Specular differentiation in one dimension: a quasi-mean value theorem, regularity, and discontinuities

Kiyuob Jung

REVIEW 3 major objections 5 minor 1 cited by

Specular differentiation in one dimension: a quasi-mean value theorem, regularity, and discontinuities

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper introduces a generalized derivative that averages the angles of forward and backward slopes, and proves that the resulting specular Euler scheme of Type 5 is second-order convergent for smooth initial-value problems.

desk verdict SE5 has a clean consistency analysis and the quasi-MVT proof is genuinely new, but the regularity theorem is circular and the convergence proof assumes solvability; worth refereeing, not accepting yet. read the letter →

arxiv 2601.09900 v5 pith:VO2A5OBD submitted 2026-01-14 math.NA cs.NA

Kiyuob Jung This is my paper · ORCID

classification math.NAcs.NA MSC 49J5265L0565L2065L12

keywords specularderivativegeneralizeddifferentiationquasi-meanvaluetheoremEulerschemesecond-orderconvergenceinitialproblemC^1regularityangleaveraging

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper defines a specular derivative on the real line by averaging the arctangents of the right and left difference quotients, then uses it to build new numerical schemes for first-order ODEs. The central result is that one of these schemes, the specular Euler scheme of Type 5, is second-order consistent and, under a Lipschitz source function, second-order convergent when the exact solution is $C^3$. This means a nonlinear angle-averaged Euler-type method can match the accuracy of the classical Crank-Nicolson scheme. The paper also develops a theory of specular differentiation, including a quasi-mean value theorem and the result that a continuous function with a second specular derivative is classically $C^1$. A sympathetic reader would care because the scheme offers a new, simple route to high-order accuracy and because the derivative theory extends classical calculus to functions with one-sided or even infinite slopes.

What carries the argument

The key object is the specular derivative $f^{\wedge}(x) = \lim_{h\to 0} \tan(\frac{1}{2} \arctan((f(x+h)-f(x))/h)+\frac{1}{2} \arctan((f(x)-f(x-h))/h))$, equivalently the smooth averaging function $A(\alpha,\beta)=\tan(\frac{1}{2} \arctan \alpha+\frac{1}{2} \arctan \beta)$. The paper relies on the expansion $A(\alpha,\alpha+h_2)=\alpha+h_2/2+O(h_2^2)$, which cancels the first-order terms in the local truncation error and produces the second-order consistency bound for SE5. The same function $A$ carries the geometric insight that the scheme blends the right and left slopes by their angles rather than by arithmetic mean, which remains defined even when one slope is infinite.

What would settle it

Implement SE5 for a smooth scalar equation such as $u' = u^2$, $u(0)=1$ on $[0,1]$ with $h = 2^{-k}$, and compute the empirical order $R(N,\infty)=\log_2(E(N/2,\infty)/E(N,\infty))$; if $R$ does not approach 2 as $k$ grows, the claimed second-order convergence fails.

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Extended reading notes

Core claim

The paper's central claim is that replacing the ordinary derivative in Euler's method with the specular derivative—defined as the tangent of the average of the arctangents of the forward and backward difference quotients—yields a one-step scheme, SE5, given by $u_{n+1}=u_n+h A(F(t_{n+1},u_{n+1}),F(t_n,u_n))$, which is second-order consistent when the exact solution is $C^3$ and second-order convergent when $F$ is Lipschitz and the solution has the same regularity. The same derivative framework establishes a quasi-Fermat theorem, a quasi-mean value theorem, Lipschitz and $C^1$ regularity theorems for continuous specularly differentiable functions, and the sharp statement that discontinuities at which

Load-bearing premise

The convergence proof assumes, without proving, that at each step the implicit equation defining $u_{n+1}$ has a unique attainable solution and that the averaging function's steepness stays bounded as the step size shrinks.

Editorial extensions

If this is right

SE5 is a one-step, second-order convergent scheme for smooth ODEs, giving an alternative to Crank-Nicolson that uses a nonlinear blend of endpoint slopes rather than a symmetric average.
The scheme reduces to explicit Euler when the two slopes coincide, so it is a one-parameter family that connects explicit and implicit Euler in a single formula.
The theory implies that continuous functions with a bounded specular derivative are Lipschitz, those with a continuous specular derivative are C^1, and existence of a second specular derivative forces classical differentiability.
Specular differentiability can be defined for discontinuous and semicontinuous functions, and its nonzero-derivative discontinuities are at most countable, so the derivative concept extends beyond classical calculus.
The paper proposes using the specular Euler framework as a component in Runge-Kutta schemes, with Type 1 for the first stage and Type 5 for subsequent stages.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the second-order convergence of SE5 holds generally, it suggests a broader design principle: replacing a derivative with an angle-averaged slope of one-sided differences can raise a one-step method's order without adding stencil points; this could be tested on higher-order Runge-Kutta families.
The quasi-mean value theorem provides a Rolle-like statement for a nonadditive, nonlinear derivative, and may be the right tool for proving error bounds for other angle-averaged implicit schemes.
The at-most-countable discontinuity result suggests a connection to regulated functions; one could investigate whether the class of specularly differentiable functions carries a Banach-space or BV-like structure, which the paper leaves open.
Because the convergence proof relies on uniform control of the local Lipschitz constant of A, practical use may require step-size adaptation that keeps L_M L h small; testing SE5 on stiff problems would reveal whether this assumption is restrictive.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is not ready as received, but it is not empty. The quasi-mean value theorem proof is genuinely new, at least to me, and the consistency computation for SE5 is clean. The extension of the specular derivative to infinite one-sided derivatives is a real—if modest—addition over the authors' earlier work. The numerical experiments are honest and reasonably extensive: SE5 and SE6 reach second order on smooth problems, and the authors admit that on the Dahlquist test SE5 does not beat Crank-Nicolson. That is fair presentation.

The soft spots are load-bearing. Proposition 2.20's proof in Appendix B begins by writing `lim_{w->x} f^(w) =: sigma`, which is exactly the limit the proposition is supposed to establish. The proof then never uses the assumed second specular derivative f^^, and it assumes one-sided derivatives of f exist without proving them. Theorem 2.21 inherits the failure. The abstract also promises results on countable discontinuities and a class strictly between C^1 and C^2; those results are not in the body. If they exist in another version, they are not in this manuscript.

The convergence proof for SE5 (Theorem 3.7) has a standard but real gap. SE5 is an implicit scheme, and no lemma establishes that u_{n+1} is well-defined for each n. The proof subtracts (3.8) from (3.18) and forms |e_{n+1}| as if the sequence already existed. This is fixable with a contraction-mapping argument using the local Lipschitz constant of A and a small-h condition, but the theorem as stated is incomplete. The part (a) proof has the same implicit-existence issue.

The reliance on the authors' prior papers is acceptable for an incremental extension, though the claim that the paper is independent of previous works is overstatement. There is a public software package, which is a point in the paper's favor.

Verdict: send to referees. A desk reject would be too harsh because the quasi-MVT proof and the SE5 idea are worth expert scrutiny. But the revision should be asked to repair or excise the regularity claims, add a well-posedness lemma for SE5, and reconcile the abstract with the body.

Referee Report

3 major / 5 minor

Summary. The manuscript develops a notion of "specular derivative" in one dimension, obtained by angle-averaging forward and backward difference quotients, and applies it to first-order ODE initial value problems. The theoretical part establishes quasi-Fermat and quasi-mean value theorems, claims that twice specular differentiability of a continuous function implies classical C^1 regularity, and announces results on discontinuities of specular derivatives. The numerical part proposes seven specular Euler-type schemes, focuses on the implicit "Type 5" scheme, and states that this scheme is consistent of order 2 and convergent of order 2 (Theorem 3.5 and Theorem 3.7), with first-order convergence under weaker regularity. Numerical experiments compare the schemes with classical Euler and Crank-Nicolson methods.

Significance. If the convergence claim for SE5 were fully established, it would be a notable result: a simple angle-averaged implicit Euler-type scheme achieving Crank-Nicolson-like second order for nonlinear ODEs. The consistency computation in Theorem 3.5 is clean, the numerical experiments are extensive, and the paper ships a publicly available Python package, which are concrete strengths. The quasi-mean value theorem is also a potentially interesting contribution. However, the main convergence theorem is not proved as stated because the implicit scheme is never shown to define a unique discrete sequence, and the proof of Proposition 2.20 is circular. These gaps are load-bearing, so the advertised results are not currently established.

major comments (3)

[§3.3, Theorem 3.7 / Definition 3.3] The theorem never proves that the implicit equation (3.8), u_{n+1}=u_n+hA(F(t_{n+1},u_{n+1}),F(t_n,u_n)), has a unique solution for every n. The proof of part (b) subtracts (3.8) from (3.18), forms |e_{n+1}|, and divides by (1-L_M L h)>0 as though well-posedness were automatic. Part (a), which is used to bound M_2=max|F_n|, likewise presupposes that the sequence {u_n} exists. The hypotheses (unique C^3 solution, F Lipschitz) do not by themselves guarantee a well-defined discrete sequence, and no h-threshold for solvability is given. Consequently the global error e_n is not well-defined under the stated hypotheses and the convergence theorem is vacuous as written. A solvability lemma for (3.8), together with uniform a priori bounds on the numerical solution, is required.
[Appendix B, proof of Proposition 2.20] The proof begins "Write lim_{w→x} f^∧(w)=:σ", i.e. it assumes exactly the limit whose existence is the content of the proposition. The subsequent argument only shows that if the limit exists, then it equals f^∧(x). The proof also applies Corollary 2.6 to assert (B.4) without checking the hypothesis that at least one one-sided derivative ∂±f(x) exists, and later uses ∂±f(x) in (B.6). Therefore Proposition 2.20 is not proved, and Theorem 2.21, which relies on it, inherits the gap. A non-circular proof, or a revision that downgrades these claims, is needed.
[Abstract and Section 2] The abstract advertises results that do not appear in the body: the at-most-countability of discontinuities at which the specular derivative is nonzero, a nowhere-continuous function with identically zero specular derivative, and a second-order class lying strictly between C^1 and C^2. None of these is stated or proved in the full text, whose actual title and content are about numerical ODE schemes. This mismatch makes the manuscript's advertised scope inaccurate and must be fixed by aligning the abstract with the body.

minor comments (5)

[Theorem 3.5] Typo: "If the equation (1.1) with admits" should read "If the equation (1.1) admits".
[Algorithm 3.1] The fixed-point iteration returns u_guess after M iterations even when the tolerance η has not been met. An explicit failure flag or an assertion that the iteration has converged would improve reproducibility and clarify the theorem's relation to the algorithm.
[Tables 2 and 3] For SE5, the ℓ∞ errors at small N are non-monotone (e.g., Table 2, N=32,64,128). The statement that numerical results confirm second-order convergence should be restricted to the asymptotic range where the error ratios stabilize.
[Definition 3.1] The specular trigonometric scheme requires u_1; the text says the exact solution is used to obtain u_1 in the examples. This is a test device, not a numerical method; it should be stated explicitly.
[References] The paper relies on [4,5,6] for properties of the "regular specular derivative" and for the definitions used in Section 3.4, but the relevant definitions are not reproduced. A self-contained statement of the imported results would help the reader.

Circularity Check

1 steps flagged · score 4.0 of 10

Proposition 2.20's proof assumes the limit whose existence is the theorem's content; the SE5 numerical analysis is otherwise self-contained.

other [Appendix B, Proof of Proposition 2.20 (Section 2.3)]
"Let ε>0 be arbitrary. Write lim_{w→x} f^∧(w) =: σ."

Proposition 2.20 asserts that f^∧ is continuous in (a,b), i.e., that lim_{w→x} f^∧(w) exists and equals f^∧(x). The proof begins by writing “lim_{w→x} f^∧(w) =: σ”, thereby postulating the existence of the very limit to be established. It then uses estimates based on (B.5) to conclude f^∧(x)=σ. No argument is given that the hypothesis f^∧∧(x) exists implies the two-sided limit exists. Thus the proof derives the conclusion only under the assumption that its left-hand side exists—exactly the content of the proposition. Theorem 2.21 inherits this unsupported step.

full rationale

The numerical half of the paper is not circular: SE5's consistency (Theorem 3.5) is a Taylor expansion of A(α, α+h_2) (Corollary A.2, proved in Appendix A) with the h^2 term cancelling, and the convergence proof (Theorem 3.7) is a standard Grönwall-type discrete inequality. Those arguments do not quote a fitted value or presuppose the desired error bound. The reliance on [4],[5],[6] is for auxiliary algebraic properties (e.g., Lemma A.3) and for an illustrative ODE, not for the main reduction; those lemmas are elementary or re-proved here. The genuine circular step is in Proposition 2.20, where the proof posits “lim_{w→x} f^∧(w) =: σ” although the proposition's conclusion is precisely that this limit exists and equals f^∧(x). This makes the claimed C^1-regularity from second specular differentiability (Theorem 2.21) unsupported. However, this does not affect the independent SE5 convergence derivation, so the overall circularity is partial rather than pervasive. The absence of a discrete well-posedness lemma in Theorem 3.7 is a correctness/rigor gap, not a circular reduction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central load-bearing assumptions are the new definition itself, standard real analysis, the smoothness/Lipschitz hypotheses in the numerical theorems, and an unjustified limit-existence premise in Proposition 2.20. No free numerical parameters enter the consistency/convergence claims.

assumptions (4)

standard math Standard real analysis: continuity, IVT, Taylor's theorem, arctan/tan identities
Used throughout Section 2, Lemma A.1, and Theorem 3.5.
ad hoc to paper Definition of the specular derivative via B and A (Def. 1.1, (1.2)-(1.4))
The entire framework rests on this new definition; its value is judged by ensuing theorems, not by external evidence.
domain assumption In the numerical theorems, u in C^2/C^3 and F is Lipschitz (Thm 3.5, 3.7)
Stated as hypotheses; the claims do not apply beyond them.
ad hoc to paper In Proposition 2.20, existence of the limit lim_{w->x} f^∧(w) is assumed before being proven
This is an unjustified load-bearing premise: the proof writes sigma for a limit it never establishes.

invented entities (1)

Specular derivative f^∧ and auxiliary functions A and B
purpose: Generalized derivative averaging the angles of forward/backward difference quotients; used to build ODE integrators (SE5).
A new mathematical definition with no empirical handle outside the paper; its justification is entirely internal to the theorems proved from it.

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Pith. "Pith review of Specular differentiation in one dimension: a quasi-mean value theorem, regularity, and discontinuities." pith.science (2026). https://pith.science/paper/VO2A5OBD

@misc{pith2026260109900,
  author       = {Pith},
  title        = {Pith review of: Specular differentiation in one dimension: a quasi-mean value theorem, regularity, and discontinuities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VO2A5OBD}},
  note         = {Machine review of arXiv:2601.09900}
}

abstract

We develop a theory of specular differentiation on real intervals. The specular derivative is defined by averaging the angles associated with the forward and backward difference quotients and extends classical differentiation. For specularly differentiable and continuous functions, we establish a quasi-mean value theorem and use it to show that continuity of the specular derivative implies $C^1$-regularity. Without assuming continuity, we prove that the discontinuities at which the specular derivative is nonzero form an at most countable set; a nowhere-continuous example with identically zero specular derivative shows that this restriction is sharp. In the second-order theory of specular differentiation, we impose additional conditions on twice specularly differentiable functions to obtain a class that lies strictly between the classes $C^1$ and $C^2$.

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Cited by 1 Pith paper

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Specular differentiation in normed vector spaces: Quasi-Mean Value and Quasi-Fermat Theorems
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Introduces specular differentiation generalizing Gâteaux and Fréchet derivatives, proves weak Mean Value and Fermat theorems in this sense, and identifies a distinguished element of the Fréchet subdifferential for con...

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Works this paper leans on

2 extracted references · cited by 1 Pith paper

[1]

[1]C. E. Aull,The first symmetric derivative, Amer. Math. Monthly, 74 (1967), pp. 708–711, https://doi.org/10.2307/2314269. [2]J. C. Butcher,Numerical methods for ordinary differential equations, John Wiley & Sons, Ltd., Chichester, 3rd ed., 2016, https://doi.org/10.1002/9781119121534. [3]K. Jung,specular-differentiation, v1.0.0, Zenodo, 2026, https://doi...

arXiv 1967
[2]

Publishing, River Edge, NJ, 1998, https://doi.org/10.1142/9789812816047

doi:10.1142/9789812816047 1998

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

[1]C. E. Aull,The first symmetric derivative, Amer. Math. Monthly, 74 (1967), pp. 708–711, https://doi.org/10.2307/2314269. [2]J. C. Butcher,Numerical methods for ordinary differential equations, John Wiley & Sons, Ltd., Chichester, 3rd ed., 2016, https://doi.org/10.1002/9781119121534. [3]K. Jung,specular-differentiation, v1.0.0, Zenodo, 2026, https://doi...

arXiv 1967

[2] [2]

Publishing, River Edge, NJ, 1998, https://doi.org/10.1142/9789812816047

doi:10.1142/9789812816047 1998