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REVIEW 4 major objections 4 minor 33 references

Main problems in constructing quantum theory based on finite mathematics

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Pith's one-line read Quantum theory built on finite rings puts particles and antiparticles in a single irreducible representation, making the distinction only approximate.

desk verdict A clear but non-advancing exposition of the FQT program; the central claim fails because the paper never defines a finite-p measurement rule. read the letter →

arxiv 2412.01846 v1 pith:KRPTHGXF submitted 2024-11-26 physics.gen-ph

Felix M. Lev This is my paper · ORCID
classification physics.gen-ph MSC 11Axx11Txx13Mxx16Gxx81R05
keywords finitemathematicsquantumtheorystandardcharacteristicpmodularrepresentationsDiracsupersingletonparticle-antiparticledistinctionadditivenumbers
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 argues that a quantum theory built on finite rings, where arithmetic runs modulo a huge but finite characteristic p, is more fundamental than standard quantum theory, which it treats as the p→∞ limit. Its central new claim is that in finite quantum theory every irreducible representation necessarily contains states with both positive and negative energies, so particles and antiparticles cannot be separated and additive conserved charges such as electric charge or baryon number lose their meaning. The author's two toy models—a superalgebra with two fermionic operators and the Dirac supersingleton—show this phenomenon explicitly without the lengthy technical proofs of earlier work. If correct, the standard particle/antiparticle picture and its superselection rules are only accurate at late cosmic times because p is currently enormous, and the ultimate theory must rest on different principles.

What carries the argument

The central object is the finite ring R_p with modulo-p arithmetic and its quadratic extension R_{$p^{2}$}, together with modular irreducible representations of the dS/AdS algebra or the osp(1,4) superalgebra. The key identity is the recurrence a(n)=q0+n−1−a(n−1) for the ladder coefficients in the two-operator superalgebra: its standard solution stays positive for all n, while the modulo-p solution vanishes at n=2p+1−2q0, forcing a finite irrep whose spectrum necessarily contains both negative and positive f(λ_n) and a zero eigenvalue. This wrapping-around-the-ring mechanism is what makes the positive and negative energy sectors cohabit in a single representation.

What would settle it

Pick any normalized state in a standard Hilbert space, such as a Gaussian wave packet, and compute the minimal approximation error with integer coefficients bounded well below p; if the error does not go to zero as p grows, the reproduction step fails. Alternatively, construct the modular irrep of the two-operator superalgebra for a small odd p, say p=7, and check whether its spectrum always contains a zero eigenvalue and both positive and negative f(λ_n); finding a q0 whose irrep is purely one-signed would refute the claim.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that modular representations over the ring R_p of characteristic p are finite-dimensional and irreducible in a way that always mixes positive and negative energy sectors, whereas the corresponding standard representations split into two infinite-dimensional irreps distinguished by the sign of the energy. In the model with two operators d' and d'' defined by h={d', d''}, standard theory gives two irreps with eigenvalues n+q0 and −(n+q0), while finite theory gives one irrep whose spectrum wraps around a circle of p points, ends at the exact negative of the starting eigenvalue, and contains a zero eigenvalue. Applied to Dirac supersingletons, the four standard objects (Di, Rac, and their antiparticles) collapse in FQT into one object, and the single finite-dimensional irrep has dimension $p^{2}$. Because the finite spectrum reduces to the standard one only when expansion coefficients are much smaller than p, the paper concludes that standard quantum theory is a degenerate limit rather than the fundamental theory.

Load-bearing premise

The argument depends on the paper's own definition of 'more general'—A is more fundamental than B if A can reproduce B's results in a limit while B cannot reproduce all of A—and on the claim that any physical state can be approximated by integer-coefficient vectors; if either fails for physically relevant states, the conclusion that FQT is the fundamental theory collapses.

Editorial extensions

If this is right

  • In FQT, superpositions of electron and positron states are allowed; particle–antiparticle superselection rules disappear at finite p.
  • Electric charge, baryon number, and similar additive quantum numbers become approximate labels valid only when all relevant expansion coefficients are much smaller than p.
  • The baryon asymmetry problem dissolves if p was smaller in the early universe, because equal baryon and antibaryon numbers cannot even be defined then.
  • The Dirac, Rac, and antiparticle singletons of standard AdS theory merge into a single finite-dimensional supersingleton irrep of dimension p^2.
  • dS and AdS symmetries are equivalent in FQT, so the absence of observed supersymmetry need not signal a fundamental preference for dS over AdS.

Reading between the lines

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

  • Because the sign-mixing mechanism is driven by the modular recurrence a(n)=q0+n−1−a(n−1) wrapping around the ring, the same one-irrep-with-both-energy-signs effect should arise in any ladder-generated representation over R_p; explicit modular constructions of Poincaré-type irreps at moderate p would show whether the two toy models are representative.
  • If p is not a constant but tied to the universe's changing state, then dimensionless couplings and masses could carry a tiny time dependence proportional to d(ln p)/dt, which might be searched for in high-precision spectroscopy or cosmological observations.
  • A natural next check is whether the zero-eigenvalue vector that appears in every finite irrep corresponds to a physical state with no additive charges; if so, it might play the role of a neutral state that standard theory mislabels as a forbidden superposition.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper argues that quantum theory based on finite rings of characteristic p (FQT) is more general than standard quantum theory (SQT), with SQT obtained as the degenerate p → ∞ limit. The central claims are that in FQT all irreducible representations necessarily contain states of both positive and negative energy, and that as a result the fundamental theory should not contain particle-antiparticle or additive-quantum-number concepts. The paper presents two model calculations: a two-operator superalgebra (Sec. 6) and the Dirac supersingleton (Sec. 8), showing that modular irreducible representations over Rp are finite-dimensional and contain both positive and negative eigenvalues of the relevant operator, whereas the corresponding SQT representations split into separate positive- and negative-energy irreducibles. The paper also discusses the equivalence of dS and AdS symmetries in FQT, the absence of a scalar product in FQT, and speculative cosmological consequences involving a time-dependent characteristic p.

Significance. If the general claim were established, it would constitute a substantial conceptual revision of quantum foundations: standard particle/antiparticle notions and additive conservation laws would be only approximate low-energy artifacts of a finite modular theory. The toy-model calculations in Secs. 6 and 8 are explicit, transparent, and easy to verify, and they do illustrate an interesting algebraic difference between modular and ordinary representations of the same superalgebra. However, the paper does not prove the general 'all IRs' statement, and, more seriously, it does not provide a measurement rule or positive-definite inner product for FQT. Since probabilities and expectation values are central results of SQT, the paper's own criterion for 'more general' cannot be applied to the claimed FQT-SQT relationship. The significance is therefore conditional: the examples may be useful illustrations of modular representation theory, but the paper does not establish the foundational conclusions announced in the abstract.

major comments (4)
  1. [Abstract; Sec. 9] The abstract states that 'in FQT, all IRs necessarily contain states with both signs of energy' and draws conclusions about particle-antiparticle concepts. The manuscript, however, proves this only for two toy models (Secs. 6 and 8), and the general statement is not stated as a precise theorem, nor is a proof sketched; the reader is referred to the author's monographs [1,2]. If the paper is intended as an expository companion to those works, it should say so explicitly and clearly separate the illustrative model results from the previously claimed general theorems. As written, the title and abstract promise more than the presented derivations establish.
  2. [Sec. 4, Eq. (2)] The claim that FQT can reproduce all results of SQT is not established. Eq. (2) only shows that every vector in a separable Hilbert space can be approximated by a finite linear combination with integer coefficients. To reproduce the results of SQT one also needs inner products, transition amplitudes, expectation values, and a probability rule. The paper itself states that over Rp2 no scalar product with (x,x)>0 for all nonzero x exists, that Hermitian conjugation has limited applicability, and that 'in FQT such a requirement cannot be imposed.' Without a positive-definite inner product and a Born-rule analogue, the criterion in Sec. 1 that 'A can reproduce any result of B' cannot be satisfied. This is a load-bearing gap in the central degeneracy claim.
  3. [Sec. 1, Definition] The paper's conclusion that FQT is more fundamental than SQT depends on a definition of 'more general' proposed by the author. The definition itself is not the issue, but its application requires a rigorous meaning for 'B is obtained from A in the formal limit when the parameter goes to infinity.' For the limit p → ∞ of modular representations, no topology, inverse system, or representation-theoretic limit is specified; the spaces Rp2 are not nested in a complex Hilbert space. A concrete test would be to exhibit, for each SQT transition amplitude or expectation value, a sequence of well-defined FQT expressions converging to it. The paper does not provide such a construction, and the toy models do not fill this gap.
  4. [Sec. 6.2, Eq. (15); Sec. 8.2] The modular IR calculations are plausible, but the claimed relationship to the SQT limit is not demonstrated. For a fixed SQT parameter q0, the finite-dimensional FQT IR has dimension that grows with p (roughly 2p−2q0 in the Sec. 6 model), and no argument is given that this family of representations converges to the direct sum of the positive- and negative-energy SQT IRs. The picture of 'one IR in FQT splitting into two IRs in SQT at p → ∞' is therefore a suggestive analogy, not a proven statement. Either a representation-theoretic limit theorem should be provided, or the claim should be weakened to apply only to the specific models.
minor comments (4)
  1. [Sec. 6.1, Eq. (10)] In Eq. (10), the condition 'h f0 = −q0 e0' should presumably read 'h f0 = −q0 f0'; otherwise the subsequent eigenvalue computation for fn is inconsistent with the stated basis.
  2. [Sec. 8.1, Eq. (19) and Eq. (21)] Eq. (19) writes 'h_j f0 = −1/2 e0' and Eq. (21) writes 'M04 fjk = −(1+j+k) ejk'; both should refer to f0 and fjk, respectively, on the right-hand side. These are typographical errors, but they are confusing in the derivation.
  3. [Sec. 2, Figure 1] The text refers to 'Figure 1' for the relation between Rp and Z, but the figure is not reproduced in the manuscript text; please include it or remove the reference.
  4. [Secs. 6.2 and 8.2] The calculations assume p is odd; the paper should explicitly state whether the claims extend to even p, since Sec. 2 gives a definition covering both cases.

Circularity Check

1 steps flagged · score 7.0 of 10

The central claim that FQT is more general than SQT is imported from the author's own prior works and is not derived in this paper; the new toy models are self-contained but do not establish the generality claim.

  1. self citation load bearing [Sec. 4, after Eq. (2)]
    "As shown in [1, 2, 3, 4], by using Definition and the above results one can prove that FQT is more general (fundamental) than SQT and the latter is a special degenerate case of the former in the formal limit p → ∞: when the numbers ( a j, b j) are such that ∀j, |f (a j)| and |f (b j)| are much less than p then FQT reproduces all results of SQT but SQT cannot reproduce all results of FQT if some of the numbers ( a j, b j) are comparable to p."

    The paper's advertised result, that FQT is more general than SQT, is not derived in this paper: it is delegated to four prior works by the same author, [1]-[4]. The only supporting argument shown, Eq. (2), is an approximation theorem about state vectors with integer coefficients; it does not by itself show that inner products, probabilities, or expectation values of SQT can be reproduced, because Sec. 4 states that no positive scalar product exists over Rp2 and that Hermitian conjugation has limited applicability. The assertion 'FQT reproduces all results of SQT' is therefore the unverified premise of a self-citation chain, and the conclusion that SQT is a degenerate case of FQT is forced by that chain plus the author's own Definition, not by the toy models in this paper.

full rationale

The algebraic examples in Secs. 6 and 8 are self-contained: the recurrence a(n)=q0+n-1-a(n-1) and its modular solution are derived explicitly, and they genuinely show that a finite-dimensional modular IR has eigenvalues n+q0 mod p, which wrap through the arbitrary positive and negative halves of Rp. Those model statements are not circular and are not fitted to data. The circularity is at the level of the paper's central conclusion. Sec. 1 defines 'more general' using a criterion proposed in the author's earlier works, and Sec. 4 then asserts, citing [1,2,3,4], that FQT satisfies the criterion. The paper explicitly concedes the absence of a scalar product and of a Born-rule analog in FQT, so the claimed reproduction of 'all results of SQT' cannot be read off from Eq. (2) and is not demonstrated here. Because that claim is load-bearing for the abstract's conclusion that particle-antiparticle and additive quantum numbers are not fundamental, the circularity burden is substantial. At the same time, the paper is transparent that the full proofs are in the cited monographs, and the new toy derivations are independent mathematical content, so the analysis is not one of total circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central argument rests on the author's Definition, the integer-coefficient approximation theorem, and the choice of modular IRs. The toy examples use a hand-selected q0 and the speculative assumption that p changes over cosmic time. No new particles or forces are introduced.

free parameters (2)
  • q0 = q0=(p+1)/2+a in Sec. 6.2; q0=(p+1)/2 in Sec. 8.2
    Representation label chosen by hand. The toy derivation of mixed-energy IRs uses this particular choice, and the paper does not show the property for all allowed q0 values.
  • p (ring characteristic) = not fitted in this paper; argued in [1,2] to be of order exp(10^80)
    Fundamental parameter of FQT. This paper assumes p exists and is finite, and cites prior work for its magnitude and for the claim that p changes over cosmic time.
assumptions (4)
  • ad hoc to paper Definition in Sec. 1: if theory A contains a finite parameter and reduces to B in a limit while able to approximate any B result, A is more general and B is degenerate.
    This criterion is the author's own and is used to conclude FQT is more fundamental than SQT.
  • domain assumption Every SQT state can be approximated with integer coefficients (Eq. 2), so FQT over Rp2 can reproduce SQT results.
    Cited to [24] and [4]. This is load-bearing for the claim that FQT contains SQT as a limit.
  • domain assumption The dS/AdS algebras defined by Eq. (1) are the relevant symmetry algebras, with no need for a spacetime background.
    The paper assumes physics is described by these algebra representations and rejects classical spacetime background as fundamental.
  • ad hoc to paper p was smaller at early stages of the universe and changes with cosmic evolution.
    This is a speculative cosmological assumption used to explain baryon asymmetry and the nature of time. No evidence is given in this paper.

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Pith. "Pith review of Main problems in constructing quantum theory based on finite mathematics." pith.science (2026). https://pith.science/paper/KRPTHGXF

@misc{pith2026241201846,
  author       = {Pith},
  title        = {Pith review of: Main problems in constructing quantum theory based on finite mathematics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KRPTHGXF}},
  note         = {Machine review of arXiv:2412.01846}
}
abstract

As shown in our publications, quantum theory based on a finite ring of characteristic $p$ (FQT) is more general than standard quantum theory (SQT) because the latter is a degenerate case of the former in the formal limit $p\to\infty$. One of the main differences between SQT and FQT is the following. In SQT, elementary objects are described by irreducible representations (IRs) of a symmetry algebra in which energies are either only positive or only negative and there are no IRs where there are states with different signs of energy. In the first case, objects are called particles, and in the second - antiparticles. As a consequence, in SQT it is possible to introduce conserved quantum numbers (electric charge, baryon number, etc.) so that particles and antiparticles differ in the signs of these numbers. However, in FQT, all IRs necessarily contain states with both signs of energy. The symmetry in FQT is higher than the symmetry in SQT because one IR in FQT splits into two IRs in SQT with positive and negative energies at $p\to\infty$. Consequently, most fundamental quantum theory will not contain the concepts of particle-antiparticle and additive quantum numbers. These concepts are only good approximations at present since at this stage of the universe the value $p$ is very large but it was not so large at earlier stages. The above properties of IRs in SQT and FQT have been discussed in our publications with detailed technical proofs. The purpose of this paper is to consider models where these properties can be derived in a much simpler way.

Figures

Figures reproduced from arXiv: 2412.01846 by the authors.

Figure 1
Figure 1. Relation between Rp and Z degenerate theory because in Z there are no operations modulo a number. In FQT, states are elements of linear spaces over Rp. One might think that SQT is more general than FQT because in SQT one can work not only with integers but also with rational and real numbers. However, as noted in [1, 2, 3, 4] and Sec. 4, since in SQT the states are projective, for describing wave functions with any … view at source ↗

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