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allaun 0e3e8d4017 feat(proto): computational witness — 8-strand Rossby/Kelvin energy verification

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

2026-06-30 19:15:39 -05:00

2.4 KiB

Raw Blame History

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 regimes)
❌ 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 exotic diffeomorphism classes bound the Rossby convergence regimes. 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

2.4 KiB Raw Blame History