Multiplicative Excursion Walls in Finite-Scale Weil Operators: Rational Ghosts, Boundary-Rank Laws, and Dense Arithmetic Curvature

Abstract

We develop a higher-order extension of the arithmetic wall structure arising from compactly supported finite-scale Weil operators. For products of truncated two-sided shifts, an exact sign-path expansion shows that each overlap cell is controlled by the range of the partial-sum path, rather than by positive subset sums. Along the physical Weil curve \(\tau_q = \log(q)/L\), the resulting wall parameter is the multiplicative excursion \[X = \frac{\max_m Q_m}{\min_m Q_m},\] where \(Q_m\) is the signed partial product of prime powers, and \(X\) is generally rational. The monotone integer sector recovers the ordinary Dirichlet coefficient \[\frac{\Lambda^{*k}(n)}{\sqrt{n}},\] but higher mixed sectors produce quotient walls. In the cubic case we classify the new spectrum exactly: besides integers with three distinct prime factors, the new walls are \[x = \frac{qs}{r},\] with prime powers \(r < q,s\) and \(qs/r\) nonintegral; the first rational wall is \(9/2\). For a nonactivation wall \(x\) we define its physical excursion order \(\mu(x)\) as a bounded multiplicative graph distance and prove that it equals the first moment order at which the wall can occur. Integer walls satisfy \[\mu(n) = \omega(n),\] whereas rational walls exhibit a finite-scale barrier defect. In particular, if \[8 < \frac{3^a}{2^b} < 9,\] then \[\mu\!\left(\frac{3^a}{2^b}\right) = 2a - 1;\] the associated defect is locally unbounded, and pure higher-order rational ghost walls are dense in the finite band \[\frac{3}{2}\log 2 < L < \log 3.\] At the spectral level, every elementary critical path contributes a rank-one wall atom. Pure first-appearance ghosts remain rank one in a fixed Legendre parity sector, while prime-power activation jets admit a boundary-history factorization of rank at most \(k\) at order \(k\). Finally, for every fixed finite Legendre truncation, the all-order resolvent and exponential completions have bounded-variation first derivative on activation-free compact chambers; their distributional second derivatives contain finite matrix-valued atomic Radon measures. In the even block these measures can have dense arithmetic support while finite total variation is retained. These are finite-scale structural results and make no claim toward the Riemann Hypothesis.