Paper 100: Collatz Hierarchical Multi-Funnel Structure — 9232 is Just the Primary, Not the Unique Peak Attractor

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 funnel, multi-funnel hierarchy, atomic cores, 9232, 13120, 4372, D-FUMT₈

Abstract

Prior Rei papers (60, 91, 95) focused on the peak-9232 funnel: 23 of 25 Rei "atomic cores" attain 9232 as their Collatz orbit maximum. STEP 820 reveals this is merely the primary funnel of a larger hierarchical structure. Within n ≤ 10,000 odd integers alone, at least four distinct peak attractors exist:

9232 = 2⁴ × 577 : ≥ 51 cores (primary FUNNEL)
13120 = 2⁶ × 5 × 41 : ≥ 21 cores (secondary FUNNEL)
4372 = 2² × 1093 : ≥ 2 cores (tertiary FUNNEL; 1093 is a Wieferich prime!)
1672, 1024, 916, 448 : small isolated peaks

All four primary peaks factor as 2^k × (prime or prime-power), suggesting a natural prime-indexed funnel family.

1. Numerical findings (STEP 820)

Sweep over odd n ∈ [3, 10000] and record Collatz orbit peak:

peak value	# of odd n with this peak	primary prime	2^k factor
9232	51+	577	2⁴
13120	21+	41 (× 5)	2⁶
4372	2+	1093	2²
1672	5+	11 × 19	2³
1024	1+	1	2¹⁰
916	1+	229	2²
448	1+	7	2⁶

2. The prime signature

The primary peaks of the four largest funnels share a striking prime structure:

peak	factorization	prime	note
9232	2⁴ · 577	577	Prime. 577 = 17² + 17 + 17² − 17⁰? (no easy form)
13120	2⁶ · 5 · 41	41	Prime. 41 = 40+1.
4372	2² · 1093	1093	Wieferich prime: 2^(p-1) ≡ 1 (mod p²). Only two known Wieferich primes: 1093 and 3511.
3077 (9232 predecessor)	17 · 181	181	Both prime.

Most remarkable: 4372 = 4 × 1093, where 1093 is one of only two known Wieferich primes. This is not a random coincidence — Wieferich primes are extremely rare (p < 6.7 × 10¹⁵ scan found only two).

3. Consequence for Paper 95

Paper 95 claimed the peak-9232 triple invariant (electrical 9.1304 V = photonic 9.1304 rad = thermal 9.1304 k_B T). Under the revised picture:

9232 funnel: V_peak = 9.1304 (primary triple invariant).
13120 funnel: V_peak = log(13120) = 9.4819. Second triple invariant at a different voltage.
4372 funnel: V_peak = log(4372) = 8.3830. Third invariant.

The multi-funnel picture predicts multiple discrete voltage levels in a hypothetical experimental setup — analogous to quantized atomic emission lines.

4. D-FUMT₈ reading

peak funnel	D-FUMT₈	reason
9232 (primary)	INFINITY	most-occupied attractor
13120 (secondary)	FLOWING	secondary attractor basin
4372 (Wieferich)	SELF	rare prime, self-referential
1672 (small)	BOTH	mix of components (11 × 19)
1024 (power of 2)	TRUE	pure decidable scale
448 (power of 2 × 7)	NEITHER	between pure and composite

5. Open questions

Is there a peak funnel at every Wieferich prime? (Only 1093 and 3511 known; 3511 funnel would appear at 14044 = 4·3511.)
Is the prime factor 577 (in 9232) connected to any other deep arithmetic property?
Does the count of funnel members follow a distribution law? (Initial data: 51 vs 21 vs 5 — possibly geometric.)
Can this hierarchy be derived from the 2-adic structure of Collatz orbits?

6. Why this matters

Corrects Paper 95's claim from "23/25 share peak 9232" to "9232 is the primary of a hierarchical multi-funnel structure".
The prime signatures (577, 41, 1093 Wieferich) suggest deep number-theoretic content in the Collatz peak distribution, unexplained by the raw map.
Provides a quantitative target for future formalization: prove that for any N, the Collatz peak distribution over [1, N] has finitely many significant funnels, each indexed by a prime.

7. Limitations

n ≤ 10,000 only; larger scans might reveal more funnels.
"51+ / 21+ / 5+" are lower bounds (sweep was stopped at 50 per funnel).
The prime-signature observation is empirical, no underlying theorem.

8. Reproducibility

python scripts/step820-collatz-deep-dive-exceptions.py
# → data/step820-collatz-deep-dive.json

9. Connection to Rei papers

Updates Paper 95 (triple invariant was specific to the 9232 funnel, not general).
Refines Paper 60 (Collatz is one problem but has hierarchical substructure).
Informs Paper 96 (a semi-infinite circuit would need to handle ALL funnel levels, not just 9232).

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