# Rossby Energy + E8 Sidon — Completion Roadmap

## Current State

### Rossby Energy (BraidStateN.lean)
- ✅ `crossingEnergy` — defined (Q16_16 weighted phase sum)
- ✅ `rossby_convergence_bound` — proven (step count increases)
- ⚠️ `rossby_energy_monotone` — axiom (energy decreases under crossStep)
- ⚠️ `regime_classification` — trivial (28 = C(8,2) coupling pairs)
- ❌ `crossingEnergy_invariant` — not yet defined
- ❌ `rossby_faster_than_kelvin` — not yet defined

### E8 Sidon (E8Sidon.lean)
- ✅ `sigma3`/`sigma7` — defined
- ✅ `E8LevelSet` — defined
- ⚠️ `sigma3_multiplicative` — 1-line fix (blocked on Mathlib `Nat.divisors_mul`)
- ⚠️ `e8_levelset_sidon` — computational N≤200, structural blocked
- ❌ `erdos30_bound` — not yet computed

## Completion Paths

### Phase 1: Rossby Energy (n=8, computational)

```
Step 1.1: Define a concrete test state (8-strand with specific chiral labels)
Step 2.1: Compute crossingEnergy(s) and crossingEnergy(crossStep(s))
Step 3.1: native_decide the difference (16-20 Q16_16 comparisons)
Step 4.1: Extract #eval witness to rossby_energy_decrease
```

**Goal**: One computational receipt proving energy decreases for a concrete Rossby (chiral) state vs staying constant for a Kelvin (achiral) state.

### Phase 2: E8 Sidon (N≤200, computational)

```
Step 2.1: Unblock sigma3_multiplicative:
  import Mathlib.Data.Nat.Divisors
  Use Nat.divisors_mul (a * b) (ha : a ≠ 0) (hb : b ≠ 0) (hcop : Coprime a b)
  → key lemma: sum over divisors of product = product of sums

Step 2.2: Build concrete level sets for N=8, 16, 32, 64, 128
  For each N, compute σ₃(n) for n ≤ N via native_decide
  Verify pairwise sums are unique (Sidon property)

Step 2.3: Extract #eval witness:
  #eval e8_levelset_sidon 64
  → output: "Sidon verified for N=64 (σ₃ constraint)"
```

### Phase 3: Integration — Rossby ↔ E8 Sidon bridge

The 28 coupling pairs (C(8,2) combinatorial) bound the possible crossing configurations.
The E8 Sidon construction improves the density bound.
Together: ε ≥ 1/4 with at most 28 iteration patterns.

### Phase 4: Generalization (future work)

- `sigma3_multiplicative` → full Mathlib dependency → PR upstream
- Dickman function density estimates → smooth number theory
- CrossStep contractiveness → needs Q16_16 inequality lemmas
- `rossby_faster_than_kelvin` → needs comparison lemma for energy dissipation rates