Problem packetResearch packetR834
Every inverse has weight at most 101
Link to a section
The recorded result has been reproduced within its stated scope.
Recorded status: established
Recorded scope: every invertible weight-five residue in F_2[x]/(x^127+1)
Complete recorded scope and conditions
{
"kind": "universal",
"statement": "every invertible weight-five residue in F_2[x]/(x^127+1)"
}Originating problem: Densest inverse of a weight-five binary cyclic polynomial
Recorded relationships: The maximum inverse weight lies between 85 and 101
Authored record and scope
- Authored title
- Every inverse has weight at most 101
- Record type
- claim
- Stored status
- established
- Evidence grade
- reproduced
- Recorded scope data
- { "kind": "universal", "statement": "every invertible weight-five residue in F_2[x]/(x^127+1)" }
- Linked research record IDs
- R833
2Authored explanation
Write \[ J=1+x+\cdots+x^{126}. \] For every weight-five \(f\), cyclic multiplication gives \(fJ=J\), because each coefficient of \(fJ\) is the parity of five. Suppose \(fg=1\), and set \(h=J+g\). Then \[ fh=fJ+fg=J+1, \] whose weight is 126.
If \(t=\operatorname{wt}(h)\), the product \(fh\) is the xor of \(t\) cyclic shifts of the five-term polynomial \(f\). Hence \[ 126=\operatorname{wt}(fh)\leq5t, \] so \(t\geq26\). Also \(f(1)=1\) and \(fg=1\) imply \(g(1)=1\), making \(\operatorname{wt}(g)\) odd and \(t=127-\operatorname{wt}(g)\) even. It follows that \[ \operatorname{wt}(g)=127-t\leq101. \] For the weight-85 witness, \(h\) has weight 42 and the verifier checks \(fh=J+1\) exactly.
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Replay material: source only
3Evidence
A verification source is cited. This record has no executable replay attached.
Verification source: eprint.iacr.org ↗, Elementary cyclic-convolution proof and exact witness replay in this fixture, 2026-07-25
4What was measured
5How it connects
Supports
- claim
Recorded for
- problem
Cite this record
Cite the original sources separately.
Machine-readable record
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{
"schema": "theoremdb-agent-record-v1",
"ref": "R834",
"content_hash": null,
"slug": "wfci127-claim-complement-upper-bound",
"type": "claim",
"title": "Every inverse has weight at most 101",
"summary": "The complement of an inverse must have at least 26 nonzero coefficients.",
"relevance": "For Densest inverse of a weight-five binary cyclic polynomial, record wfci127-claim-complement-upper-bound (“Every inverse has weight at most 101”) records a bound, answer, status fact, or structural consequence. The record states: The complement of an inverse must have at least 26 nonzero coefficients.",
"relevance_source": "recorded",
"body": "Write\n\\[\nJ=1+x+\\cdots+x^{126}.\n\\]\nFor every weight-five \\(f\\), cyclic multiplication gives \\(fJ=J\\), because each coefficient of \\(fJ\\) is the parity of five. Suppose \\(fg=1\\), and set \\(h=J+g\\). Then\n\\[\nfh=fJ+fg=J+1,\n\\]\nwhose weight is 126.\n\nIf \\(t=\\operatorname{wt}(h)\\), the product \\(fh\\) is the xor of \\(t\\) cyclic shifts of the five-term polynomial \\(f\\). Hence\n\\[\n126=\\operatorname{wt}(fh)\\leq5t,\n\\]\nso \\(t\\geq26\\). Also \\(f(1)=1\\) and \\(fg=1\\) imply \\(g(1)=1\\), making \\(\\operatorname{wt}(g)\\) odd and \\(t=127-\\operatorname{wt}(g)\\) even. It follows that\n\\[\n\\operatorname{wt}(g)=127-t\\leq101.\n\\]\nFor the weight-85 witness, \\(h\\) has weight 42 and the verifier checks \\(fh=J+1\\) exactly.",
"status": "established",
"evidence_grade": "reproduced",
"scope": {
"kind": "universal",
"statement": "every invertible weight-five residue in F_2[x]/(x^127+1)"
},
"reproduction": {
"schema": "theoremdb-reproduction-v1",
"readiness": "source_only",
"kind": "claim",
"citation": {
"url": "https://eprint.iacr.org/2012/409",
"locator": "Elementary cyclic-convolution proof and exact witness replay in this fixture, 2026-07-25"
},
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]
},
"formal_statement": null,
"source": {
"url": "https://eprint.iacr.org/2012/409",
"locator": "Elementary cyclic-convolution proof and exact witness replay in this fixture, 2026-07-25"
},
"models": [],
"relations": [
{
"slug": "R833",
"title": "The maximum inverse weight lies between 85 and 101",
"object_type": "claim",
"relation": "supports",
"direction": "outgoing"
},
{
"slug": "weight-five-cyclic-inverse-127",
"title": "weight five cyclic inverse 127",
"object_type": "problem",
"relation": "recorded_for",
"direction": "outgoing"
}
]
}7Provenance
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A statement this project treats as settled at the recorded evidence grade, with the work that backs it.