An Arithmetic-Fluctuation Hilbert–Pólya Mechanism: Canonical Principal-Response Mode Closure

This preprint develops an arithmetic-fluctuation Hilbert--P\'olya mechanism based on the completed localized Weil form. Prime powers are represented as repeated returns of prime delay-line colligations, while the endpoint, archimedean, and prime sectors are assembled before the positivity step. A Hilbert--P\'olya operator is intended to arise only after positivity of the completed arithmetic form has been established. The manuscript reports validated finite-window results for the first three prime-power support windows. At the endpoint \(L=log 7\), whose active von Mangoldt labels are \(2,3,4,5\), outward-rounded interval certificates establish positivity of the finite low compression, high-sector coercivity, and the residual-controlled Schur complement within the stated numerical trust boundary. The cofinal analysis develops exact factor-pair resonance laws, source compression, parity reduction, first-order temporal annihilation, cubic residual-tail bounds, canonical Kreĭn shorting, finite response-rank reduction, monotone Ritz response approximation, signed spectral deflation, and semidefinite Schur closure. Revision v66 removes the dependence of the preceding response-shell formulation on an arbitrarily selected Ritz exhaustion. For each prime-power endpoint \(m\), let\[ M_m={pmatrix}Q_m & B_m*_m & C_m{pmatrix},_m≥ c_mI>0,\]and define the exact low-sector Schur operator\[Q_m-B_m*C_m⁻¹B_m.\]When \(Q_m0\), set\[ 1(0,∞)(Q_m)\]and define the canonical principal-response load\[Q_m/2B_m*C_m⁻¹B_mQ_m/2,\]where \(Q_m/2\) is the Moore--Penrose inverse square root on \(RanQ_m\), extended by zero on \( Q_m\). Revision v66 proves the exact finite equivalence\[Σ_m0\]if and only if\[Q_m0,_m Q_m=\{0\}, L_mᵖʳ P_mQ.\]Equivalently, the largest singular value of\[C_m-1/2B_mQ_m/2\]must be at most one. A singular value equal to one is an admissible GNS-null saturation; only a value greater than one produces a negative completed endpoint direction. A singular-value-adapted high-form Ritz basis yields at most\[rank A_mᵖʳ≤ L_m\]nonzero response modes, whose shell leverages are exactly the squared principal singular values. Thus the arbitrary response-shell ledger is replaced by one finite, intrinsic, exhaustion-independent principal-response load. The remaining unresolved theorem is the cofinal arithmetic estimate\[Qm,±0,m,± Qm,±=\{0\},λₘₐₓ( Lm,±ᵖʳ)≤1\]for all sufficiently large prime-power endpoints and in both parity sectors. This cofinal estimate is not proved in the present manuscript. Accordingly, the paper does not establish global Weil positivity and does not claim a proof of the Riemann Hypothesis.