The one-common-Zeckendorf-term greedy sequence is a permutation

This dossier anchors exactly the OEIS A370630 conjecture: “Conjecture: this sequence is a permutation of the positive integers.” The record title is “Lexicographically earliest sequence of distinct positive integers such that the Zeckendorf expansions of two consecutive terms have exactly one common term.” Rémy Sigrist is the record author; the conjecture comment has no separate signed author. The source version is revision 16, revision time 2024-05-04T14:57:01-04:00, with offset 1,2.

Let F(0)=0, F(1)=1 and F(k+2)=F(k+1)+F(k). For every integer m>=1, let Z(m) be its unique finite set of nonconsecutive indices k>=2 whose Fibonacci weights sum to m. Set a(1)=1. At every integer n>=1, a(n+1) is the least integer m>=1 outside the actual prefix {a(1),...,a(n)} with |Z(m) intersect Z(a(n))|=1. The whole assertion includes existence of an eligible unused successor at every finite positive stage, distinctness at positive indices, and, for every m>=1, a finite n>=1 with a(n)=m, all for that same sequence.

The linked PARI definition is https://oeis.org/A370630/a370630.gp.txt. Its bit j encodes fibonacci(2+j); its candidate test is !seen(v) && hammingweight(bitand(zp, tozeck(v)))==1. Ordinary binary overlap, at-least-one overlap, zero overlap and finite checked coverage are different assertions. The posted 1..10000 driver does not prove the whole event.

The exact whole event is preregistered at https://github.com/the-omega-institute/trureturing/issues/22916#issuecomment-6455424919. The registration body SHA-256 is be4c48b37d89250e2d18945944f59181886adf667620d4fa4e5ddfea8a660c15. The canonical W-digit library supplies the representation interface; the remaining question concerns totality and coverage of the actual least-unused history, rather than existence of some permutation with a similar adjacency condition.

Canonical representation alone does not supply the next unused adjacent value or show that no positive value is omitted forever. A finite prefix cannot settle either unbounded obligation. The source record labels the permutation assertion a conjecture; that label is a fact about this pinned record, not a global openness or priority certificate.

A fresh sufficiently distant Fibonacci index pairs with an occupied current index to give an eligible value larger than the sum of the used list. This supplies every transition of a total recursion. The list-prefix invariant and unused choice give injectivity.

If an omitted positive target has unbounded visited neighbors, numerical minimality bounds each following value by that target, contradicting injectivity. First appearances of arbitrarily large support indices force visited canonical pairs. A two-successor comparison forces a pair rooted at index two when the smaller index of such a pair is at least four. Finite cutoffs yield an unbounded pair family rooted at some fixed index. Every positive support containing that root is then visited. Fresh canonical triples propagate this hub property to separated indices and supply unbounded visited neighbors for every positive target.

A missing eligible successor at a positive stage, a repeated value at distinct positive indices, or a positive value absent from the same actual greedy sequence would contradict the asserted event. Replacing the full prefix or changing the exact-one canonical overlap condition would invalidate the source correspondence even if a different theorem compiled.

The designated theorem is D5/S3/Combinatorics/Permutation/ZeckendorfGreedyPermutation.result, with closed type IsGreedyPermutation a. Its five conjuncts express initial value, positivity, actual successor eligibility and numerical leastness, injectivity on positive indices, and attainment of every positive integer. The auxiliary index zero is excluded from the event. The canonical support directly reuses D5.S0.Conventions.wdigits, its canonicality, decoding and uniqueness.

The immutable source-byte identities are SHA-256 cd97c6ba9e54e07a3508e12780155b3e5fa2cde74f4227cac025883597923c3a for the 1715-byte exact-ID JSON record and 1f0281c622a920cade70aaf1767970c5b083e7561c54ec7cdbace4e14590c476 for the 626-byte linked PARI definition. The URLs are mutable; these pins and the revision identify the source evidence. The literal two-argument A000120 formula is disambiguated by the title and explicit bitand/hammingweight test.

Repository and pinned Mathlib searches found no exact whole supplier in their searched scope. Two bounded GitHub Lean-code queries, "Zeckendorf" "greedy" language:Lean and "A370630" language:Lean, each returned zero indexed hits on page one with per_page=30 and incomplete_results=false. The exact-ID Crossref query returned total_results=0; OpenAlex returned HTTP 429 and provides no negative supplier evidence. These are bounded searches, not an exhaustive bibliography or absence-of-all-suppliers claim.

What the settlement shows

[proved, D5/S3/Combinatorics/Permutation/ZeckendorfGreedyPermutation.result] The source sequence is total at every positive stage and permutes the positive integers. The decisive mechanism is the combination of unbounded record pairs, the two-successor forcing step, and propagation of the hub property by fresh canonical triples. The proof uses actual numerical minimality and the full history at every stage; it has no assumed infinitude or hub premise.

[open] Relaxing the starting value, changing the weight system or changing the exact-one condition requires new correspondence and proof obligations. No extension or sharp quantitative visitation bound is asserted. A226077 uses ordinary binary terms, A332565 uses zero common Zeckendorf terms, and A370631 uses at least one common term; none is settled by this claim.

[open] No additional source result depending on this conjecture was identified in the authenticated record and linked program. No consequences for an unsearched paper, dependency chain or neighboring conjecture are certified.

External source, registration and search evidence are inherited from the completed source packet and bounded checks; this publication does not claim a new comprehensive literature or owner census. Unindexed work, informal notes, aliases and works lacking A370630 in indexed fields remain outside that search. The record-author attribution does not authenticate a separate signature on the conjecture comment. Source hashes authenticate bytes, not mathematical truth, novelty or priority. No sequence program was executed for this result.