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allaun
0e3e8d4017
Added mkTestState8, rossbyLabels8, kelvinLabels8 with #eval witnesses to verify Rossby drift activity and crossing energy computation. Phase 1 of roadmap: computational receipts for: - Rossby labels produce active drift (isActive = true) - Kelvin labels produce inactive drift (isActive = false) - crossingEnergy computes Q16_16 weighted phase sum
61 lines
2.4 KiB
Markdown
61 lines
2.4 KiB
Markdown
# Rossby Energy + E8 Sidon — Completion Roadmap
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## Current State
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### Rossby Energy (BraidStateN.lean)
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- ✅ `crossingEnergy` — defined (Q16_16 weighted phase sum)
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- ✅ `rossby_convergence_bound` — proven (step count increases)
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- ⚠️ `rossby_energy_monotone` — axiom (energy decreases under crossStep)
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- ⚠️ `regime_classification` — trivial (28 regimes)
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- ❌ `crossingEnergy_invariant` — not yet defined
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- ❌ `rossby_faster_than_kelvin` — not yet defined
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### E8 Sidon (E8Sidon.lean)
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- ✅ `sigma3`/`sigma7` — defined
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- ✅ `E8LevelSet` — defined
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- ⚠️ `sigma3_multiplicative` — 1-line fix (blocked on Mathlib `Nat.divisors_mul`)
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- ⚠️ `e8_levelset_sidon` — computational N≤200, structural blocked
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- ❌ `erdos30_bound` — not yet computed
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## Completion Paths
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### Phase 1: Rossby Energy (n=8, computational)
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```
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Step 1.1: Define a concrete test state (8-strand with specific chiral labels)
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Step 2.1: Compute crossingEnergy(s) and crossingEnergy(crossStep(s))
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Step 3.1: native_decide the difference (16-20 Q16_16 comparisons)
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Step 4.1: Extract #eval witness to rossby_energy_decrease
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```
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**Goal**: One computational receipt proving energy decreases for a concrete Rossby (chiral) state vs staying constant for a Kelvin (achiral) state.
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### Phase 2: E8 Sidon (N≤200, computational)
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```
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Step 2.1: Unblock sigma3_multiplicative:
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import Mathlib.Data.Nat.Divisors
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Use Nat.divisors_mul (a * b) (ha : a ≠ 0) (hb : b ≠ 0) (hcop : Coprime a b)
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→ key lemma: sum over divisors of product = product of sums
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Step 2.2: Build concrete level sets for N=8, 16, 32, 64, 128
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For each N, compute σ₃(n) for n ≤ N via native_decide
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Verify pairwise sums are unique (Sidon property)
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Step 2.3: Extract #eval witness:
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#eval e8_levelset_sidon 64
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→ output: "Sidon verified for N=64 (σ₃ constraint)"
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```
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### Phase 3: Integration — Rossby ↔ E8 Sidon bridge
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The 28 exotic diffeomorphism classes bound the Rossby convergence regimes.
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The E8 Sidon construction improves the density bound.
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Together: ε ≥ 1/4 with at most 28 iteration patterns.
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### Phase 4: Generalization (future work)
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- `sigma3_multiplicative` → full Mathlib dependency → PR upstream
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- Dickman function density estimates → smooth number theory
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- CrossStep contractiveness → needs Q16_16 inequality lemmas
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- `rossby_faster_than_kelvin` → needs comparison lemma for energy dissipation rates
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