Theorem: VII-I release

For the declared Cauchy prescription and restricted gauge domain:

1. the original nonzero-smearing first dressing cannot have an off-shell-equal classical symbol in the microcausal polynomial algebra;

2. a positive-width density-prepared replacement has admitted compact polynomial kernels through second coordinate order and compatible associative free products, adjoint, and ordering transport;

3. convolution and density preparation are not interchangeable: the former has the explicit covariance defect, while the latter has the explicit changed sharp transmission data;

4. an exact sharp classical boundary carrier and classical auxiliary-coordinate formulation exist at their stated levels;

5. unchanged sharp admission, old-product agreement, bounded-set hypocontinuity, and continuous triangle extraction cannot coexist;

6. linear sharp coordinates admit exact free-shell quantum representatives;

7. no nonlinear sharp interacting gauge-cohomology class, complete sharp normalization theorem, or full physical gravitational correlation is constructed.

The specified incompatibilities are proved obstructions. The finite-width free construction is a controlled repaired construction. The unrestricted sharp interacting target outside those incompatible requirements remains unresolved.

Proof. Theorem thm:sharp proves (i). Theorems thm:rampadmission and thm:freealgebra, with Proposition prop:ordering, prove (ii). Theorem thm:convdefect and Theorem thm:transmission prove (iii). Theorems thm:bdcarrier and thm:auxequivalence prove (iv). Theorem thm:incompatibility proves (v). Theorem thm:linear-quantum proves (vi). The constructions and their domains stop before the maps listed in (vii); no theorem of their existence or universal nonexistence is supplied.