The Diophantine Alignment Problem as a Spectral–Informational Problem: From Kronecker Flow to KS Reconstruction Threshold
Fuente:
Zenodo
Saved in:
| Main Author: | |
|---|---|
| Format: | Recurso digital |
| Language: | English |
| Published: |
Zenodo
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866901120543096832 |
|---|---|
| author | Maréchal, Thierry |
| author_facet | Maréchal, Thierry |
| contents | <p>We reformulate the Diophantine Alignment hypothesis (DA) — the key obstruction to proving the Riemann Hypothesis via the Battery bound zero-free region — in spectral and informational frameworks.</p> <p>The central discovery is that DA is fractally self-similar: through iterated application of the Schmidt Subspace Theorem, the full k-dimensional DA reduces through a cascade DA(k) → DA(k−1) → ... → DA(1) to a single irreducible question — the irrationality measure of log 2/log 3. The complete chain is: RH ⇐ Battery bound + DA ⇐ μ(log 2/log 3) ≤ 2.</p> <p>The spectral formulation constructs an alignment operator H_k on the k-torus T^k with Kronecker flow ω = (log 2, ..., log p_k). The Fourier representation reveals the small denominator structure: diagonal elements are (m·ω)² (kinetic energy encoding prime arithmetic), off-diagonal elements are a_j/2 (potential coupling). Numerical verification at k=1..7 confirms the spectral equivalence.</p> <p>Three frameworks are deployed on the μ ≤ 2 problem. The KS reconstruction threshold (from the Lifting Wall) models the Fourier mode lattice as a broadcasting tree, with per-edge information bounds from Baker's theorem. The Mahler product reformulation (Tool 39) transforms μ ≤ 2 into a statement about Birkhoff sums of log|2cos(πx)| along the doubling map T(x) = {2x}, connecting to the spectral gap of a weighted transfer operator. The Roth-Kolmogorov conjecture (Tool 40) proposes a circuit-complexity lower bound for Chebyshev iterates that would complete the chain.</p> <p>The paper surveys the known bounds: Baker gives μ < ∞ (effective but enormous), the Schmidt cascade gives μ ≤ 3, and computational work gives μ < 4.31. The gap between μ ≤ 3 and the target μ = 2 is documented as a fundamental wall of transcendental number theory — no known technique can bridge it for any specific transcendental number.</p> <p>Since this paper was completed, the companion paper (Transverse Localization Obstruction) has shown that Open Problem 1 of the Spectral Program (θ_eff ≥ c > 0) is false due to rank-one kinetic structure permitting free transverse localization, and that Open Problem 2 (restricted Weil positivity) is the sole surviving path. The μ ≤ 2 obstruction documented here (Open Problem 3) remains a wall but is no longer on the critical path to RH. This paper serves as the detailed technical record of Path 28 in the cartography of failed approaches initiated in "Twenty-Five Ways Not to Prove the Riemann Hypothesis."</p> <p>Independent contributions regardless of RH: the fractal self-similarity of DA via Schmidt cascade, the Birkhoff/Mahler dynamical reformulation connecting irrationality measures to transfer operator spectra, the Roth-Kolmogorov conjecture linking Diophantine approximation to circuit complexity, and the explicit KS analysis of Fourier mode broadcasting trees.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19221897 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Diophantine Alignment Problem as a Spectral–Informational Problem: From Kronecker Flow to KS Reconstruction Threshold Maréchal, Thierry Diophantine alignment irrationality measure log 2/log 3 Riemann Hypothesis Battery bound Schmidt Subspace Theorem Kesten-Stigum threshold Kronecker flow spectral operator small denominators Baker's theorem Mahler product Birkhoff sums doubling map transfer operator Chebyshev polynomials circuit complexity Roth-Kolmogorov conjecture Lifting Wall failed approaches research program <p>We reformulate the Diophantine Alignment hypothesis (DA) — the key obstruction to proving the Riemann Hypothesis via the Battery bound zero-free region — in spectral and informational frameworks.</p> <p>The central discovery is that DA is fractally self-similar: through iterated application of the Schmidt Subspace Theorem, the full k-dimensional DA reduces through a cascade DA(k) → DA(k−1) → ... → DA(1) to a single irreducible question — the irrationality measure of log 2/log 3. The complete chain is: RH ⇐ Battery bound + DA ⇐ μ(log 2/log 3) ≤ 2.</p> <p>The spectral formulation constructs an alignment operator H_k on the k-torus T^k with Kronecker flow ω = (log 2, ..., log p_k). The Fourier representation reveals the small denominator structure: diagonal elements are (m·ω)² (kinetic energy encoding prime arithmetic), off-diagonal elements are a_j/2 (potential coupling). Numerical verification at k=1..7 confirms the spectral equivalence.</p> <p>Three frameworks are deployed on the μ ≤ 2 problem. The KS reconstruction threshold (from the Lifting Wall) models the Fourier mode lattice as a broadcasting tree, with per-edge information bounds from Baker's theorem. The Mahler product reformulation (Tool 39) transforms μ ≤ 2 into a statement about Birkhoff sums of log|2cos(πx)| along the doubling map T(x) = {2x}, connecting to the spectral gap of a weighted transfer operator. The Roth-Kolmogorov conjecture (Tool 40) proposes a circuit-complexity lower bound for Chebyshev iterates that would complete the chain.</p> <p>The paper surveys the known bounds: Baker gives μ < ∞ (effective but enormous), the Schmidt cascade gives μ ≤ 3, and computational work gives μ < 4.31. The gap between μ ≤ 3 and the target μ = 2 is documented as a fundamental wall of transcendental number theory — no known technique can bridge it for any specific transcendental number.</p> <p>Since this paper was completed, the companion paper (Transverse Localization Obstruction) has shown that Open Problem 1 of the Spectral Program (θ_eff ≥ c > 0) is false due to rank-one kinetic structure permitting free transverse localization, and that Open Problem 2 (restricted Weil positivity) is the sole surviving path. The μ ≤ 2 obstruction documented here (Open Problem 3) remains a wall but is no longer on the critical path to RH. This paper serves as the detailed technical record of Path 28 in the cartography of failed approaches initiated in "Twenty-Five Ways Not to Prove the Riemann Hypothesis."</p> <p>Independent contributions regardless of RH: the fractal self-similarity of DA via Schmidt cascade, the Birkhoff/Mahler dynamical reformulation connecting irrationality measures to transfer operator spectra, the Roth-Kolmogorov conjecture linking Diophantine approximation to circuit complexity, and the explicit KS analysis of Fourier mode broadcasting trees.</p> |
| title | The Diophantine Alignment Problem as a Spectral–Informational Problem: From Kronecker Flow to KS Reconstruction Threshold |
| topic | Diophantine alignment irrationality measure log 2/log 3 Riemann Hypothesis Battery bound Schmidt Subspace Theorem Kesten-Stigum threshold Kronecker flow spectral operator small denominators Baker's theorem Mahler product Birkhoff sums doubling map transfer operator Chebyshev polynomials circuit complexity Roth-Kolmogorov conjecture Lifting Wall failed approaches research program |
| url | https://doi.org/10.5281/zenodo.19221897 |