Two Minuses, One Plus, and a Bare Trace Under Oath

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.