REVIEW 2 major objections 5 minor 4 references
Some properties of the simple nonlinear recursion $y(\ell + 1) = [1-y(\ell)]^p$ with $p$ an arbitrary positive integer
T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read For every integer $p$, each real solution of $y_{n+1}=(1-y_n)^p$ is equilibrium, 2-cycle, or unbounded.
desk verdict The p=3 divergence claim is false—y(0)=1.5 converges to the {0,1} cycle—so the paper's central classification and the odd-p extension are wrong as written. 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
The carrying object is the one-dimensional map $f_p(y)=[1-y]^p$ together with the fixed-point equation $y=[1-y]^p$, whose real roots divide the real line into regions with different fates. On the unit interval $f_p$ is a decreasing fold that swaps the two sides of the small fixed point; iterating the map repeatedly pushes a non-fixed initial value alternately toward the two boundary values $0$ and $1$. For even $p$, the presence of a second real root larger than $2$ supplies the threshold: initial values whose orbit climbs beyond that root are carried monotonically to $+\infty$, whereas lower seeds fall into the same $0\leftrightarrow 1$ alternation.
What would settle it
Numerically iterate $y_{n+1}=(1-y_n)^7$ from $1/2$: the dichotomy predicts the orbit approaches the alternating pair $\{0,1\}$, while for $p=8$ the equation $y=(1-y)^8$ must have exactly two real roots. Finding a different period, or an extra real root in either count, would kill the classification.
Extended reading notes
Core claim
For odd $p$, the equation $y=(1-y)^p$ has exactly one real root, a number inside the unit interval; the paper shows that every real orbit starting in $(0,1)$, other than that fixed point, jumps from one side of the fixed point to the other at each step and is asymptotically periodic with period $2$, approaching $0$ and $1$ in alternation, while every orbit starting outside $(0,1)$ escapes to infinity with alternating signs. For even $p$, the same fixed-point equation has exactly two real roots, one in $(0,1)$ and one larger than $2$; orbits that land in $(0,1)$ other than the small fixed point follow the same $0\leftrightarrow 1$ approach, while orbits that pass beyond the larger root diverge monotonically to $+\infty$. The paper asserts, with approximate roots listed for a few cases, that this same dichotomy holds for all odd $p>4$ and all even $p>5$.
Load-bearing premise
The assumption that every larger odd or even exponent has the same real-root pattern and the same invariant interval as the small cases is asserted by analogy and numerical evidence rather than proved; if a larger $p$ introduced an extra real fixed point, an extra invariant interval, or a different attracting cycle, the claimed complete classification would fail.
Editorial extensions
If this is right
- For every odd $p$, all non-equilibrium seeds in $(0,1)$ are asymptotically periodic with period $2$, approaching $0$ and $1$; all seeds outside $(0,1)$ are unbounded.
- For every even $p$, any seed whose orbit does not exceed the larger real root eventually approaches the same $0\leftrightarrow 1$ alternation, while any seed whose orbit passes that root diverges to $+\infty$.
- The complete long-time behavior of the family is thereby determined by the parity of $p$ and by the real solutions of the algebraic equation $y=(1-y)^p$; no numerical simulation is needed once those roots are located.
- No bounded orbit other than a fixed point or the two exact $0\leftrightarrow 1$ cycles can exist, so the family displays no other periodic attractors.
Reading between the lines
- Editorial inference: the rate of approach to the $0\leftrightarrow1$ alternation is set by the second iterate near the boundaries; for small $y$, $f_p(f_p(y))$ behaves like $p^p y^p$, so the even and odd subsequences converge superlinearly, at order $p$, to their limiting values.
- Editorial inference: because $f_p$ is a continuous monotone map on $(0,1)$ with all periodic points confined to period 2, the dichotomy suggests a topological endpoint, namely that no higher-period cycles can appear for any integer $p$; the paper states this picture informally but does not prove it as a theorem.
- Editorial inference: a testable extension is to real exponents $p>0$, where one can check numerically whether the same two-basin structure survives and where, if anywhere, periodic orbits of higher period first appear.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript studies the real dynamics of the first-order nonlinear recursion y(l+1)=(1-y(l))^p for arbitrary positive integers p. It claims a complete classification: for initial data in (0,1), except the unstable fixed point, the orbit is asymptotically 2-periodic with values approaching 0 and 1; for initial data outside [0,1], the orbit diverges in modulus, with the only persistent exception being the equilibrium value. Sections 3 and 4 treat p=3 and p=4 in detail; Sections 5 and 6 assert by analogy that all odd p>4 and even p>5 behave in the same way, supported only by lists of approximate fixed points. The paper also records the trivial period-2 solutions and the equilibrium equation y=(1-y)^p.
Significance. If the classification were correct, the paper would provide a simple and complete description of all real orbits of a one-parameter family of interval maps, which could be useful in elementary applications and in constructing solvable systems of recursions. The paper is honest about the elementary nature of the findings and does not introduce fitted parameters. However, the central claim is false already in the base case p=3: there exist initial data with y(0)>1 that are attracted to the period-2 cycle {0,1}, contradicting the asserted dichotomy. Since the claimed extension to all odd p rests on this case, the main result as stated is incorrect. The treatment of p=4 is plausible, and the approximate fixed points listed in Sections 5 and 6 may be useful data, but they do not compensate for the failure of the central classification.
major comments (2)
- [Section 3, Eq. (4)] The claim that every solution with y(0) outside the interval 0<y(0)<1 diverges in modulus is false. For the initial datum y(0)=1.5, the recursion y(l+1)=(1-y(l))^3 gives y(1)=-0.125, y(2)=1.125^3 ≈ 1.4238, y(3)=(-0.4238)^3 ≈ -0.0761, y(4)≈1.246, and the even subsequence tends to 1 while the odd subsequence tends to 0. Thus this orbit is asymptotically 2-periodic, converging to the cycle {0,1}. Analytically, the second iterate F(y)=(1-(1-y)^3)^3 satisfies F(1)=1 and F'(1)=0, so y=1 is an attracting fixed point of F, and its basin extends into y>1. This is a concrete counterexample to the classification of all real solutions stated in Section 3.
- [Sections 5 and 6] The paper offers no proof that the dynamics for p>4 are qualitatively identical to those for p=3 (odd p) or p=4 (even p). The sentence 'it is easy to convince oneself' is not a mathematical argument, and the listed approximate roots do not rule out additional invariant intervals, additional attracting cycles, or a different stable manifold structure for some p. Since the base case p=3 already exhibits a basin of attraction outside [0,1] that contradicts the stated dichotomy, the extrapolation to arbitrary p is unsupported and, for odd p, inherits the error identified above. A rigorous treatment of these cases would be needed before the paper's central claim could stand.
minor comments (5)
- [Section 2, after Eq. (3b)] The sentence 'at least 1 real solution of this equation always exists iff p is an odd positive integer' is incorrect: for even p, e.g. p=2 and p=4, the equation y=(1-y)^p has two real solutions, so at least one real solution exists for every positive integer p. The intended statement is presumably that exactly one real solution exists when p is odd.
- [Section 4, Eq. (7c)] The final inequality in Eq. (7c) reads '> y (1)', but the comparison should be with y(2), i.e. '> y (2)', since the goal is to show that the orbit starting just above y(2) increases. As printed, the inequality compares an expression to itself and is nonsensical.
- [Section 4, first paragraph] The text says 'In this Section 3 we restrict our consideration' but the current section is Section 4; the reference should be corrected.
- [Section 4, first paragraph] The statement that for p=4, 'whatever the value is of the initial datum y(0), for all subsequent values ... the dependent variable y(l) shall be positive, y(l)>0 (except for the special case with y(0)=1)' is false: for y(0)=2 one gets y(1)=1, y(2)=0, y(3)=1, so y(2)=0. The claim should be nonnegativity, allowing the value 0.
- [Throughout] The phrase 'asymptotically isochonous' in Section 4 should be 'asymptotically isochronous'; also, the equations are typeset with OCR artifacts (e.g. missing minus signs and exponents), though these likely originated in the arXiv extraction rather than the author's original manuscript.
Circularity Check
No significant circularity; the paper's p=3 and p=4 derivations are self-contained, and the cited prior work is external rather than load-bearing, though Sections 5-6 contain unproved generalizations.
full rationale
The paper's claimed results for p=3 and p=4 are derived by direct iteration and elementary monotonicity/local expansion arguments from the recursion y(ell+1)=(1-y(ell))^p. For example, Section 3 gives explicit iterates for y(0)=1/2 and y(0)=2, while Section 4 linearizes near the real root y(2) to argue divergence for y(0)>y(2). No parameter is fitted to data and then renamed a prediction, and no object is defined in terms of the behavior it is claimed to explain. The self-citations are to the author's earlier paper [1] for the p=2 case and to follow-up papers in the Outlook; these are prior external works rather than the present derivation assuming its own conclusion, and the p=3/p=4 analysis does not rest on them. Sections 5 and 6 generalize by assertion ('it is easy to convince oneself that the phenomenology is quite analogous') rather than by proof; this is an evidentiary gap, and the p=3 outside-[0,1] divergence claim is in fact contradicted by iterates such as y(0)=3/2, but a false or unsupported mathematical assertion is not the same as a circular derivation. No load-bearing step reduces by the paper's own equations to its inputs. Accordingly the circularity score is 0.
Assumptions & free parameters
assumptions (3)
- domain assumption The real-valued iteration of f_p(y)=(1-y)^p is considered; all quantities are finite real numbers for real initial data.
- standard math The intermediate value theorem and monotonicity of f_p are used to assert the number of real fixed points of y=(1-y)^p.
- ad hoc to paper For p>4, the dynamics are asserted to be qualitatively identical to the p=3 or p=4 cases without a complete proof.
Cite this review
Pith. "Pith review of Some properties of the simple nonlinear recursion $y(\ell + 1) = [1-y(\ell)]^p$ with $p$ an arbitrary positive integer." pith.science (2026). https://pith.science/paper/UN7H2QAP
@misc{pith2026250109463,
author = {Pith},
title = {Pith review of: Some properties of the simple nonlinear recursion $y(\ell + 1) = [1-y(\ell)]^p$ with $p$ an arbitrary positive integer},
year = {2026},
howpublished = {\url{https://pith.science/paper/UN7H2QAP}},
note = {Machine review of arXiv:2501.09463}
}
abstract
It is shown that the behavior of the solutions of the nonlinear recursion $y(\ell + 1) = [1-y(\ell)]^p$ -- where the dependent variable $y(l)$ is a real number, $\ell= 0; 1; 2...$ is the independent variable, and $p$ is an arbitrary positive integer -- is easily ascertainable.
Reference graph
Works this paper leans on
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[1]
Simple recursions displaying interesting evolutions
F. Calogero, "Simple recursions displaying interesting evolutions", arXiv:2405.00370v1 [nlin.SI] 1 May 2024
work page Pith review arXiv 2024
-
[2]
Solvable nonlinear systems of 2 recursions displaying interesting evolutions
F. Calogero, "Solvable nonlinear system of 2 recursions displaying inter- esting evolutions", arXiv:2407.18270v1 [nlin.SI] 20 Jul 2024
work page Pith review arXiv 2024
-
[3]
Interesting system of $3$ first-order recursions
F. Calogero, "Interesting system of 3 rst-order recursions", arXiv:2409.05074v1 [nlin.SI] 8 Sep 2024
work page Pith review arXiv 2024
-
[4]
F. Calogero, "A simple approach to identify systems of nonlinear recur- sions featuring solutions whose evolution is explicitly ascertainable and which may be asymptotically isochronous as functions of the independent variable (a ticking time)", arXiv:2410.14448v1 [nlin.SI] 18 Oct 2024. 6
Reviewed August 10, 2026 · model on record in the stance chip above.
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