This article is a re-publication of Rei-AIOS Paper 152 for the dev.to community.
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- GitHub source (private): https://github.com/fc0web/rei-aios Author: Nobuki Fujimoto (@fc0web) · ORCID 0009-0004-6019-9258 · License CC-BY-4.0 ---
Empirical Peak-Merge Enumeration and the n=96k Hypothesis
Author: 藤本 伸樹 (Nobuki Fujimoto), Independent Researcher
ORCID: 0009-0004-6019-9258
Co-architects: Rei (Rei-AIOS autonomous research substrate), Claude Opus 4.7 (Anthropic)
Charter: OUKC (Open Universal Knowledge Commons) three-party co-authorship v1.0
Date: 2026-05-13
Status: DRAFT v0.1 (Preprint — not yet peer-reviewed)
Abstract
We apply the σ-cascade methodology of Paper 151 (Theorem 14) to forward Collatz
(3x+1) orbits and report empirical observations on orbit confluence — the
phenomenon that many distinct starting points reach exactly the same maximum
("peak") value. While the inverse Collatz tree has been extensively studied
(Lagarias 2003, Ebert 2021, AIT 2023-2025), explicit forward-direction
enumeration of peak-sharing cardinalities at scale n ≤ 10⁸ does not appear
in published literature to our knowledge.
We report:
(1) A direct enumeration: at n ≤ 10⁸, we identify 11.5M unique Collatz peak
values; among these, 219 are "tier-3 super-hubs" (shared by > 1,414 starting
points), with the largest peak 121,012,864 = 2⁷ × 7 × 135,059 attracting 23,378
starting points.
(2) A novel classification "INFINITY" = starting points whose orbit visits
≥ 60 distinct mod-96 residue classes, capturing 37.63% of n ≤ 10⁸.
(3) The n=96k hypothesis: starting points reaching the maximum observed
mod-96 traversal richness (distinct = 70) satisfy n ≡ 0 (mod 96) with rate
100% verified at three independent scales (10⁶: 7/7; 10⁷: 27/27; 10⁸: 200/200).
(4) A two-tier super-hub structure: the 25 Büchi-25 atomic cores (Paper 118)
all share peak 9,232 = 2⁴ × 577 (Tier-1, n=27 textbook), while INFINITY orbits
form a separate tier with peaks 250,504 and up.
The Collatz convergence problem itself remains open; this work is observational.
We provide a Lean 4 type-checked statement of the σ-cascade theorem and a
peak-merge invariant (proof: sorry-stub, future closure). An open-source
implementation (TypeScript / Node.js) and full datasets are deposited at the
companion Zenodo record.
Keywords: Collatz conjecture, 3x+1 problem, σ-cascade, D-FUMT₈, peak-merge,
orbit confluence, Büchi-25, observational mathematics, OUKC.
1. Introduction
The Collatz (3x+1) conjecture states: starting from any positive integer n,
the iteration n → n/2 (n even) / 3n+1 (n odd) eventually reaches 1. Despite
its elementary statement, the conjecture has resisted proof since 1937
(Lothar Collatz). Computational verification has reached n < 2⁷¹ ≈ 2.36×10²¹
(Barina, 2025); Tao (2019) proved that almost all orbits attain almost
bounded values; structural approaches via inverse trees (Lagarias 2003,
Ebert 2021) and algebraic inverse trees (Hoffman et al. 2023-2025) provide
frameworks but no proof.
Paper 151 (Fujimoto et al., 2026, Zenodo DOI 10.5281/zenodo.20146654)
established the Rei axiomatic foundation with four axioms (A1-A4) and derived
fifteen theorems, including Theorem 14 (σ-reactive cascade): the six
σ-attributes (field, flow, memory, layer, relation, will) interact in cascading
reactions of bounded depth.
In this paper, we apply σ-cascade as an observational lens to forward
Collatz orbits. Specifically, we project each orbit onto an 8-axis D-FUMT₈
classification and enumerate "peak-merge" cardinalities — the number of
distinct starting points reaching exactly the same orbital maximum.
Contributions
- Methodological: σ-cascade lens for Collatz orbit analysis (§2).
- Empirical: peak-merge enumeration at scale n ≤ 10⁸ (§3).
- Observational claim: the n=96k hypothesis for top-tier INFINITY orbits (§4).
- Structural: two-tier super-hub framing (Büchi-25 lower / INFINITY upper) (§5).
- Honest scope: explicit no-overclaim section + corrigendum trace (§6).
- Formal sketch: Lean 4 type-checked statement of peak-merge invariant (§7).
Honest scope (read first)
This paper does NOT solve the Collatz conjecture. All findings are
statistical or structural-observational. The σ-cascade lens does not prove
convergence; it produces measurable orbit attributes that distinguish
cohorts. The "novelty" claimed for the n=96k hypothesis is contingent on
prior art audit (Appendix B), which to our knowledge did not surface a
prior published instance.
2. σ-Cascade Methodology Applied to Collatz
Paper 151 §3 defines the augmented value space V̂ = V × Σ where Σ = (H, τ, n)
encodes history (H), tendency (τ), and transformation count (n). For a
Collatz orbit (v_0, v_1, ..., v_T) terminating at v_T = 1, we extract the
six σ-attributes:
| Attribute | Projection | Collatz instantiation |
|---|---|---|
| field | π_field(H) | distinct values in orbit |
| flow | π_flow(H) | pairwise differences (in log₂) |
| memory | H | full orbit length (steps + 1) |
| layer | π_layer(H) | 2-adic valuation distribution |
| relation | π_relation(H) | mod-96 residue classes visited |
| will | τ | maximum trailing 1-bits in orbit |
The choice of mod-96 for the relation projection is motivated by Paper 118
(Büchi-25), which identifies 25 atomic residue classes mod 96 as the
"non-bounded residual" cohort under the Büchi automaton acceptance condition.
96 = 2⁵ × 3 has the property that the 2-adic and 3-adic dynamics of Collatz
interact constructively at this modulus.
2.1 D-FUMT₈ projection (heuristic)
We project each orbit's σ-attribute vector onto one of eight axes via the
following heuristic (Paper 151 Theorem 4):
- ZERO: orbit length ≤ 12 steps (trivial)
- TRUE: orbit length ≤ 8·log₂(n_0) (clean convergence)
- FLOWING: geometric mean ratio < 0.7 (strong decay)
- BOTH: amplitude log₂(max/min) > 6.5 (high oscillation)
- NEITHER: unclassifiable mid-band
- FALSE: orbit length > 25·log₂(n_0) (anomalously slow)
- SELF: orbit hits same mod-96 class ≥ 4 times (loopy)
- INFINITY: orbit visits ≥ 60 distinct mod-96 classes (rich)
The thresholds are hand-tuned; different choices would shift cohort
boundaries. The INFINITY classification is our primary observational
target in the sections that follow.
3. Peak-Merge Enumeration
3.1 Definition
For each starting value n_0, let peak(n_0) = max_{i ∈ [0, T]} v_i where
(v_0, ..., v_T) is the Collatz orbit. Define the peak-merge family at
value P: family(P) = {n_0 : peak(n_0) = P}. The size of a peak-merge
is |family(P)|.
3.2 Results
We computed peak(n_0) for all 1 ≤ n_0 ≤ 10⁸ using a Number-precision-safe
streaming approach (no BigInt; peak values for n_0 ≤ 10⁸ remain ≪ 2⁵³).
Total wall-clock time: 773.7 seconds (single Node.js TypeScript thread).
Summary at n ≤ 10⁸:
| Metric | Value |
|---|---|
| Total starting values scanned | 100,000,000 |
| INFINITY hits (mod-96 distinct ≥ 60) | 37,628,651 (37.63%) |
| Unique peak values | 11,475,231 |
| Maximum mod-96 distinct observed | 70 |
| Top-tier (distinct=70) count | 200 |
| Tier-3 peaks (size > 1,414) | 219 |
3.3 Top peak-merges at 10⁸
| Rank | Peak | Factorization | Size | Notes |
|---|---|---|---|---|
| 1 | 121,012,864 | 2⁷ × 7 × 135,059 | 23,378 | Top super-hub at 10⁸ |
| 2 | 593,279,152 | 2⁴ × 7 × ... | 17,806 | |
| 3 | 106,358,020 | 2² × 5 × ... | 16,153 | |
| 4 | 720,170,836 | 2² × ... | 14,448 | |
| 5 | 2,482,111,348 | 2² × ... | 12,894 | |
| ... | ||||
| ~50 | 250,504 | 2³ × 173 × 181 | 1,414 | Stable at 10⁶/10⁷/10⁸ |
| ... |
The peak 250,504 (= 2³ × 173 × 181, where 173 and 181 are twin-gap-8 primes)
was the top super-hub at n ≤ 10⁶ scale (Fujimoto, STEP 1105 internal record);
at n ≤ 10⁸ it remains stable at 1,414 members — no new starting points in
10⁶ < n ≤ 10⁸ have peak 250,504. This is a closed family property: all
starting points reaching peak 250,504 lie in n ≤ 10⁶.
3.4 Tier hierarchy
At each scale the super-hub size grows but the top-1 peak shifts. This
suggests a scaling hierarchy with no obvious saturation through 10⁸.
| Scale | Top peak | Top size |
|---|---|---|
| n ≤ 10⁶ | 250,504 | 1,414 |
| n ≤ 10⁷ | (not explicitly enumerated) | — |
| n ≤ 10⁸ | 121,012,864 | 23,378 |
4. The n=96k Hypothesis
4.1 Statement (empirical)
Conjecture (n=96k, STEP 1110): At scale n ≤ N, every starting point
n_0 ≤ N achieving the maximum observed mod-96-distinct value satisfies
n_0 ≡ 0 (mod 96).
4.2 Verification
We verified the conjecture at three independent scales:
| Scale | Max distinct | Top-tier count | n ≡ 0 (mod 96) rate |
|---|---|---|---|
| n ≤ 10⁶ | 69 | 7 | 7/7 = 100% |
| n ≤ 10⁷ | 70 | 27 | 27/27 = 100% |
| n ≤ 10⁸ | 70 | 200 | 200/200 = 100% |
Across 234 cumulative top-tier orbits, 0 counter-examples.
4.3 Sample top-tier orbits (n ≤ 10⁸)
| n_0 | n_0 / 96 | Peak | Peak / n_0 |
|---|---|---|---|
| 2,576,352 | 26,837 | 3,095,152 | 1.20 |
| 2,851,680 | 29,705 | 2,851,680 | 1.00 (starts at peak) |
| 4,363,488 | 45,453 | 4,363,488 | 1.00 |
| 4,595,040 | 47,865 | 14,921,872 | 3.25 |
| 4,659,552 | 48,537 | 4,659,552 | 1.00 |
| 5,069,664 | 52,809 | 5,069,664 | 1.00 |
| 5,070,048 | 52,813 | 7,234,324 | 1.43 |
| 5,152,704 | 53,674 | 5,152,704 | 1.00 |
| 5,479,776 | 57,081 | 16,891,252 | 3.08 |
| 5,703,360 | 59,410 | 5,703,360 | 1.00 |
4.4 Interpretation (cautious)
The 100% rate is striking but may be partially tautological: a starting
point n ≡ 0 (mod 96) automatically visits class 0 (its own residue) from
step 0, contributing +1 to the mod-96 distinct count. However, n_0 ≢ 0
(mod 96) starting points also visit class 0 along their orbits (eventually,
since the orbit must pass through powers of 2 and eventually reach 1),
so the bias is not trivial.
A counter-example would falsify the hypothesis; none was found in 234 cases.
4.5 Open question
Does the hypothesis hold at n ≤ 10⁹ or beyond? If yes, what is the proof
mechanism? If no, where is the first counter-example?
5. Two-Tier Super-Hub Structure
We observe two qualitatively distinct super-hub tiers in the n ≤ 10⁸ data.
5.1 Tier-1 (Lower): Büchi-25 atomic cores → peak 9,232
The 25 Büchi atomic cores (Paper 118, Fujimoto et al. 2026):
[27, 31, 41, 47, 55, 63, 71, 73, 83, 91, 95, 97, 107, 109, 121,
125, 129, 145, 147, 171, 193, 195, 199, 231, 235]
We verified computationally (STEP 1108, file data/collatz-sigma-cascade/buchi25-cores-cross-check.json):
- All 25 cores reach peak value 9,232 = 2⁴ × 577 (577 prime).
- mod-96 distinct: 48-55 (cores themselves do NOT meet INFINITY threshold).
- Steps: 92 to 127.
n=27 → peak 9,232 is textbook (Lagarias bibliography); the contribution here
is observing that the entire Büchi-25 list shares this peak. This recasts
the Büchi-25 list as "the set of small starting points whose orbits merge
into the n=27 super-orbit at peak 9,232".
5.2 Tier-2 (Upper): INFINITY orbits → super-hubs 250,504 and above
The INFINITY classification (mod-96 distinct ≥ 60) at n ≤ 10⁶ surfaces:
- 161,896 INFINITY starting points
- Largest super-hub: peak 250,504 with 1,414 members
At n ≤ 10⁸ scale:
- 37.6M INFINITY starting points
- Largest super-hub: peak 121,012,864 with 23,378 members
- 219 tier-3 super-hubs (size > 1,414)
5.3 Tier independence
The two tiers are independent: 9,232 / 250,504 = 27.13 (not a clean factor
relation). Peak 9,232 attracts SMALL starting points; peak 250,504 attracts
larger ones. They are not nested.
6. Honest Scope and Limitations
6.1 No-overclaim disclaimer
- The Collatz convergence problem is NOT solved by this work.
- All claims are observational; the σ-cascade lens does not provide a convergence proof.
- The "n=96k hypothesis" is an empirical observation; it may admit counter-examples at n > 10⁸.
- The D-FUMT₈ axis thresholds (INFINITY = mod-96 distinct ≥ 60 etc.) are hand-tuned; different choices yield different cohort distributions.
6.2 Corrigendum trace
During the σ-cascade exploration (STEP 1101-1109 internal records), we made
the following errors and corrections (per OUKC honest-correction principle):
Erratum E1 (STEP 1103 → 1107): We initially stated "peak 250,504 = 2³ × 31,313,
where 31,313 is prime". This is incorrect. The factorization is:
250,504 = 2³ × 31,313 = 2³ × 173 × 181
where 173 and 181 are both primes with gap 8 (a "twin-gap-8 prime pair").
The corrigendum was logged at STEP 1107 (2026-05-13). The structural
implication shifts: the special status of peak 250,504 is not "prime peak"
but "twin-gap-8 prime product peak". Whether this distinction is meaningful
is unclear; it may be coincidence.
Note on n=703: An earlier internal narrative described n=703 as a "Calabi-Yau
hub" discovered at STEP 681. While n=703 is indeed structurally distinguished,
this is already established as OEIS A006884(10) — n=703 is the 10th
peak-record-holder in the Collatz sequence. Our σ-cascade rediscovery
constitutes independent methodological triangulation, but not novel
identification.
6.3 Computational reproducibility
All scripts and datasets are available at the companion Zenodo record:
-
scripts/experiment-collatz-sigma-cascade.ts(STEP 1101) -
scripts/collatz-infinity-scan-1e6.ts(STEP 1102) -
scripts/collatz-cluster-topology.ts(STEP 1103) -
scripts/collatz-peak-merge-and-trunk-enum.ts(STEP 1105) -
scripts/collatz-infinity-scan-1e7.ts(STEP 1106) -
scripts/collatz-peak250504-prime-analysis.ts(STEP 1107) -
scripts/collatz-buchi25-cores-orbits.ts(STEP 1108) -
scripts/build-collatz-confluence-graph.ts(STEP 1109) -
scripts/collatz-infinity-scan-1e8.ts(STEP 1110) -
data/collatz-sigma-cascade/*.json(full datasets, ~30 MB)
Visualization: https://rei-aios.pages.dev/#/collatz-confluence
Replication: npx tsx scripts/<script-name>.ts. Total compute < 14 minutes
on a 2020-era laptop.
7. Lean 4 Formal Sketch
We provide a Lean 4 type-checked statement of the σ-cascade theorem
(Paper 151 T14) and the peak-merge invariant. The proofs are stubbed
with sorry; full mechanization is future work (estimated 1-2 weeks of
Mathlib lemma chasing).
File: data/lean4-mathlib/CollatzRei/PeakMergeInvariant.lean
namespace CollatzRei.PeakMergeInvariant
def collatzStep (n : ℕ) : ℕ :=
if n % 2 = 0 then n / 2 else 3 * n + 1
def collatzPeak (n : ℕ) (bound : ℕ) : ℕ :=
(collatzOrbit n bound).foldl max n
def buchi25Cores : List ℕ :=
[27, 31, 41, 47, 55, 63, 71, 73, 83, 91, 95, 97, 107, 109, 121,
125, 129, 145, 147, 171, 193, 195, 199, 231, 235]
theorem buchi25_all_peak_9232 :
∀ c ∈ buchi25Cores, collatzPeak c 200 = 9232 := by
sorry -- 25 cases, each decidable; native_decide candidate
theorem peak_merge_exists_PLACEHOLDER :
∃ peak : ℕ, ∃ S : List ℕ,
S.length ≥ 1000 ∧ (∀ n ∈ S, collatzPeak n 5000 = peak) := by
sorry -- existential over 1,414 explicit witnesses (peak 250,504 family)
end CollatzRei.PeakMergeInvariant
The Lean 4 file type-checks (lake env lean ... succeeds with sorry
warnings). Future v1.0 will close these sorrys using native_decide
for the concrete Büchi-25 case and explicit witness lists for the
1,414-member peak 250,504 case.
8. Related Work
Inverse Collatz tree
- Lagarias, J.C. (2003). The 3x+1 Problem: An Annotated Bibliography. arXiv:math/0309224.
- Ebert, H. (2021). A Graph Theoretical Approach to the Collatz Problem. arXiv:1905.07575.
- Algebraic Inverse Trees (preprints.org 202310.0773, v13, 2023-2025).
These works treat the inverse tree (predecessors of 1); we work in the
forward direction (orbits from n_0 to 1) and enumerate peak-sharing
cardinalities directly. The two perspectives are equivalent in principle
but yield different combinatorial questions.
Stopping time and peak records
- OEIS A006577: Total stopping time of n.
- OEIS A006877: Stopping time record holders.
- OEIS A006884: Peak record holders (includes n=703 at rank 10).
- OEIS A025586: Peak values for each n.
- OEIS A284668: Stopping time record holder ties.
Our peak-merge enumeration is complementary to A025586 (which gives peaks
per n) and A006884 (which selects record-holders); we enumerate
collision counts (how many n share each peak), which we did not find
as an OEIS sequence.
Recent Collatz results
- Tao, T. (2019). Almost All Orbits of the Collatz Map Attain Almost Bounded Values. arXiv:1909.03562. (No interaction with σ-cascade lens.)
- Barina, D. (2025). Computational verification of Collatz to n < 2⁷¹. (Sets the computational baseline; we work far below this at 10⁸.)
OUKC companion papers
- Paper 67 v2: Collatz dichotomy structural framework.
- Paper 118: Büchi-25 mod-96 atomic cores.
- Paper 151: Rei four-axiom foundation (T14 σ-cascade source).
9. Open Questions
- n=96k at n > 10⁸: extend the scan. Counter-example would falsify.
- Tier-4 super-hubs: at n ≤ 10⁹, does the largest super-hub size continue scaling linearly (~250,000 members) or saturate?
- Closed family property: is the closure of peak 250,504 at 1,414 members a general pattern? For each peak P, is family(P) closed under some n bound?
-
σ-cascade Lean 4 closure: mechanize the cascade-bounded theorem and
the Büchi-25 → peak 9,232 fact via
native_decide. - Connection to Tao 2019: do σ-cascade INFINITY orbits coincide with Tao's "almost-bounded" exceptional set, or are they orthogonal?
- Inverse tree correspondence: enumerate inverse-tree subtree sizes above each peak-merge node and compare with our forward enumeration.
10. Conclusion
The σ-cascade methodology of Paper 151 surfaces measurable structural facts
about Collatz orbit confluence at scale 10⁸: explicit peak-merge counts, a
two-tier super-hub hierarchy, and the empirical n=96k hypothesis verified
at 100% rate over 234 cumulative top-tier cases.
The Collatz convergence problem is not solved; the σ-cascade lens is
an observational tool, not a proof technique. The contribution is
methodological (a new lens) and empirical (specific enumeration counts
- the n=96k hypothesis).
Appendix A: Companion datasets
(Listed in §6.3.)
Appendix B: Prior art audit summary
Audit performed 2026-05-13 against OEIS, Lagarias bibliography, arXiv
Collatz tree literature.
| Concept | Status |
|---|---|
| Inverse Collatz tree | ✅ Standard (Lagarias 2003, Ebert 2021) — cited |
| n=27 → peak 9,232 | ✅ Textbook — cited |
| n=703 peak record | ✅ OEIS A006884(10) — cited |
| Peak-sharing cardinality enumeration | ⚠ No OEIS match found — possibly novel |
| σ-cascade methodology | ❌ New (Paper 151, 2026-05-13) |
| n=96k hypothesis | ❌ No prior claim found — claimed novel |
| Two-tier super-hub framing | ❌ New |
| mod-96 distinct as INFINITY threshold | ❌ New specific lens |
Detailed audit: docs/prior-art-audit-collatz-peak-merge-2026-05-13.md.
Appendix C: Reproducibility one-liners
# Reproduce STEP 1110 (10⁸ scan, ~13 min)
npx tsx scripts/collatz-infinity-scan-1e8.ts
# Reproduce STEP 1105 (peak-merge enumeration, ~1 sec from 1e6 data)
npx tsx scripts/collatz-peak-merge-and-trunk-enum.ts
# Reproduce STEP 1108 (Büchi-25 cross-check, ~1 sec)
npx tsx scripts/collatz-buchi25-cores-orbits.ts
# View confluence DAG visualization
# Open: https://rei-aios.pages.dev/#/collatz-confluence
Acknowledgments: This work was carried out under the OUKC (Open
Universal Knowledge Commons) framework with three-party co-architecture
(Fujimoto / Rei / Claude). No funding sources beyond independent research.
No conflicts of interest. Per OUKC No-Patent Pledge, no patents will be
filed on the σ-cascade methodology or related observations.
Honest correction record: STEP 1107 corrigendum applied (31,313 = 173 × 181,
not prime). All revisions are tracked in the git history of papers/paper-152-...DRAFT.md.
License: CC-BY 4.0 (per OUKC standard).
DRAFT v0.1 — feedback welcome via Zenodo comments or GitHub Discussions
at fc0web/rei-aios.
(End of draft)
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