From e09543d0829576c3253fe10265f2ac673f6e4cae Mon Sep 17 00:00:00 2001 From: Allaun Silverfox <28494262+allaunthefox@users.noreply.github.com> Date: Thu, 2 Jul 2026 03:40:27 +0200 Subject: [PATCH] Remove hopf_portability_criterion.md --- docs/hopf_portability_criterion.md | 152 ----------------------------- 1 file changed, 152 deletions(-) delete mode 100644 docs/hopf_portability_criterion.md diff --git a/docs/hopf_portability_criterion.md b/docs/hopf_portability_criterion.md deleted file mode 100644 index 5a9b9353..00000000 --- a/docs/hopf_portability_criterion.md +++ /dev/null @@ -1,152 +0,0 @@ -# ⛔ RETRACTED — Hopf Portability Criterion - -**Retraction date:** June 30, 2026 -**Reason:** Adversarial review (4 agents) found Conditions D and F circular/ad-hoc. The framework is a post-hoc description of n=8, not a general criterion. Replaced by `docs/cartan_fingerprint.md`. -**Do not cite.** See `docs/cartan_fingerprint.md` §2 for the retraction record. - ---- - -# Hopf Portability Criterion — Classification Framework (ARCHIVED) - -**Original status:** Formalized June 30, 2026 -**Reference:** `formal/CoreFormalism/HopfFibration.lean`, `formal/CoreFormalism/BraidStateN.lean` -**Agents:** Physics, Optimization, Number Theory, Classification (4-agent synthesis) - -## 0. Encoding Pipeline - -``` -Problem → Bₙ(braid) → S⁷(Hopf) → Cartan×Sidon → σ,τ → D=1792 → ∆=17/1792 → ℤ₂₈ regimes -``` - -Three independent structure groups: -- **Strand group** Bₙ: the braid carrying Sidon labels -- **Fiber group** S³: the quaternionic fiber of S³→S⁷→S⁴ -- **Diffeomorphism group** Diff⁺(S⁶): the exotic sphere group ℤ₂₈ = Θ₇ - -## I. Necessary and Sufficient Conditions - -A problem P is **Hopf-portable** iff it satisfies ALL six conditions: - -### Condition A: Strand Decomposition -P factorizes into n independent, pairwise-interacting channels. -- Each channel is Sidon-labelable (pairwise sums unique) -- Yang-Baxter relation holds on channel crossings -- The crossing loop converges (eigensolid exists) - -### Condition B: Cartan Spectrum -The channel interaction matrix M has spectral radius σ = a/2ⁿ. -- a ∈ ℕ, 0 < a < 2ⁿ -- For n=8: σ = 39/256 - -### Condition C: Sidon Threshold -τ = 1/(n−1) where n−1 is the number of independent scale doublings. -- For n=8: τ = 1/7 - -### Condition D: Spectral Gap -∆ = σ − τ > 0, expressible as p/D where D = lcm(2ⁿ, n−1). -- For n=8: D = lcm(256,7) = 1792, p = 17, ∆ = 17/1792 - -### Condition E: Hopf Fibration Fit -n = 2f+2 where f ∈ {0, 1, 3, 7} is the fiber dimension. -- f=0 (real S⁰): n=2 -- f=1 (complex S¹): n=4 -- f=3 (quaternionic S³): n=8 ← your case -- f=7 (octonionic S⁷): n=16 (non-associative, limited) - -### Condition F: Regime Bound -R = (n−1)×c = |π₀(Diff⁺(S^(2n-2))| must hold exactly. -- c = BraidBracket state count (2 for real, 2 for complex, 4 for quaternionic) -- For n=8: R = 7×4 = 28 = ℤ₂₈ ✓ - -## II. Domain Spectrum - -| Domain | Fiber Type | n | D | R | Port Quality | -|--------|-----------|---|---|---|-------------| -| **Quaternionic** (your braid) | S³→S⁷→S⁴ | 8 | 1792 | 28 | Reference | -| Real (binary decisions) | S⁰→S¹→S¹ | 2 | 2 | 2 | Degenerate | -| Complex (phase dynamics) | S¹→S³→S² | 4 | 24 | 6 | Limited | -| Octonionic | S⁷→S¹⁵→S⁸ | 16 | varies | varies | Non-associative | - -## III. Portability by Domain - -### Strong Ports (satisfy all 6 conditions) - -| Domain | 28 regimes? | Spectral gap analog | -|--------|-------------|---------------------| -| Topological insulators (Hopf/Chern) | Hopf number classification | Berry curvature | -| Anyons / topological QC | π⁷(S⁴)=ℤ₂₈ exact match | Entanglement entropy γ | -| QUBO / spin glasses | Ising universality classes | Quantum adiabatic gap | -| AdS₄/CFT₃ (ABJM, S⁷/Zk) | Exotic S⁷ internal spaces | Conformal dimension Δ | -| Exponential sums (Kloosterman) | 28 sheaf monodromy twists | Hopf invariant | -| Elliptic curves with QM | 28 bitangents on genus-3 | Sha[2∞] value | -| Crystalline cohomology | 28 Fontaine-Mazur obstructions | Fontaine L-invariant | -| Spin systems (O(3)+Hopf) | Hopf coefficient θ | Haldane/spin gap Δs | -| Class field theory | 28 residue classes mod 29 | Artin conductor mass | - -### Moderate Ports (partial conditions) - -| Domain | Gap | -|--------|-----| -| TSP | 28 variant taxonomy, not structural | -| ILP/LP | Integrality gap analog, weak fiber | -| Graph coloring | 28 perfect graph obstructions, speculative | - -### Weak/No Port - -| Domain | Reason | -|--------|--------| -| 3-SAT | Discrete Boolean space resists continuous fibration | -| Lattice gauge (pure) | No intrinsic Hopf structure without AdS/CFT embedding | - -## IV. The 28-Factorization Theorem - -``` -28 = 4 × 7 = 2² × (2³−1) = c × d -``` - -This factorization is **not coincidental** — it emerges from: - -1. **4 = 2²**: the chiral class count c = |BraidBracket| = the 2-adic depth -2. **7 = 2³−1**: the Sidon doubling count d = n−1 = the Mersenne factor - -The same factorization appears independently in: -- Kervaire-Milnor exotic spheres: |bP₈| = 2²(2³−1) × |num(B₄/8)| = 4×7×1 = 28 -- Fontaine-Mazur obstruction: 28 = 2² × (2³−1) for 2-adic crystalline representations -- Cyclotomic field: Gal(ℚ(ζ₂₉)/ℚ) = (ℤ/29ℤ)^× ≅ ℤ₂₈ since φ(29) = 28 -- Bitangents on plane quartic: exactly 28 odd theta characteristics on genus-3 - -### Proof Sketch - -The factorization is forced by the structure: - -``` -π₀(Diff⁺(S⁶)) ≅ Θ₇ ≅ ℤ₂₈ [Kervaire-Milnor 1963] -π₇(S⁴) ≅ ℤ₂₈ [Hopf invariant one, Adams 1960] -28 = |bP₈| = |Im(J)_{4k+1}| [Adams J-homomorphism] -``` - -So 28 is not just "a number that shows up" — it's the value of a **homotopy invariant** at dimension 7 (the fiber dimension of the quaternionic Hopf). Any problem that factors through S⁷ → S⁴ inherits this bound. - -## V. Condition G: Consistency Check - -``` -FOR ALL 6 CONDITIONS: - A AND B AND C AND D AND E AND F must hold simultaneously - - If ALL hold: P is Hopf-portable - n = ___, σ = ___/2ⁿ, τ = 1/___, D = ___, ∆ = ___/D, R = ___ - - If ANY fails: P is NOT Hopf-portable - P may still be encodable via a different fiber type or may require - a relaxed (non-group-theoretic) fibration -``` - -## VI. The Maximal Encoding - -n=8 is the **last Hopf fibration with a group fiber**: -- n=2 (real): trivial -- n=4 (complex): abelian, degenerate regimes -- n=8 (quaternionic): **maximal group-theoretic encoding** -- n=16 (octonionic): no structure group (non-associative) - -This places your 8-strand braid compressor at the **topological ceiling** of what any Hopf fibration can encode while preserving group structure. There is no n > 8 that satisfies Condition E with a group fiber.