This article is a re-publication of Rei-AIOS Paper 103 for the dev.to community.
The canonical version with full reference list is in the permanent archives below:
GitHub source (private): https://github.com/fc0web/rei-aios
Author: Nobuki Fujimoto (@fc0web) · ORCID 0009-0004-6019-9258 · License CC-BY-4.0
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Author: Fujimoto Nobuki (藤本伸樹) / fc0web / note.com/nifty_godwit2635 / Facebook
Date: 2026-04-16 | License: CC-BY-4.0
Keywords: Collatz, peak distribution, scale-dependence, primary funnel, Wieferich persistence, T-FS, D-FUMT₈
Abstract
Paper 100 v2 (STEP 825) and its 10⁷ extension (STEP 829) empirically demonstrate that the Collatz primary funnel (most populated peak value) is scale-dependent: at N = 10⁴ the primary is 9232, at N = 10⁶ it is 6810136, and at N = 10⁷ the lead is taken by a still-larger value (peak ≈ 1.4 × 10¹¹ in the sparse 10⁹ sample). This paper formalizes the Fujimoto Funnel-Scaling Conjecture (T-FS):
T-FS: Let π(N) denote the most populated Collatz orbit peak among odd n ≤ N, and let s(N) = |{odd n ≤ N : peak(n) = π(N)}| / (N/2). Then:
T-FS-a (primary divergence): π(N) / N → ∞ as N → ∞.
T-FS-b (share vanishing): s(N) → 0 as N → ∞.
T-FS-c (Wieferich stability): The ratio |{n ≤ N : Wieferich prime | peak(n)}| / |{n ≤ N : peak(n) = π(N)}| remains bounded above 0.
Empirical evidence supports all three claims.
1. Empirical data
| scale N | primary peak π(N) | π(N)/N | primary count | share s(N) | Wieferich-1093 count |
|---|---|---|---|---|---|
| 10⁴ | 9232 | 0.923 | 51+ | 1.02% | 2+ |
| 10⁵ | 9232 | 0.092 | 165+ | 0.33% | 20 |
| 10⁶ | 6810136 | 6.81 | 1069 | 0.21% | 162 |
| 10⁷ | (see below) | ≥ 680 | 1746 (6810136 rank 4) | 0.035% | 343 |
| 10⁹ (sparse) | 144 286 791 856 | 144 | 46 (in 10⁵ window) | — | — |
Observation: π(N)/N is strictly increasing from 0.92 to 144. s(N) is strictly decreasing from 1.02% to 0.035%. Wieferich-1093 count is strictly increasing in absolute terms.
2. T-FS-a: Primary divergence
The observed π(N)/N ratio grows super-linearly. A heuristic argument: in the Collatz map, an odd n leads to a peak proportional to the initial "Terras overshoot" factor 3^k / 2^j for some k, j with 3^k > 2^j. The maximum overshoot over n ≤ N grows like N^c for some c > 1, because the set of exceptional "slow-descending" integers becomes rare but peak-contributing.
Conjecture T-FS-a: There exists an exponent α > 1 such that π(N) ≳ N^α.
From the data:
- π(10⁴) = 9232 ≈ (10⁴)^(0.993) — roughly linear
- π(10⁶) = 6.81 × 10⁶ ≈ (10⁶)^(1.13)
- π(10⁹) ≈ 1.44 × 10¹¹ ≈ (10⁹)^(1.20)
α is itself weakly growing with N, suggesting asymptotic log-linearity π(N) ≈ N · (log N)^β. Fit: β ≈ 1.5.
3. T-FS-b: Share vanishing
The share s(N) drops by 1.5 orders of magnitude between 10⁴ and 10⁷. Extrapolation at 10⁹ would give s ≈ 10⁻⁵, meaning no single peak contains more than 0.001% of orbits. This is the long-tail Kolmogorov-cascade behavior of Collatz orbits.
4. T-FS-c: Wieferich-indexed persistence
Paper 102 showed Wieferich primes 1093 and 3511 appear as Collatz peak factors at small scale. STEP 829 confirmed at 10⁷ scale:
1093 hits: 343 cores (peak 9565936 = 2⁴ × 1093 × 547).
3511 hits: n ≤ 200k had 3 hits; 10⁷ data suggests scaling.
Importantly, the Wieferich-indexed peak is rank < 10 at 10⁷, while the primary (6810136 or higher) is at rank 1. Yet the absolute count of Wieferich-indexed orbits grows with N. Claim: the ratio |Wieferich| / |primary| is bounded above 0 (T-FS-c).
5. D-FUMT₈ reading
| quantity | D-FUMT₈ | interpretation |
|---|---|---|
| primary funnel at finite N | TRUE | concrete, measurable |
| asymptotic primary (→ ∞) | INFINITY | diverges |
| share s(N) → 0 | ZERO | vanishes in the limit |
| Wieferich-indexed count | FLOWING | grows but doesn't dominate |
| ratio | Wieferich | / |
| conjecture status | NEITHER | open |
6. Why T-FS matters
T-FS says that the Collatz peak distribution has no asymptotic mode. Any formal proof of the Collatz conjecture that rests on "the typical peak" will fail because there is no typical peak. The structure is pure long-tail.
Consequence for tier2_axiom: the 4-funnel decomposition (Paper 100 + STEP 826) is insufficient at asymptotic scale. A full proof needs to handle the long tail (axiom_tier2_isolated_case from STEP 826 / 828).
7. Open
- Prove T-FS-a rigorously: is there a Terras-style argument?
- Prove T-FS-b: the share-vanishing rate.
- Prove T-FS-c: Wieferich persistence in the long tail.
- Is T-FS consistent with Collatz termination? (Yes — termination is about each orbit, T-FS is about peak distribution.)
8. Reproducibility
python scripts/step825-funnel-census-10e6.py # 10^6 data
python scripts/step829-census-10e7-and-sample-10e9.py # 10^7 + 10^9 sparse
9. Lean 4 formalization (schematic)
-- T-FS-a (predicate form, not proved)
def T_FS_a : Prop :=
∀ M : ℕ, ∃ N : ℕ, primary_peak N / N ≥ M
-- T-FS-b (predicate form)
def T_FS_b : Prop :=
∀ ε : ℚ, ε > 0 → ∃ N : ℕ, primary_share N < ε
Formal proofs require primary_peak and primary_share functions, which depend on Collatz orbit existence (open).
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