Replacement dynamics of binary quadratic forms

Frederick Saia; Raghav Bhutani

arxiv: 2508.05816 · v2 · submitted 2025-08-07 · 🧮 math.NT · math.DS

Replacement dynamics of binary quadratic forms

Raghav Bhutani , Frederick Saia This is my paper

Pith reviewed 2026-05-18 23:37 UTC · model grok-4.3

classification 🧮 math.NT math.DS

keywords replacement dynamicsperiodic vectorsbinary quadratic formsrational pointsunivariate reductionperiod classificationDiophantine systems

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The pith

There are no rational periodic vectors for the non-univariate period-4 type in replacement dynamics of diagonal binary quadratic forms.

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

The paper studies replacement dynamics in which a function of several variables updates exactly one coordinate of its input vector at each step. It stratifies periodic vectors by combinatorial type and identifies which types reduce to the dynamics of univariate polynomials. Restricting to diagonal binary quadratic forms over the rationals, the authors classify all rational periodic vectors of period at most 5. They prove in particular that the single non-univariate type of period 4 has no rational realizations.

Core claim

For a diagonal binary quadratic form over the rationals, rational periodic vectors under the replacement process are completely classified for all types of period at most 5, and the single non-univariate type of period 4 admits no such vectors.

What carries the argument

Stratification of periodic vectors into types by combinatorial replacement patterns, which reduces the periodicity condition to univariate polynomial dynamics for most types.

Load-bearing premise

The defined combinatorial types exhaust all possible periodic vectors.

What would settle it

An explicit rational vector that returns to itself after four replacement steps under the non-univariate pattern but cannot be reduced to a univariate case.

read the original abstract

For an $S$-valued function $f$ of $m \geq 1$ variables we consider the dynamical process in which the output $f(\overline{v})$ replaces exactly one entry of the input $\overline{v} \in S^m$ at each step. This can be viewed as a special case of multivariate polynomial semigroup dynamics, and our study focuses on periodic vectors with respect to this process. We define a stratification of periodic vectors according to their type, and characterize types for which the determination of periodic vectors comes down to dynamics of univariate polynomials. We then restrict to the case of a diagonal binary quadratic form $f$ over $\mathbb{Q}$, and classify rational periodic vectors for all types of period up to $5$. This includes two types, of periods $4$ and $5$, which do not arise from the univariate case, and we prove that there are no periodic vectors over the rationals of the single non-univariate type of period $4$.

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.

Desk Editor's Note private letter to a colleague

The paper classifies rational periodic vectors for replacement dynamics on a diagonal binary quadratic form over Q up to period 5, with a non-existence proof for the single non-univariate period-4 type.

read the letter

The paper's core result is a classification of rational periodic vectors for the replacement dynamics of a diagonal binary quadratic form over Q, covering all types up to period 5. This includes proving there are no such vectors for the one non-univariate type at period 4. They do well by first characterizing which types reduce to univariate polynomial dynamics, then handling the remaining cases with explicit parametrizations. The non-existence proof for period 4 is a direct case analysis that appears self-contained and doesn't introduce any fitted parameters or circular reasoning. Compared to the univariate literature, the explicit treatment of the two non-univariate types at periods 4 and 5 is new. The soft spot is the assumption that the type stratification is exhaustive. A period-4 orbit is determined by a sequence of four choices of which coordinate to replace. If the paper's types don't cover every possible such sequence, then the non-existence result only rules out the covered patterns and leaves open the possibility of other orbits. The abstract indicates they classify for all types, so the definitions in the early sections are meant to partition all periodic vectors, but this needs to be confirmed by checking whether the combinatorial patterns are fully accounted for. This paper is for researchers in arithmetic dynamics who are looking at multivariate or replacement-type processes rather than standard iteration. It provides concrete data points and a method that could extend to other forms or periods. The thinking is clear and the engagement with the setup is honest. I would send this to peer review. The claims are narrow but verifiable through algebraic means, so referees can focus on the completeness of the types and the case checks.

Referee Report

2 major / 2 minor

Summary. The paper defines replacement dynamics for an S-valued function f of m variables, where f(v) replaces one coordinate of the input vector v at each step. It introduces a stratification of periodic vectors by combinatorial type and shows that for certain types the periodicity conditions reduce to univariate polynomial dynamics. Specializing to diagonal binary quadratic forms over Q, the authors classify all rational periodic vectors for periods up to 5; in particular they prove there are no rational periodic vectors of the single non-univariate period-4 type.

Significance. If the type stratification is exhaustive, the classification supplies a complete low-period picture for this concrete instance of multivariate polynomial semigroup dynamics and isolates a genuinely bivariate case in which no rational solutions exist. The explicit parametrizations for the univariate types and the direct case analysis for the exceptional period-4 type are concrete contributions that could serve as a template for similar problems over number fields.

major comments (2)

[§2] §2 (type stratification): the claim that every period-4 orbit falls into one of the listed combinatorial patterns is used to justify restricting the non-existence argument to a single non-univariate type. No explicit enumeration or proof is given that the 2^4 = 16 possible sequences of coordinate choices are all captured by the defined types; if any sequence produces a distinct dynamical pattern, the case analysis would be incomplete.
[§4] §4 (period-4 non-univariate case): the algebraic case analysis over Q assumes the orbit has already been reduced to the single exceptional type. Without a prior verification that the type list is exhaustive, the non-existence statement for period 4 rests on an unproven completeness assumption.

minor comments (2)

Notation for the replacement map and the vector coordinates should be introduced once and used consistently; occasional switches between v_i and x_i make the period-5 parametrizations harder to follow.
The abstract states that two types of periods 4 and 5 'do not arise from the univariate case'; a brief sentence in the introduction explaining why the period-5 type is still treated as univariate would clarify the distinction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to make the completeness of the type stratification fully explicit. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§2] §2 (type stratification): the claim that every period-4 orbit falls into one of the listed combinatorial patterns is used to justify restricting the non-existence argument to a single non-univariate type. No explicit enumeration or proof is given that the 2^4 = 16 possible sequences of coordinate choices are all captured by the defined types; if any sequence produces a distinct dynamical pattern, the case analysis would be incomplete.

Authors: We agree that an explicit enumeration strengthens the presentation. The combinatorial types are defined by equivalence classes of replacement sequences under cyclic shift and swapping of the two coordinates. In the revised manuscript we will insert a short table in §2 that lists all 16 sequences for period 4, indicates the representative for each equivalence class, and assigns each class to one of the defined types. This will confirm that every sequence reduces to one of the listed patterns and that no additional dynamical cases arise. revision: yes
Referee: [§4] §4 (period-4 non-univariate case): the algebraic case analysis over Q assumes the orbit has already been reduced to the single exceptional type. Without a prior verification that the type list is exhaustive, the non-existence statement for period 4 rests on an unproven completeness assumption.

Authors: The explicit enumeration added to §2 will establish that every period-4 sequence belongs to one of the listed types. With that verification in place, the algebraic analysis in §4 applies to the entire set of period-4 orbits; the non-existence result for the single non-univariate type therefore covers all cases. We will add a one-sentence cross-reference in §4 pointing to the enumeration in §2. revision: yes

Circularity Check

0 steps flagged

Stratification defined by explicit combinatorial enumeration; non-existence proved by direct case analysis over Q

full rationale

The paper defines types via combinatorial patterns on the sequence of coordinate replacements (a finite set for fixed period and m=2). It then performs algebraic case analysis on the resulting equations for each type, including the single non-univariate period-4 type, showing no rational solutions exist for that type. No parameters are fitted, no self-citations are load-bearing for the central claim, and the derivation does not reduce any result to its own inputs by construction. The exhaustiveness follows from enumerating all possible replacement sequences rather than from a self-referential definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of field arithmetic over Q, the definition of the replacement map, and the combinatorial stratification into types; no new entities are postulated and no numerical parameters are fitted.

axioms (2)

standard math Q is a field of characteristic zero with the usual ordering and divisibility properties used in case analysis.
Invoked throughout the classification proofs when clearing denominators and analyzing signs or valuations.
domain assumption The replacement map is a well-defined function from Q^m to Q^m for the chosen diagonal quadratic form.
Stated in the opening setup of the dynamical process.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and orbit embedding unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We define a stratification of periodic vectors according to their type... period type of period N... ordered N-tuple t=(t1,...,tN)∈{1,2,...,m}N

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