DCCQ: From Ordered Bernoulli Levels to Critical-Line Geometry: Integer Quantization, Bernoulli Residual Phase, and Prime-Power Spectra

Y. Kenan Y{\i}lmaz

REVIEW

DCCQ: From Ordered Bernoulli Levels to Critical-Line Geometry: Integer Quantization, Bernoulli Residual Phase, and Prime-Power Spectra

Y. Kenan Y{i}lmaz

Machine Learning · Submitted 3 Sep 2026

Pith's verdict: A careful, honestly scoped diagnostic paper: the main exact identity is universal rounding arithmetic, the critical-line geometry comes from an explicitly labeled continuation assumption, and the numerical controls are negative but reported as such.

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arXiv ID2609.03801 (version 2)

Pith IDI27FCKKR, Pith's permanent record of this paper and its review

CitationY. Kenan Y{\i}lmaz. "DCCQ: From Ordered Bernoulli Levels to Critical-Line Geometry: Integer Quantization, Bernoulli Residual Phase, and Prime-Power Spectra." arXiv:2609.03801 (2026). https://pith.science/paper/I27FCKKR

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Editor's note

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Short version: this is a legitimate, carefully written mathematics paper, and I agree with both the reader and the stress-test note. The genuinely new thing is the package: applying the nearest-integer decomposition to L_k = 1/4 + gamma_k^2, writing the residual exactly as the periodic Bernoulli function Btilde_1(gamma_k^2 + 3/4), and then circularizing to Z_k = i exp(2 pi i gamma_k^2). Equation (17) is exact, elementary, and independent of RH. What the paper does well is state that exactness plainly, separate exact identities from classical connections and from numerical controls, and report the negative controls as negative. Remark 3.3 is particularly good: it says explicitly that the continuation rule 1 - z = conjugate(z) is an additional assumption, not a consequence of the level equation.

The soft spots are real but not load-bearing. The title and opening abstract could let a casual reader think the Bernoulli level geometry alone forces the critical line, whereas the paper itself repeatedly restricts the claim to a geometry obtained under a specified continuation rule. That is a presentation issue, not a mathematical flaw. The zeta-specific content is diagnostic and currently negative: the prime-shell count matches the PNT heuristic, the Weyl moments show no convincing dependence, and the correlations with omega(N) and Omega(N) are small. So the significance is potential rather than established, and the paper says so. The complex-s lift in Sections 7 and 8 is correct but mostly a re-encoding of log m; Corollary 6.5 is honest about what the scalar projection hides. Section 9 is background context, not load-bearing.

I have no major objection. The strongest exact claim is universal rounding arithmetic applied to a specific coordinate, and the paper does not pretend it derives zero locations or proves RH. The preliminary statistical tests are genuinely preliminary and labeled as such. The citation pattern looks sane, and the relevant literature appears to be cited.

The paper deserves a serious referee. It is the kind of work where a referee’s main job is to check that the exact statements and the numerical controls really are separated as cleanly as the author claims; on my reading, they are. I would not cite it in my own work soon, because it produces no new result about the zeros themselves, but I would bring it to a reading group as an example of disciplined framing around an elementary identity.

Referee report

Significance

If the paper's intent is to provide an exact, elementary decomposition of the quadratic coordinate $1/4+\gamma_k^2$ in terms of nearest-integer rounding and a periodic Bernoulli residual, then it succeeds straightforwardly. The core identities (Eqs. (16)–(18)) are exact and universal, involving no zeta-specific content beyond using the tabulated zero ordinates. The paper's strengths are its transparency: it clearly marks the continuation rule (7) as an assumption, disclaims any RH proof, and reports negative numerical controls without overinterpretation. The conceptual link between the quadratic coordinate, integer shells, and the periodic Bernoulli phase is cleanly presented. The main limitation is that the critical-line geometry follows from an imposed continuation rule rather than from the level equation alone, a point the authors acknowledge. As a synthesis of elementary number-theoretic and analytic objects, the paper may be of interest to readers exploring structural decompositions of zero ordinates, though it does not produce new results about the zeta function itself.

Minor objections (4)

[Title and Abstract] The title ``From Ordered Bernoulli Levels to Critical-Line Geometry'' may suggest to a casual reader that the critical line $\Re s=1/2$ is a theorem derived solely from the Bernoulli level structure, whereas the paper relies on the additional continuation rule $1-z=\overline{z}$ (Eq.~7) to force $\Re z=1/2$. While Remark~3.3 addresses this, the title and the opening sentence of the abstract are somewhat stronger than the actual derivation. Consider a more guarded phrasing such as ``Ordered Bernoulli Levels and a Continuation-Induced Critical-Line Geometry''.
[Table 1] The column ``factorization of $N_k$'' writes prime factorizations compactly (e.g., $2^3\cdot5^2$). For readers unfamiliar with the conventions of factorion notation, a brief note in the caption explaining that exponents are indicated by superscripts would be helpful.
[Figure 4 caption] The reference line labelled $1/\sqrt{200}$ is described only as ``dashed reference'' with no explanation of what it represents. If it is the expected standard deviation of $|W_h|$ under uniformity for $h\neq0$, this should be stated; if it is something else, clarify.
[§5.4 (page 8)] The term ``conditional circular moment'' in Eq.~22 is introduced without a brief definition of what ``conditional'' refers to (namely, conditioning on $N_k\in\mathcal A$). A short sentence linking the notation to the classical Weyl criterion would improve readability.

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The paper

The paper claims

An exact decomposition of zeta zero levels reveals a periodic Bernoulli residual without approximation.

The reading

This paper establishes that the quadratic coordinate Q(z)=z(1-z), derived from a conjugated Bernoulli kernel, forces the critical line's real part 1/2 before any zeta input. For each tabulated nontrivial zeta zero ordinate gamma_k, the level L_k = 1/4+$gamma_k^{2}$ decomposes exactly into a nearest integer shell N_k and a centered residual delta_k that equals the periodic first Bernoulli function of $gamma_k^{2}$+3/4. This decomposition is exact, not asymptotic, and does not depend on the Riemann Hypothesis. Circularization yields a lossless integer/phase pair (N_k, Z_k), and the integer shell factorizes into prime-generator coordinates. The paper separates these exact identities from conditional Weyl statistics that test whether arithmetic classes of N_k modulate the residual phase.

What carries the argument

The paper's argument rests on three linked objects: (1) the complement-preserving continuation rule 1-z = conjgate(z), which forces the real part 1/2 for solutions of the Bernoulli level equation; (2) the quadratic coordinate Q(z)=z(1-z)=1/4+$u^{2}$, which supplies a real level geometry with a sharp minimum at the central point; and (3) the periodic first Bernoulli function B̃_1(x) = {x} - 1/2, which exactly captures the centered residual of the zero-induced levels L_k = 1/4+$gamma_k^{2}$.

Extended reading notes

Core claim

The central discovery is that for every nontrivial zeta zero ordinate gamma_k, the real number L_k = 1/4 + $gamma_k^{2}$ admits an exact nearest-integer decomposition L_k = N_k + delta_k, where N_k = floor(L_k + 1/2) and delta_k = B̃_1($gamma_k^{2}$ + 3/4) with B̃_1(x) = {x} - 1/2 being the periodic first Bernoulli function. Consequently, the residual delta_k is not a rounding error but an exact periodic Bernoulli readout. This identity is derived from the Bernoulli-level geometry and is independent of the truth of the Riemann Hypothesis.

Load-bearing premise

The load-bearing premise is that the Bernoulli kernel is continued to complex parameters by imposing the rule 1-z = conjugate(z), which forces the real part 1/2; the level equation alone does not force every complex parameter onto this line.

Editorial extensions

If this is right

The integer shell N_k and the circular phase Z_k = e^{2π i delta_k} form a lossless pair encoding each zero's level, isolating gamma_k^2 mod 1 as the modular variable.
Prime factorization of N_k separates zeros by arithmetic type (prime shell, prime power, composite) purely from the integer coordinate, without input from the zero ordinate beyond the shell number.
The first Bernoulli residual provides a local periodic structure tied to gamma_k^2 mod 1, decoupled from the higher Bernoulli numbers in Euler-Maclaurin expansions of zeta.
Conditional Weyl statistics on Z_k conditioned on arithmetic classes of N_k become a sharp statistical test for whether the prime-factor content of the integer shell modulates the residual phase, going beyond the classical explicit formula's global coupling.

Reading between the lines

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

If the decomposition is exact and the continuation rule is accepted as natural, then analogous constructions for other Dirichlet L-functions could yield similar integer–residual decompositions, potentially revealing arithmetic structure in their zero ordinates.
The paper's explicit separation of exact identities from conditional statistical tests offers a methodological template for studying other suspected relations between combinatorial geometry and prime distributions: state the algebraic structure exactly, then test the unsolved statistical dependence.
The weak assumption that the continuation rule forces ℜz=1/2 means that the critical line is a consequence of the choice of continuation, not an inherent property of the level equation; different continuations could generate other vertical lines for different L-functions.

Original abstract

We present the discrete--complex complement quotient (DCCQ) framework through the ordered Bernoulli-word kernel $f(p,n,k)=p^k(1-p)^{n-k}$ and its inverse-integer level sets. The binary level $2^{-n}$ selects $p=1/2$ as the unique real split-independent anchor. Imposing the additional continuation rule $1-z=\overline z$ gives the conjugation-symmetric line $z=1/2+iu$. On the central normalized branch, the inverse-level split $\kappa=k/n$ also has real part $1/2$. The quadratic coordinate $Q(z)=1/4+u^2$ identifies complementary conjugate points, has minimum $1/4$, and has an explicit two-sign inverse; restricting its values to integers defines an arithmetic quantization. For supplied positive critical-line zero ordinates $\gamma_k$, the induced level has the exact decomposition \[ L_k=\frac14+\gamma_k^2=N_k+\delta_k,\qquad \delta_k=\widetilde B_1\!\left(\gamma_k^2+\frac34\right), \] where $N_k=\lfloor L_k+1/2\rfloor$ and $\widetilde B_1(x)=\{x\}-1/2$. The integer--phase pair $(N_k,e^{2\pi i\delta_k})$ retains the level and the positive ordinate. Prime-exponent coordinates describe the factorization of the integer component, while conditional Fourier moments formulate an open question about the residual phases. The finite numerical diagnostics are descriptive. A distinct complex exponent $s$ gives an inverse-level representation of the supplied term $m^{-s}$; direct substitution cancels the auxiliary coordinates. These terms assemble into the classical Dirichlet series and Euler product for $\Re s>1$. The framework organizes exact coordinate identities, classical zeta connections, and numerical diagnostics; it establishes neither conditional equidistribution nor a zero-location theorem.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,060 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.96 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.30 · compares with prior results 0.11 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study the ordered Bernoulli-word kernel f(p,n,k)=p^k(1-p)^(n-k) and the geometry generated by its inverse-integer level sets. The binary level 2^(-n) selects p=1/2 as the unique real split-independent anchor. Under complement-preserving complex continuation, the pair becomes z=1/2+iu and 1-z=1/2-iu, producing a conjugation-symmetric vertical geometry before any zeta-function input is introduced. The quadratic coordinate Q(z)=z(1-z)=1/4+u^2 has a sharp minimum at the central point and admits an exact integer quantization. For critical-line zero ordinates gamma_k, the induced levels L_k=1/4+gamma_k^2 are decomposed exactly as L_k=N_k+delta_k, where N_k is the nearest integer and delta_k is a periodic first-Bernoulli residual. Circularization gives Z_k=exp(2 pi i delta_k), isolating gamma_k^2 mod 1 as the residual phase variable. Unique factorization resolves the integer shells into prime-generator coordinates, while a distinct complex exponent s lifts the same construction to the Dirichlet atoms m^(-s), linking the Dirichlet-series and Euler-product assemblies. Exact identities, classical zeta connections, numerical controls, and open conditional Weyl tests are kept explicitly separate. No proof of the Riemann Hypothesis is claimed.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the ordered Bernoulli-word kernel f(p,n,k)=p^k(1-p)^(n-k) and the geometry generated by its inverse-integer level sets.	contribution 1.00	0.85	0.06	asserted 0.88
s1	The binary level 2^(-n) selects p=1/2 as the unique real split-independent anchor.	result 0.95	0.60	0.07	asserted 1.00
s2	Under complement-preserving complex continuation, the pair becomes z=1/2+iu and 1-z=1/2-iu, producing a conjugation-symmetric vertical geometry before any zeta-function input is introduced.	method 0.47	0.70	0.05	asserted 1.00
s3	The quadratic coordinate Q(z)=z(1-z)=1/4+u^2 has a sharp minimum at the central point and admits an exact integer quantization.	result 0.76	0.64	0.08	asserted 1.00
s4	For critical-line zero ordinates gamma_k, the induced levels L_k=1/4+gamma_k^2 are decomposed exactly as L_k=N_k+delta_k, where N_k is the nearest integer and delta_k is a periodic first-Bernoulli residual.	result 0.56	0.70	0.18	asserted 1.00
s5	Circularization gives Z_k=exp(2 pi i delta_k), isolating gamma_k^2 mod 1 as the residual phase variable.	method 0.87	0.72	0.06	asserted 1.00
s6	Unique factorization resolves the integer shells into prime-generator coordinates, while a distinct complex exponent s lifts the same construction to the Dirichlet atoms m^(-s), linking the Dirichlet-series and Euler-product assemblies.	contribution 0.70	0.64	0.17	asserted 1.00
s7	Exact identities, classical zeta connections, numerical controls, and open conditional Weyl tests are kept explicitly separate.	method 0.52	0.59	0.03	asserted 0.96
s8	No proof of the Riemann Hypothesis is claimed.	limitation 1.00	0.44	0.05	asserted 0.97

What kind of paper the classifier reports this is: theoretical result 0.86 · evidence from proof or derivation 0.87.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Centered residuals δk = Lk − Nk for the first forty zeros. The interval [−1/2, 1/2) is intrinsic to nearest-integer centering. The visual absence of a simple trend motivates circular/Fourier diagnostics rather than ordinary linear regression. 14

**Figure 2.** Unit-circle representation Zk = e 2πiδk for the first forty zeros. Labels 1–10 identify the first ten levels. Circularization removes the artificial discontinuity between residuals near +1/2 and −1/2. A.3 Fourier/Weyl diagnostics and coarse factorization controls

**Figure 3.** Histogram of the first 200 centered residuals. The dashed line is the density of

**Figure 4.** First ten Weyl magnitudes for e 2πim(γ 2 k+3/4), using the first 200 zeros. The dashed reference is 1/ √ 200

**Figure 5.** Residual versus total number of prime factors

**Figure 6.** Exact sign/parity skeleton − cos(πn/2) of the Bernoulli-number sequence for n ≥ 2

**Figure 7.** True versus finite-window reconstructed zero ordinates for

**Figure 8.** Circular reconstruction error measured in turns. The wraparound behavior

Figures reproduced from arXiv:2609.03801 by the authors.

References

Discussion

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