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 \( D_{Q5} = C[V(Q5)] \) 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 \( d^{adm}(ψ_Y) \) 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 \( ε_Y(ψ)² = Σ_v |d_v − d_v^{adm}|² \) 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 \( D_Y(Π_Yψ) − d^{adm}(Π_Yψ) \), 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 d^{adm} remains the primary open computation on which the explicitness of D and ε_Y depends.
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 ( D_{Q5} = 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 ( d^{adm}(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_v^{adm}|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) − d^{adm}(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 d^{adm} remains the primary open computation on which the explicitness of D and ε_Y depends.
TA32 identifies the structural object tested by the barrier comparator of TA30/TA31. The admitted helical phase is not merely passed through the Y-gate as a scalar phase: it is deposited as an admissible Hamming-weight occupancy profile on the 32-vertex Q5 transport graph. Any mismatch between the deposited profile and the inversion-compatible Hamming target produces a global closure defect, which is forced into the residual sector through the extraction block D. This identifies D as the Hamming-deposition defect extraction operator and provides the missing bridge from the TA31 block scaffold to the forced block computation of TA36. The argument proceeds in three layers: deposition → defect → residual forcing. The deposition map DY:HM→C[V(Q5)] assigns to each admitted transport state an amplitude profile d_v on the 32 vertices. The total deposition defect εY(ψ)=(Σv∣dv−dvadm∣2)1/2 measures deviation from the inversion-compatible admissible target profile d^{adm}, determined jointly by Gray transport order, Hamming- weight compatibility, barrier inversion consistency, and Y-gate admissibility. The closure defect δC(ψ)=minφ‖UCψ−eiφψ‖ measures global recursive closure consistency, with the minimization over φ factoring out trivial global phase. The main theorem establishes the defect bound δC(ψ)≥κCεY(ψ) for a structural constant κ_C > 0, provided the admissible closure subspace is spectrally isolated - that is, U_C possesses a positive spectral gap away from the admissible eigenspace. The finiteness of Q5 guarantees a discrete spectrum; spectral isolation is the operative assumption yielding κC>0 and is explicit. Consequences: if εY(ψ)=0 (Hamming-compatible deposition), the state remains in the coherent carrier sector with δC=0.IfεY(ψ)>0, recursive closure is defective and the non-admissible component is forced into the residual sector through D:HM→HR. Three corollaries establish the identification of D as the defect extraction operator (with ‖DψM‖≥κCεY(ψM)); the defect-free passage of fully admissible transport; and monotone residual forcing (larger deposition mismatch produces larger residual extraction). The explicit form of D from Q5 geometry is deferred to TA36. The admissible target profile dadm is structurally characterized but not yet computed vertex-by-vertex; its explicit realization is a flagged future proof obligation.
TA31 converts the TA30 barrier-comparator picture into an explicit operator form. The barrier is modelled as a five-channel scattering system with Hilbert space H5=spanBL,L,M,R,BR, decomposed into a three-channel coherent triadic carrier sector HM=spanL,M,R and a two-channel asymmetric residual-balancing sector HR=spanBL,BR. The admissible Y-gate split ψPQ=ΠYψPQ+KψPQ (T96/T122), together with the three-mode helical leakage chain u_2 ↔ u_3 ↔ u_4 with no direct u_2 ↔ u_4 coupling (T127), forces a (3+2)×(3+2) block scattering structure. The barrier scattering operator takes block form S_B = [[A, C], [D, E]], where A: H_M → H_M governs coherent triadic carrier transport; D: H_M → H_R extracts phase-defective structure into residual fibres; C: H_R → H_M reinjerts circulated residual as coherent reinforcement; and E: H_R → H_R governs residual redistribution and balancing circulation. Four corollaries establish: the residual extraction channel (DM_in is the extraction term); the residual reinforcement channel (CR_in is the reinjection term); the residual circulation channel (E as the operator-level complement-sector balancing); and the coherence maintenance condition, repeated coherent transport remains stable when ‖DMin‖ and ‖CRin‖remain bounded and ρ(E)≤1. The spectral radius condition classifies three dynamical regimes:ρ(E)<1 damps residual mismatch, ρ(E)=1gives conservative circulation, and ρ(E)>1 produces instability and possible coherence rupture. Unitarity of S_B is not asserted. Explicit computation of block entries A, C, D, E as functions of Q5 geometry (w_H, Gray, Π_Y, K, Θ, A_L) is deferred to TA35 (geometric block-forcing definitions), TA36 (forced extraction block D), TA37 (forced circulation block E), and TA38 (forced reinjection block C). TA31 provides the structural scaffold that those theorems fill in. Physical interpretation beyond the transport formalism remains conditional.
TA38 derives the reinjection block CB:HR→HM of the barrier-scattering matrix from the Q5 admissibility architecture. Given the residual state circulated by TA36 and TA37, only the component of residual return lying in the Y-admissible gate subspace may re-enter the coherent carrier sector. This forces the reinjection block to take the formCB=ΠYUreturnHR.The theorem also gives the exact nonvanishing condition: reinjection occurs precisely when the residual return transport has nonzero overlap with im(ΠY). The resulting carrier output consists of direct carrier transport plus a lagged residual reinforcement term,ψMout(t)=ABψM+CBetEBDBψM.Together with TA36 and TA37, this completes the forced off-diagonal block trilogy: extraction, mediated circulation, and admissible reinjection. The result shows that the residual sector of the barrier-scattering matrix is geometrically constrained by the Q5 architecture rather than introduced as a phenomenological model. Status: solid for the forced structure of CB; conditional for nontrivial reinjection pending explicit computation of the leakage generator AL.
TA30 is a structural synthesis theorem unifying the post-T90 barrier machinery into a single operator interpretation. Earlier theorems established admissible Y-gate projection, complementary leakage sectors, antisymmetric transport survival, helical leakage propagation, lag-return phase bias, and residual phase redistribution as separate results. TA30 compresses these into one operative principle: the barrier acts as a coherence-selective inversion comparator rather than merely a separator. The full transport pipeline (T97) evolves the fibre state as ψout=DQMQΠYRDNMNψfull, acting on states in H5×5. Every reduced transport state decomposes into an admissible crossing component ΠYψPQand a complementary residual component KψPQ. From T88/T89, symmetric dressing contributions cancel under inversion conjugacy (Sn+Sd=0), so only antisymmetric, order-sensitive transport structure survives barrier comparison. Residual transport propagates helically through the three-mode chain u_2 ↔ u_3 ↔ u_4 under the adjacency operator AL=(1/√80)⋅[phasematrix] (T127), with no direct u_2 ↔ u_4 coupling. The middle mode acts as a mandatory balancing intermediary. Residual return (T94/T121) induces lag-phase observable correction δP=κsin(2θ)ΔY(t)cosφ, with coupling strength κ=1/√40 closing consistently with the derived transport normalization (T126). The residual scattering architecture is formalized through a block operator S_B = [[A, C], [D, E]], where A governs coherent carrier transport, D extracts phase-defective structure into residual fibres, C reinjerts circulated residual into coherent transport, and E governs residual redistribution and balancing circulation. The spectral radius of E classifies the dynamical regime: ρ(E) < 1 damps residual circulation, ρ(E) = 1 gives conservative circulation, ρ(E) > 1 produces rupture. The complement sector can function as a stabilizing coherence reservoir provided ρ(E) ≤ 1. The central structural result is that coherent transport is stabilized not by annihilating phase incompatibility but by exporting it into dynamically active residual-balancing fibres. The complement sector is not discarded leakage but an active balancing ecology. Explicit computation of block entries A, C, D, E from Q5 geometry is the primary remaining open computation, addressed in TA31–TA35. Physical interpretations (coherence-selective scattering, open-system phase stabilization, structured decoherence redistribution) remain conditional.
This Theorem closes the second open assumption of the T39 bridge lemma by deriving the leg-matching identity a = √2‖v_L‖ + O(δ²_eff) from Q5 carrier structure. Two independent Q5-native mechanisms force the leading-order identity. The complement involution ι: x ↦ 1 − x maps the Hamming-weight-1 subspace isometrically onto the Hamming-weight-2 subspace, intertwines the T24 ambient su(2) action by sign-reversing Ĵ_z and exchanging Ĵ_±, and satisfies ι(v_L) = v_R with ‖v_R‖ = ‖v_L‖. Within the spin-1 module V₁ carried by the Hamming-weight-1 sector, the raising operator Ĵ₊ acts with matrix element √2 between adjacent weight states, a standard spin-1 Clebsch-Gordan coefficient. Combining: at leading order a = √2‖v_L‖, where the √2 is the canonical spin-1 matrix element, not a normalization convention. The Y-transport correction is quadratic not linear: since θ = π/2 + O(δ_eff) from T43, the amplitude scales as sin θ = cos(O(δ_eff)) = 1 − O(δ²_eff), giving a = √2‖v_L‖ + O(δ²_eff). The leading-order result is stable to first order in the off-support leakage. Together with T43, both open assumptions in the T39 bridge lemma are now derived results. Status: Complement isometry solid: direct computation on the residual carrier basis. Spin-1 matrix element solid: standard result. Leading-order leg-matching conditional on identification of transport leg amplitude with Ĵ₊ matrix element on V₁ is structurally natural within the T24–T25 framework but not yet derived from explicit kernel-level transport generators. Quadratic correction inherits T43 parametric control conditionality. All results inherit T22–T25 and T39 conditionality. Dependencies: T22, T24, T25, T39, T43.
T93 extends the reduced Q5 interference framework of T91–T92 to an enlarged leakage-state space Hext=C2⊕HL, where C2 is the coherent crossing sector and HL collects non-crossing leakage modes. The theorem distinguishes between visible coherent weight C(t)=∥ΠPQΨ(t)∥2, leakage weight D(t)=∥(I−ΠPQ)Ψ(t)∥2, and gate-admission weight G(t)=∥ΠYΨ(t)∥2, where ΠY projects onto the Y-admissible crossing mode. A key result is that visible coherence and gate alignment are distinct quantities. A state may retain nonzero amplitude in the observable P/Q sector while failing admission into the Y-crossing mode. This leads to a geometric definition of decoherence: DY(t)=1−∥ΠYΨ(t)∥2. Decoherence is therefore interpreted as loss of overlap with the admissible crossing channel rather than destruction of phase information. The theorem further establishes that interference visibility is governed by the gate-admitted component of the state: V(t)=∥ΠYΨ(t)∥, yielding the extended interference law Pr(P)=21(1+V(t)sin(2t)cosϕ). Perfect visibility corresponds to complete gate alignment, while reduced visibility reflects either leakage into non-crossing modes or misalignment with the crossing sector. An extended generator G=(AK−K†B) is introduced to couple the coherent crossing sector to the leakage sector. Under the skew-adjoint conditions \[A^{\dagger}=-A,\qquadB^{\dagger}=-B,\] the evolution operator etG preserves total norm, implying C(t)+D(t)=1. Leakage therefore represents redistribution of phase weight rather than irreversible loss. The theorem provides a geometric interpretation of decoherence within the Q5 framework: phase information may remain globally conserved while becoming inaccessible to coherent crossing dynamics. In this picture, coherence is determined not merely by the presence of phase amplitude, but by its alignment with the unique Y-admissible transport mode.