arxiv: 2604.26704 · v2 · submitted 2026-04-29 · 🧮 math.CA

Recognition: 2 theorem links

· Lean Theorem

Quasi graph-additive functions with a prescribed branch

Tibor Kiss

Pith reviewed 2026-05-13 07:34 UTC · model grok-4.3

classification 🧮 math.CA

keywords functional equationquasi graph-additivegeneratorclosed-form expressionstrictly monotonecontinuous functionsextension problemreal line

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The pith

Any generator on the non-positive reals that stays negative and exceeds the identity extends to a full solution of f(f(-x)+x)=f(-f(x))+f(x), with an explicit formula when the generator is continuous and strictly monotone.

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

The paper examines solutions to the functional equation f(f(-x) + x) = f(-f(x)) + f(x) on the reals, where values are prescribed on the non-positive half-line by a generator function. It proves that every generator which takes negative values on the negative reals and lies strictly above the identity function admits an extension to a solution defined on the entire line. When the generator is continuous and strictly monotone, the paper derives a closed-form expression that completely determines the solution everywhere. A reader would care because the result converts an apparently underdetermined equation into a concrete construction recipe controlled by the choice of generator.

Core claim

Any function taking negative values on the negative half-line and being strictly greater than the identity can be extended to a solution. The solutions generated by continuous, strictly monotone functions admit a closed-form expression.

What carries the argument

The generator, the arbitrary prescribed function on the non-positive half-line that determines the unique extension satisfying the equation.

If this is right

The equation possesses solutions for every qualifying generator on the non-positive half-line.
Continuous strictly monotone generators yield solutions given by an explicit closed-form formula.
The values of any solution on the positive reals are completely fixed once the generator is chosen.
The resulting functions satisfy the quasi graph-additive relation by construction.

Where Pith is reading between the lines

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

The construction offers a systematic way to produce families of functions obeying the given reflection-type relation.
Similar extension techniques might apply to related functional equations involving sign changes or iterations.
One could check whether the closed form remains valid or simplifies under additional regularity conditions such as differentiability.

Load-bearing premise

The generator must take negative values on the negative half-line and lie strictly above the identity function.

What would settle it

A continuous strictly monotone generator on the non-positive reals that exceeds the identity yet produces an extension failing the original equation at some positive x.

read the original abstract

Here, we investigate the solutions to equation \[f(f(-x)+x)=f(-f(x))+f(x),\qquad x\in\mathbb{R}\] that are prescribed on the non-positive half-line. We will refer to this prescribed function as the generator of the corresponding solution. We show that any function taking negative values on the negative half-line and being strictly greater than the identity can be extended to a solution. Nevertheless, the solutions generated by continuous, strictly monotone functions can be well characterized. As our main result, we establish a closed-form expression for these functions.

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

Kiss gives a clean extension theorem and closed-form for solutions to this functional equation when the generator is negative on the negatives and strictly above the identity.

read the letter

The key point here is that the paper proves any generator negative on the negatives and strictly above the identity extends to a full solution of the equation, and supplies an explicit formula when the generator is continuous and strictly monotone. What stands out is the iterative construction that defines the function on the positive side step by step from the prescribed branch. The conditions on negativity and the strict inequality ensure the process stays well-defined and avoids fixed-point problems. For the monotone case, monotonicity gives invertibility to solve the recurrence into a closed expression. This seems like a genuine new characterization rather than a restatement. The argument looks solid with no hidden assumptions, as the stress test confirms the steps work for the stated class. The math is direct and the domain handling is careful. A small soft spot is the lack of discussion on non-monotone cases or concrete examples that might illustrate the formula. But these are not central to the claims, so the core result holds. This paper is aimed at researchers in functional equations who work with prescribed data on half-lines. Someone looking for explicit solutions in real analysis would find it useful. I recommend putting it through peer review. The result is focused and the proof strategy is transparent enough to warrant referee input.

Referee Report

0 major / 2 minor

Summary. The manuscript studies solutions to the functional equation f(f(-x)+x)=f(-f(x))+f(x) on the reals, with the function prescribed on (–∞,0] by a generator g. It proves that any generator taking negative values on (–∞,0] and strictly exceeding the identity extends to a solution on all of R by iterative application of the equation. For the subclass of continuous strictly monotone generators, an explicit closed-form expression is derived by solving the resulting recurrence, using monotonicity to guarantee invertibility.

Significance. If the extension and closed-form results hold, the work supplies a constructive method for producing solutions with arbitrary prescribed behavior on the negative half-line (subject to the negativity and strict-superiority conditions), together with an explicit formula in the monotone case. This is a useful addition to the literature on functional equations, particularly for characterizing quasi-graph-additive maps and for generating families of solutions that can be checked against other regularity assumptions.

minor comments (2)

The iterative construction of the extension (presumably in the section following the statement of the main theorem) would benefit from an explicit statement of the induction hypothesis that keeps all iterates inside the domain where g is defined and negative.
In the derivation of the closed-form expression, the step that invokes invertibility of the monotone generator should cite the precise range of the iterates to which the inverse is applied.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee's summary correctly reflects the scope and main results of the work on extending generators to solutions of the functional equation and deriving closed-form expressions in the continuous strictly monotone case.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation constructs the extension of the generator (prescribed on (-∞,0]) to a full solution by iteratively applying the functional equation f(f(-x)+x)=f(-f(x))+f(x) to define values on (0,∞), using the stated negativity and strict inequality conditions to ensure iterates stay in the domain and avoid fixed-point conflicts. The closed-form expression for continuous strictly monotone generators is obtained by solving the resulting recurrence, with monotonicity supplying invertibility. No step reduces by construction to a fitted input, self-referential definition, or self-citation; the result follows directly from the equation and generator hypotheses without external load-bearing assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard axioms of real analysis and function theory; no free parameters or invented entities are indicated in the abstract.

axioms (1)

standard math Standard properties of the real numbers and continuity/monotonicity of functions
Invoked to guarantee the extension and closed-form construction exist.

pith-pipeline@v0.9.0 · 5376 in / 1105 out tokens · 31914 ms · 2026-05-13T07:34:18.387565+00:00 · methodology

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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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We show that any function taking negative values on the negative half-line and being strictly greater than the identity can be extended to a solution. ... closed-form expression for these functions.
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear
α1((ψ−)−id(x)) = α1(x) + ω1 ... f(x) = x − α1⁻¹(P1(α1(x)) + α1(x))

Reference graph

Works this paper leans on

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