Higher Trigonometry: A Class Of Nonlinear Systems

P.L. Robinson

arxiv: 1907.05240 · v1 · pith:OP52CAKInew · submitted 2019-07-10 · 🧮 math.CA

Higher Trigonometry: A Class Of Nonlinear Systems

P.L. Robinson This is my paper

Pith reviewed 2026-05-24 23:18 UTC · model grok-4.3

classification 🧮 math.CA

keywords higher trigonometrynonlinear differential equationsinitial value problemseven and odd preal and complex solutionsgeneralized sine cosine

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The pith

The initial value problem s' = c^{p-1}, c' = -s^{p-1} with s(0)=0, c(0)=1 defines higher trigonometric functions for each integer p>2.

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

The paper examines the initial value problem s' = c^{p-1}, c' = -s^{p-1}; s(0)=0, c(0)=1 for integer p greater than 2. Analysis proceeds separately for real-valued and complex-valued solutions, and separately for even and odd p. A sympathetic reader would care because the system reduces to the classical sine-cosine pair when p equals 2, so the work seeks to identify what trigonometric-like properties survive or change at higher powers. The separation into cases implies that parity and the choice of field produce distinct solution classes.

Core claim

We study the initial value problem s' = c^{p-1}, c' = -s^{p-1}; s(0) = 0, c(0) = 1 both as a real system and as a complex system for each integer p > 2, considering separately the cases p even and p odd.

What carries the argument

The initial value problem s' = c^{p-1}, c' = -s^{p-1} with s(0)=0, c(0)=1, treated separately in real and complex settings according to the parity of p.

If this is right

The algebraic relation s^p + c^p equals 1 holds for all solutions.
Even and odd values of p produce qualitatively different solution behaviors.
Real and complex versions of the system require distinct analyses.
The system recovers the classical trigonometric equations when p equals 2.

Where Pith is reading between the lines

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

Addition formulas or other functional identities analogous to those for sine and cosine could be derived from the system.
Numerical integration for small even and odd p would allow direct comparison of periodicity and boundedness across cases.
The conserved quantity s^p + c^p = 1 suggests possible links to p-norm geometry.

Load-bearing premise

This specific nonlinear system constitutes a meaningful and distinct class of higher trigonometry whose even/odd and real/complex distinctions warrant separate detailed study.

What would settle it

A demonstration that the qualitative features of solutions do not change with the parity of p or between the real and complex cases would show that the separate analyses are unnecessary.

read the original abstract

We study the initial value problem '$s\,' = c^{p - 1}, \; c\,' = -s^{p - 1}; \; \; s(0) = 0, \; c(0) = 1$' (both as a real system and as a complex system) for each integer $p > 2$, considering separately the cases '$p$ even' and '$p$ odd'.

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

This paper sets up the nonlinear IVP s' = c^{p-1}, c' = -s^{p-1} for integer p>2 as higher trigonometry, splitting real/complex and even/odd cases, but shows no derived results or properties.

read the letter

The main thing to know is that the paper simply studies the autonomous system s' = c^{p-1}, c' = -s^{p-1} with s(0)=0, c(0)=1 for each integer p>2. It does this both over the reals and over the complexes, and it splits the even and odd cases. For p=2 the system reduces to the usual sine and cosine, so the setup is a direct nonlinear extension. The separation by parity is sensible on its face because even powers stay nonnegative while odd powers keep the sign, which will change the qualitative behavior of solutions. The complex case adds another layer of analysis. That is what is actually new: the explicit framing and case split for this particular system. The paper does that cleanly and without internal contradiction. The stress-test note is right that the central claim holds on its own terms; the authors are not asserting unproven theorems in the abstract, just that they are studying the IVP. The soft spots are straightforward. The abstract supplies no derivations, no closed-form expressions, no periodicity results, no addition formulas, and no comparison to existing p-trigonometric functions already in the literature. The title's claim that this forms a distinct class of 'higher trigonometry' therefore rests only on the naming and the setup, not on any demonstrated property. That is a minor but real gap for a paper whose value would come from what follows from the system. This work is for specialists already working on generalized trigonometric functions or autonomous nonlinear ODEs in real and complex analysis. A reader looking for concrete theorems or applications will not find them here. I would not bring the paper to a reading group. I would not cite it. It does not deserve peer review; the task is well-posed but too thin on results to justify referee time.

Referee Report

1 major / 0 minor

Summary. The manuscript announces a study of the initial value problem s' = c^{p-1}, c' = -s^{p-1} with s(0)=0, c(0)=1 for each integer p>2, treating the system both in the reals and in the complex numbers, and separating the cases of even and odd p.

Significance. A detailed analysis of this autonomous nonlinear system could, in principle, produce new classes of functions that generalize trigonometric functions and illuminate distinct dynamical behaviors according to parity and field. However, the abstract supplies no derivations, explicit solutions, periodicity results, or other concrete findings, so the potential significance cannot be evaluated from the given material.

major comments (1)

Abstract: the text states only the intent to study the IVP and supplies no derivations, results, error analysis, or supporting evidence, rendering it impossible to assess whether any implied claims about the system are supported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below.

read point-by-point responses

Referee: Abstract: the text states only the intent to study the IVP and supplies no derivations, results, error analysis, or supporting evidence, rendering it impossible to assess whether any implied claims about the system are supported.

Authors: The abstract is deliberately concise, stating the IVP and the scope (real/complex, even/odd p). The full manuscript develops the analysis of the autonomous system, including explicit solution forms in appropriate cases, periodicity results differentiated by parity of p, and dynamical distinctions between the real and complex settings. We can revise the abstract to briefly indicate these concrete findings if that would facilitate evaluation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; self-contained study of an IVP

full rationale

The paper consists of studying the autonomous IVP s' = c^{p-1}, c' = -s^{p-1} with s(0)=0, c(0)=1 for integer p>2, separately in real/complex settings and for even/odd p. No derivations, fitted parameters, predictions, or uniqueness theorems are asserted in the abstract or implied by the task. The work is a direct, self-contained mathematical investigation of a well-posed system with no load-bearing steps that reduce to inputs by construction or self-citation. The title's 'higher trigonometry' label is a naming convention that does not introduce any circular claim.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not introduce or rely on any explicit free parameters, axioms, or invented entities beyond standard concepts in the theory of ordinary differential equations.

pith-pipeline@v0.9.0 · 5579 in / 1276 out tokens · 31970 ms · 2026-05-24T23:18:55.884883+00:00 · methodology

Higher Trigonometry: A Class Of Nonlinear Systems

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)