---
name: Collatz F-Entropy Search (STEP 673)
description: Systematic search for a Perelman-style monotone functional φ for Collatz, with AME evolutionary search and counter-example mining
type: project
originSessionId: cd413b52-217d-49a2-ae30-92af7ffedf9b
---
STEP 673: Collatz F-Entropy Search Engine — `src/axiom-os/collatz-f-entropy-search.ts`

**Why:** STEP 672 (Perelman-Dot Approach) identified that the missing piece for a Collatz proof is a Perelman-style F-entropy: a single functional that is provably monotone along every orbit. This is the breakthrough analog to Perelman's F-functional for Ricci flow.

**How to apply:** When exploring new approaches to Collatz, this engine provides:
- 11 baseline F-entropy candidates (log₂, bit_length, trailing-ones corrected, popcount, "00"-density, etc.)
- AME evolutionary search over 5-coefficient linear combinations
- Counter-example mining (find n where a candidate fails monotonicity)
- D-FUMT₈ classification of each candidate

## Empirical findings (test corpus: 112 starts, up to 500 steps each)

| Candidate | mean Δφ | max Δφ | mono % |
|-----------|---------|--------|--------|
| φ₀ = log₂(n) (Syracuse) | -0.3073 | +0.74 | 47.2% |
| φ₃ = log₂(n) + log₂(3)·t₁ | -0.4097 | +12.26 | 81.1% |
| φ₈ = log₂(n) + log₂(3)·t₁ - log₂(3)·v₂(3n+1) | -0.4189 | +13.85 | 72.0% |
| φ₉ = log₂(syracuse(n)) lookahead | -0.2942 | +0.65 | 49.9% |
| AME-evolved (25 gen, pop 24) | -0.0732 | +1.04 | 63.6% |

**KEY RESULT (negative): NO simple state functional φ(n) of (log₂, t₁, v₂, popcount, "00") was strictly monotone for all 2399+ Syracuse transitions tested.**

This is consistent with theory: the existence of such a state functional would immediately prove Collatz. Therefore the Perelman F-entropy analog **must be path-dependent**:
  φ(orbit_so_far) = log₂(n_now) + Σ_{i<now} correction_i(orbit_i)

This is exactly what Perelman's F is — an **integral over the trajectory**, not a pointwise function.

## Worst counter-examples discovered
- n=9663 (Δ=+1.04 for evolved candidate)
- n=7, 9, 15, 23, 25, 41 (small failures across all trailing-ones-corrected candidates)

These are precise locations where new mathematics is needed.

## SEED_KERNEL theories added
- T-FE-1: Collatz F-Entropy Search [FLOWING]
- T-FE-2: Syracuse Logarithm Theorem [TRUE] — E[Δlog₂] = log₂(3) - 2 ≈ -0.415
- T-FE-3: Trailing-Ones Correction Conjecture [NEITHER]
- T-FE-4: Counter-Example Mining as Roadmap [BOTH]
- T-FE-5: AME-Evolved F-Entropy [SELF]

Test: `npx tsx test/step673-f-entropy-search-test.ts` (4 passed, 0 failed)