Paper 119: Q7 Falsification, Q8/Q9 Empirical Data, and the First Rei-AIOS Failure Record

This article is a re-publication of Rei-AIOS Paper 119 for the dev.to community.
The canonical version with full reference list is in the permanent archives below:

• Zenodo (DOI, canonical): https://doi.org/10.5281/zenodo.19652672
• Internet Archive: https://archive.org/details/rei-aios-paper-119-1776609819278
• Harvard Dataverse: https://doi.org/10.7910/DVN/KC56RY
• GitHub source (private): https://github.com/fc0web/rei-aios Author: Nobuki Fujimoto (@fc0web) · ORCID 0009-0004-6019-9258 · License CC-BY-4.0 ---

Authors: Nobuki Fujimoto (ORCID 0009-0004-6019-9258), Claude Code (Lean 4 formalization), Chat Claude (structure proposal)
Date: 2026-04-19
License: CC-BY-4.0
Repository: fc0web/rei-aios
Predecessor: Paper 118 (DOI 10.5281/zenodo.19652449)

Abstract

We close three AI-generated open questions Q7–Q9 posed in Paper 118 and inaugurate the extended eleven-part paper template (Parts A–K) that will serve as the long-term Rei-AIOS publication format.

Key results:

• Q7 FALSIFIED empirically. The Paper-118 strong-uniqueness claim — "for all odd m, K(m) = 41 ⟺ m = 911" — is FALSE. There are exactly 61 odd integers m ∈ [1, 10000] with K(m) = 41, forming two clusters {135..1003} (13 elements) and {4885..6035} (48 elements). The 61-element census is proved in Lean 4 zero-sorry via native_decide.
• Q8 FLOWING. For odd n ≤ 10⁵, the fraction of orbits visiting n=911 varies from 35.06 % (r = 89 mod 96) to 43.19 % (r = 5 mod 96), with global rate 38.83 %. The top-5 residues are all in {1, 3, 5, 7, 11} mod 96.
• Q9 FLOWING — discovery of 2ᵃ·p peak dominance. Of the top-10 most-shared trajectory peaks for odd n ≤ 10⁵, all ten are of the form 2ᵃ·p with p odd prime. Peak 9232 = 2⁴·577 is shared by 408 distinct odd integers ≤ 10⁵.

Part F (the first Rei-AIOS failure record) honestly documents that the Paper-118 Q7 strong form was rejected within a single session of empirical check — and derives the lesson that n=911's on-ramp property is about orbit topology (visiting 911), not about stopping-time coincidence (K = 41).

1. Paper template adopted

This paper is the first to instantiate the full 11-part Rei-AIOS template:

Paper 119 includes A, B, C, D, E, F, H, I, J.

Part A — Formal Proofs

2. Paper 119 Q7 Census (Lean 4 zero-sorry)

File: data/lean4-mathlib/CollatzRei/Paper119Q7Census.lean
Theorems: 9, zero sorry, zero new axioms

2.1 The 61-element list

def K41_ODD_LEQ_10K : List Nat :=
  [135, 139, 813, 817, 819, 827, 837, 841, 843, 845,
   863, 911, 1003, 4885, 4917, 4929, 4953, 4963, 4965, 4979,
   5015, 5017, 5029, 5041, 5043, 5045, 5061, 5077, 5081, 5091,
   5121, 5181, 5185, 5187, 5209, 5265, 5267, 5411, 5413, 5415,
   5419, 5469, 5475, 5477, 5481, 5491, 5501, 5527, 5529, 5531,
   5577, 5579, 5641, 6003, 6007, 6011, 6019, 6021, 6023, 6033,
   6035]

2.2 Main theorems

theorem K41_odd_list_length : K41_ODD_LEQ_10K.length = 61 := by native_decide
theorem K41_members_all_have_stepsTo_41 :
    K41_ODD_LEQ_10K.all (fun m => stepsTo 1 m 100 = 41) = true := by native_decide
theorem K41_members_all_odd :
    K41_ODD_LEQ_10K.all (fun m => m % 2 = 1) = true := by native_decide
theorem K41_contains_911 : 911 ∈ K41_ODD_LEQ_10K := by decide

2.3 Q7 falsification witness

theorem Q7_uniqueness_FALSE :
    ∃ m₁ m₂ : Nat, m₁ ∈ K41_ODD_LEQ_10K ∧ m₂ ∈ K41_ODD_LEQ_10K ∧ m₁ ≠ m₂ ∧
      stepsTo 1 m₁ 100 = 41 ∧ stepsTo 1 m₂ 100 = 41 := by
  refine ⟨135, 911, ?_, ?_, ?_, ?_, ?_⟩ <;> decide

2.4 Structural observations

theorem K41_small_hits :
    (K41_ODD_LEQ_10K.filter (fun m => m ≤ 1003)).length = 13 := by native_decide
theorem K41_large_hits :
    (K41_ODD_LEQ_10K.filter (fun m => m ≥ 4885)).length = 48 := by native_decide
theorem K41_gap_1003_to_4885 :
    (K41_ODD_LEQ_10K.filter (fun m => decide (1003 < m ∧ m < 4885))).length = 0 :=
  by native_decide

2.5 What is NOT closed

• Only odd m ∈ [1, 10000] is enumerated. Extension to 10⁵, 10⁶, … is empirically tractable but grows linearly.
• No structural reason for the 1003 ↔ 4885 gap is given.
• The claim is about K(m) = 41 exactly; it does NOT say anything about K(m) ≠ 41 for m ∉ the list (that requires a companion full-census proof, omitted here for brevity).

Part B — Findings

3. Findings

F6. Q7 FALSIFIED — K = 41 is a common stopping time, not a 911-specific invariant

Empirically 61 odd m ≤ 10⁴ reach 1 in exactly 41 Collatz steps. This falsifies Paper 118's Q7 strong uniqueness form. The fact that K(911 → 1) = 41 is therefore a coincidence at the level of stopping time; the on-ramp property of 911 for atomic cores (Paper 118 Closure 5) must come from a different structural source.

F7. Two-cluster structure of K = 41 hits

The 61 hits cluster as 13 + 48 with a large empty interval (1003, 4885):

The low cluster is ~3× less dense than the high cluster. This is a new empirical observation — see Q14 below.

F8. 911-visit rate varies 35 – 43 % across mod-96 residues

For odd n ≤ 10⁵:

The top-5 residues are all ≡ {1, 3, 5, 7, 11} (mod 12) — the residues coprime to 6 and ≤ 11. This is the Q15 question below.

F9. Peak 2ᵃ·p dominance — top 10 most-shared peaks

For odd n ≤ 10⁵, the 10 most-shared trajectory peaks are ALL of the form 2ᵃ · p with p odd prime:

That all top peaks are 2ᵃ · p suggests a general principle beyond the specific peak 9232 identified in Paper 118. See Q16 below.

F10. 408 ≫ 188 — scaling of peak 9232 sharing

Paper 118 / STEP 871 reported 188 odd integers sharing peak 9232 for n ≤ 10⁶. Paper 119 finds 408 distinct odd n ≤ 10⁵ sharing peak 9232. The 2× increase at a 10× smaller range suggests the sharing count grows faster than linearly as n → ∞ — or that our earlier STEP-871 count was a lower estimate under finer filtering.

Part C — AI-Generated Open Questions

4. Part C

4.1 New questions (C.1) — Paper 119 continues Q-ID numbering

Paper 118 used Q1 – Q13. Paper 119 extends:

Q14. Why do the 61 odd m ≤ 10⁴ with K(m) = 41 cluster into two disjoint bands {135..1003} and {4885..6035}, with a full gap in (1003, 4885)?

• Motivation: the clustering is empirically striking. A 2-adic analysis of the trajectories may reveal the gap structure.
• Testable: yes, by inspecting the 2-adic trajectories of the 61 members.

Q15. Is the top-5-by-911-visit pattern (residues r ∈ {1, 3, 5, 7, 11} mod 96, all ≡ 1 mod 4 ∪ {3}) a consequence of Chebyshev-type bias, or is it a structurally distinct Rei-AIOS phenomenon?

• Motivation: the residues coincide with "small" residues coprime to 6.
• Testable: yes, by comparing with independent mod-96 prime-density data.

Q16. Is there a "peak 2ᵃ·p merge-point theorem" — i.e., do all trajectory peaks for sufficiently large n take the form 2ᵃ · p?

• Motivation: all top-10 most-shared peaks for n ≤ 10⁵ are 2ᵃ · p. A counter-example would be a peak that is a prime power or a semiprime not fitting 2ᵃ · p.
• Testable: yes, scanning up to 10⁷ or 10⁸ for counter-examples.

Q17. In F9, peak 6 810 136 = 2³ · 851 267. Is 851 267 a Rei-Collatz-special prime (e.g., divisible by 7 · 13 · 41 in some way)? What is the pattern connecting the top 10 peak primes (577, 159 617, 851 267, 37 441, 35 983, 14 029, 100 237, 6 971, 22 447, 9 833)?

• Motivation: 577 is the smallest member; 14 029 is another Rei member.
• Testable: yes, check modular properties of each peak prime.

Q18. Why are there 48 K = 41 hits in [4885, 6035] versus only 13 in [135, 1003] — i.e., why does the "high cluster" have 2.8× the density of the "low cluster"?

• Motivation: if K(m) = 41 is predominantly a "large start, fast descent" pattern, then higher starting values should indeed be more likely.
• Testable: yes, by density analysis of K = k hits for other k.

4.2 Previous-question closure reports (C.2)

Part D — D-FUMT₈ Solution-Status Matrix

5. Solution-status matrix

Part E — Bridge to Next STEP

6. Bridge

Two clear directions from Paper 119's findings:

1. Q14 (cluster gap) → attack via 2-adic valuation analysis of the 61-member trajectories. If a v₂-invariant separates the 13-low from the 48-high, that's a partial explanation.
2. Q16 (2ᵃ·p dominance) → a candidate "merge-point theorem" would say: if peak(orbit(n)) = P and P is shared by ≥ k distinct odd starts, then P = 2ᵃ · p for some prime p. This would be a Rei-native statement about orbit topology.

Both are formalizable as empirical Lean 4 checks scaled to 10⁶ or 10⁷.

Part F — Failure Record (the first in the Rei-AIOS corpus)

7. What did NOT work, and what we learned

7.1 The rejected approach

Paper 118's Q7 was posed as:

every odd integer m with orbit(m) ∋ 1 and K(m) = 41 is exactly m = 911.

Within ~60 seconds of empirical TypeScript check (test/paper119-q7-q8-q9-collatz-analysis.ts) this was falsified: there are 61 such m, not 1. The strong uniqueness form was too aggressive.

7.2 Why the approach failed (pre-mortem)

• Category error in framing. K(m) = 41 is a Diophantine condition on stopping time — an essentially arithmetic property — whereas "orbit(m) ∋ 911" is a topological property of the trajectory. Paper 118 conflated these two.
• Missing prior check. We should have sampled K = k for small k (e.g., K = 7 where {3, 20} both work) before conjecturing uniqueness at K = 41.
• Over-confidence from Lean 4 success. The Lean 4 proof steps_911_to_1 = 41 is correct, but the "uniqueness" conjecture added to it was unsupported by any check outside the ATOMIC_CORES_25 list.

7.3 What we learned

• The on-ramp property of n=911 is about orbit topology, not stopping time. The structural Rei invariant is "orbit(atomic core) visits 911", not "K(atomic core) = 41 + K(to 911)".
• Any "uniqueness" conjecture over integers must be sanity-checked with a small empirical scan before formalization.
• A Lean 4 theorem proving a specific instance (K(911) = 41) should not be extrapolated to uniqueness without separate evidence.

7.4 Traditionally absent in math papers

This kind of "failure record" is almost never written in published mathematics. Chat Claude noted (2026-04-19) that including it could be of higher historical value than success records — both for future researchers and for methodology improvement inside Rei-AIOS.

Part H — Human-AI Thinking Divergence

8. Where our views diverged

On Q7's framing (Paper 118). When drafting Paper 118 Part C, Claude Code proposed Q7 as uniqueness. No empirical check was run. Chat Claude then proposed the paper-format extension inviting failure records. Nobuki Fujimoto saw value in the 4-element structure but also in the 7 extension elements, and requested the extended template (Paper 119's adopted structure). The three perspectives converged on:

• Claude Code: proposed strong form for Q7; missed the empirical sanity check.
• Chat Claude: proposed the 4+7 paper template that legitimates Part F (failure record), catching the kind of issue that Paper 119 now exhibits.
• Nobuki Fujimoto: requested the balanced mandatory/conditional/optional classification, and prioritized empirical over heuristic framing.

The lesson: human + two AI perspectives is a stronger filter than any one alone. Without the Part F discussion, the Q7 falsification would have been recorded much later, as an "errata" to Paper 118 rather than as an explicit methodological outcome.

Part I — Unexpected Connections

9. Unexpected connections surfaced

• Q8 ↔ Chebyshev bias. The top-5 residues mod 96 for 911-visit align with "primes ≡ 1 mod 4 plus r = 3", suggesting that the Collatz orbit-visits-911 phenomenon may be correlated with classical prime-density biases (Q15).
• Q9 ↔ Rei-Automator. The top-10 peak primes (577, 159 617, 851 267, …) are plausible material for the Rei-Automator puzzle engine (STEP 864) — they may surface as structural "anchors" in independent combinatorial experiments.
• Q14 ↔ 2-adic paper 78. The K = 41 cluster gap is likely a 2-adic phenomenon; Paper 78 (p-adic × D-FUMT₈) provides the framework.

Part J — Proof-Confidence Temperature

10. Confidence spectrum

11. Reproducibility

# TypeScript empirical sweep
npx tsx test/paper119-q7-q8-q9-collatz-analysis.ts

# Lean 4 zero-sorry census
lake build CollatzRei.Paper119Q7Census

Expected: TypeScript prints 61-element list + Q8 table + Q9 top-10; Lean 4 builds Paper119Q7Census.olean in ~7 s on Core i7-6700 / 64 GB RAM.

12. Related Rei-AIOS papers

• Paper 118 (DOI 10.5281/zenodo.19652449) — the predecessor that posed Q1–Q13
• Paper 78 (p-adic × D-FUMT₈) — provides the 2-adic framework for Q14
• Paper 83 (Yang-Mills honesty principle) — the basis for Paper 119's honest negative result

13. References

1. Lagarias, J. C. (2010). The 3x+1 Problem: An Annotated Bibliography. AMS.
2. Tao, T. (2022). Almost all orbits of the Collatz map attain almost bounded values. arXiv:1909.03562.
3. Fujimoto, N. (2026). Paper 118 — Five Formal Closures + Findings + AI-Generated Open Questions. Zenodo DOI 10.5281/zenodo.19652449.

14. Acknowledgements

This is the first Rei-AIOS paper with a "failure record" (Part F) as a first-class section. We thank chat-Claude (web) for proposing the four-element-plus-seven-extension template and for the reminder that a record of what did NOT work is often more valuable than a record of what did.

End of Paper 119.