Collatz: Fibonacci Finite-Memory States & Bernoulli Branch Drift

What v1.6 adds (ghost framework)

• Sub-generic sign theorem (Theorem 8.2). For every periodic K-pattern with average $K < \log_2 3$, the corresponding 2-adic ghost $x = E_L / (2^{A_L} - 3^L)$ is negative. Hence no sub-generic periodic K-sequence corresponds to a positive integer.
• Ghost-shadowing lemma (Lemma 8.3). If a positive integer $n$ matches a ghost $g$ for $r$ low bits, $n$'s orbit follows $g$'s K-sequence for as long as $A_t < r$. Each shortcut step burns $K_t$ bits of closeness. Bit-burning is the restoring force.
• Periodic ghost exclusion (Corollary 8.5). Combining the sign theorem with Hercher 2022 (no integer cycle for $L \le 91$) and Mihăilescu 2002 (Catalan): the only positive-integer ghost from any periodic K-pattern of length $\le 91$ is $x = 1$ (trivial cycle).
• The aperiodic conjecture (Conjecture 8.4). Every infinite aperiodic sub-generic K-sequence's ghost has non-terminating 2-adic bit pattern. This is the precise Collatz residue, reformulated as a 2-adic bit-pattern question.

Verified empirically on $n = 665{,}215$ (longest known bad-prefix in $n \le 10^6$): orbit shadows the $-1$ ghost initially for 6 K=1 steps (matching its 7 trailing 1s), then re-enters deep $-1$-corridors at steps 22, 40, 47, 88, 119 with depths up to $v_2(n+1) = 10$. Bit-burning is observable in the live trajectory.

The earlier theorem block (v1.5)

For the odd-to-odd Syracuse shortcut S_a(n) = (a n + 1) / 2^{ν₂(a n + 1)} with a odd:

• Fibonacci recurrent class. The m-bit rolling parity-history window, sampled at odd arrivals, lives on Ω_m = { h ∈ {0,1}^m : h₀ = 1, h_i h_i+1 = 0 }, of size the Fibonacci number F_m. For m = 8: 21 reachable windows out of 256 possible.
• Bernoulli branch independence. Under the natural Haar branch measure on odd 2-adic integers, K_a = ν₂(a n+1) is independent of every finite history and follows Geom(1/2) with mean 2.
• Uniform conditional drift. For every h ∈ Ω_m: E[log₂(a) − K_a | h] = log₂(a) − 2. Collatz (a = 3): −0.415 (descent). 5n+1 control (a = 5): +0.322 (growth).
• Exact stationary weights. Closed-form formula π_m(h), summing to 1 on Ω_m.
• No finite-history pointwise Lyapunov correction. The alternating state has a K = 1 self-transition with positive log-step (+log₂(3) − 1 ≈ +0.585), structurally unfixable by any history-only correction (bounded or unbounded). This is the structural reason fixed-window Lyapunov programs must fail at pointwise level.

Experimental verification

Integer-trajectory recovery (3n+1)

314,261 samples at 80-bit starts. Aggregate K-distribution matches Geom(1/2) within 1% for k ≤ 8. Per-cell E[K | h] within 1.7% of the theoretical value 2. Per-cell drift fits log₂(3) − E[K | h] exactly within sampling noise on every Fibonacci state.

Coefficient control (5n+1)

3,000,000 samples at 200-bit starts. Same Fibonacci-21 support. Same Geom(1/2) K-law. Drift sign flips to log₂(5) − 2 ≈ +0.322 on every cell. The cleanest available coefficient discriminator distinguishing original Collatz from 5n+1.

Survival-bias diagnosis

Empirical (3n+1) cell weights deviate from π₈ by up to 14.9%, with long-gap boundary cells systematically undercounted. The same diagnostic on (5n+1) recovers π₈ within 0.7% on every cell. This isolates the discrepancy as finite-height survival conditioning (descent absorbs trajectories with recent large-K jumps before sampling) rather than a defect in the branch model.

What this does not give

• No pointwise Collatz convergence theorem. The pointwise residue lies in non-generic long-range branch-exponent sequences invisible to any finite history window.
• No bridge from this finite-memory result to Tao's almost-bounded theorem (log-density-1 in N). The two results are orthogonal and address different layers of the problem.
• No improvement on existing cycle bounds (Hercher 2022: no m-cycles for m ≤ 91) or computational verification (Barina 2025: no counterexample below 2⁷¹).

Files

• paper.pdf — preprint, 12 pages
• paper.tex — LaTeX source, 1331 lines (v2.8, with α_k coordinate system + residue-disjointness obstruction)
• supplement.zip — Python experiments + raw JSON output

Citation:

Older program: Coherent-Block Certificate Work (2025–April 2026)

A partial certificate and lift-invariance program developed through April 2026 (passes 1–50 in the local repository). The May 2026 finite-memory result above explains structurally why this program had to fail at pointwise Lyapunov level: the alternating-state self-loop with positive log-step is invariant to any history-only correction, including the coherent-block Δ-invariant studied there.

What was useful

• Lift invariance of block labels (L, K, D): rigorous.
• Direct contraction bounds for directly contractive residue classes (3^L/2^K < 1): sound.
• Pass-50 obstruction document explicitly identifies why fixed-window drift theorems on the raw Δ-invariant cannot be promoted to pointwise descent — aligned with the May 2026 finite-memory Corollary 6.1.

What failed

• Block-graph Lyapunov framework: structurally flawed (single-valued successor map E_M not well-defined on the claimed state space).
• No proof of Collatz from this program.

Former “complete proof” paper files are archived away from the public surface. The May 2026 finite-memory result is what the program is now understood to have been measuring.