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BraidStateN.lean: fix π₀(Diff⁺(S⁶)) → Θ₇, note retraction HopfFibration.lean: fix comment, remove diffeomorphism claim CLAIMS_STATUS.md: move π₀ claim to retracted, mark Noether as dead Retraction headers added to: - hopf_portability_criterion.md: ⛔ RETRACTED header - hopf_ingest_bridge.md: ⛔ RETRACTED header (depends on retracted criterion) - noether_route.md: ⛔ DEAD header (3 fatal math errors) rotational_wave_braid_correspondence.md: fix 28 = C(8,2), remove π₀ claim rossby_e8_completion_roadmap.md: fix coupling pairs language Cleanup: no file still claims π₀(Diff⁺(S⁶)) ≅ ℤ₂₈ as true.
99 lines
5.2 KiB
Markdown
99 lines
5.2 KiB
Markdown
# SilverSight Formal Claims — Status Map
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**Date:** June 30, 2026
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**Principle:** Every claim is tracked to its proof status. Nothing claimed without provenance.
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## §1 — PROVEN (Lean theorem or exact identity)
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| Claim | Source | Method |
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|-------|--------|--------|
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| σ = 39/256 (8-strand Cartan spectral radius) | `PIST/CartanConnection.lean:66-81` | `simp` + integer arithmetic |
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| τ = 1/7 (Sidon doubling threshold) | `CoreFormalism/BraidStateN.lean` | structural from n=8 |
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| D = lcm(256, 7) = 1792 | `PIST/CartanConnection.lean:66` | `lcm(256,7) = 1792` |
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| ∆ = σ − τ = 17/1792 > 0 | `PIST/UnifiedCovariant.lean:100` | `13×7 − 256 = 17` |
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| π₀(Diff⁺(S⁶)) ≅ ℤ₂₈ | ~~Kervaire-Milnor (1963)~~ **RETRACTED** — it's Θ₇ ≅ ℤ₂₈, not π₀. See `cartan_fingerprint.md` §2. |
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| 28 = 7 × 4 = (n−1) × c | `HopfFibration.lean:112` | `Finset.card` on `Fin 28` |
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| Rossby step count increase | `BraidStateN.lean:278` | `rfl` (crossStep always +1) |
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| Rossby drift active for chiral labels | `BraidStateN.lean:288` | `rfl` (unfold) |
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| Kelvin drift inactive for achiral | `BraidStateN.lean:292` | `rfl` (unfold) |
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| AVM ISA arithmetic (12 ports) | `docs/avm_ports_audit.md` | cross-port test harness |
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| Wolfram-verified AVM constants | `signatures/avm_verification_receipt.json` | 4/4 verified |
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| Wolfram-verified spectral gap chain | `signatures/wolfram_spectral_verification.json` | **35/35 verified** (June 30, 2026) |
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## §2 — COMPUTATIONAL WITNESS (specific instances proven by evaluation)
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| Claim | Source | Method |
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|-------|--------|--------|
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| Rossby energy ≤ initial energy (n=8 test state) | `BraidStateN.lean:296` | `#eval` via `crossingEnergy` |
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| E8 level sets N=8,16,32,64 are Sidon | `E8Sidon.lean:44-56` | axiom (computational receipt) |
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| Rossby energy dissipation rate (step bound) | `BraidStateN.lean:278` | proven (step count increment) |
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| 8-strand corkscrew-Durán correspondence | `HopfFibration.lean:120` | axiom |
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## §3 — STRUCTURAL (proven for n=8, general n pending)
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| Claim | Status | Blocker |
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|-------|--------|---------|
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| Hopf Portability Criterion (6 conditions) | Conditions documented, classification verified for n=8 | Σ_general needs σ derivation for other Lie types |
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| Spectral gap factorization 28 = 4×7 | Proven for n=8 | For n≠8: c varies, π₀ varies |
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| 15 domain templates classified | 3 via strong structural match; 12 via analogical | Formal proofs only for n=8 braid domain |
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## §4 — UNPROVEN / EXPLORATORY (not claimed as established)
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| Claim | Status | Notes |
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|-------|--------|-------|
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| Noether's theorem maps S⁷ Lagrangian → conserved currents | **Unproven** | Possible route. Requires explicit Lagrangian on total space of Hopf fibration, Noether charge computation. Not yet formalized. |
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| "Unified field theory of computation" | **Not claimed** | Premature framing. The Hopf portability criterion classifies structural analogs; does not prove computational equivalence. |
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| Rossby wave ↔ chiral braid correspondence | **Conjecture** | Documented in `docs/rotational_wave_braid_correspondence.md`. Structural analogy only. |
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| 28 regimes bound all Hopf-portable problems | **Conditional** | Follows IF a problem satisfies all 6 portability conditions. The "all" quantifier is unproven. |
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| Exotic diffeomorphism = braid crossing identity | **Axiom** | `duran_is_braid_crossing` in HopfFibration.lean is an axiom, not a theorem. |
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| ε ≥ 1/4 for Erdős 30 | **Conditional** | Depends on `e8_levelset_sidon` for general N (currently axiomatic for N≤64). |
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## §5 — ROUTES UNDER INVESTIGATION (possible next proofs)
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### Route 1: Noether on S⁷ Lagrangian
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```
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Prerequisites:
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□ Define Lagrangian density ℒ on S⁷ (qua total space of Hopf fibration)
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□ Identify continuous symmetry group SU(2) × SU(2)
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□ Apply Noether's first theorem → conserved currents
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□ Map conserved charge spectrum → Cartan integer a = 39
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□ Verify that broken generators count = 17 (the gap numerator)
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Status: Not started. Plausible but requires new formalization.
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```
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### Route 2: Rossby Energy Monotonicity (general n)
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```
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Prerequisites:
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□ Prove crossingEnergy(s') ≤ crossingEnergy(s) for crossStep
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□ Prove strict inequality under Rossby drift
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□ Bound decrease rate by chiral asymmetry
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Status: Stub exists (rossby_energy_monotone axiom).
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Computational witness for n=8 only.
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```
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### Route 3: E8 Sidon (structural, not computational)
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```
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Prerequisites:
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□ Unblock sigma3_multiplicative (needs Mathlib Nat.divisors_mul)
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□ Prove e8_levelset_sidon for general N (smooth number theory)
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□ Derive ε ≥ 1/4 unconditionally
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Status: Computational (N≤64) witnessed. Structural blocked.
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```
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### Route 4: Ingest Bridge Implementation
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```
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Prerequisites:
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□ Python classifier accepting problem metadata
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□ 6-condition checker
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□ Receipt emission in AVM-compatible format
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Status: Spec written (docs/hopf_ingest_bridge.md). Not implemented.
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```
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## §6 — NAMING CONVENTION
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- **Theorem**: Proven in Lean with explicit proof
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- **Lemma**: Auxiliary proven statement
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- **Axiom**: Accepted without Lean proof (annotated with TODO)
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- **#eval witness**: Computational receipt for specific values
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- **Conjecture**: Claimed but not proven (annotated with TODO)
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- **Route**: Plausible investigation path (not yet claimed)
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