From a97e39a239f8fd408d96c307df026f2caa8f8e20 Mon Sep 17 00:00:00 2001 From: allaun Date: Thu, 18 Jun 2026 22:28:50 -0500 Subject: [PATCH] docs(rrc): add topological braid adapter spec for fibonacci anyons 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) --- .../specs/TOPOLOGICAL_BRAID_ADAPTER_SPEC.md | 138 ++++++++++++++++++ 1 file changed, 138 insertions(+) create mode 100644 6-Documentation/docs/specs/TOPOLOGICAL_BRAID_ADAPTER_SPEC.md diff --git a/6-Documentation/docs/specs/TOPOLOGICAL_BRAID_ADAPTER_SPEC.md b/6-Documentation/docs/specs/TOPOLOGICAL_BRAID_ADAPTER_SPEC.md new file mode 100644 index 00000000..375d5d57 --- /dev/null +++ b/6-Documentation/docs/specs/TOPOLOGICAL_BRAID_ADAPTER_SPEC.md @@ -0,0 +1,138 @@ +# Topological Braid Adapter Specification: Fibonacci Anyon Isomorphism + +**Status:** PROPOSED +**Applies to:** `Semantics.HydrogenicPhiTorsionBraid`, `Semantics.SLUG3`, `Semantics.UnitQuaternion`, `Semantics.GoldenRatioSeparation` +**Reference Papers:** +1. **Zhang et al. (2406.08320v2):** "Quantum Gates on Symmetric Tetrahedron Geometry" (Dual Quaternions / X-type Braid Gates). +2. **Rouabah (2008.03542v1):** "Hadamard Approximation and Braid Word Representation". +3. **Hadjiivanov & Georgiev (2404.01778v4):** "Fibonacci Anyon Braid Matrices and n-Strand Recursions". +4. **Gu et al. (2112.07195v2):** "KZ Singularities, Routing Protocols, and Topological Protection". + +--- + +## 1. Overview & Mathematical Isomorphism + +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**. + +The adapter maps the geometric and pressure-based constructs of the compiler to the algebraic and topological coordinates of anyon fusion and braiding. + +```mermaid +graph TD + subgraph Compiler State + CR[ColorRope C,M,Y,K] + SI[stairIndex Sequence] + TC[tensegrityCoherent] + PP[phi_pow recurrence] + ZB[Zeckendorf Bits] + end + + subgraph Fibonacci Anyon System + DQ[Dual Quaternions Q1, Q2] + SW[B_3 Braid Word / SLUG3State] + YB[Yang-Baxter Consistency] + BM[n-Strand Braid Matrices] + FT[Fusion Tree Basis Vectors] + end + + CR -->|Isomorphism 1| DQ + SI -->|Isomorphism 2| SW + TC -->|Isomorphism 3| YB + PP -->|Isomorphism 4| BM + ZB -->|Isomorphism 5| FT +``` + +--- + +## 2. The Five Key Isomorphisms + +### Isomorphism 1: `colorRope` to `DualQuaternion` (Zhang et al. 2406.08320v2) + +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. + +The `ColorRope` channels \((C, M, Y, K)\) map to the coordinates of the dual quaternions \((Q_1, Q_2)\) as follows: +- **\(Q_1\) (Real/Rotational Quaternions):** Encodes the spatial constraint and evidence mass. + \[ + Q_1 = \left( \cos\left(\frac{\theta_C}{2}\right), 0, 0, M \cdot \sin\left(\frac{\theta_C}{2}\right) \right) + \] + where \(\theta_C\) is the angle derived from the constraint channel \(C\). +- **\(Q_2\) (Dual/Translational Quaternions):** Encodes the residual risk (fray) and active/admissible movement. + \[ + Q_2 = \left( K, Y, 0, 0 \right) + \] +- **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. + +### Isomorphism 2: `stairIndex` to `SLUG3State` B_3 Word (Rouabah 2008.03542v1) + +We map the sequence of crossing events tracked by `stairIndex` to generator words in the braid group \(B_3\). +- Each step \(i\) in the sequence is classified into a generator \(\sigma_i\) or its inverse \(\sigma_i^{-1}\): + - If the step transition has a positive phase velocity: \(\sigma_1\) (braiding strand 1 & 2). + - If the step transition has a negative phase velocity: \(\sigma_2\) (braiding strand 2 & 3). + - If no crossing occurs: \(e\) (identity). +- These map directly to `SLUG3State` where \((y, u, v) \in \{-1, 0, 1\}^3\) represents the three ternary states. +- 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. + +### Isomorphism 3: `tensegrityCoherent` to Yang-Baxter Equation + +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**: +\[ +\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} +\] +- 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. +- 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. + +### Isomorphism 4: `phi_pow` to n-Strand Braid Matrices (Hadjiivanov & Georgiev 2404.01778v4) + +The quantum dimension of the Fibonacci anyon \(\tau\) is the golden ratio \(\varphi \approx 1.618034\), satisfying: +\[ +d_\tau^2 = d_\tau + 1 +\] +The recurrence relation in `phi_pow n` computes the Fibonacci coefficients \((a_n, b_n)\) representing: +\[ +\varphi^n = a_n \varphi + b_n +\] +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. + +### Isomorphism 5: Zeckendorf Bits to Fusion Tree Basis Vectors + +The Fibonacci anyon fusion rules are: +\[ +\tau \otimes \tau = 1 \oplus \tau +\] +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. +- This constraint maps exactly to the **Zeckendorf representation** (no adjacent 1s). +- A valid phinary digit sequence (e.g. `1010010`) represents a physically allowed path through the anyon fusion tree. +- 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 \(F_n\). + +--- + +## 3. Implementation Steps for the Bridge API + +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`: + +```lean +import Semantics.HydrogenicPhiTorsionBraid +import Semantics.SLUG3 +import Semantics.UnitQuaternion + +namespace Semantics.TopologicalBraidAdapter + +open Semantics.HydrogenicPhiTorsionBraid +open Semantics.SLUG3 +open Semantics.UnitQuaternion + +/-- Maps a ColorRope to a pair of UnitQuaternions representing the Zhang et al. coordinates. -/ +def ropeToDualQuaternions (r : ColorRope) : UnitQuaternion × UnitQuaternion := + -- Implementation maps C, M to Q1, and Y, K to Q2 + sorry + +/-- Maps a sequence of stair indices to a B3 braid word (List SLUG3State). -/ +def stairIndexToBraidWord (indices : List Nat) : List SLUG3State := + sorry + +/-- Theorem proving that if tensegrity is coherent, the braid word satisfies the Yang-Baxter relation. -/ +theorem coherent_implies_yang_baxter (p : HardProblemState) (s : BraidSample) : + tensegrityCoherent p s = true → BraidWord.satisfiesYangBaxter (stairIndexToBraidWord [s.stairIndex]) := + sorry + +end Semantics.TopologicalBraidAdapter +```