Remembering Solomon Marcus
math.HO · cs.AI
Quick verdict
Memorial essay for Solomon Marcus mixing memoir, surrealism and unproofed math sketches (universal logs, Euler formula in racks); evocative but short on rigor.
Summary
This arXiv preprint (math.HO) is the latest installment in Florin Nichita's 'Memories with Solomon Marcus' series, honoring the Romanian mathematician around the centenary of his birth. Framed by Andre Breton's surrealist manifesto and Nicolescu's transdisciplinarity, it interleaves personal recollections (Marcus's remarks on colleagues such as Nicusor Dan and Marian Aprodu), a report on Laura De Marco's June 2026 Romanian Academy talk on the Mandelbrot set, and the 2026 AMS second-prize poem 'Pantoum for the Mandelbrot set'. The mathematical content comprises: (i) 'universal logarithms' L2, L3 (e.g., L2(4)=2, L2(27)=3) and a generalized radical R2; (ii) new operations on the reals claimed to 'approximately generate' addition; (iii) stated inequalities linking these to sqrt(ab) and the arithmetic mean for a,b >= 4; (iv) an Euler-type formula for (R union {-infinity}, +, a new operation); and (v) an appendix developing 'trigonometry' in racks and weak racks, proving an Euler-formula analogue (Theorem 4.8, e(x,x) = (cos x, sin x)) and asserting, without proof, that the rack exponential, cosh and sinh solve the quantum Yang-Baxter equation (Theorem 4.12). Main limitations: nearly all mathematical claims are stated without proofs, definitions are imprecise or internally inconsistent (e.g., L3(1)=2 as printed), and the memoir, poetry commentary and mathematics are only loosely integrated.
Review complete
Assessment at a glance
5
risk flags identified
● high 2
● medium 2
● low 1
4 strengths · 5 weaknesses · 5 author questions
Levels reflect verification complexity — not a quality score. Evolve never rates your paper.
Key claims under review
- The paper defines order-k 'universal logarithms' (e.g., L2(4)=2, L2(27)=3) and a generalized radical R2, and asserts mean-type inequalities such as L2(ab) <= sqrt(ab) <= (a+b)/2 <= a(circle)b - 2 for a, b >= 4.
- New binary operations on the reals 'approximately generate' ordinary addition, and the structure (R union {-infinity}, +, oplus) admits an Euler-type formula.
- In racks, the maps cos x = e*x and sin x = x (diamond) e satisfy product and co-product homomorphism properties and the fundamental identity sin(cos x) = cos(sin x) = x.
- Theorem 4.8: the rack exponential obeys an Euler-formula analogue, e(x,x) = (cos x, sin x), with the identity e(Pi,Pi) = (U, O); this is proved by direct substitution.
- Theorem 4.12 (unproved): the rack exponential, cosh and sinh functions solve the quantum Yang-Baxter equation, linking the tribute to braid-group representations.
Strengths
Elegant unifying result with a complete proof
Theorem 4.8 (exp_e composed with the diagonal equals [cos x sin] composed with the diagonal, i.e. e(x,x) = (cos x, sin x)) packages the rack 'trigonometry' of section 4.1 into a clean Euler-formula analogue; the two-line proof by direct substitution is correct given the definitions cos x = e*x and sin x = x (diamond) e.
Accessible novel constructs fitting the math.HO genre
Universal logarithms (L2(4)=2, L2(27)=3, L3(10^10^10)=10), the generalized radical R2 and symmetric exponentiation are elementary, playful definitions in the spirit of recreational mathematics, matching the paper's cultural framing and the cited Ramanujan-style pedagogy of Bradshaw-Vignat [10].
Genuine interdisciplinary documentation
Connects the tribute to live research threads
Weaknesses
Most mathematical claims lack proofs
Imprecise or internally inconsistent definitions
Fragmented structure with weak internal connection
Self-referential and rhetorical citation practice
Copyediting deficiencies suggesting no referee pass
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Frequently asked questions
Who was Solomon Marcus and why is this paper written about him?
Solomon Marcus (1925-2016) was a Romanian Academy mathematician known for crossing disciplinary boundaries — mathematical analysis, mathematical linguistics, poetics and semiotics. This arXiv note (2609.18224) by Florin Felix Nichita is part of a memorial series published around the centenary of Marcus's birth, mixing personal recollections of their meetings with short mathematical observations inspired by them.
What is a 'universal logarithm' as defined in this paper?
It is a logarithm-like function with no fixed base: the order-two universal logarithm L2 maps a number n to the largest m satisfying n = m^m, so L2(4) = 2 and L2(27) = 3. The order-three version L3 uses three-level power towers (e.g., L3 of 10^10^10 is 10). The paper pairs it with a 'generalized radical' R2 and states inequalities relating both to the geometric and arithmetic means, though without proofs.
Is this a research paper with proven theorems?
Largely no. It is a commemorative essay: the only fully proved result is the rack-theoretic Euler formula (Theorem 4.8, e(x,x) = (cos x, sin x)), a short direct verification. Other claims — the mean-type inequalities in section 2.3 and Theorem 4.12 on Yang-Baxter solutions — are stated without proof. Readers should treat the mathematical sections as sketches or research announcements within a memorial essay.
What are racks and what does the paper do with them?
Racks are algebraic structures with two binary operations satisfying self-distributivity-type axioms, originally studied in knot theory. The paper defines 'trigonometric' functions on a rack (cos x = e*x and sin x = x (diamond) e for chosen elements e and O), shows they obey product and co-product rules, proves an Euler-formula analogue e(x,x) = (cos x, sin x), and remarks that weak racks relate to Boolean algebras and that the rack exponential purportedly solves the quantum Yang-Baxter equation.
About this review
This is an AI-generated preliminary review by Evolve.Pub, produced from the openly available text of the paper. It is not peer review and does not represent the authors' views. The canonical record of the paper is at the publisher link above.
- Text extraction severely mangled the formula-heavy sections (2.1-2.4, 4.1-4.3): power towers render as 'L3(222)', exponent examples as 'b22 = 4', and the operation formulas as concatenated strings, so several apparent errors (e.g., L3(1)=2, the -2/-3 inequality constants) may be extraction artifacts rather than authorial mistakes.
- The review is based on arXiv v1 (16 Sep 2026) extracted text only; no figures were present or analyzable, and references [2] and [4] are 'in press', so their content could not be verified.
- The section 2.4 displayed formula is only partially legible ([log2(cos^2 ...), log2(sin^2 ...)]), so the characterization of that 'Euler formula' claim may be inaccurate in this review.
- The rack axioms in Definition 4.1 match the standard literature (Wikipedia [18] is cited), but the paper cites no primary rack/YBE references, so novelty of Theorem 4.8 relative to prior work could not be assessed.
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