Geometric Block-Forcing Definitions

TA35 is the definitional hinge between the block-operator scaffold of TA31-TA34 and the explicit forced block computations of TA36-TA38. TA31 named the blocks A, C, D, E and specified their operator domains. TA35 fixes the precise geometric objects from which those blocks will be derived, establishing that all four are functions of Q5 geometry with no free parameters: \[ (A, C, D, E) = F(Q5, w_H, Gray, Π_Y, K, Θ, A_L) \]. Ten definitions are provided. The carrier--residual split \( H_5 = H_M ⊕ H_R \) partitions span{L, M, R} (coherent carrier) from span{B_L, B_R} (residual balancing). The Q5 deposition space \( DQ5 = C[V(Q5)] \) assigns each admitted transport state an amplitude profile on the 32 vertices. The Hamming-weight deposition profile \( P(d) = (P_0,...,P_5) \) records amplitude distribution across the six Hamming-weight layers. The admissible target profile \( dᵃᵈᵐ(ψ_Y) \) is characterized by four simultaneous conditions (Gray transport order, Y-gate admissibility, Θ-inversion consistency, and Hamming-weight closure ) with explicit vertex-by-vertex computation deferred as a primary open obligation. The deposition defect \( ε_Y(ψ)² = Σ_v |d_v − d_vᵃᵈᵐ|² \) measures failure of the admitted phase to deposit cleanly onto the required profile. The residual extraction subspace is defined as the span over all \( ψ ∈ H_M \) of the deposition mismatch \( D_Y(Π_Yψ) − dᵃᵈᵐ(Π_Yψ) \), making it a proper subspace rather than a state-dependent expression. The extraction block D: H_M → H_R is characterized as the projection of deposition mismatch onto H_R, with explicit computation in TA36. The residual circulation kernel E: H_R → H_R is induced by the helical leakage chain u_2 ↔ u_3 ↔ u_4 (no direct u_2 ↔ u_4 coupling), with spectral abscissa ω(E) determining stability regime and explicit computation in TA37. The reinjection-compatible subspace is defined via Rein(r) (the Y-admissible projection of residual boundary output) with \( C := Rein|R_rein \) computed in TA38, avoiding circularity in the definition. The direct carrier block A is the restriction of barrier-admissible transport to the clean carrier sector after removal of residual-defective deposition. The theorem states that these definitions reduce the block-forcing problem to fixed geometric data so that no phenomenological degrees of freedom remain once the pending explicit computations are carried out. This is a programmatic claim, not a completed derivation: the admissible target profile dᵃᵈᵐ remains the primary open computation on which the explicitness of D and ε_Y depends.