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Analysis and Spectral Theory

Edition 1

RH, Approximately: Certified Lower Bounds for Odd Weil Positivity

Scott Schubr

Primary subject: Mathematics
Subject subfields: Analysis and Spectral Theory · Number Theory and Special Functions · Numerical Analysis
Paper type: Research Article
Manuscript stage: Preprint

We prove the complete lower bound q_H[h] ≥ −1.5 × 10⁻¹¹ ‖h‖² on the original constrained minimal odd domain in its reciprocal-theta norm. The certificate retains the full joint covariance of 4080 phase columns and five exact polar/null rows. Shorting that certified 4085-dimensional receiving Gram gives a directed 4080-dimensional exact-row Schur certificate. Its proof includes the complete arithmetic remainder, infinite frequency and physical tails, source conversion errors and the original-domain return. A real Bessel suppression argument and complete spatial-endpoint control sharpen the approximation without changing the frame. Three complementary compact-support certificates, in the distinct ordinary physical L² norm, give A₈ ⪰ 10⁻⁴⁰I + 4.75 × 10⁻¹¹Pₑ₇,₈, A₉ ⪰ 10⁻⁵⁰I + 4 × 10⁻¹²Pₑ₈,₉ and A₁₀ ⪰ 10⁻⁶⁰I + 10⁻¹³Pₑ₉,₁₀, with explicit sharp directional upper trials and joint eleven-, thirteen- and fifteen-feature receivers. An exact ground transformation preserves signed arithmetic pairing through closure even where the two positive edge energies separately diverge. It yields a norm-convergent paired tail, a sharper localized sign bound and exact compatibility with polar repair. Further results give fixed-null-prefix funding, all-support gamma approximation, whole complementary-space bounds and an explicit fixed-cutoff exterior obstruction. These are certified original-domain and physical receiving results with their respective norms and hypotheses.

October 26, 2026Read paper

Numerical Analysis

Edition 1

Certified negative-band approximation for odd Weil forms

Scott Schubr

Primary subject: Mathematics
Subject subfields: Numerical Analysis · Functional Analysis · Number Theory
Paper type: Research Article
Manuscript stage: Preprint

We construct certified finite approximations to the complete negative Fourier observation of an odd Weil form in its reciprocal-theta norm. Weighted Chebyshev features on a rigorous negative-symbol cover give a positive remainder with an explicit trace bound after exact orthogonal constraints. Discrete cosine orthogonality provides a degree-independent ridge charge. At arithmetic cutoff eleven, a 1366-feature construction admits ridge parameter 10^-17 with complete trace error below 1.121 × 10^-14. A support-sensitive paired arithmetic-tail estimate yields a nonnegative addition to the localized positive reference while preserving the same negative observation. We derive the corresponding finite inverse conditions and a compact physical-to-weighted certificate transfer, retaining the complete covariance and tail costs.

October 26, 2026Read paper

Functional Analysis

Edition 1

Certified source elimination for localized odd Weil forms

Scott Schubr

Primary subject: Mathematics
Subject subfields: Functional Analysis · Number Theory · Numerical Analysis
Paper type: Research Article
Manuscript stage: Preprint

We give a quantitative local correction and a form-domain continuation method for the complete odd Weil form. On the support |u| < (log 8)/2 we verify a positive scalar floor together with a 3 × 10⁻¹¹ directional floor along the normalized first-collar polar profile. The sharp coefficient of that profile lies between 3 × 10⁻¹¹ + 10⁻⁴⁰ and 5.641396988 × 10⁻¹¹. The lower certificate compresses to every intermediate support between (log 7)/2 and (log 8)/2. For a compact physical source with a certified gap, we construct an explicit logarithmic Fourier reference and a finite inverse upper using form-core trials and exterior and constraint annihilators. The resulting certificates are complete at every strict finite cap. A rational example exhibits a strict cap certified by the completed receiver although every fixed-extension primal trial fails. Directional inverse metrics, exact polar elimination and portable interval and rational verification provide the local application and its continuation inputs.

October 26, 2026Read paper

Number Theory

Edition 1

Relative finite optimization of arithmetic approximation distances

Scott Schubr

Primary subject: Mathematics
Subject subfields: Number Theory · Numerical Analysis
Paper type: Research Article
Manuscript stage: Preprint

We give finite rational approximations to the complete least-squares distance from the constant sequence to fractional-part sequences {n/k} in the norm sum over n >= 1 of |f(n)|^2/[n(n+1)]. The exact periodic mean is a positive gcd covariance form. Montgomery–Vaughan’s circular Hilbert inequality yields a coefficient-uniform complete-tail error with constant D=N(N−1). This gives rational lower and upper optimization problems and a midpoint optimizer whose actual squared error is at most (1−beta^2)^(-1) times the complete optimum, where beta=2D(2D+1)/[(R+1)(R+2)]<1. The latter is a Kantorovich application and gives a sufficient cutoff R=O(N^2 epsilon^(-1/4)) for relative 1+epsilon quality of the optimizer. The computed optimum-value brackets are separate and retain their explicitly stated widths. An adjacent arithmetic wavelet shows that no bounded uniform prefix extension is possible at R asymptotic to N log N; at R asymptotic to N^(1+alpha) its full-to-prefix ratio tends to 1/alpha. The mean, Hilbert inequality, wavelet head identity and quadratic optimization tools are classical or previously established; their combination gives explicit native-norm truncation guarantees. A standard-library exact rational implementation is included.

October 26, 2026Read paper