REVIEW 2 major objections 15 references
A Novel Property of Generalized Fibonacci Sequence in Grids
T0 review · 2 major / 0 minor · reviewed 2026-05-23 · grok-4.3
Pith's one-line read In odd-order grids filled with generalized Fibonacci numbers, the main-to-sub-diagonal sum ratio depends only on the grid order.
desk verdict The paper claims a ratio of diagonal sums in odd-order generalized-Fibonacci grids depends only on grid order, but the abstract leaves the filling rule and sub-diagonal undefined, so the independence claim cannot be checked. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
Formulas for sums of equidistant subsequences applied to the diagonals of the grid.
What would settle it
Fill a 3 by 3 grid with one generalized Fibonacci sequence, compute the main and sub-diagonal sums and their ratio; repeat with a different initial terms or growth rate and check whether the ratio stays identical.
Extended reading notes
Core claim
Leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of equidistant subsequences, the investigation of odd-order grids shows that the ratio of the sum of numbers along the main-diagonal and sub-diagonal is determined exclusively by the order of the grid.
Load-bearing premise
The grid must be populated with consecutive terms from the generalized Fibonacci sequence in an order that makes the diagonals correspond to equidistant subsequences.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript claims that in odd-order grids filled with consecutive terms of a generalized Fibonacci sequence, the ratio of the sum of entries along the main diagonal to the sum along a sub-diagonal depends only on the grid order n (independent of the recurrence coefficients and initial conditions), and derives a closed-form identity for this ratio by applying summation formulas for equidistant subsequences of generalized Fibonacci numbers.
Significance. If the central claim holds under explicit definitions, the result would constitute a clean, parameter-free identity extending classical Fibonacci summation properties to grid arrangements; this could be of interest in combinatorial number theory for its simplicity and potential for generalization.
major comments (2)
- [Abstract] Abstract: the assertion that the ratio 'is solely dependent on the order of the grid' cannot be verified because the manuscript supplies neither the explicit coordinate-to-index map for filling the grid (row-major, column-major, or other) nor the precise definition of which anti-diagonal constitutes the 'sub-diagonal'; these choices are load-bearing for the applicability of equidistant-subsequence summation formulas and for independence from sequence parameters.
- [Abstract] The abstract states that a proof exists via equidistant-subsequence identities but provides neither the derivation steps, the resulting closed-form expression, nor any numerical check (e.g., for n=3 or n=5 with specific generalized Fibonacci parameters); without these, the central claim that the ratio reduces to a function of n alone remains uncheckable against the paper's own data or equations.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments highlighting the need for greater explicitness. We address each point below and will make the corresponding revisions.
read point-by-point responses
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Referee: [Abstract] Abstract: the assertion that the ratio 'is solely dependent on the order of the grid' cannot be verified because the manuscript supplies neither the explicit coordinate-to-index map for filling the grid (row-major, column-major, or other) nor the precise definition of which anti-diagonal constitutes the 'sub-diagonal'; these choices are load-bearing for the applicability of equidistant-subsequence summation formulas and for independence from sequence parameters.
Authors: We agree this information is essential for verification. The revised manuscript will explicitly state that the grid is filled in row-major order with consecutive terms of the generalized Fibonacci sequence beginning at position (1,1), and will precisely define the main diagonal (i = j) together with the specific sub-diagonal used (the unique parallel diagonal whose entries form an equidistant subsequence under the chosen indexing). These additions will make the applicability of the summation formulas transparent. revision: yes
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Referee: [Abstract] The abstract states that a proof exists via equidistant-subsequence identities but provides neither the derivation steps, the resulting closed-form expression, nor any numerical check (e.g., for n=3 or n=5 with specific generalized Fibonacci parameters); without these, the central claim that the ratio reduces to a function of n alone remains uncheckable against the paper's own data or equations.
Authors: The body of the manuscript already contains the derivation via the equidistant-subsequence summation formulas and arrives at a closed-form identity depending only on n. Nevertheless, we accept that the abstract is too terse and that explicit numerical checks would strengthen verifiability. The revision will (i) include the closed-form expression in the abstract and (ii) add a short section presenting concrete numerical checks for n=3 and n=5 using both the standard Fibonacci sequence and another generalized Fibonacci sequence with distinct coefficients. revision: yes
Circularity Check
No circularity; derivation applies standard summation identities to grid diagonals
full rationale
The paper derives the claimed ratio of main- and sub-diagonal sums by applying known closed-form identities for equidistant subsequences of generalized Fibonacci sequences. No self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation is present in the abstract or described chain; the independence from recurrence coefficients follows directly from the summation formulas once the (unspecified) filling map is fixed. The result is therefore self-contained against external benchmarks rather than reducing to its own inputs.
Assumptions & free parameters
Cite this review
Pith. "Pith review of A Novel Property of Generalized Fibonacci Sequence in Grids." pith.science (2026). https://pith.science/paper/2407.05369
@misc{pith2026240705369,
author = {Pith},
title = {Pith review of: A Novel Property of Generalized Fibonacci Sequence in Grids},
year = {2026},
howpublished = {\url{https://pith.science/paper/2407.05369}},
note = {Machine review of arXiv:2407.05369}
}
read the original abstract
Fibonacci sequence, generated by summing the preceding two terms, is a classical sequence renowned for its elegant properties. In this paper, leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of equidistant subsequences, we investigate the ratio of the sum of numbers along main-diagonal and sub-diagonal of odd-order grids containing generalized Fibonacci sequences. We show that this ratio is solely dependent on the order of the grid, providing a concise and splendid identity.
Figures
Lean theorems connected to this paper
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IndisputableMonolith/Constants.leanphi_golden_ratio, phi_fixed_point echoes?
echoesECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.
P_{k=0}^{2n} G_{2k(n+1)} / P_{k=0}^{2n} G_{2n(k+1)} = c(n) ... solely dependent on the value of n
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel echoes?
echoesECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.
Gn = ½(−A+2B)Fn + ½ALn; sum formula (2.2) via a=φ, b=1−φ
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
Works this paper leans on
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work page 2023
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[15]
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Reviewed May 23, 2026 · model on record in the stance chip above.
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