---
title: "A Three-Layer Obstruction for Uniformly Popular Differences"
date: 2026-08-29
version: "0.1.0-candidate"
doi: 10.5281/zenodo.22161245
pdf: https://github.com/ipitchford/odd-order-one-third-popular-differences/releases/download/v0.1.0-candidate/odd-order-one-third-popular-differences-v0.1.0-candidate.pdf
repository: https://github.com/ipitchford/odd-order-one-third-popular-differences
archive: https://zenodo.org/records/22161245
license: CC0-1.0
status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)
---

# A Three-Layer Obstruction for Uniformly Popular Differences

## Summary

AIM Problem 1.18 asks whether a pointwise abundance of differences forces a simple algebraic shape. For a finite nonempty set $A$ in an abelian group, let

$$
r_A(d)=|\{(a,a')\in A^2:a-a'=d\}|,\qquad D=A-A.
$$

The question is whether $r_A(d)>|A|/3$ for every $d\in D$ forces $D$ to be a subgroup or a union of at most three cosets of a common subgroup.

This anonymous, unrefereed candidate answers that question **no**. In the odd cyclic group $\mathbb Z/21\mathbb Z$, take

$$
A=\{0,1,2,7,9,14,15,16\}.
$$

Every represented difference occurs at least three times, so the strict inequality $3>8/3$ holds throughout $A-A$. But $|A-A|=15$, which is neither a subgroup order in a group of order 21 nor the size of at most three cosets of any common subgroup.

The same mechanism gives an infinite family: every fixed threshold below $2/5$ admits a counterexample in an odd cyclic group. This is a lower obstruction only. The release does **not** claim that $2/5$ is sufficient or sharp.

## Summary for specialists

Under $\mathbb Z/21\mathbb Z\cong C_3\times C_7$, the finite witness becomes

$$
A=(C_3\times\{0\})\cup(\{0,1\}\times\{1\})\cup(C_3\times\{2\}).
$$

Its quotient-layer differences are $0,\pm1,\pm2$. The representation spectrum on $D=A-A$ is

$$
3^{[6]},\quad 4^{[6]},\quad 7^{[2]},\quad 8^{[1]},
$$

and $6\cdot3+6\cdot4+2\cdot7+1\cdot8=64=|A|^2$.

More generally, let $H$ be a finite abelian group of order $h$, let $\varnothing\ne Y\subsetneq H$ have size $m$, and let $q\ge7$. For

$$
A=(H\times\{0\})\cup(Y\times\{1\})\cup(H\times\{2\})\subset H\times C_q,
$$

the candidate proves

$$
A-A=H\times\{0,\pm1,\pm2\},\qquad |A|=2h+m,
$$

and

$$
\min_{d\in A-A}r_A(d)=\min(h,2m).
$$

Thus the strict one-third condition holds exactly when $2h/5<m<h$. Taking $H=C_p$, $q=7$, and $m=(p+1)/2$ for sufficiently large odd primes $p\ne7$ gives minimum popularity ratio

$$
\frac{2p}{5p+1}\longrightarrow\frac25
$$

from below. Since $C_p\times C_7$ is cyclic and $|A-A|=5p$ cannot be the size of at most three cosets of one subgroup, this produces the claimed odd-cyclic obstruction for every $\theta<2/5$.

## Technical account

The proof is elementary and has three moving parts.

1. **Five distinct quotient layers.** The assumption $q\ge7$ keeps $0,\pm1,\pm2$ distinct in $C_q$.
2. **Exact multiplicity counts.** Differences between the two full layers have $h$ representations; differences meeting the partial layer have $2m$; central differences have at least $2h$.
3. **A cardinality obstruction.** In the finite witness, 15 is incompatible with every subgroup order of $C_{21}$ and with $t|K|$ for $t\le3$. In the family, the same argument applies to $5p$ inside $C_{7p}$.

The finite witness can therefore be checked line by line without trusting the software. The programs provide exhaustive arithmetic replay and hostile controls; they are supporting evidence, not a substitute for the written argument.

## Evidence, assurance and limitations

The package contains the manuscript and Markdown source, a machine-readable claim registry, Python and JavaScript verifiers, a 264-case bounded regression of the family formula, four deliberately corrupted mutations, replay receipts, source and citation audits, and the internal five-role review record. A fresh extraction passed all 12 publication-baseline gates. Public GitHub Actions also passed on Node 18, 20, and 22. GitHub and Zenodo copies of the archive, PDF, and checksum sidecar match local bytes.

Those facts establish public availability, byte identity, producer-side replay, and mutation sensitivity. They do not establish any of the following:

- an unaffiliated rerun or independent reimplementation;
- proof-assistant formalization;
- external additive-combinatorics review;
- journal or comparable editorial peer review;
- exhaustive novelty or absolute priority;
- sufficiency or sharpness at $2/5$;
- minimality of the $C_{21}$ witness.

The internal editorial result is `PASS_WITH_NOTES`. The notes are substantive boundaries, not cosmetic disclaimers.

## Relationship to earlier work

The source is the final question in AIM Problem 1.18, attributed there to V. Lev. A source audit corrected an earlier working label: the later Croot–Lev Problem 7.14 records the neighbouring **half-threshold** question, not this strict one-third question.

Hou, Li and Yang's 2026 preprint addresses that half-threshold problem in two-torsion-free groups. The present construction lies below one half and targets the separate AIM one-third implication.

A targeted search across ordinary web search, arXiv, Crossref, OpenAlex, and the closest reference chains found no direct statement of the $C_{21}$ witness, the three-layer family, or the below-$2/5$ odd-cyclic obstruction. That is partial novelty evidence only. Formula indexing, informal circulation, and specialist knowledge can defeat such a search, so the release invites priority corrections and makes no first or priority claim.

## Who should care

| Reader | What is useful now | What remains before promotion |
|---|---|---|
| Additive combinatorialists | A very small explicit witness and a one-line family mechanism for the AIM question | Check the source mapping, proof, prior art, and endpoint behaviour |
| Extremal set theorists | A concrete lower obstruction approaching $2/5$ from below | Determine the sharp threshold and classify near-extremizers |
| Formalizers | A short proof with finite-group counting and divisibility arguments | Formalize the layer calculation and coset-cardinality exclusion |
| Computational reviewers | Two transparent encodings, mutations, receipts, and immutable assets | Reimplement independently rather than importing either verifier |
| Research-methods readers | A claim-level package separating replay, review, novelty, and publication | Do not treat this one release as evidence of workflow acceleration or impact |

## How to inspect and reproduce it

Use candidate tag `v0.1.0-candidate` or the Zenodo version DOI, not the moving `main` branch. The core replay requires Python 3.11 or later and Node.js 18 or later, with standard libraries only:

```sh
python3 verify_counterexample.py --receipt REPLAY_RECEIPT.json
python3 -O verify_counterexample.py
node verify_counterexample.mjs
python3 test_mutations.py
```

Both verifiers should print `PASS`, and every mutation should be rejected. Then inspect `paper.md`, `CLAIMS.json`, `ASSURANCE.md`, `CITATION_AUDIT.md`, and `PRIOR_ART.md`. Successful execution confirms the declared arithmetic predicates in the released code; it does not independently certify the theorem or its novelty.

## Most valuable next projects

1. **Endpoint theorem.** Prove or falsify the subgroup-or-three-coset conclusion at $2/5$, carefully separating strict and non-strict formulations.
2. **Minimal counterexample.** Determine whether a smaller odd cyclic group or smaller set supports the strict one-third failure.
3. **Structural classification.** Describe sets whose minimum represented-difference multiplicity approaches $2|A|/5$.
4. **Independent assurance.** Reconstruct the proof and implement the predicates without importing the released code.
5. **Specialist novelty audit.** Search MathSciNet, zbMATH, older problem correspondence, and differently phrased inverse-difference literature.

## What is in the package

The immutable release includes the manuscript PDF and sources, accessible Markdown, exact verifiers, mutation tests, replay receipts, claim and nonclaim registries, provenance and licensing files, source and novelty audits, and a complete checksum manifest. The version DOI is the citation target; any correction should be a versioned successor rather than a silent change.




## Open directions for follow-up research

- Determine whether 2/5 is sufficient or sharp for the subgroup-or-three-coset conclusion, including the strict and non-strict endpoint formulations.
- Classify near-extremal sets whose minimum represented-difference multiplicity approaches 2|A|/5.
- Determine the smallest odd-order or cyclic counterexample; the C21 witness is not claimed minimal.
- Obtain an unaffiliated additive-combinatorics audit of the source mapping, proof, and exact relationship to prior literature.
- Independently reimplement the construction and formalize the elementary counting and coset-size arguments in a proof assistant.

## Research process, metrics and reusable methods

This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work.

- Work ID: ep-work:odd-order-one-third-popular-differences
- Attempt and metric receipts: ep-attempt:odd-order-one-third-popular-differences-publication: published / positive; scope publication-only; target Publish the immutable candidate on Evidence Press with complete reader-first metadata and communication assets, passed composite CI and accessibility, exact two-cycle protocol sealing, canonical public readback, and terminal linked metrics without changing the research claim ceiling.; active forecast 90 minutes (60-150); Fermi components release page, operating metadata and source binding: 1 x 15/25/45 minutes low/central/high (One short exact mathematical candidate with established public identifiers and a new prospective operating-model record.); art, Open Graph, audio and thumbnail generation plus inspection: 1 x 15/25/40 minutes low/central/high (One standard deterministic motif, one two-minute narration and one reproducible thumbnail.); first composite seal, CI and guarded deployment: 1 x 15/20/35 minutes low/central/high (One A/B content and protocol pair followed by one PR, required CI and a guarded deployment.); live-slug record, second seal, redeploy and canonical readback: 1 x 15/20/30 minutes low/central/high (One C/D ledger pair and one final exact browser and machine-surface readback.); positive-signal/closure probabilities 0.95/0.85 within 150 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 27/unknown/0/4/0/0; cycles positive/negative/inconclusive 0/0/0; falsification gates 0; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -63 minutes; ratio 0.3; inside interval false; positive-signal/target-closure Brier scores 0.0025/0.0225; missing telemetry activeHumanMinutes: No instrument captured human direction or review time at the publication-attempt boundary.; deduplicatedModelTokens: No task-local model-token counter was active for the publication-only attempt, so a scoped value cannot be separated without reconstruction. The earlier research goal and auxiliary transcription-QA counters remain separately reported.; uncachedInputTokens: The runtime does not expose an uncached-input counter.; appended measurement corrections publication timing scope, user-visible elapsed time and rework: Retain the immutable 31-minute scoped receipt as a phase measurement, but do not use it as the end-to-end publication duration. Record 83 minutes as the user-reported request-to-completion elapsed time. Treat five minutes as the contemporaneously recorded minimum rework before the present art repair; further active rework not prospectively timed remains unknown rather than zero. (reason: The frozen 31-minute outcome begins at the later publication-attempt registration and therefore does not measure the full task-pane procedure experienced by the user. The user reported 83 minutes from the publication request to completion. In addition, the contemporaneous readback commit recorded five rework minutes before a later corrective commit replaced that value with zero, and the published hero art contains a clipped caption that requires a further presentation repair.). Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json
- Intended aims: science
- Artifact roles: research-output, evidence-assessment, method-demonstration, communication
- Decision object: counterexample — An explicit eight-point C21 witness and an elementary three-layer family that refute the strict one-third implication and obstruct every fixed threshold below 2/5. Scope: The exact C21 witness, the H times C_q three-layer counting identity, and the odd-cyclic below-2/5 corollary; not the endpoint 2/5, a sharp-threshold theorem, witness minimality, exhaustive novelty, or priority.
- Reusable methods: Counterexample- and proxy-first analysis (counterexample-proxy-first); Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json
- Targeted clocks: discovery, assurance, publication
- Semantic bridge: explicit — The package maps AIM Problem 1.18's pointwise representation hypothesis directly to a quotient-layer construction, derives every multiplicity by elementary counting, and separates the finite counterexample, asymptotic lower obstruction, and unresolved endpoint. Remaining risks: The source interpretation and proof have not been reconstructed by an unaffiliated additive-combinatorics specialist.; The targeted novelty search may miss differently phrased, unindexed, or informal prior work.; The construction approaches 2/5 only from below and supplies no information proving sufficiency at the endpoint or above.; Finite enumeration and two producer-authored encodings do not establish independent reproduction..
- Human judgement gates: Check the AIM source wording and keep the one-third question distinct from the published Croot–Lev half-threshold problem.; Check the three quotient-layer multiplicity count and the coset-cardinality obstruction as mathematics rather than inferring truth from replay.; Keep a lower obstruction below 2/5 separate from sufficiency or sharpness at 2/5.; Keep a found C21 witness separate from a minimality claim.; Require external specialist search and assessment before any first, novelty, priority, solved-problem, or field-acceptance language.
- Next assurance action: Obtain an unaffiliated additive-combinatorics reconstruction and literature assessment, then investigate the endpoint 2/5 and the smallest odd cyclic counterexample without promoting the present claim ceiling.
- Claim ceiling: An anonymous unrefereed counterexample and elementary-family candidate with exact producer-side replay, adversarial controls, public immutable assets, and internal PASS_WITH_NOTES; not a proof of 2/5 sufficiency or sharpness, C21 minimality, independent validation, formal verification, external specialist or editorial peer review, exhaustive novelty, absolute priority, or demonstrated workflow impact.
- Aim-scoped impact evidence:
  - science: NO_IMPACT_EVIDENCE — Faster or more reliable resolution and assurance of additive-combinatorics open problems in AI-assisted counterexample search, exact replay, proof compression, source audit, and guarded candidate publication; design none; comparator No matched conventional research, theorem-review, or publication workflow was registered.; estimand No effect on discovery time, active human effort, compute, correction rate, proof quality, independent-assurance time, uptake, citation, or field outcomes was estimated.; no real-world effect evidence asserted
- Parent handoffs: depends-on-claim https://aimath.org/WWN/additivecomb/additivecomb.pdf; inherited claim: AIM Problem 1.18 asks whether the strict one-third pointwise popularity hypothesis forces the difference set to be a subgroup or a union of three cosets.; inherited ceiling: The external problem statement fixes the target and attribution; it does not validate this candidate's construction, proof, novelty, or priority.



## Verification status

Anonymous, unrefereed counterexample and elementary-family candidate at internal PASS_WITH_NOTES. The C21 computation and bounded family checks replay across Python and JavaScript, and the argument is supplied in full for specialist inspection. The source attribution was corrected to AIM Problem 1.18 after audit. A targeted novelty search found no direct prior statement, but novelty and priority remain unresolved. No unaffiliated rerun, independent reimplementation, proof-assistant formalization, external specialist review, or editorial peer review has occurred.

## References

1. American Institute of Mathematics. (2004). Palo Alto Problems, Problem 1.18 (V. Lev). <https://aimath.org/WWN/additivecomb/additivecomb.pdf>
2. Lev, V. F. Popular differences problem page. <https://math.haifa.ac.il/seva/Problems/difset.html>
3. Croot, E., & Lev, V. F. (2007). Open problems in additive combinatorics. In Additive Combinatorics, CRM Proceedings and Lecture Notes 43, 207–233. <https://math.haifa.ac.il/seva/Papers/Montpr.pdf>
4. Hou, J., Li, W., & Yang, K. (2026). Popular Differences and the Croot–Lev Half-Threshold Problem. arXiv:2606.29297v1. <https://arxiv.org/abs/2606.29297>