Finite Gilbreath extension component maximum

Leila Muney, Holes in Valid-Extension Sets of Finite Gilbreath Sequences, arXiv:2606.23721v2, Section 16(4), asks:

Is max_{S∈G_n} #comp(K_S)=2^(n-4) for all n≥5? Verified for n≤10.

The integer triangle is s_a^0=s_a and s_a^b=|s_(a+1)^(b-1)-s_a^(b-1)| at every valid index. A prefix is Gilbreath when every positive-row left entry is one. The family G_n consists of every strictly increasing Gilbreath integer prefix of length n beginning with two and three. K_S contains every integer k for which appending k is Gilbreath; no increasing condition is imposed on this appended integer. Components are literal maximal runs of common difference two in the finite one-parity extension set.

D5/S3/Combinatorics/FiniteDifferences/GilbreathExtension.result gives the universal upper bound and eligible attainment for every n≥5, together with finiteness of the full extension set, its single parity and finiteness of its literal maximal-run set. The statement is exactly componentMaximum.

The source's Theorem 35 already supplies attainment, while Section 16(4) asks for the matching universal upper bound beyond the computed lengths. The bounded literature qualification covers the complete original v2, official arXiv feeds with one and two records, and complete visible MathDB searches with twenty and seven results. No exact whole resolution was found in those scopes. No worldwide absence or priority claim is made.

The distinct maximum-cardinality and uniqueness question, Conjecture 30, has a posted Shivam Patel proof at MathDB 375513 and is excluded.

For a nonempty prefix of length at least two, its actual appended triangle identifies K_S with the preimage of K_(difference S) under k ↦ |k-last(S)|. This identity includes every integer k. Finite absolute-value preimages preserve finiteness and a single parity. The literal endpoint-run bridge enumerates a,a+2,...,a+2m and excludes the adjacent points.

A parity interval has at most two parity intervals in an absolute-value preimage, with its nonnegative part clipped in the correct parity. Each maximal-run start in an interval cover must be the left endpoint of one of its covering intervals; a run start has a unique right endpoint. Thus a cover by r intervals gives at most r runs.

The actual bottom rows (1,x,y),(1,z),(1) force (y,z)=(0,0),(2,2),(2,0),(0,2), or (4,2). Their reverse fiber is the even interval from zero through y+z+2. Consequently every four-entry Gilbreath prefix has a one-interval full extension set. Applying the exact preimage recursion n-4 times produces a cover by at most 2^(n-4) parity intervals.

For attainment, the source family V_n is used with attribution to Definition 31 and Theorem 35. Its first entry is two, its entries at positions 2 through n-1 are 2^(i-1)+1, and its last entry is 2^(n-1)-1. Internally it is indexed by m=n-4 and includes the seed V_4=(2,3,5,7), whose full extension set is the single run from three through eleven. The difference row of V_(n+1) is V_n shifted down by one. The full extension set of the m-indexed family lies between 2m+3 and 2^(m+4)-2m-5. Every extension after shifting down by one is therefore at least two. Its absolute-value preimage consists of two separated reflected copies of its maximal runs, so the count doubles at each step and equals 2^m. Strict increase, the initial entries, exact length and the Gilbreath property are all proved for this family.

A valid increasing prefix of any length n≥5 with more than 2^(n-4) literal maximal step-two runs would refute the upper bound. A failure of eligibility or of the exact run count in the attributed family would invalidate the attainment part. The formal statement assumes neither bound nor attainment.

The canonical Lean theorem is GilbreathExtension.result; all supporting triangle, finiteness, parity, interval-cover, base and family proofs are in its source module. Finite numerical checks are not used as the all-length proof. The theorem uses ordinary integer and natural arithmetic and no new axiom.

Tier 1 explicit recent external closing question, prospectively registered in issue 17115. Admission basis: open-problem-resolution. The whole theorem has proof_shape: content; its live proof uses the new uniform interval-cover estimate and the unconditional formal reconstruction of the known attaining family. Bind-only helpers are consumed by these proofs. Attainment earns no separate open-problem credit. The universal bound and attainment form one external-question result.

What the settlement shows

The decisive mechanism is interval-cover doubling from a realizable four-entry seed, rather than a count of the points in the extension set. The internal upper-bound lemma applies to every Gilbreath list of length at least four, without increase or the specified initial pair. This is a consequence in the same development, not another resolved external question. The maximum follows from a universal bound and one witness per length; no finite enumeration of the eligible family is assumed.

The bounded literature search establishes no worldwide priority or perpetual freshness guarantee. Other questions of the source, including the distinct cardinality and uniqueness question and the prime Gilbreath conjecture, are outside this result. The source license is an arXiv perpetual non-exclusive license, not a code license; the Lean and Scribe are independently authored, and no author program is copied or executed.