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Holes in Valid-Extension Sets of Finite Gilbreath Sequences

T0 review · 0 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper proves that the set of valid extensions of a finite Gilbreath sequence can fail to fill its natural parity interval, gives the exact order-sensitive condition under which it fills the interval, and identifies the first failure at

desk verdict A solid, correct repair of Gatti's interval claim, with an exact extension-set theory that deserves serious refereeing. read the letter →

arxiv 2606.23721 v2 pith:AGQEHOWQ submitted 2026-06-18 math.CO math.NT

Leila Muney This is my paper · ORCID
classification math.COmath.NT MSC 05A1511B75
keywords Gilbreathsequencesvalid-extensionsetdifferencetriangleabsolutedifferencesintervalcompletenesssubsetsumsreverse-treealgorithmholes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies the local extension problem for finite Gilbreath sequences: given such a sequence, which integers can be appended to it while preserving the Gilbreath property? A previously proposed characterization in the literature predicted that the set of valid extensions always fills a parity-compatible interval around the last term. The paper shows this is false: the valid-extension set can have interior holes, and the smallest failure occurs at length 5 for the sequence (2,3,5,9,15). It replaces the interval claim with an exact membership criterion, a reverse-tree algorithm that computes the extension set, and a sharp condition—order-sensitive, unlike the classical subset-sum completeness criterion—that determines exactly when the set does fill the interval. If the paper is right, it settles the finite structure of these extension sets and refutes the previously proposed interval characterization.

What carries the argument

The load-bearing object is the right anti-diagonal (e_1,...,e_{n-1}) of the difference triangle, together with the folding map F_S that composes the absolute-value steps x -> |x-e_i| in that fixed order. The reverse tree inverts these folds step by step from the required apex value 1, yielding the full set of valid distances. The interval-completeness criterion compares each e_i with the sum of the later anti-diagonal entries plus 1, exactly the no-gap condition that keeps the reverse tree from falling apart; unlike the sorted subset-sum criterion, the order is forced by the geometry of the triangle.

What would settle it

Write a direct, triangulation-based program that, for every increasing sequence in G_5, appends each candidate integer and recomputes the full difference triangle; if any sequence other than (2,3,5,9,15) has a hole in its valid-extension set, or if (2,3,5,9,15) admits the value 15 as a valid extension, the central claims fail. Independently recomputing N_11 (the number of G_11 sequences) via a different method would test the enumeration.

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Extended reading notes

Core claim

The central claim is that the valid-extension set K_S is governed by the right anti-diagonal of the difference triangle through the iterated absolute-value map F_S(d)=|...||d-e_1|-e_2|...-e_{n-1}|: a value k is valid exactly when F_S(|k-s_n|)=1. On this basis the paper proves (Theorem 20) that K_S equals the candidate interval C_S—all parity-compatible integers within distance A(S)+1 of the last term—if and only if the anti-diagonal entries satisfy e_i <= 1 + sum_{j>i} e_j for all i <= n-2. This is an ordered analogue of the classical subset-sum completeness criterion. The first failure occurs at n=5 for S=(2,3,5,9,15), whose only hole is the last term itself, 15. The paper also proves the m

Load-bearing premise

The theorem-level proofs are hand-argument, but the enumeration data for lengths up to 11, the uniqueness of the first counterexample as stated in Table 1, and the evidence for the maximum-width conjecture all rest on the correctness of the included program, supported only by the statement that it was checked against an independent implementation.

Editorial extensions

If this is right

  • The previously proposed interval-filling characterization of valid extensions is refuted: some Gilbreath sequences admit fewer valid extensions than the candidate interval, and the gap can be measured by a defect h(S).
  • For any individual sequence, K_S can be computed exactly by the reverse-tree algorithm, and membership in the valid-extension set can be decided by evaluating one folding map.
  • Interval completeness is now a checkable, finite condition on the anti-diagonal; every sequence of length at most 4 is interval-complete, and the only length-5 exception is (2,3,5,9,15).
  • The minimal sequence (2,3,5,7,...,2n-1) is the unique minimizer of extension width, with width 5 for all n>=3, while the doubling sequence (2,3,5,9,17,...,2^{n-1}+1) attains width 2^{n-1}+1 and is conjectured to be the unique maximizer.
  • There exist sequences (the V_n family) whose valid-extension set splits into 2^{n-4} connected components in the parity lattice, so extension sets can be exponentially disconnected.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The order-sensitive completeness criterion may apply beyond Gilbreath sequences: it holds for any tuple of nonnegative integers with a final 1 and otherwise even entries, suggesting a general theory of fibers of folding maps where classical subset-sum completeness is the unordered shadow.
  • The defect h(S) measures how much the nested absolute-value structure constrains independent sign choices; one could test whether the fraction of defective sequences tends to a nonzero limit, as the data hint.
  • The V_n family suggests an extremal principle: the gap between the candidate interval and the true extension set grows exponentially when the leading anti-diagonal entry is large, so sequences with widely spread gaps might be the natural extremal objects.
  • A direct check: implement a naive triangle-building extension test independently of the paper's reverse-tree code, and verify the counts N_2...N_11 and the first-hole example.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper studies finite strictly increasing Gilbreath sequences beginning with (2,3). For a sequence S, it defines K_S, the set of integers k such that appending k preserves the Gilbreath property, and analyzes K_S through the nested absolute-value map F_S. The main contributions are: an exact membership criterion and a reverse-tree algorithm for computing K_S (Proposition 18); a comparison of K_S with the candidate interval C_S and the independently signed-sum set S^±; and a sharp order-sensitive condition, e_i ≤ 1 + Σ_{j>i} e_j for 1 ≤ i ≤ n−2, characterizing when K_S = C_S (Theorem 20). The paper proves that this interval-completeness fails first at n = 5 for S = (2,3,5,9,15), with a single hole at 15 (Theorem 24); determines the unique minimum-width sequence L_n (Theorem 25); computes the width of the doubling sequence (Theorem 29); and constructs a family V_n with exponentially many parity-lattice components (Theorem 35). Enumeration data through n = 11 and Python code are included.

Significance. The central structural theorem is a genuine correction to the previously claimed interval-filling characterization and replaces it with an exact ordered analogue of Brown's subset-sum completeness criterion. The hand proofs are self-contained and I found no gaps in the key arguments: the reverse-tree computations, the interval-completeness algebra, the first-hole proof, and the component-doubling construction are all consistent. The paper is also careful in its claims: it explicitly disclaims novelty for the OEIS enumeration and for the extremal-growth identifications, and it includes reproducible code. The computational parts, while not a substitute for a repository or raw output, are corroborated by the OEIS match and are not load-bearing for the main theorems.

minor comments (4)
  1. [Section 17] The code is embedded in the PDF and is stated to reproduce Table 1, but no repository, commit hash, or raw output is provided. For the n = 11 count and the n ≤ 10 verification of Conjecture 30, an independent reader must re-implement from the printed code. Please consider hosting the code and the computed values (or at least checksums) in a public repository.
  2. [Sections 1.1 and 5] The refutation of Gatti [10] depends on the assertion that Gatti treats the signs in the unfolded absolute-value expression as freely independent. Since the paper does not reproduce Gatti's exact wording, please verify this reading against the original and, if necessary, add a direct quotation or qualifying sentence so that the correction is not vulnerable to a straw-man objection.
  3. [Section 14, Table 1] The n = 11 row uses dashes for all columns except N_11. The text explains that only the count was computed from the G_10 frontier, but the caption could repeat this to avoid ambiguity, and it would be useful to state that per-sequence statistics at n = 11 were not attempted.
  4. [Section 8, Definition 17] When e = 0, the preimage step P_e(T) does not double the set; the lower branch is active only for t = 0. A one-line example or remark here would help readers avoid a common misinterpretation of the reverse-tree cardinality behavior.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorems are proved from definitions via the reverse-tree characterization, and external benchmarks are used only for corroboration.

full rationale

The paper's derivation chain is self-contained. Proposition 2 reformulates the definition of a valid extension as the fiber condition F_S(|k-s_n|)=1; this is bookkeeping from the difference triangle, not a fitted or assumed result. The central structural theorem (Theorem 20) is proved from the reverse-tree characterization (Proposition 18) using a self-contained normalized preimage lemma; the criterion e_i <= 1 + sum_{j>i} e_j is derived, not assumed. Brown's completeness criterion is both cited and proved in Theorem 11, and it is used only to analyze the intermediate signed-sum set S^±, not to define K_S. The first-hole theorem (Theorem 24) is verified by explicit enumeration of G_4 and G_5 and by a hand-computed reverse tree. The minimum-width theorem, the doubling-sequence width, and the exponentially disconnected family are each proved by induction from the anti-diagonal structure. No fitted parameters appear anywhere; no load-bearing self-citation occurs; no uniqueness theorem is imported from the author's prior work. Computational enumeration data are corroborated by OEIS A080839 and by the included Python program, and those computations are not load-bearing for the main theorems. The statements that computational verification was checked independently and that originality claims are based on literature searches are appropriate limitations, not circular steps. Therefore the paper receives a circularity score of 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters: the paper's objects (K_S, C_S, F_S, e_i) are definitions, and every stated constant (A(S)+1, widths, component counts) is derived. The central criterion of Theorem 20 is proved from the definitions; the only external inputs are classical subset-sum theory (Brown's criterion, itself proved in the paper), OEIS corroboration, and the correctness of the shipped code. No new postulated entities of the graviton type appear: the sets T_i, the family V_n, and the folding map F_S are constructions inside the defined theory, not entities requiring independent evidence.

assumptions (3)
  • domain assumption Correctness of OEIS A080839 values used to corroborate N_n
    Section 14.1: the enumeration column of Table 1 is cross-checked against OEIS A080839; an error in the database entry would affect only the corroboration — the hand theorems stand alone.
  • domain assumption Shift-invariance: |K_S| and the structure of K_S depend only on the gap sequence (Section 7.1)
    Stated without proof as 'a shift argument shows…'; true because translating every entry by c leaves the triangle below row 0 unchanged, but the argument is not displayed.
  • domain assumption The independent verification of the computational claims (Acknowledgments) is accurate
    The paper states the computational claims were verified against an independent implementation, but no artifact (code, log, hash) of that verification is provided; for the Table-1 claims the reader must trust this assertion or re-run the shipped code.

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Cite this review

Pith. "Pith review of Holes in Valid-Extension Sets of Finite Gilbreath Sequences." pith.science (2026). https://pith.science/paper/AGQEHOWQ

@misc{pith2026260623721,
  author       = {Pith},
  title        = {Pith review of: Holes in Valid-Extension Sets of Finite Gilbreath Sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AGQEHOWQ}},
  note         = {Machine review of arXiv:2606.23721}
}
read the original abstract

Given a finite sequence of integers, form its difference triangle by repeatedly taking absolute differences of consecutive entries. We call the sequence Gilbreath if the leftmost entry of every row below the top is 1. The Gilbreath conjecture, which remains open, asserts that every initial segment of the primes is a Gilbreath sequence. This paper studies the local extension problem: given a Gilbreath sequence, which integers can be appended to it while preserving the Gilbreath property? We call the set of such admissible values the valid-extension set of the sequence. A previously proposed characterization in the literature predicts that this set always fills a natural parity interval around the last term. We show that this fails in general: the valid-extension set can have interior holes, with the smallest failure occurring at length 5 for the sequence (2,3,5,9,15). The paper develops a corrected theory of valid extensions. We give an exact criterion for membership in the valid-extension set, an algorithm that computes it, and a sharp condition determining exactly when the set fills the candidate interval. This last condition is an order-sensitive analogue of the classical Brown completeness criterion for subset sums. We also establish endpoint validity and reflection symmetry, determine the exact minimum size of the valid-extension set together with its unique minimizer, exhibit a family whose valid-extension set has exponentially many components, and provide enumeration data through length 11.

Figures

Figures reproduced from arXiv: 2606.23721 by the authors.

Figure 1
Figure 1. Difference triangle of an arbitrary sequence of length [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Anatomy of the difference triangle of S = (2, 3, 5, 9, 15). The left diagonal (red) consists of 1’s, which is the defining Gilbreath condition. The right anti-diagonal (blue) is (e1, e2, e3, e4) = (6, 2, 0, 1). When a proposed extension k is appended, the new right-edge entries are computed by comparing the previous new entry with these anti-diagonal entries. The bottom apex 1 belongs to both structures, since en−1 … view at source ↗
Figure 3
Figure 3. The reverse-tree process on the sequence (2 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The mechanism behind the interval-completeness criterion (Theorem 20). With [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: The first hole. For S = (2, 3, 5, 9, 15), the candidate set CS consists of all odd integers in [sn − A(S) − 1, sn + A(S) + 1] = [5, 25]. The valid-extension set KS (blue dots) contains all of these except the center value k = 15 (marked ×). Thus HS = {15} and h(S) = 1.…
Figure 6
Figure 6. Figure 6: Component doubling in the family Vn (Theorem 35). Each horizontal block represents a maximal run of even integers spaced by 2. The recursion Dn = PEn (Dn−1), with En = 2n−2 − 2, sends Dn−1 to two separated reflected copies En − Dn−1 and En + Dn−1. Thus the number of co…

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Cited by 1 Pith paper

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Reference graph

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