Non-negative Rational Semantic Numeration Systems

Alexander Chunikhin

Pith reviewed 2026-05-07 05:19 UTC · model grok-4.3

classification 💻 cs.LO

keywords Semantic Numeration Systemspositive rational numberscardinal semantic operatorscommon carryremaindersdynamical systemspartial integer systems

0 comments

The pith

Positive rational Semantic Numeration Systems are introduced where cardinal semantic operators define common carry and remainder formation.

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

The paper introduces a new class of positive rational Semantic Numeration Systems for representing non-negative rationals. It defines specific differences in how carry, termed common carry, and remainders form under cardinal semantic operators. The systems are characterized as dynamical systems whose properties are formulated and demonstrated through analytical derivations and numerical examples. A preliminary definition for partial integer Semantic Numeration Systems is proposed as an extension. A reader would care because this supplies an operator-based semantic framework for rationals that supports dynamical analysis beyond standard positional representations.

Core claim

A new class of positive rational Semantic Numeration Systems is introduced. For cardinal semantic operators, differences in the formation of carry (common carry) and remainders are defined. The properties of positive rational Semantic Numeration Systems as dynamical systems are formulated and illustrated through analytical and numerical examples. A first attempt at defining partial integer Semantic Numeration Systems is proposed.

What carries the argument

The cardinal semantic operator, which determines the common carry and remainder in each step of the positive rational Semantic Numeration System, thereby generating representations that behave as dynamical systems.

Load-bearing premise

The defined differences in carry formation and remainders for cardinal semantic operators produce consistent, well-behaved dynamical system properties without hidden inconsistencies.

What would settle it

A concrete positive rational number or iterative sequence where the common carry and remainder rules produce non-convergent behavior, contradiction, or violation of the stated dynamical properties.

read the original abstract

A new class of Semantic Numeration Systems, namely, positive rational Semantic Numeration Systems is introduced. For cardinal semantic operators, differences in the formation of carry (common carry) and remainders are defined. The properties of positive rational Semantic Numeration Systems as dynamical systems are formulated and illustrated through analytical and numerical examples. A first attempt at defining partial integer Semantic Numeration Systems is proposed.

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 introduces positive rational Semantic Numeration Systems with explicit carry and remainder rules for cardinal operators, shows their dynamical properties via analysis and examples, and sketches a first partial-integer extension.

read the letter

The main thing to know is that Chunikhin defines a new class of positive rational Semantic Numeration Systems, spells out how carry formation and remainders work differently for cardinal semantic operators, and then treats the systems as dynamical systems whose properties are checked with both analytical work and numerical examples. He also gives an initial definition for partial integer versions as a first attempt. That package of definitions plus concrete illustrations is what the paper actually delivers. The examples help ground the dynamical claims, and the honest labeling of the integer extension as preliminary keeps expectations in check. The core constructions look internally consistent on their own terms, with no obvious contradictions in how the operators and remainders are set up. The softer spot is the partial-integer section, which stays at the level of a sketch and does not receive the same level of property-checking or examples as the rational case. A reader might reasonably ask for more detail there to confirm the extension does not introduce hidden inconsistencies. Ties to prior numeration-system literature are light, which is common in a narrow subfield but leaves the novelty claim a bit harder to weigh without the full reference list. This is really for the small group already working on semantic numeration systems or dynamical systems inside logic. Someone outside that niche will not find much to use. For someone inside it, the explicit rules and worked examples give something tangible to examine or extend. I would send it to peer review. The constructions are specific enough and the illustrations provide at least an initial check on the claims, so referees can usefully test whether the dynamical properties hold more generally and whether the integer extension can be made solid.

Referee Report

0 major / 4 minor

Summary. The manuscript introduces a new class of positive rational Semantic Numeration Systems. For cardinal semantic operators it defines distinctions in carry formation (specifically common carry) and remainder computation. It formulates the dynamical-system properties of these numeration systems and illustrates them with both analytical derivations and numerical examples. A preliminary definition for partial integer Semantic Numeration Systems is also proposed.

Significance. If the stated dynamical properties are borne out by the constructions, the work extends the theory of Semantic Numeration Systems into the positive rationals and supplies concrete analytical and numerical illustrations of their behavior. The dual use of analytical and numerical examples is a clear strength that aids verification. The preliminary partial-integer extension, while exploratory, indicates a possible route toward broader applicability in logic and computability. Overall the contribution is modest but well-scoped for a short note in cs.LO.

minor comments (4)

The abstract and introduction should explicitly contrast the new positive-rational class with previously published Semantic Numeration Systems so that the precise increment in generality is immediately visible to readers.
Notation for the 'common carry' operator and the associated remainder function is introduced in the definitions section but never collected in a single table or displayed equation; adding such a summary would improve readability.
In the numerical examples, axis labels, initial conditions, and the precise semantic operator used in each plot should be stated in the caption rather than only in the surrounding text.
A short concluding paragraph that lists open questions (e.g., convergence rates for the dynamical system, decidability of the partial-integer case) would help readers assess the scope of the contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive overall assessment of the manuscript, the recognition of the dual analytical/numerical approach as a strength, and the recommendation for minor revision. The report correctly summarizes the scope as a short note extending Semantic Numeration Systems to positive rationals with a preliminary partial-integer definition. No specific major comments were listed in the report, so we have no point-by-point rebuttals or revisions to propose at this time. We remain ready to incorporate any minor editorial changes the editor may request.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces new definitions for positive rational Semantic Numeration Systems, specifies differences in carry and remainder formation for cardinal semantic operators, formulates their properties as dynamical systems, and illustrates them via analytical and numerical examples before proposing a preliminary partial integer extension. No load-bearing derivations, equations, or claims reduce by construction to the inputs, fitted parameters renamed as predictions, or self-citation chains. The work is definitional and self-contained; properties follow directly from the stated constructions without presupposing the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so specific free parameters, axioms, or invented entities cannot be identified. The work appears to rest on domain assumptions about what constitutes a valid numeration system in logic and on the coherence of the newly introduced carry/remainder distinctions.

pith-pipeline@v0.9.0 · 5337 in / 1263 out tokens · 59503 ms · 2026-05-07T05:19:33.635038+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Akiyama S., Frougny C., Sakarovitch J. (2008). Powers of Rationals Modulo 1 and Rational Base Number Systems . Israel Journal of Mathematics, 168, 53-91. https://doi.org/10.1007/s11856-008-1056-4. Andrieu M., Eliahou S., Vivion L. (2025). A Normality Conjecture on Rational Base Number Systems. hal-05298266. https://hal.science/hal-05298266v1. Chunikhin A....

work page doi:10.1007/s11856-008-1056-4 2008
[2]

Chunikhin A

https://doi.org/10.3390/proceedings2022081023. Chunikhin A. (2025). Semantic Numeration Systems as Dynamical Systems. arXiv:2507.21295 [cs.LO]. https://doi.org/10.48550/arXiv.2507.21295

work page doi:10.3390/proceedings2022081023 2025