UFFT Paper #75 — The Born Rule from Imprint Statistics and the Entangled Pair State from the Antiunitary Twin Map

Paper #45 gave the foam two propagation channels: the wall channel (the face Laplacian L, local, causal) and the void channel (ηV, antipodal, non-local, no signalling). This paper supplies the two pieces of quantum mechanics that the corpus had until now imported rather than derived: the Born rule and the entangled pair state. First, the Born rule is reduced to two structural premises about the void-pair imprint event. Phase-blindness (the imprint probability depends only on a channel's amplitude modulus) and fine-graining additivity (a local refinement of detector channels cannot change coarse probabilities) force pₖ = |cₖ|² uniquely, by the Cauchy functional equation. Both premises are then discharged in turn. Fine-graining additivity is equivalent to no-signalling, which Paper #45 §8 derives from bulk incompressibility. Phase-blindness follows from substrate stationarity: a channel's phase is a time offset of its carrier, and no threshold functional driven by stationary substrate fluctuations can depend on a time offset; the same argument kills the linear response identically, so the trigger rate is quadratic in the amplitude at leading order. The Born exponent thus traces to two equation-of-state properties of the foam, incompressibility of the bulk and equilibration of the substrate. On a discrete substrate the counting measure is literal: channel k holds exactly N|cₖ|² micro-quanta (Parseval). Second, the pair state of a displacement event is derived rather than selected. The twin map Θ = V∘K (antipodal map composed with conjugation, both existing corpus operators) is antiunitary and flips the torsion charge exactly. The pair state Σₖ |k⟩⊗Θ|k⟩ is proved basis-independent if and only if the twin map is antiunitary, which is the operator content of "opposite in every direction simultaneously". The resulting state is unique, maximally entangled, saturates the Tsirelson bound (CHSH = 2√2), carries exactly zero total torsion charge, and is one local unitary from the singlet of Paper #2. Third, the laboratory qubit is unique by symmetry: orientation-averaged observables commute with the cell's Oₕ action, Schur's lemma confines two-state structure to the two multiplicity-two sectors, the gauge pair is excluded, and the chiral doublet with routing observable C is all that remains. The η-tension between Papers #2 and #45 is also resolved: the void coupling scales the formation probability of entangled pairs (∝ η²) and never the strength of their correlations.