Layer 0 — Core Concepts Note

Date: 2026-01-31 • ID: layer-0-core-concepts • Scope: definitions + symbols + core equations

Compact reference for terminology, notation, and the core “existence-style” equations used throughout the work. Each item includes an inline source pointer.

A. Core Terms

Observable (fixed observable)

A single logarithmically localized exponential sum, fixed once and for all, on which all upper and lower bounds are imposed. No auxiliary or varying observables are introduced.

Source: extracted from Layer 0 — Core Proof Engine, Ch.1.

Same-object principle

All estimates—pointwise, mean-square, lower or upper—are applied to the same fixed observable with the same normalization and intrinsic energy scale. Bounds obtained for different objects are not compared.

Source: extracted from Layer 0 — Core Proof Engine, Ch.1–2.

Mean–square incompatibility (collision)

The structural situation in which an exponential lower bound and a uniform polynomial mean–square upper bound are forced to apply to the same observable, leading to a contradiction as the window length grows.

Source: extracted from Layer 0 — Core Proof Engine, Ch.1 and Ch.5.

Filtered set

A subset of the logarithmic axis obtained by restricting the support of the sum to remove rigid arithmetic resonances, while leaving the observable and its normalization unchanged.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2.

Uniformity of constants

A bound is said to be uniform if its implicit constants do not depend on the window length L, nor on the window location U or frequency β, unless explicitly stated otherwise.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2 and App.~A.3.

B. Symbols & Objects

$u = \log n$

Logarithmic coordinate on the positive integers. All arithmetic sums are interpreted as sums over subsets of the real line indexed by the variable u.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2.

$\Lambda \subset \mathbb{R}$

A filtered point set on the logarithmic axis. Typical examples include square-free filters, regular model sets, or incommensurate samplings, chosen to suppress spurious arithmetic resonances.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2.

$U$

The location parameter of a logarithmic window. Translation in U corresponds to shifting the observation window along the log-axis.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2.

$L$

The length of the logarithmic window. Asymptotic statements are considered in the regime L → ∞.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2.

$ \beta $

Frequency (dual) variable conjugate to the logarithmic coordinate u. Resonance occurs when β matches the imaginary part of a zeta zero.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2–3.

$\phi_L(x) = \eta(x/L)e^{x/2}$

Normalized window function. The exponential factor aligns the observable with the critical line normalization in the explicit formula.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2 and App.~A.2.

$w(u)$

An admissible weight function, assumed to satisfy boundedness and regularity conditions sufficient for all stated estimates.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2.

C. Core Equations

C.1 Definition of the observable

\[ A_\phi(U,L;\beta) = \sum_{u \in \Lambda \cap [U,U+L]} w(u)\,\phi_L(u-U)\,e^{-i\beta u} \]

Logarithmically localized exponential observable. The window location U and length L determine the region of observation, while β is the dual frequency variable. This observable is fixed once and for all throughout the analysis.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2.

C.2 Intrinsic energy scale

\[ M_\phi(U,L) = \sum_{u \in \Lambda \cap [U,U+L]} w(u)^2\,|\phi_L(u-U)|^2 \]

Intrinsic energy associated with the observable. All upper and lower bounds are measured relative to this scale. Uniform estimates require constants that do not depend on the window length L.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2.

C.3 Normalization of the window

\[ \phi_L(x) = \eta(x/L)\,e^{x/2}, \qquad \eta \in C_c^\infty([0,1]),\ \eta \ge 0,\ \int_0^1 \eta(x)\,dx = 1 \]

The exponential factor e^{x/2} aligns the observable with the critical line normalization in the Weil–Guinand explicit formula. This choice separates exponential amplification from polynomial background growth.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2 and App.~A.2.

C.4 Existence-style lower bound (forced amplification)

\[ \text{If there exists a nontrivial zero }\rho_0=\sigma_0+i\gamma_0\text{ with }\sigma_0>\tfrac12, \text{ then for sufficiently large }L\text{ there exists }U=U(L)\text{ such that} \] \[ |A_\phi(U,L;\gamma_0)|^2 \ge c_\phi^2\,M_\phi(U,L)\,e^{2(\sigma_0-1/2)L}. \]

Off–critical-line zeros force exponential amplification of the fixed observable at the resonant frequency β = γ₀. The constant c_φ is independent of L.

Source: extracted from Layer 0 — Core Proof Engine, Ch.3.

C.5 Uniform mean–square upper bound

\[ \int_{U_0}^{U_0+L} |A_\phi(U,L;\beta)|^2\,dU \le C_{\phi,\varepsilon}\,L^{1+\varepsilon}\, \sup_{U} M_\phi(U,L) \qquad (\forall\,\beta \in \mathbb{R}). \]

Uniform mean–square control of the same observable over window translations. The implicit constant C_φ,ε is independent of L, U, and β.

Source: extracted from Layer 0 — Core Proof Engine, Ch.4.2.

C.6 Mean–square collision principle

Exponential lower bounds and uniform polynomial mean–square upper bounds cannot simultaneously hold for the same fixed observable as L → ∞.

Source: extracted from Layer 0 — Core Proof Engine, Ch.5.

D. Reading Rules

Fixed-object rule.

All statements, bounds, and comparisons refer to the same fixed observable A_φ(U,L;β) defined in Section C.1. No result obtained for a different observable may be transferred to it.

Source: extracted from Layer 0 — Core Proof Engine, Ch.1–2.
Same normalization rule.

Every estimate is measured relative to the intrinsic energy scale M_φ(U,L). No re-scaling, smoothing, or renormalization is permitted between bounds.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2.
Uniformity rule.

Unless explicitly stated otherwise, all implicit constants are independent of the window length L. Dependence on U or β must be stated explicitly.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2 and App.~A.3.
Filter scope rule.

Filters restrict only the support set Λ. They do not alter the observable, its normalization, or the form of the explicit formula.

Source: extracted from Layer 0 — Core Proof Engine, Ch.2.
Existential interpretation rule.

Statements asserting the existence of a window location U are to be read as existence claims only. No uniformity in U is implied unless explicitly stated.

Source: extracted from Layer 0 — Core Proof Engine, Ch.3.
Collision reading rule.

The mean–square collision principle applies only when the exponential lower bound and the polynomial mean–square upper bound are imposed on the same observable with the same normalization.

Source: extracted from Layer 0 — Core Proof Engine, Ch.5.