- 1 file changed, 289 deletions(-) delete mode 100644 docs/REDERIVATION.md diff --git a/docs/REDERIVATION.md b/docs/REDERIVATION.md deleted file mode 100644 index 6c41ec05..00000000 --- a/docs/REDERIVATION.md +++ /dev/null @@ -1,289 +0,0 @@ -# Rederivation: Research Stack from DNA First Principles - -**Starting point:** The Hachimoji DNA encoding, monotone LUT, braid sort, -and 8×8 surface. Nothing else. Everything else is derived. - ---- - -## 1. The Imaginary Axis (from DNA) - -A DNA sequence is a string over 8 bases: `{A, B, C, G, P, S, T, Z}`. -Each base is a digit 0–7. A sequence of length k is a point in ℤ/8^kℤ. - -This is a **discrete imaginary axis**. It carries no physical units. -It is pure information: a coordinate in a symbolic space. - -**Definition.** The *semantic coordinate* of a QUBO solution is its rank -in the monotone LUT. Rank 0 = optimal. Rank n-1 = worst. - -``` -S : Solution → ℕ -S(x) = position of x in energy-sorted order -``` - -The semantic coordinate is observer-independent. It depends only on the -energy function and the sort. No physical units. No conversion factor. -Pure information count. - -**Theorem (Monotonicity).** For any two solutions x₁, x₂: -``` -S(x₁) < S(x₂) ⟺ E(x₁) ≤ E(x₂) -``` -*Proof.* By construction of the monotone LUT. ∎ - -This is the imaginary axis. The DNA sequence encodes it. -The LUT maps it. The sort preserves it. - ---- - -## 2. The Real Axis (from Energy) - -The QUBO energy E(x) = x^T Q x is a scalar. It has no units — it's a -pure number. But it acts like a physical quantity: it determines which -solutions are "heavier" (higher energy) and which are "lighter" (lower). - -**Definition.** The *physical projection* of a solution is its energy: -``` -P : Solution → ℝ -P(x) = E(x) = x^T Q x -``` - -The physical projection IS the observer's measurement. The QUBO matrix Q -is the observer. Different Q matrices are different observers measuring -the same solution space through different lenses. - -**Theorem (Observer Dependence).** Two QUBO matrices Q₁, Q₂ produce -different energy orderings of the same solution set. -Different observers see different physical projections of the same -semantic coordinates. - -*Proof.* Let Q₁ = diag(1,2,3) and Q₂ = diag(3,2,1). -Solution [1,0,0] has E₁=1, E₂=3 under the two observers. -The semantic coordinate S([1,0,0]) is observer-independent, -but the physical projection P differs. ∎ - ---- - -## 3. Imaginary Semantic Time (from LUT Structure) - -The monotone LUT has two orderings: -- **Semantic ordering:** by sequence (lexicographic, observer-independent) -- **Physical ordering:** by energy (QUBO-dependent, observer-dependent) - -These two orderings are the real and imaginary axes of a complex plane: - -``` -T_semantic(x) = S(x) [imaginary axis: information count] -T_physical(x) = E(x) [real axis: observer measurement] -``` - -The full state of a solution is: -``` -T(x) = (E(x), S(x)) = physical + i·semantic -``` - -**Theorem (Semantic Period Ratio).** For a banded QUBO with n variables, -the ratio of consecutive semantic periods is exactly: -``` -T_semantic(k+1) / T_semantic(k) = 8 -``` -where 8 is the base count (the number of Hachimoji states). - -*Proof.* Each additional base multiplies the address space by 8. -The semantic coordinate space grows as 8^k. The ratio of consecutive -levels is 8^k / 8^(k-1) = 8. ∎ - -This is the DNA analog of the Research Stack's period ratio = 3. -The base count (8) plays the role of the Menger factor (3). - ---- - -## 4. Sieve Observers (from Base Encoding) - -Each DNA base is a digit mod 8. A sequence of length k is a point -mod 8^k. The encoding is a **sieve** with modulus 8. - -**Definition.** A *sieve observer* is an information-processing system -with a native modulus ℓ. It sees: -``` -observation(x) = S(x) mod ℓ -``` - -The DNA encoder is a sieve observer with ℓ = 8. -It sees the residue class of the semantic coordinate mod 8. - -**Theorem (No Privileged Sieve).** No modulus ℓ is "correct." -Different moduli reveal different aspects of the same coordinate. -ℓ = 8 (DNA bases) is one choice. ℓ = 2 (binary) is another. -ℓ = 16 (hex) is another. All are valid projections. - -*Proof.* The semantic coordinate S(x) is the ground truth. -Any modulus ℓ ≥ 1 produces a valid residue class. -No ℓ is privileged because the coordinate exists independently -of any particular representation. ∎ - -**Theorem (CRT Reconciliation).** Two sieve observers with coprime -moduli ℓ₁, ℓ₂ can reconstruct the coordinate mod ℓ₁·ℓ₂. - -*Proof.* By the Chinese Remainder Theorem. -If ℓ₁ ⊥ ℓ₂, then the pair (S(x) mod ℓ₁, S(x) mod ℓ₂) uniquely -determines S(x) mod ℓ₁·ℓ₂. ∎ - -**Application to DNA.** A DNA sequence of length 7 encodes up to 8^7 -= 2,097,152 unique coordinates. Two observers, one reading the first -3 bases (mod 8³ = 512) and one reading the last 4 bases (mod 8⁴ = 4096), -can reconcile via CRT to recover the full 7-base coordinate -(mod 512·4096 = 2,097,152). This is exactly 8^7. ∎ - ---- - -## 5. Semantic Mass (from Energy Landscape) - -The QUBO energy determines a "mass" for each solution. - -**Definition.** The *semantic mass* of a solution x is: -``` -m(x) = E(x) - E_min -``` -where E_min is the optimal energy. Mass is zero at the optimum -and increases as solutions become worse. - -**Theorem (Mass is Non-Negative).** For all x: m(x) ≥ 0. -*Proof.* E(x) ≥ E_min by definition of minimum. ∎ - -**Theorem (Mass Controls Inertia).** Solutions with higher semantic mass -are harder to reach from the optimal state. The "distance" from the -optimal to solution x is proportional to m(x). - -*Proof.* In the monotone LUT, the semantic coordinate S(x) is the rank. -The mass m(x) = E(x) - E_min increases monotonically with S(x) -(because the LUT is sorted by energy). Therefore, higher mass = -higher rank = farther from optimal. ∎ - -**Definition.** The *semantic energy* at a solution is: -``` -E_s(x) = m(x) · c_s² -``` -where c_s is the "semantic coherence speed" — the maximum rate at -which meaning can propagate through the manifold without losing -coherence. For the DNA encoding, c_s = 8 (the base count). - -This is the DNA analog of E = mc². The "speed of light" in -semantic space is the base count. - ---- - -## 6. Braid Sort (from DNA Operations) - -The braid sort kernel operates on DNA sequences via compare-swap -operations. Each operation is a **braid crossing**. - -**Definition.** A *braid crossing* is: -``` -cross(a, b) = (min(a,b), max(a,b)) -``` -If a > b, the crossing swaps them. If a ≤ b, no change. - -**Definition.** The *eigensolid* is the state where no crossings -remain — the array is sorted. - -**Theorem (Convergence).** After at most n-1 passes of odd-even -transposition sort, any array of n elements reaches the eigensolid. - -*Proof.* Odd-even transposition sort is a comparison sort that -performs exactly ⌈n/2⌉ compare-swap operations per pass. -After n-1 passes, the array is sorted. This is a standard result. ∎ - -**Theorem (FAMM Gate).** The braid crossing is the minimal operation -that preserves the eigensolid invariant: after each crossing, -the array is "more sorted" (the number of inversions is non-increasing). - -*Proof.* A crossing at positions (i, i+1) either: -1. Fixes an inversion (a[i] > a[i+1]) → inversions decrease by 1 -2. Leaves a non-inversion (a[i] ≤ a[i+1]) → inversions unchanged -In neither case do inversions increase. ∎ - -This is the FAMM gate from BraidTreeDIATPIST: it filters inadmissible -configurations (inversions) at each step. - ---- - -## 7. The 8×8 Surface (from Manifold Projection) - -The QUBO solution is rendered as an 8×8 pixel grid. Each pixel -represents one variable. The color encodes the variable's value. - -**Definition.** The *eigenvalue fingerprint* of a QUBO solution is -the 8×8 surface where: -``` -pixel(row, col) = HachimojiColor(x[row·8 + col]) -``` - -**Theorem (Fingerprint Uniqueness).** Two different solutions produce -different surfaces (when n_vars ≤ 64). - -*Proof.* Each pixel encodes one variable. If two solutions differ -in any variable, the corresponding pixel differs. ∎ - -**Theorem (Mass Visualization).** The number of green pixels (x=1) -is proportional to the number of active variables. For diagonal -QUBOs with positive entries, more green = higher energy = higher mass. - -*Proof.* For Q = diag(q₁, ..., qₙ) with qᵢ > 0: -E(x) = Σ qᵢ·xᵢ. Each xᵢ=1 adds qᵢ to the energy. -More 1s = more energy = more mass. The surface renders this directly. ∎ - ---- - -## 8. The Unified Picture - -Everything derives from three primitives: - -1. **DNA sequence** (symbolic coordinate, imaginary axis) -2. **QUBO energy** (physical projection, real axis) -3. **Monotone LUT** (the bridge between them) - -From these, we derive: - -| Concept | DNA Origin | -|---|---| -| Imaginary Semantic Time | Sequence rank vs energy | -| Sieve Observers | Base encoding mod 8 | -| CRT Reconciliation | Multi-base sequence decomposition | -| Semantic Mass | Energy minus optimum | -| Semantic Energy | E_s = m · 8² | -| Braid Sort | Compare-swap on base-8 digits | -| FAMM Gate | Inversion filtering | -| Eigensolid | Sorted array (convergence) | -| 8×8 Surface | Variable-to-pixel mapping | - -The Research Stack formalized these concepts in Lean 4. -SilverSight built the engine in Python. -The DNA language is the interface between them. - -The imaginary axis is the sequence. -The real axis is the energy. -The LUT is the bridge. -The braid is the algorithm. -The surface is the answer. - ---- - -## 9. What This Enables - -With these derivations, we can now: - -1. **Encode any optimization problem** as DNA sequences -2. **Sort on GPU** via braid crossings (triangle math) -3. **Render solutions** as 8×8 surfaces (eigenvalue fingerprints) -4. **Reconcile observers** via CRT (multi-resolution analysis) -5. **Measure semantic mass** (energy landscape navigation) -6. **Prove convergence** (eigensolid = sorted = solved) - -The Millennium Prize problems remain unsolvable at scale. -But the encoding is correct. The math is sound. The engine works. -The freeze point is the boundary of what's computationally accessible. -Beyond it lies the unknown. - -The DNA language maps the known. The manifold extends into the unknown. -The LUT bridges them.