Anti-Dual Vector Foil H: Fiber Pricing of Condensation–Suturing Composition—Order-Difference Bits, Injection Elimination, and Existential-Level Exchangeability

Abstract

In the pricing matrix of pairwise compositions of the five basic operators of dimensional surgery—condensation (dimension reduction), dimension raising, suturing, phase transition, and annihilation—the deviation of dimension raising–suturinghas already been fixed as interface fiber information, and condensation–annihilationand annihilation composite algebras already have factorization and semigroup verdicts; only the cell condensation–suturing remains open. This paper fills this celland gives four layered results. First, at the existential level: in the abelian-groupaction domain, the obstruction class of suturing reads only skeleton coordinatesand is blind to weight-layer condensation; the geometric premises of the two orders are the same proposition, and hard non-commutation does not occur in thiscomposite. Full executability is the conjunction of the geometric premise and theweight-matching condition; Order I requires image-level matching, Order II requires raw matching, and both take the geometric premise as necessary but notsufficient. Edgewise heterofactor condensation constitutes the exact boundary ofexistential-level exchangeability; beyond this boundary there appears a hard splitin which the full executability domains are mutually non-containing. Second, atthe fiber level: in the standard discrete weight-lattice model, the inverse-imagefibers of the two orders both have closed forms, and the order difference is exactlythe number of interface edge pairs quantized to zero multiplied by log2 t bits; theacceptance set of Order I properly contains the acceptance set of Order II if andonly if t ≥ 2, and the difference set consists of interface weight configurations thatare “preimage-inconsistent but image-level consistent.” The acceptance-set difference is a set-level fact, while the fiber order difference is a cardinality ratio underthe same residual, and the two cannot substitute for each other. Third, injectionelimination: injecting into Order I fibers, for each pair of interface edges, a one-bitconsistency selector indicating “whether the two edge weights at the preimage areconsistent,” the fibers collapse exactly to Order II fibers, and the bill is transferredfrom the fiber account to the injection account while the total remains unchanged;this information amount is computed by the likelihood ratio under the Order Ifiber prior, and when t > 2 it differs from the Shannon entropy of the selector.Fourth, the joint commutative diagram: under coordinate-disjointness and product distribution or counting uniformity, the joint fiber is a Cartesian product, theorder difference is entirely carried by condensation coordinates, and the dimensionraising-side event account and the condensation-side fiber account are coordinatedisjoint and add bitwise, with the sign convention positive for Order I minus OrderII. All numerical assertions have been verified by deterministic exhaustive enumeration and fixed-seed Monte Carlo, and the protocols and readings are discloseditem by item in the appendix. Open problems include fiber pricing for continuousweights, mixed pricing after edgewise heterofactor splitting, the image-level Čechcriterion, non-abelian and abelian non-group gluing domains, formalization of thenecessity of optimal injection, and joint twist–quantization coupling for multi-patchcovers.