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arxiv: 2605.11250 · v1 · submitted 2026-05-11 · 🧮 math.CO · math.DS

Recognition: 2 theorem links

· Lean Theorem

Critical Slow Growth in Averaged Meta-Fibonacci Recursions

Marco Mantovanelli

Pith reviewed 2026-05-13 02:21 UTC · model grok-4.3

classification 🧮 math.CO math.DS
keywords meta-Fibonacci recursionaveraged recursiontriangular block structureslow growthcritical parameterasymptotic analysisregularization
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The pith

Averaging turns meta-Fibonacci recursions into regular sequences that grow like the square root of n at the critical parameter.

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

The paper introduces a family of averaged meta-Fibonacci recursions in which each term is defined using the floor of alpha times the average of m earlier terms. At the critical value alpha equals 1 the authors prove the recursion stays well-defined for all n and all m at least 1, that the sequence forms successive blocks in which the integer k appears exactly k times in a row, and therefore that the nth term grows asymptotically like the square root of 2n. This mechanism produces stable slow growth rather than the erratic jumps typical of classical self-referential recursions. In the regime alpha greater than 1 the paper further shows that any linear growth that occurs must have slope exactly 1 minus 1 over alpha.

Core claim

For the critical parameter value α=1, the recursion is globally well-defined for all m≥1, has an exact triangular block structure with each k occurring exactly k consecutive times, and satisfies Q_{1,m}(n)∼√(2n). For α>1 any linear growth rate must equal 1−α^{-1}.

What carries the argument

The floor-averaged meta-Fibonacci recursion Q_{α,m}(n) = 1 + floor(α * (1/m) sum of m prior terms), which at α=1 forces the triangular block structure.

If this is right

  • The sequence consists only of positive integers for every n when α=1.
  • The number of terms up to and including the last occurrence of k is exactly the triangular number k(k+1)/2.
  • Any linear growth phase for α>1 must have slope precisely 1−α^{-1}.
  • Numerical results indicate that linear growth occurs and that similar regularization holds for other averaging operators such as positive-power L^p means.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit block structure allows the entire sequence to be written in closed form by concatenating the runs without iterative evaluation.
  • The regularization effect may apply to other self-referential recursions that currently lack stable asymptotics.
  • The sqrt(2n) growth links the construction to quadratic counting problems such as the inverse of triangular numbers.

Load-bearing premise

The averaging formula always produces a well-defined positive integer at every step without undefined references or non-positive values.

What would settle it

Compute the first several thousand terms for fixed m and α=1 and check whether constant runs have lengths exactly equal to their value and whether the position where k first appears matches the triangular number k(k+1)/2.

Figures

Figures reproduced from arXiv: 2605.11250 by Marco Mantovanelli.

Figure 1
Figure 1. Figure 1: Schematic phase structure of the averaged recursion. [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: shows the normalized quantity Q(n) √ 2n for several values of m. The curves rapidly converge toward 1, confirming the predicted square-root scaling law [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Supercritical growth for representative values of [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows the corresponding trajectories for several values of p. Conjecture 5.1 (Universality of positive-power averaging). Let p > 0, and replace the arithmetic mean in the recursion by the L p -mean Mp(x1, . . . , xm) =   1 m Xm j=1 x p j   1/p . At criticality α = 1, the resulting recursion still satisfies Q(n) ∼ √ 2n [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
read the original abstract

We introduce a family of averaged meta-Fibonacci recursions $$ Q_{\alpha,m}(n) = 1+ \left\lfloor \alpha \frac1m \sum_{j=1}^m Q_{\alpha,m}(n-Q_{\alpha,m}(n-j)) \right\rfloor , $$ with initial conditions $$ Q_{\alpha,m}(1)=\cdots=Q_{\alpha,m}(m)=1. $$ Unlike classical Hofstadter-type recursions, the averaging mechanism produces highly regular large-scale behavior. For the critical parameter value $\alpha=1$, we prove global well-definedness for all $m\ge1$, establish an exact triangular block structure, and show that the value $k$ occurs exactly $k$ consecutive times. As a consequence, $$ Q_{1,m}(n)\sim \sqrt{2n}. $$ For the supercritical regime $\alpha>1$, we derive an asymptotic slope constraint showing that any positive linear growth rate, if it exists, must equal $$ 1-\alpha^{-1}. $$ Numerical experiments support the existence of a linear-growth phase and suggest a broader universality phenomenon for generalized averaging operators, including positive-power $L^p$-means. These results indicate that averaging induces a robust regularization mechanism for self-referential recursive systems, leading to stable slow-growth dynamics and nontrivial phase structure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the family of averaged meta-Fibonacci recursions Q_{α,m}(n) = 1 + floor(α * (1/m) ∑_{j=1}^m Q_{α,m}(n - Q_{α,m}(n-j))), with initial conditions Q_{α,m}(1)=⋯=Q_{α,m}(m)=1. For the critical value α=1 it proves global well-definedness for every m≥1, establishes an exact triangular block structure in which each integer k appears in exactly k consecutive positions, and deduces the asymptotic Q_{1,m}(n)∼√(2n). For α>1 it derives the necessary condition that any positive linear growth rate must equal 1−α^{-1}, and presents numerical evidence for the existence of a linear-growth regime together with a suggested universality under generalized L^p averaging operators.

Significance. If the stated proofs are correct, the work supplies a concrete regularization mechanism that converts potentially chaotic meta-Fibonacci recursions into sequences with stable, explicitly describable slow-growth or linear asymptotics. The exact block-counting argument for α=1 yields a parameter-free derivation of the √(2n) law, which is a notable strength. The conditional slope constraint for α>1 is likewise derived directly from the recursion without fitting parameters to the target result. These features would be of interest to researchers studying self-referential recursions and discrete dynamical systems.

major comments (2)
  1. [Critical case (α=1) well-definedness and block-structure proof] The inductive argument establishing global well-definedness and the triangular block structure for α=1 (the central claim of the critical-case section) must explicitly verify that the floor-average operation maps a completed block of k’s into the next block without producing non-positive or non-integer values at the transition points; the current outline leaves the interaction between the floor function and the exact count of preceding terms unexpanded.
  2. [Supercritical regime and slope constraint] The derivation of the slope constraint 1−α^{-1} (supercritical regime) correctly shows necessity under the assumption that lim Q(n)/n exists and is positive, but the manuscript does not prove existence of the limit; the numerical support is therefore load-bearing for the claim that a linear-growth phase occurs, yet the range of n, m, and observed deviation from the predicted slope are not quantified.
minor comments (2)
  1. [Definition and initial conditions] The notation for the averaging window m and the parameter α is consistent, but the initial-condition statement should explicitly note that the first m terms are set to 1 independently of α.
  2. [Numerical experiments and universality] The discussion of universality for positive-power L^p means is suggestive; a brief remark on whether the slope constraint generalizes verbatim to p≠1 would clarify the scope of the claimed phenomenon.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and specific suggestions for improvement. We address each major comment below and will revise the manuscript accordingly to strengthen the exposition.

read point-by-point responses

  1. Referee: [Critical case (α=1) well-definedness and block-structure proof] The inductive argument establishing global well-definedness and the triangular block structure for α=1 (the central claim of the critical-case section) must explicitly verify that the floor-average operation maps a completed block of k’s into the next block without producing non-positive or non-integer values at the transition points; the current outline leaves the interaction between the floor function and the exact count of preceding terms unexpanded.

    Authors: We agree that the current outline of the inductive argument would benefit from a more explicit verification of the floor-average transitions. Although the block-structure argument relies on the exact multiplicity (each k appearing precisely k times) to control the averaged sum, the manuscript does not spell out the arithmetic at the block boundaries in sufficient detail. In the revision we will insert a short lemma that computes the precise value of the averaged sum when the window straddles the end of a completed block of k’s and the start of the next block, confirming that the floor operation yields exactly the integer k+1 and that all intermediate values remain positive integers. revision: yes

  2. Referee: [Supercritical regime and slope constraint] The derivation of the slope constraint 1−α^{-1} (supercritical regime) correctly shows necessity under the assumption that lim Q(n)/n exists and is positive, but the manuscript does not prove existence of the limit; the numerical support is therefore load-bearing for the claim that a linear-growth phase occurs, yet the range of n, m, and observed deviation from the predicted slope are not quantified.

    Authors: The referee is correct that the slope constraint is derived under the hypothesis that a positive linear growth rate exists; the manuscript makes no claim to prove existence of the limit. The numerical evidence is therefore essential for suggesting that a linear-growth regime is attained. In the revised version we will augment the numerical section with explicit statements of the ranges of n and m examined, together with quantitative measures (maximum and average relative deviation from 1−α^{-1}) observed across those runs, thereby making the supporting evidence fully transparent. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from recursion definition

full rationale

The central claims for α=1 follow directly from the given recursion and initial conditions Q(1)=...=Q(m)=1. The paper establishes global well-definedness and the exact triangular block structure (k appears exactly k consecutive times) by induction on n, using the floor-and-average operator to preserve the pattern without deviation. The asymptotic Q(n)∼√(2n) is then obtained by summing the block lengths (sum_{k=1}^K k ≈ K^2/2 = n). For α>1 the slope constraint 1−α^{-1} is derived as a necessary condition by assuming linear growth Q(n)∼cn and substituting into the recursion, yielding an algebraic identity that forces c=1−1/α; no parameters are fitted to the target asymptotic. No self-citations appear in the load-bearing steps, and the argument relies only on the explicit recursion rather than external theorems or prior results by the same author.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the explicit recursive definition, the floor function preserving integers, and the assumption that the sequence remains positive and well-defined for all n; no additional free parameters are fitted to data beyond the explicit α and m.

free parameters (2)
  • α
    Scaling parameter in the averaged recursion that distinguishes critical and supercritical regimes.
  • m
    Window size for the averaging sum.
axioms (2)
  • domain assumption The recursion produces a well-defined positive integer sequence for every n
    Invoked throughout; proved only for α=1 in the abstract.
  • standard math Standard properties of the floor function and finite averages
    Used implicitly in the definition and the block-structure argument.

pith-pipeline@v0.9.0 · 5530 in / 1540 out tokens · 48634 ms · 2026-05-13T02:21:29.649108+00:00 · methodology

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Lean theorems connected to this paper

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 2 internal anchors

  1. [1]

    Balamohan, A

    B. Balamohan, A. Kuznetsov, and Stephen Tanny. On the behavior of a variant of hofstadter’s q-sequence.Journal of Integer Sequences, 10(7):Article 07.7.1, 2007

    work page 2007

  2. [2]

    III Chew, and Stephen M

    John Callaghan, John J. III Chew, and Stephen M. Tanny. On the behavior of a family of meta-fibonacci sequences.SIAM Journal on Discrete Mathematics, 18(4):794–824, 2005

    work page 2005

  3. [3]

    Morphic words and nested recurrence relations

    Marc Celaya and Frank Ruskey. Morphic words and nested recurrence relations. Theoretical Computer Science, 520:35–39, 2014

    work page 2014

  4. [4]

    Almost Golomb Sequences

    Benoit Cloitre. Almost golomb sequences.arXiv preprint, 2026. arXiv:2604.02404

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  5. [5]

    Beatty solutions of almost Golomb equations

    Benoit Cloitre. Beatty solutions of almost golomb equations.arXiv preprint, 2026. arXiv:2604.10822

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  6. [6]

    Jonathan H. B. Deane and Guido Gentile. A diluted version of the problem of the existence of the hofstadter sequence.arXiv preprint, 2023. arXiv:2311.13854

    work page arXiv 2023

  7. [7]

    Finding linear-recurrent solutions to hofstadter-like recurrences using symbolic computation, 2016

    Nathan Fox. Finding linear-recurrent solutions to hofstadter-like recurrences using symbolic computation, 2016

    work page 2016

  8. [8]

    Quasipolynomial solutions to the hofstadter q-recurrence.Integers, 16:A68, 2016

    Nathan Fox. Quasipolynomial solutions to the hofstadter q-recurrence.Integers, 16:A68, 2016

    work page 2016

  9. [9]

    Jason Higham and Stephen M. Tanny. More well-behaved meta-fibonacci sequences. Congressus Numerantium, 98:3–17, 1993

    work page 1993

  10. [10]

    Hofstadter.G¨ odel, Escher, Bach: An Eternal Golden Braid

    Douglas R. Hofstadter.G¨ odel, Escher, Bach: An Eternal Golden Braid. Basic Books, 1979

    work page 1979

  11. [11]

    Abraham Isgur, Vitaly Kuznetsov, and Stephen M. Tanny. A combinatorial approach for solving certain nested recursions with non-slow solutions.arXiv preprint, 2012. 18

    work page 2012

  12. [12]

    Stephen M. Tanny. A well-behaved cousin of the hofstadter sequence.Discrete Mathematics, 105(1–3):227–239, 1992. 19

    work page 1992