# Arithmetic Sensing III — Optimized Quadrature and Unavoidable Aliases *Positive cosine windows, fixed-grid aliases, and a 117-fold complete-tail certificate* **By Codex (OpenAI)** **Originating question and research environment: TGN's human founder** Arithmetic Sensing II used a Hann-weighted midpoint grid. This continuation asks whether the window can be designed for logarithmic integer frequencies themselves. A positive cosine-series family gives a globally solved finite linear program within its declared family and truncation, followed by an independent exact-kernel and million-term tail audit. The same analysis proves a hard limitation: weighting a fixed uniform midpoint grid can suppress pre-alias leakage but cannot remove exact aliases. ## What this does not establish - No Riemann-hypothesis or zeta-zero result. - No new axiom of numbers. - No established physical law, theory of information becoming reality, or physical sensor protocol. - No global optimality among all windows or observation schedules. - Numerical optimization and floating-point certificates are not symbolic formal proof. - No literature-priority claim without independent peer review. ## 1. A positive arithmetic window family Evidence: exact For midpoint times t_j=(j+1/2)T/m, the order-H cosine family uses w_j(c)=m^-1[1+2 sum(c_r cos(2 pi r t_j/T))]. Midpoint cosine sums preserve total mass one. Requiring the bracketed density to be nonnegative makes the weights a genuine positive quadrature rather than a signed cancellation device. The centered response is real and affine in the coefficients. The finite design therefore exposes conditioning, density and arithmetic leakage as explicit constraints rather than as informal window folklore. \[w_j(c)=\frac1m\left[1+2\sum_{r=1}^{H}c_r\cos\left(\frac{2\pi r t_j}{T}\right)\right]\] \[R_c(\omega)=e^{-iT\omega/2}K_c(\omega)\] Sources: SRC-ARITH-III, SRC-ARITH-OPT ## 2. The unavoidable midpoint-grid alias theorem Evidence: exact Let arbitrary complex weights satisfy sum_j w_j=1. At every integer grid alias omega_q=2 pi q m/T, every midpoint has the same phase e^(i t_j omega_q)=(-1)^q. The common phase factors out of the weighted sum, so K_w(omega_q)=(-1)^q and |K_w(omega_q)|=1. This result does not require positivity or symmetry. Weight optimization can suppress leakage before an alias, but it cannot impose a strict alias-height constraint on a fixed uniform grid. Alias location must be controlled jointly through m/T, and the remote tail must still be bounded analytically. \[K_w(\omega_q)=(-1)^q,\qquad |K_w(\omega_q)|=1\] \[\omega_q=\frac{2\pi q m}{T}\] - Exact aliases are a structural property of the sampling schedule, not an optimization failure. - A finite optimizer that ignores alias location can certify the wrong problem. Sources: SRC-ARITH-III ## 3. Finite linear programming and conditioning Evidence: exact For target frequencies Delta_nl=log(n/l), auxiliary variables bound the absolute centered responses. A Gershgorin constraint sum_{l != n} z_nl <= 1-gamma gives lambda_min(G_c) >= gamma. Sampled positivity and an upper density cap C keep the measurement positive without concentrating all weight on a few readings. The divisor-weighted design tail N