| _version_ | 1866902081185513472 |
|---|---|
| author | Perisic, Aleksandar |
| author_facet | Perisic, Aleksandar |
| contents | <p>This paper defines zeros on the $1 \leftrightarrow s$ edge as intrinsic spectral obstructions of the completed multiplicative object, and sharpens that definition by importing the explicit blur and synthesis notation of Paper II. The point is not to imitate the classical zeta picture without naming zeta. The point is to say, in native multiplicative-spectral language, what exactly is overlapping, what exactly fails, and how contact with zero occurs once the multiplicative skeleton, the blur law, and the host ambient have already been fixed.</p> <p>For each prime $p$, Paper II attaches primitive spectral data $G_p$ (or equivalently a measurevalued primitive spectrum $\rho_p$ ), a blur law $\mu_b$, a local blurred factor</p> <p>$$<br>F_{p, b}(s)=\exp \left(\left(\mathcal{T}_b G_p\right)(s)\right)<br>$$</p> <p>and hence a finite or infinite Eulerized object</p> <p>$$<br>S_b(s)=\prod_{p \in \mathcal{P}} F_{p, b}(s)<br>$$</p> <p>on an assembly region and, when admissible, on a larger spectral domain. Formally, this may be written as</p> <p>$$<br>S_b(s)=\exp \left(\mathcal{H}_b(s)\right), \quad \mathcal{H}_b(s):=\sum_{p \in \mathcal{P}}\left(\mathcal{T}_b G_p\right)(s)<br>$$</p> <p>whenever the overlap field $\mathcal{H}_b$ is defined in the chosen ambient. This field is the first precise answer to the question of where zeros come from: zeros do not arise from one prime in isolation, but from the cumulative overlap of many prime-local demands inside a single completed host.</p> <p>The paper then defines the ambient distance-to-failure functional</p> <p>$$<br>\Delta_b(s):=\operatorname{dist}\left(0, \operatorname{spec}_{\mathcal{A}}\left(S_b(s)\right)\right)<br>$$</p> <p>and calls $s$ a zero/obstruction point exactly when $\Delta_b(s)=0$, or when the corresponding completed object fails even to exist. This turns the older slogan "zeros are where factorization fails globally" into a more precise statement: zeros are points where the multi-prime overlap field has driven the completed object into spectral contact with zero, or into a failure of admissible completion, or into a breakdown of the inverse Mellin-side law.</p> <p>The paper develops four obstruction classes in this stronger notation: symbol obstruction, ambient obstruction, operator obstruction, and deformation obstruction. It proves a symbolprotection theorem, a symbol-to-ambient obstruction theorem, and a deformation theorem for blurred families. The final picture is that zeros are not mystical analytic accidents. They are measurable structural signals that the prime-local overlap can no longer be fused into a globally invertible multiplicative whole.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18926516 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Toward a Native Multiplicative-Spectral Theory III: Zeros as Obstructions on the 1↔s Edge Perisic, Aleksandar <p>This paper defines zeros on the $1 \leftrightarrow s$ edge as intrinsic spectral obstructions of the completed multiplicative object, and sharpens that definition by importing the explicit blur and synthesis notation of Paper II. The point is not to imitate the classical zeta picture without naming zeta. The point is to say, in native multiplicative-spectral language, what exactly is overlapping, what exactly fails, and how contact with zero occurs once the multiplicative skeleton, the blur law, and the host ambient have already been fixed.</p> <p>For each prime $p$, Paper II attaches primitive spectral data $G_p$ (or equivalently a measurevalued primitive spectrum $\rho_p$ ), a blur law $\mu_b$, a local blurred factor</p> <p>$$<br>F_{p, b}(s)=\exp \left(\left(\mathcal{T}_b G_p\right)(s)\right)<br>$$</p> <p>and hence a finite or infinite Eulerized object</p> <p>$$<br>S_b(s)=\prod_{p \in \mathcal{P}} F_{p, b}(s)<br>$$</p> <p>on an assembly region and, when admissible, on a larger spectral domain. Formally, this may be written as</p> <p>$$<br>S_b(s)=\exp \left(\mathcal{H}_b(s)\right), \quad \mathcal{H}_b(s):=\sum_{p \in \mathcal{P}}\left(\mathcal{T}_b G_p\right)(s)<br>$$</p> <p>whenever the overlap field $\mathcal{H}_b$ is defined in the chosen ambient. This field is the first precise answer to the question of where zeros come from: zeros do not arise from one prime in isolation, but from the cumulative overlap of many prime-local demands inside a single completed host.</p> <p>The paper then defines the ambient distance-to-failure functional</p> <p>$$<br>\Delta_b(s):=\operatorname{dist}\left(0, \operatorname{spec}_{\mathcal{A}}\left(S_b(s)\right)\right)<br>$$</p> <p>and calls $s$ a zero/obstruction point exactly when $\Delta_b(s)=0$, or when the corresponding completed object fails even to exist. This turns the older slogan "zeros are where factorization fails globally" into a more precise statement: zeros are points where the multi-prime overlap field has driven the completed object into spectral contact with zero, or into a failure of admissible completion, or into a breakdown of the inverse Mellin-side law.</p> <p>The paper develops four obstruction classes in this stronger notation: symbol obstruction, ambient obstruction, operator obstruction, and deformation obstruction. It proves a symbolprotection theorem, a symbol-to-ambient obstruction theorem, and a deformation theorem for blurred families. The final picture is that zeros are not mystical analytic accidents. They are measurable structural signals that the prime-local overlap can no longer be fused into a globally invertible multiplicative whole.</p> |
| title | Toward a Native Multiplicative-Spectral Theory III: Zeros as Obstructions on the 1↔s Edge |
| url | https://doi.org/10.5281/zenodo.18926516 |