PaperlayerView exact revision on arXiv

Source-linked AI summary

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

Y. Kenan Yılmaz

arXiv:2609.03801v1cs.LGmath.NT

TL;DR

The paper asks what geometry arises from inverse-integer levels of an ordered Bernoulli-word kernel, then constructs a conjugation-symmetric line and an integer-shell/residual-phase description of zero ordinates. It organizes the shells by prime factors and relates a distinct complex exponent to both classical zeta assemblies, while keeping the RH question open.

  • Problem

    The paper asks what geometry inverse-integer levels of the ordered Bernoulli kernel produce, and whether arithmetic classes of integer shells leave a Fourier signature in residual phases.

    Source: §1 · p. 2, §5.4 · p. 8

  • Method

    It maps critical-line zero ordinates into quadratic levels with integer shells and residual phases, organizes shells in prime coordinates, and lifts the construction to Dirichlet atoms m^-s.

    Source: §1 · p. 2, §11 · p. 13

  • Results

    The construction identifies γ_k^2 mod 1 as the modular variable, classifies integer shells by prime-generator occupations, and distinguishes negative coarse shell–phase tests from the open conditional Weyl question.

    Source: §1 · p. 2, §11 · p. 13

  • Takeaways & Limitations

    The shared atom m^-s presents Dirichlet-series enumeration and Euler-product prime-power generation as complementary zeta assemblies.

    Source: §11 · p. 13

    The 1/2 geometry belongs to the Bernoulli and normalized split coordinates; the separate exponent s is unrestricted, so no RH proof follows.

    Source: §10 · p. 12

Abstract

from arXiv · show

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.

1 Introduction

The inverse-integer Bernoulli construction starts from the unique real split-independent anchor p = 1/2 and, under complement-preserving continuation, produces a conjugation-symmetric 1/2 line. Its quadratic coordinate connects that geometry to integer shells, zero-ordinate phases, prime-factor coordinates, and Dirichlet atoms, without claiming RH.

1 Introduction

  • Complement-preserving continuation gives z = 1/2 ± iu and a conjugation-symmetric vertical line before zeta or prime classification enters.The normalized central split coordinate κ = k/n inherits the same real part on the central phase branch.Source: §1 · p. 2
  • The quadratic coordinate maps critical-line zero ordinates to levels L_k = 1/4 + γ_k^2, isolating γ_k^2 mod 1 as the residual phase variable.Each level has an exact nearest-integer shell and centered residual.Source: §1 · p. 2
  • Prime-exponent coordinates distinguish repeated single-prime occupations from multi-prime composites, while a distinct complex exponent s realizes Dirichlet atoms m^-s.The atom m^-s underlies the Dirichlet-series and Euler-product assemblies.Source: §1 · p. 2
  • The construction does not claim RH: its derived 1/2 geometry belongs to the Bernoulli and normalized split coordinates, not to an imposed condition on s.The relationship with zeta zeros is diagnostic and structural.Source: §1 · p. 3

2 Ordered Bernoulli levels and the binary anchor

  • The binary level 2^-n uniquely selects p = 1/2 as the real split-independent anchor.At this point, the ordered Bernoulli kernel has value 2^-n for every finite real or complex k.Source: §2 · p. 3

3 Complex phase and conjugation-symmetric continuation

Complex continuation supplies a vertical geometry through a chosen complement-preserving rule, while other complex level equations can yield vertical arithmetic lattices. The resulting 1/2 line depends on the continuation choice, not on the level equation alone.

3 Complex phase and conjugation-symmetric continuation

  • For p ≠ 1/2, the complex split variable k = a + iτ yields a vertical arithmetic lattice of solutions to the level equation.Its spacing 2π/log 2 is a consequence of phase closure, not a statement about zeta-zero spacing.Source: §3 · p. 4
  • The complementary Bernoulli pair is continued as conjugates, producing a vertical line with real part 1/2.This real part follows from the adopted complement-preserving continuation rule.Source: §3 · p. 4
  • The binary level alone does not force every complex parameter onto the 1/2 line; that conclusion depends on the additional continuation rule.The real anchor p = 1/2 is derived from the binary level together with split symmetry.Source: §3 · p. 4

4 Quadratic level coordinate and integer quantization

The quadratic coordinate has a unique central minimum, and restricting its levels to integers quantizes the vertical geometry. Zero ordinates can be mapped to these levels diagnostically, but the initial prime-shell count shows no preferential prime enrichment.

4 Quadratic level coordinate and integer quantization

  • Q(z) ≥ 1/4, with equality only at u = 0; restricting levels to positive integers gives an arithmetic quantization, not an externally specified spectrum.The central point is the unique minimum state.Source: §4 · pp. 5–6
  • Integer levels m ≥ 2 separate into primes, proper prime powers, and multi-prime composites, interpreted respectively as one generator, repeated occupation, and finite superposition.The integer 2 also has a prior binary-anchor role.Source: §4 · p. 6

4.1 Zeta zeros as a diagnostic of the level coordinate

  • Using the standard tabulated ordinates, the first zero γ1 = 14.134725… gives L1 = 200.0404548…; nearby integer 200 lies in the multi-generator composite class.Source: §4.1 · p. 6
  • For the first 40 tabulated zero ordinates, 5 rounded levels are prime (12.5%), close to the heuristic expectation 4.83 and not evidence of preferential prime enrichment.This negative control separates exact coordinate identities from numerical pattern-seeking.Source: §4.1 · p. 6

5 Exact integer–residual decomposition and Bernoulli phase

Critical-line zero ordinates yield an exact integer-shell and centered-residual decomposition, with circularization retaining the residual phase separately from the shell. The paper distinguishes this local Bernoulli residual from classical Bernoulli corrections and treats arithmetic-conditioned phase tests as open, with preliminary controls negative.

5 Exact integer–residual decomposition and Bernoulli phase

  • Each zero-induced level L_k has an exact decomposition into an integer shell N_k and a centered periodic-Bernoulli residual δ_k.The special input is L_k = 1/4 + γ_k^2, linking the level to both the quadratic coordinate and a zeta-zero ordinate.Source: §5 · pp. 6–7

5.1 Two distinct Bernoulli roles

  • The local first-Bernoulli residual is distinct from indexed Bernoulli numbers used as Euler–Maclaurin and Gamma–zeta correction coefficients.No identity asserts that δ_k is a sum of higher Bernoulli corrections.Source: §5.1 · p. 7

5.2 Prime-power spectrum and common-source interpretation

  • The classical explicit formula couples the global von Mangoldt prime-power spectrum to zeta zeros, not through a one-to-one matching of individual primes and zeros.The coupling is global.Source: §5.2 · pp. 7–8

5.3 Circular residual coordinate

  • Circularization removes the integer shell because exp(2πiN_k) = 1, leaving complementary integer and phase coordinates whose pair is lossless.The unit-circle coordinate avoids the residual's coordinate cut at ±1/2.Source: §5.3 · p. 8

5.4 Conditional Weyl statistics

  • Conditional Weyl moments test whether arithmetic classes of N_k leave Fourier signatures in Z_k; preliminary factorization and Bernoulli-mode controls show no statistically convincing coupling.The tested controls are retained as negative results, not as evidence for the stronger hypothesis.Source: §5.4 · pp. 8–9

6 Prime-coordinate modes and scalar collapse

The section represents integers as prime-coordinate states, then maps those states to a scalar central-branch level. That projection is affine in log m and preserves factorization uniqueness, although it hides separate coordinate entries.

6 Prime-coordinate modes and scalar collapse

  • Arithmetic states use finitely supported prime-indexed sequences with canonical basis vectors, allowing role classification before scalar projection.The prime-exponent state space is c00(P).Source: §6 · p. 9
  • Prime powers occupy one prime coordinate repeatedly, while multi-prime composites are finite superpositions of prime-coordinate occupations.Successive powers of one prime form an arithmetic trajectory with increment ∆r.Source: §6 · p. 9
  • For fixed n and u ≠ 0, the central-branch scalar level is affine in log m: km = A + B log m, with constants independent of m.Source: §6 · p. 10
  • Scalar projection hides separate prime-coordinate entries but preserves factorization uniqueness: equal levels imply m = m′ under fixed n and u ≠ 0.Source: §6 · p. 10
  • The scalar level coordinate is a specific affine logarithmic projection, whereas the Bohr lift retains the prime multi-index [3].The scalar projection does not replace that richer space.Source: §6 · p. 10

7 Continuation to a complex spectral exponent

The paper replaces the integer total exponent with a distinct complex exponent s and lifts Dirichlet atoms into a scalar k-coordinate family. The central lift is affine in log m, while s is not restricted to real part 1/2.

7 Continuation to a complex spectral exponent

  • Replacing integer total exponent n with distinct complex s gives a complex analytic extension of the multiplicative Bernoulli kernel, not a probability mass function; no Re s = 1/2 restriction is imposed.Source: §7 · p. 10
  • Proposition 7.1 treats the lift of a Dirichlet atom for u ≠ 0, complex s, and positive m under the principal-branch convention.Source: §7 · p. 10
  • For fixed s and u, the central lift is A + B log m; the scalar coordinate re-encodes the logarithmic embedding, while further arithmetic information resides in α(m).Source: §7 · p. 11

8 The two zeta faces from the common atom

The section presents two assemblies of the common atom m^-s: global enumeration over integers and local generation through prime-power occupations.

8.1 Dirichlet face: enumeration

  • For Re s > 1, global enumeration of the integer atoms yields ζ(s).The level construction produces individual integer atoms; global enumeration gives the zeta function.Source: §8.1 · p. 11

8.2 Euler face: prime-power generation

  • A local Euler factor generates the full prime-power tower 1, r, r², … through the same atom; unique factorization assigns each finite occupation choice exactly one integer.Source: §8.2 · p. 11

8 The two zeta faces from the common atom

  • The common interface between the two zeta faces is the atom m^-s.The Euler face generates integer states from prime-power occupations.Source: §8.2 · p. 11

9 Relation to probabilistic zeta constructions

The paper distinguishes its Bernoulli-kernel entry point from probabilistic zeta constructions. After complex continuation, the kernel is no longer a probability mass function, and the level-set lift alone makes no new infinite-divisibility or characteristic-function claim.

9 Relation to probabilistic zeta constructions

  • The paper starts from a Bernoulli likelihood kernel and analytically extends its exponent split.Source: §9 · p. 12
  • Once n is replaced by complex s, F is no longer interpreted as a probability mass, so the level-set lift alone yields no new infinite-divisibility or characteristic-function claim.Source: §9 · p. 12
  • The stated contribution is explicit Bernoulli level geometry mapping integer atoms into a centered complex k-coordinate family.Source: §9 · p. 12

10 Coordinate scope, limitations, and relation to the Riemann Hypothesis

The paper separates the Bernoulli and normalized split coordinates, where the real-part-1/2 geometry arises, from the later unrestricted Dirichlet exponent. Its zero comparison is diagnostic and structural, not a proof of the Riemann Hypothesis.

  • The Bernoulli coordinate z and normalized split coordinate κ lie on ℜ=1/2, but the later Dirichlet exponent s is distinct and unrestricted.The paper does not derive ℜs=1/2 for zeta zeros.Source: §10 · p. 12
  • The zero comparison does not pair individual prime powers with individual zeros; conditional Weyl statistics instead ask whether integer-shell classes condition residual phases.The classical explicit formula supplies a global prime-power/zero coupling, while the statistical question remains open.Source: §10 · p. 12
  • The principal logarithm branch specifies the level coordinates; other branches yield shifted level lattices.Source: §10 · p. 12
  • The scalar level coordinate depends only on log m, projecting the full prime-exponent state to one dimension; “superposition” means finite-support prime-exponent space, not a quantum state or asserted spectral operator.Source: §10 · p. 12

11 Conclusion

The construction begins with an ordered Bernoulli likelihood and builds conjugation-symmetric geometry, integer shells, residual phases, and prime-factor coordinates. It connects these layers to classical zeta assemblies while keeping the identities, numerical controls, and open tests distinct; it does not claim a proof of RH.

11 Conclusion

  • The binary level selects p=1/2, and complement-preserving continuation produces the vertical pair 1/2 ± iu before zeta input.The normalized central split coordinate independently has the same real part.Source: §11 · pp. 12–13
  • Prime powers occupy one prime coordinate repeatedly, while composites with multiple distinct prime divisors are finite multi-generator superpositions.The binary anchor's role is distinct from the factorization classification built on the resulting geometry.Source: §11 · p. 13
  • Q(z)=1/4+u^2 gives a sharp-minimum real level with exact integer quantization; the first forty-zero prime-hit control is negative.Source: §11 · p. 13
  • Each zero-induced level separates into an integer shell and centered periodic-Bernoulli residual, whose circularization isolates γ_k^2 mod 1 as phase.The local first-Bernoulli role differs from higher Bernoulli numbers in Euler–Maclaurin/zeta analysis; coarse shell–phase tests are negative, leaving conditional Weyl moments open.Source: §11 · p. 13
  • The distinct exponent s places every Dirichlet atom m^-s in the multiplicative architecture, where the Dirichlet series enumerates integers and the Euler product generates them from prime-power towers.The scalar level coordinate collapses full prime-exponent information through log m.Source: §11 · p. 13
  • The framework is algebraic and organizational rather than a proof of RH, keeping exact statements, known theory, finite numerical controls, and open tests distinct.Source: §11 · p. 13

A.1 Exact level decomposition

The appendix pairs the first ten quadratic levels with their nearest-integer shells, then uses plots and diagnostics to illustrate residual geometry and arithmetic controls. These finite-sample and preliminary diagnostics do not establish equidistribution or support the manuscript’s exact identities.

A.1 Exact level decomposition

  • The first ten quadratic levels are displayed beside their nearest-integer shells, without asserting that factorization determines the residual or vice versa.The table presents the two quantities side by side.Source: §A.1 · p. 13, Table 1 · p. 14

A.2 Circular residual geometry

  • For the first forty zeros, centered residuals lie in [−1/2, 1/2), and their visual lack of a simple trend motivates circular/Fourier diagnostics.The interval follows from nearest-integer centering.Source: Figure 1 · p. 14
  • Circularizing the first forty residuals onto the unit circle removes the artificial discontinuity between values near +1/2 and −1/2.The first ten levels are labeled in the figure.Source: Figure 2 · p. 15

A.3 Fourier/Weyl diagnostics and coarse factorization controls

  • The first-200 residual histogram is compared with a uniform U[−1/2, 1/2) density, but is only a finite-sample diagnostic and does not establish equidistribution.Source: Figure 3 · p. 15
  • Residuals plotted against Ω(N_k) illustrate that a coarse factor-count statistic can discard too much prime-generator structure for a stronger shell–phase hypothesis.Source: Figure 5 · p. 17

A.4 Bernoulli hierarchy illustration

  • The Bernoulli-number sequence’s exact sign/parity skeleton for n ≥2 is −cos(πn/2).Source: Figure 6 · p. 17

A.5 Preliminary prime-power reconstruction diagnostics

  • Finite-window prime-power reconstructions illustrate sensitivity and circular error, but are preliminary and are not premises of any exact manuscript statement.The figures include true-versus-reconstructed early ordinates and circular reconstruction error measured in turns.Source: §A.5 · pp. 17–18, Figure 7 · p. 18, Figure 8 · p. 18

A.6 Arithmetic shell classes for conditional Weyl tests

  • Among the first forty zero-induced shells, prime cases occur at k = 8, 14, 24, 32, and 39; these appendix diagnostics are not evidence for the exact identities.Source: §A.6 · p. 19
Loading 2609.03801v1…