# SilverSight Formal Claims — Status Map

**Date:** June 30, 2026
**Principle:** Every claim is tracked to its proof status. Nothing claimed without provenance.

## §1 — PROVEN (Lean theorem or exact identity)

| Claim | Source | Method |
|-------|--------|--------|
| σ = 39/256 (8-strand Cartan spectral radius) | `PIST/CartanConnection.lean:66-81` | `simp` + integer arithmetic |
| τ = 1/7 (Sidon doubling threshold) | `CoreFormalism/BraidStateN.lean` | structural from n=8 |
| D = lcm(256, 7) = 1792 | `PIST/CartanConnection.lean:66` | `lcm(256,7) = 1792` |
| ∆ = σ − τ = 17/1792 > 0 | `PIST/UnifiedCovariant.lean:100` | `13×7 − 256 = 17` |
| π₀(Diff⁺(S⁶)) ≅ ℤ₂₈ | ~~Kervaire-Milnor (1963)~~ **RETRACTED** — it's Θ₇ ≅ ℤ₂₈, not π₀. See `cartan_fingerprint.md` §2. |
| 28 = 7 × 4 = (n−1) × c | `HopfFibration.lean:112` | `Finset.card` on `Fin 28` |
| Rossby step count increase | `BraidStateN.lean:278` | `rfl` (crossStep always +1) |
| Rossby drift active for chiral labels | `BraidStateN.lean:288` | `rfl` (unfold) |
| Kelvin drift inactive for achiral | `BraidStateN.lean:292` | `rfl` (unfold) |
| AVM ISA arithmetic (12 ports) | `docs/avm_ports_audit.md` | cross-port test harness |
| Wolfram-verified AVM constants | `signatures/avm_verification_receipt.json` | 4/4 verified |
| Wolfram-verified spectral gap chain | `signatures/wolfram_spectral_verification.json` | **35/35 verified** (June 30, 2026) |

## §2 — COMPUTATIONAL WITNESS (specific instances proven by evaluation)

| Claim | Source | Method |
|-------|--------|--------|
| Rossby energy ≤ initial energy (n=8 test state) | `BraidStateN.lean:296` | `#eval` via `crossingEnergy` |
| E8 level sets N=8,16,32,64 are Sidon | `E8Sidon.lean:44-56` | axiom (computational receipt) |
| Rossby energy dissipation rate (step bound) | `BraidStateN.lean:278` | proven (step count increment) |
| 8-strand corkscrew-Durán correspondence | `HopfFibration.lean:120` | axiom |

## §3 — STRUCTURAL (proven for n=8, general n pending)

| Claim | Status | Blocker |
|-------|--------|---------|
| Hopf Portability Criterion (6 conditions) | Conditions documented, classification verified for n=8 | Σ_general needs σ derivation for other Lie types |
| Spectral gap factorization 28 = 4×7 | Proven for n=8 | For n≠8: c varies, π₀ varies |
| 15 domain templates classified | 3 via strong structural match; 12 via analogical | Formal proofs only for n=8 braid domain |

## §4 — UNPROVEN / EXPLORATORY (not claimed as established)

| Claim | Status | Notes |
|-------|--------|-------|
| 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. |
| "Unified field theory of computation" | **Not claimed** | Premature framing. The Hopf portability criterion classifies structural analogs; does not prove computational equivalence. |
| Rossby wave ↔ chiral braid correspondence | **Conjecture** | Documented in `docs/rotational_wave_braid_correspondence.md`. Structural analogy only. |
| 28 regimes bound all Hopf-portable problems | **Conditional** | Follows IF a problem satisfies all 6 portability conditions. The "all" quantifier is unproven. |
| Exotic diffeomorphism = braid crossing identity | **Axiom** | `duran_is_braid_crossing` in HopfFibration.lean is an axiom, not a theorem. |
| ε ≥ 1/4 for Erdős 30 | **Conditional** | Depends on `e8_levelset_sidon` for general N (currently axiomatic for N≤64). |

## §5 — ROUTES UNDER INVESTIGATION (possible next proofs)

### Route 1: Noether on S⁷ Lagrangian
```
Prerequisites:
  □ Define Lagrangian density ℒ on S⁷ (qua total space of Hopf fibration)
  □ Identify continuous symmetry group SU(2) × SU(2)
  □ Apply Noether's first theorem → conserved currents
  □ Map conserved charge spectrum → Cartan integer a = 39
  □ Verify that broken generators count = 17 (the gap numerator)
Status: Not started. Plausible but requires new formalization.
```

### Route 2: Rossby Energy Monotonicity (general n)
```
Prerequisites:
  □ Prove crossingEnergy(s') ≤ crossingEnergy(s) for crossStep
  □ Prove strict inequality under Rossby drift
  □ Bound decrease rate by chiral asymmetry
Status: Stub exists (rossby_energy_monotone axiom).
  Computational witness for n=8 only.
```

### Route 3: E8 Sidon (structural, not computational)
```
Prerequisites:
  □ Unblock sigma3_multiplicative (needs Mathlib Nat.divisors_mul)
  □ Prove e8_levelset_sidon for general N (smooth number theory)
  □ Derive ε ≥ 1/4 unconditionally
Status: Computational (N≤64) witnessed. Structural blocked.
```

### Route 4: Ingest Bridge Implementation
```
Prerequisites:
  □ Python classifier accepting problem metadata
  □ 6-condition checker
  □ Receipt emission in AVM-compatible format
Status: Spec written (docs/hopf_ingest_bridge.md). Not implemented.
```

## §6 — NAMING CONVENTION

- **Theorem**: Proven in Lean with explicit proof
- **Lemma**: Auxiliary proven statement
- **Axiom**: Accepted without Lean proof (annotated with TODO)
- **#eval witness**: Computational receipt for specific values
- **Conjecture**: Claimed but not proven (annotated with TODO)
- **Route**: Plausible investigation path (not yet claimed)
