Two Minuses, One Plus, and a Bare Trace Under Oath: Ordered Closed Traces and Whole-Ball Response Curvature

Abstract

With the identity matrix on the stand, the bare trace reports \(252\) and the originally normalized prepared state reports \(1\): the first evaluates the trace over the full matter carrier, the second the action within the fixed preparation. For the fixed \(S_{01}/dx^1\) external source with the co-directional readout operator, zero output momentum, double Laplace parameters \(z=w=6c(1-i)\), the same preparation, and an incoming/observing two-ball window of radius \(B\), we prove that in the complete five-term response both the uncentred (raw) and the centred (connected) quadratic coefficient tensors have two negative and one positive eigenvalue. Here "curvature" means the quadratic coefficient after the non-analytic linear term of the sharp window is removed, not the Hessian of the complete response at the origin. Two branches start from the same action containing an independent dual matter field; that action, its background, matter carrier, and constants are generated from common source data with no fitted parameter. Under the \(P_6\) grading, the closed trace of any finite admissible ordered gauge-propagation word equals the closed trace of its block-diagonal part; the prescribed one-directional scalar-insertion closed trace vanishes, and so does every open chain with an arbitrary admissible propagation word between two co-directional insertions. The prepared state, the boundary weight \(K\) and the complete second-quantized word give different evaluations; the actual return operator produces a nonzero reading, while an intermediate preparation projection erases it. The field-equation branch passes from \(289\) fields through auxiliary, Ward and symmetry reductions to a \(103\)-field system, counts matter feedback once, and constructs a rational \(55\)-dimensional bosonic kernel together with field/source recovery maps. The matter block at the origin has rank \(42\); keeping six light matter directions yields a \(61\)-dimensional second-order system whose constant term has rank \(56\), and its five-dimensional light kernel gives both linear and quadratic momentum time scales and is read out as propagation through the dynamical \(g_{00}\) source. The two quantum source legs, the readout contact term and the two field propagation legs form the complete five-term response. The non-analytic linear term of the true ball intersection, the full complex Gram matrix, implicit root derivatives, full-direction integrals over paired frames and outward-rounded rational remainders together determine the quadratic tensor. The raw three-axis coefficients lie in \([-1.355200,-1.309214]\), \([2.140271,2.186257]\) and \([-1.742600,-1.696614]\) times \(10^{-20}\) in the units of the original action; a uniform bound on all mixed terms makes the inertia statement apply to the complete tensor. The mean term is kept separately; its second-axis interval contains zero. The actual Fourier Hessian of the same original action is also identified with the complete 289-field matrix. Five origin directions and a 98-dimensional coordinate complement generate an exact Schur kernel for all four complex momenta, a source-generated invertibility radius and a strict cubic remainder. The actual ordered current reconstructs the denominator-cleared whole field, retaining contact and all nine null directions. For another original configuration preparation, the complete ordered five-factor word, its actual Noether contact and four Fourier keys generate a real-field half-axis response operator. A source-generated created-unit/background difference then observes the same history: within its positive source-generated nonlinear time interval, it produces the full \(289\times289\) quantum tensor \(\Pi_T\), and the finite integral of the mother-Euler amplitude derivative is exactly \([W_T H_{289}-\Pi_T]a\). Coincident clocks give \(W_T=T\); the complete linearized half-axis retains its own growth condition. The same original Green returns this preparation's response to all fields, with conditions stated separately from those of the specified two-ball window.