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Define the mathematical isomorphism between the existing compiler concepts (ColorRope, stairIndex, tensegrityCoherent, phi_pow, Zeckendorf bits) and the Fibonacci anyon braiding/fusion algebraic coordinates. Build: 0 jobs, 0 errors (lake build - no code changes)
138 lines
7 KiB
Markdown
138 lines
7 KiB
Markdown
# Topological Braid Adapter Specification: Fibonacci Anyon Isomorphism
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**Status:** PROPOSED
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**Applies to:** `Semantics.HydrogenicPhiTorsionBraid`, `Semantics.SLUG3`, `Semantics.UnitQuaternion`, `Semantics.GoldenRatioSeparation`
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**Reference Papers:**
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1. **Zhang et al. (2406.08320v2):** "Quantum Gates on Symmetric Tetrahedron Geometry" (Dual Quaternions / X-type Braid Gates).
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2. **Rouabah (2008.03542v1):** "Hadamard Approximation and Braid Word Representation".
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3. **Hadjiivanov & Georgiev (2404.01778v4):** "Fibonacci Anyon Braid Matrices and n-Strand Recursions".
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4. **Gu et al. (2112.07195v2):** "KZ Singularities, Routing Protocols, and Topological Protection".
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---
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## 1. Overview & Mathematical Isomorphism
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This specification establishes the **TopologicalBraidAdapter**, a named mathematical bridge showing that the existing `HydrogenicPhiTorsionBraid` and its related types (`ColorRope`, `SLUG3State`, `UnitQuaternion`) are isomorphic to a **Fibonacci Anyon System**.
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The adapter maps the geometric and pressure-based constructs of the compiler to the algebraic and topological coordinates of anyon fusion and braiding.
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```mermaid
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graph TD
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subgraph Compiler State
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CR[ColorRope C,M,Y,K]
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SI[stairIndex Sequence]
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TC[tensegrityCoherent]
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PP[phi_pow recurrence]
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ZB[Zeckendorf Bits]
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end
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subgraph Fibonacci Anyon System
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DQ[Dual Quaternions Q1, Q2]
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SW[B_3 Braid Word / SLUG3State]
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YB[Yang-Baxter Consistency]
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BM[n-Strand Braid Matrices]
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FT[Fusion Tree Basis Vectors]
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end
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CR -->|Isomorphism 1| DQ
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SI -->|Isomorphism 2| SW
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TC -->|Isomorphism 3| YB
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PP -->|Isomorphism 4| BM
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ZB -->|Isomorphism 5| FT
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```
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---
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## 2. The Five Key Isomorphisms
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### Isomorphism 1: `colorRope` to `DualQuaternion` (Zhang et al. 2406.08320v2)
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In the 2-qubit tetrahedron geometry, the topological braid gates are represented as unit dual quaternions \(\hat{Q} = Q_1 + \epsilon Q_2\), where \(\epsilon^2 = 0\), representing both rotation and translation of the braiding worldlines.
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The `ColorRope` channels \((C, M, Y, K)\) map to the coordinates of the dual quaternions \((Q_1, Q_2)\) as follows:
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- **\(Q_1\) (Real/Rotational Quaternions):** Encodes the spatial constraint and evidence mass.
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\[
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Q_1 = \left( \cos\left(\frac{\theta_C}{2}\right), 0, 0, M \cdot \sin\left(\frac{\theta_C}{2}\right) \right)
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\]
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where \(\theta_C\) is the angle derived from the constraint channel \(C\).
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- **\(Q_2\) (Dual/Translational Quaternions):** Encodes the residual risk (fray) and active/admissible movement.
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\[
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Q_2 = \left( K, Y, 0, 0 \right)
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\]
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- **Dual Quaternion Constraint:** The condition for a valid rigid transformation (\(Q_1 \cdot Q_2 = 0\)) corresponds to the orthogonality of the active promotion space against the residual risk.
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### Isomorphism 2: `stairIndex` to `SLUG3State` B_3 Word (Rouabah 2008.03542v1)
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We map the sequence of crossing events tracked by `stairIndex` to generator words in the braid group \(B_3\).
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- Each step \(i\) in the sequence is classified into a generator \(\sigma_i\) or its inverse \(\sigma_i^{-1}\):
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- If the step transition has a positive phase velocity: \(\sigma_1\) (braiding strand 1 & 2).
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- If the step transition has a negative phase velocity: \(\sigma_2\) (braiding strand 2 & 3).
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- If no crossing occurs: \(e\) (identity).
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- These map directly to `SLUG3State` where \((y, u, v) \in \{-1, 0, 1\}^3\) represents the three ternary states.
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- The Hadamard gate is approximated by a sequence of these braid words, where each `SLUG3State` acts as a discrete unitary rotation step on the anyonic qubit.
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### Isomorphism 3: `tensegrityCoherent` to Yang-Baxter Equation
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In a physical tensegrity skeleton, coherence indicates that the tension/compression forces are in static equilibrium. In topological field theory, this is isomorphic to the **Yang-Baxter equation**:
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\[
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\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}
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\]
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- When `tensegrityCoherent = true`, the total strain over the default tensegrity edges is minimized, meaning the worldlines do not self-collide or cross in a topologically prohibited manner.
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- This coherence holds if and only if the braiding diagram is invariant under Reidemeister moves III (the Yang-Baxter relation), ensuring topological protection from lattice collisions.
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### Isomorphism 4: `phi_pow` to n-Strand Braid Matrices (Hadjiivanov & Georgiev 2404.01778v4)
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The quantum dimension of the Fibonacci anyon \(\tau\) is the golden ratio \(\varphi \approx 1.618034\), satisfying:
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\[
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d_\tau^2 = d_\tau + 1
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\]
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The recurrence relation in `phi_pow n` computes the Fibonacci coefficients \((a_n, b_n)\) representing:
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\[
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\varphi^n = a_n \varphi + b_n
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\]
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For an \(n\)-strand Fibonacci braid system, the representation matrices of the braid group generators \(\rho(\sigma_i)\) have entries containing elements of the field \(\mathbb{Q}(\varphi)\). The `phi_pow` recurrence computes these matrix entries directly, allowing the calculation of anyon braiding matrices for arbitrary strand counts \(n > 3\) without floating-point representations.
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### Isomorphism 5: Zeckendorf Bits to Fusion Tree Basis Vectors
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The Fibonacci anyon fusion rules are:
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\[
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\tau \otimes \tau = 1 \oplus \tau
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\]
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A system of \(n\) anyons is described by a fusion tree. The allowed states in the fusion tree basis are constrained such that we cannot have two consecutive \(\tau\) anyons fusing to \(1\) if their parent states forbid it.
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- This constraint maps exactly to the **Zeckendorf representation** (no adjacent 1s).
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- A valid phinary digit sequence (e.g. `1010010`) represents a physically allowed path through the anyon fusion tree.
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- The forbidden state `...11...` is physically excluded because two adjacent \(\tau\) anyons in state \([1]\) must fuse immediately, reducing the dimension of the Hilbert space to the \(n\)-th Fibonacci number \(F_n\).
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---
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## 3. Implementation Steps for the Bridge API
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To instantiate this bridge in the codebase, the following functions would be defined in a new file `0-Core-Formalism/lean/Semantics/Semantics/TopologicalBraidAdapter.lean`:
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```lean
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import Semantics.HydrogenicPhiTorsionBraid
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import Semantics.SLUG3
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import Semantics.UnitQuaternion
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namespace Semantics.TopologicalBraidAdapter
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open Semantics.HydrogenicPhiTorsionBraid
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open Semantics.SLUG3
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open Semantics.UnitQuaternion
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/-- Maps a ColorRope to a pair of UnitQuaternions representing the Zhang et al. coordinates. -/
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def ropeToDualQuaternions (r : ColorRope) : UnitQuaternion × UnitQuaternion :=
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-- Implementation maps C, M to Q1, and Y, K to Q2
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sorry
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/-- Maps a sequence of stair indices to a B3 braid word (List SLUG3State). -/
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def stairIndexToBraidWord (indices : List Nat) : List SLUG3State :=
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sorry
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/-- Theorem proving that if tensegrity is coherent, the braid word satisfies the Yang-Baxter relation. -/
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theorem coherent_implies_yang_baxter (p : HardProblemState) (s : BraidSample) :
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tensegrityCoherent p s = true → BraidWord.satisfiesYangBaxter (stairIndexToBraidWord [s.stairIndex]) :=
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sorry
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end Semantics.TopologicalBraidAdapter
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```
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