Polygonal Gnomons and the Hessian Amplifier: A Two-Dimensional Geometric Framework from Almost-Prime Intervals to Riemann Zeta Variance

We introduce a two-parameter array of polygonal gnomon numbers $A_{m,j}, W_{m,j}$ ($m,j \ge 1$) and study the family of explicit intervals $I_{m,j} = [A_{m,j}, A_{m,j}+W_{m,j}]$ it generates. In Part I we show that this array asymptotically exceeds the exponent threshold of the classical Iwaniec--Laborde theorem on almost-primes ($P_2$-numbers) in short intervals, producing a geometric surplus of order $j^{0.55}m^{0.1}$; combined with the cited theorem, this yields, for each fixed $j$, a qualitative guarantee that $I_{m,j}$ contains a $P_2$-number for all sufficiently large $m$. In Part II we isolate a genuine obstruction to making this effective: the \emph{Narrow Oscillation Paradox}. We show that the relative width $W_{m,j}/A_{m,j}$ collapses like $2/m$, and that both the explicit Type~II bilinear route and the explicit spectral (Kuznetsov) route to an effective bound carry a constant inflation of order $m^{2/3}$, which grows strictly faster than the geometric surplus of Part I and so overwhelms it for every finite $m$. In Part III we invert the strategy: rather than seeking one explicit interval, we integrate the Riemann--von Mangoldt explicit formula over the entire two-dimensional array. We identify $\Phi(m,j) = \log\widehat A(m,j)$, for the smooth model $\widehat A(m,j)=\tfrac{j}{2}(m+j)^2$ asymptotic to $A_{m,j}$, as the natural pair-correlation phase evaluated at the array's base points, prove that its Hessian is everywhere non-degenerate with $\det H(m,j) = 2/(j^2(m+j)^2) > 0$, and show that the array's own exponent function sweeps out exactly the interval $\theta \in [1/2,2/3]$ as the relative growth rate of $j$ against $m$ varies -- which we identify as the relevant critical window for the variance of primes in short intervals. We show that this absence of a stationary point is a structural advantage rather than an obstruction: iterating the classical van der Corput first-derivative test in each variable separately, with no main term surviving at any stage, we prove that the continuous double integral of $e^{iN\Phi}$ over any dyadic block $[M,2M]\times[J,2J]$ satisfies the unconditional bound $|I(N)|\le 6J(2M+3J)/N^2$. We complement this with a numerical experiment using the first $1{,}000$ genuine non-trivial zeros of $\zeta(s)$: on the block $M=100,J=10$, the discrete analogue of $I(N)$ collapses from a trivial value of $1111$ to a median of $12.15$ over all $499{,}500$ off-diagonal pairs. We then examine directly, via Poisson summation, why this collapse occurs, and find that it is not simply an inheritance of $I(N)$'s $N^{-2}$ decay: each resonant term in the Poisson expansion is individually controlled by the same non-degenerate Hessian, but their aggregate settles, for generic $N$, at an $N$-independent floor of size $\sqrt{M^{1+\alpha}}$ -- a finding we test directly across several block sizes, not merely infer. We then formulate -- as a precise but explicitly conjectural principle, which we name the \emph{Non-Linear Maier Matrix Method} -- the extension of this floor phenomenon to the off-diagonal pair sums over zeta zeros $(\gamma_1,\gamma_2)$ governing that variance, and show this extension reduces entirely to a single, isolated conjecture about the floor's persistence, rather than to the more elaborate weight-and-decorrelation mechanism it first appeared to require. Throughout, we separate, without euphemism, what is proved (the integral bound, the Poisson decomposition, and the reduction of the main conjecture to the floor conjecture), what is numerically observed (the discrete-sum experiment and the discretization floor, tested across zero count and block size alike), and what is proposed (the floor's persistence past its transition scale, which is now the entire unresolved content of the conjecture).