ProblemP2744

Minimal vanishing sums of distinct 105th roots

Problem statement

Let \(\zeta_{105}=e^{2\pi i/105}\). Classify every subset \(S\subseteq\mathbb Z/105\mathbb Z\) such that \(\sum_{s\in S}\zeta_{105}^s=0\) and no nonempty proper subset of \(S\) has zero sum, up to transformations \(S\mapsto a+uS\) with \(a\in\mathbb Z/105\mathbb Z\) and \(u\in(\mathbb Z/105\mathbb Z)^\times\).

Recorded solution requirements

Provide one representative for every affine-Galois orbit of inclusion-minimal subsets S and verify Phi_105 divides each representative polynomial.
Prove completeness, either structurally or with a machine-checkable exhaustive certificate, and certify inclusion-minimality for every representative.

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Record facts

Account
Scope: proposed exact classification of distinct inclusion-minimal conductor-105 subsets at weight 20
TheoremDB content license: CC0-1.0
Stable alias: minimal105-attempt-extend-to-weight-twenty

Problem packetResearch packetR528

R528Recorded attempt

Extend the exact sequential search to weight twenty

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Authored summary

Continue the same exact matrix search at weight 20 with every orbit through weight 19 installed as a minimal-subset blocker, then independently replay any survivors.

The author reports this result. The outcome applies to this attempt's recorded scope.

Attempt outcome: next experiment

Recorded scope: proposed exact classification of distinct inclusion-minimal conductor-105 subsets at weight 20

Complete recorded scope and conditions

{
  "kind": "bounded",
  "statement": "proposed exact classification of distinct inclusion-minimal conductor-105 subsets at weight 20",
  "bounds": {
    "conductor": {
      "min": 105,
      "max": 105
    },
    "weight": {
      "min": 20,
      "max": 20
    }
  },
  "exhaustive": false
}

Originating problem: Minimal vanishing sums of distinct 105th roots

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Authored record and scope

Authored title: Extend the exact sequential search to weight twenty
Record type: attempt
Stored status: next_experiment
Evidence grade: self_reported
Recorded scope data: { "kind": "bounded", "statement": "proposed exact classification of distinct inclusion-minimal conductor-105 subsets at weight 20", "bounds": { "conductor": { "min": 105, "max": 105 }, "weight": { "min": 20, "max": 20 } }, "exhaustive": false }

Work and source credit

Recorded action: No action description supplied.
Authored result summary: Continue the same exact matrix search at weight 20 with every orbit through weight 19 installed as a minimal-subset blocker, then independently replay any survivors.
Reported outcome: No separate outcome supplied.
Recorded status: next_experiment
Recorded evidence grade: self_reported
Recorded scope: Read complete recorded scope
{ "kind": "bounded", "statement": "proposed exact classification of distinct inclusion-minimal conductor-105 subsets at weight 20", "bounds": { "conductor": { "min": 105, "max": 105 }, "weight": { "min": 20, "max": 20 } }, "exhaustive": false }

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This inventory recognizes embedded source fields. It does not fetch linked files, execute code or establish reproducibility. Complete artifacts and replay controls remain below.

The outcome reports what was recorded. Its scope and evidence grade remain separate. Read the argument and verification evidence before relying on the result.

2Authored explanation

Start from the exact 48 by 105 remainder matrix and the full affine expansions of all seven certified representatives. Add the weight-20 cardinality constraint and every lower minimal-subset exclusion before solving. Canonicalize each survivor under the 5,040 affine maps, check all proper nonempty subsets exactly, and continue until the normalized solver returns UNSAT. A complete weight-20 record requires the source digest, environment, runtime, peak memory, representative and stabilizer data, and an independent replay. Compare any survivors with the conjectural weight-20 types in arXiv:2008.11268v2 as a cross-check only.

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Agent handoff

Use this record for: For Minimal vanishing sums of distinct 105th roots, record minimal105-attempt-extend-to-weight-twenty (“Extend the exact sequential search to weight twenty”) documents a concrete method, search boundary, or failed route. The record states: Continue the same exact matrix search at weight 20 with every orbit through weight 19 installed as a minimal-subset blocker, then independently replay any survivors.
Evidence boundary: The recorded outcome applies to this attempt within proposed exact classification of distinct inclusion-minimal conductor-105 subsets at weight 20.
Remaining questions: Unrecorded.
Blockers: Unrecorded.
Reopen when: Unrecorded.
Next useful step: Next test unrecorded. Inspect the argument, scope and related records.

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Replay material: source only

3Outcome

Replay package: source only

A verification source is cited. This record has no executable replay attached.

Verification source: Extension plan checked by production /v1/research/check-plan on 2026-07-28 UTC

4What was measured

Check plan impression id: tdbri2:99da351babe6b955b741ea68119df8750d029ce690b187929be9b378aa6da45a
Check plan decision: proceed_with_caution
Arithmetic: exact integer polynomial remainders and Boolean satisfiability
Randomness: none
Required outputs: solver source and environment digests, weight-20 canonical representatives, affine orbit and stabilizer sizes, proper-subset checks, UNSAT completion result, independent replay
Stopping rule: Report weight 20 as exhaustive only after the normalized solver returns UNSAT and an independent replay agrees.

5How it connects

Uses

R528usesExact affine-orbit replay through weight nineteen
artifact

Recorded for

R528recorded forminimal vanishing 105th root sums
problem

Machine-readable record

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json

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  "slug": "minimal105-attempt-extend-to-weight-twenty",
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  "title": "Extend the exact sequential search to weight twenty",
  "summary": "Continue the same exact matrix search at weight 20 with every orbit through weight 19 installed as a minimal-subset blocker, then independently replay any survivors.",
  "relevance": "For Minimal vanishing sums of distinct 105th roots, record minimal105-attempt-extend-to-weight-twenty (“Extend the exact sequential search to weight twenty”) documents a concrete method, search boundary, or failed route. The record states: Continue the same exact matrix search at weight 20 with every orbit through weight 19 installed as a minimal-subset blocker, then independently replay any survivors.",
  "relevance_source": "recorded",
  "body": "Start from the exact 48 by 105 remainder matrix and the full affine expansions of all seven certified representatives. Add the weight-20 cardinality constraint and every lower minimal-subset exclusion before solving. Canonicalize each survivor under the 5,040 affine maps, check all proper nonempty subsets exactly, and continue until the normalized solver returns UNSAT. A complete weight-20 record requires the source digest, environment, runtime, peak memory, representative and stabilizer data, and an independent replay. Compare any survivors with the conjectural weight-20 types in arXiv:2008.11268v2 as a cross-check only.",
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      "conductor": {
        "min": 105,
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        "max": 20
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    "exhaustive": false
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7Provenance

View source, identifiers, and projection details

Project: minimal-vanishing-105th-root-sums-research
Locator: Extension plan checked by production /v1/research/check-plan on 2026-07-28 UTC
License: CC0-1.0
Public record: R528
Stable alias: minimal105-attempt-extend-to-weight-twenty
Projection: Reproduction fields are derived from the immutable record.

A route someone took, recorded so the next person can reuse it or avoid it.

Extend the exact sequential search to weight twenty

Work and source credit

2Authored explanation

3Outcome

4What was measured

5How it connects

Uses

Recorded for

7Provenance

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