TA38 derives the reinjection block \(C_{\mathcal B}:\mathcal H_R\to\mathcal H_M\) 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 form\[C_{\mathcal B}=\Pi_Y\,\mathcal U_{\mathrm{return}}\big|_{\mathcal H_R}.\]The theorem also gives the exact nonvanishing condition: reinjection occurs precisely when the residual return transport has nonzero overlap with \(\operatorname{im}(\Pi_Y)\). The resulting carrier output consists of direct carrier transport plus a lagged residual reinforcement term,\[\psi_M^{\mathrm{out}}(t)=A_{\mathcal B}\psi_M+C_{\mathcal B}e^{tE_{\mathcal B}}D_{\mathcal B}\psi_M.\]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 \(C_{\mathcal B}\); conditional for nontrivial reinjection pending explicit computation of the leakage generator \(A_L\).
TA38 derives the reinjection block (C_{\mathcal B}:\mathcal H_R\to\mathcal H_M) 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 form[C_{\mathcal B}=\Pi_Y,\mathcal U_{\mathrm{return}}\big|{\mathcal H_R}.]The theorem also gives the exact nonvanishing condition: reinjection occurs precisely when the residual return transport has nonzero overlap with (\operatorname{im}(\Pi_Y)). The resulting carrier output consists of direct carrier transport plus a lagged residual reinforcement term,[\psi_M^{\mathrm{out}}(t)=A{\mathcal B}\psi_M+C_{\mathcal B}e^{tE_{\mathcal B}}D_{\mathcal B}\psi_M.]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 (C_{\mathcal B}); conditional for nontrivial reinjection pending explicit computation of the leakage generator (A_L).
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.
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,wH,Gray,ΠY,K,Θ,AL). Ten definitions are provided. The carrier--residual split H5=HM⊕HR 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)=(P0,...,P5) records amplitude distribution across the six Hamming-weight layers. The admissible target profile dadm(ψY) 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 εY(ψ)2=Σv∣dv−dvadm∣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∈HM of the deposition mismatch DY(ΠYψ)−dadm(ΠYψ), 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∣Rrein 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.
T94 extends the leakage-state framework of T93 by deriving the leading-order return dynamics of phase amplitude that leaves the Y-admissible crossing sector. Rather than treating leakage as a terminal loss mechanism, the theorem models leakage as a redistribution process in which rejected amplitude enters a secondary sector, circulates internally, and subsequently returns to the observable crossing sector. The extended state space is Hext=C2⊕HL, with dynamics governed by the coupled generator G=(AK−K†B), where K transfers amplitude into the leakage sector, K† returns amplitude from the leakage sector, and B governs internal circulation within HL. The theorem establishes that leakage is redistribution rather than destruction. For an initially coherent crossing state ψPQ(0), amplitude enters the leakage sector according to λ(t)=tKψPQ(0)+O(t2), and subsequently evolves through the internal circulation operator etB. The resulting leading-order return contribution to the crossing sector is R(t)=K†etBKψPQ(0). This return term is identified as the residual surface return of the extended Q5 architecture. A central result of the theorem is that residual return is not automatically restorative. The effect of the returned phase is determined entirely by its alignment with the gate-admissible crossing component ψY(t)=ΠYΨ(t). The coherence response is governed by the real overlap ℜ⟨ψY(t),R(t)⟩. Three regimes emerge: ℜ⟨ψY,R⟩>0 produces reinforcement of coherent crossing, ℜ⟨ψY,R⟩<0 produces dephasing, and ℜ⟨ψY,R⟩=0 produces a neutral contribution. The theorem, therefore, refines the decoherence picture introduced in T93. Decoherence is not interpreted as the disappearance of phase information but as the return of redistributed phase in a state that is no longer aligned with the admissible crossing mode. Misaligned return decreases gate-admission weight and reduces interference visibility even when the total norm remains conserved. Several corollaries follow. Generic residual return does not remain phase-locked to the crossing mode and therefore fails to restore coherence automatically. Exact recovery requires reconstruction of the full leakage-sector evolution etB, making reversibility information-demanding even when the underlying dynamics remain unitary. Effective irreversibility is thus attributed to the information cost of reconstruction rather than to fundamental non-unitarity. T94 serves as the first formal return-transport theorem in the Q5 sequence. Together with T93, it provides a complete first-order picture of leakage dynamics: crossing sectorKleakage sectoretBsurface circulationK†return transport. The theorem establishes that the coherence consequences of leakage are determined not by the existence of returned phase, but by the alignment of that returned phase with the unique Y-admissible crossing mode.
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.
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.
T97 unifies the previously developed Q5 crossing, leakage, and gate-admission frameworks into a single operator composition describing coherent barrier transport. The theorem establishes that barrier crossing is not free propagation but a constrained sequence of transformations acting on the full incoming fibre state. Starting from an incoming packet ψfull∈H5×5, The theorem identifies five sequential stages of transport: \[M_N\rightarrowD_N\rightarrow\mathcal{R}\rightarrow\Pi_Y\rightarrowD_QM_Q.\] These correspond respectively to native-side kernel interaction, native-side transport, dimensional reduction into the reduced crossing sector, gate-admissibility selection, and dual-side propagation. The resulting transport law is ψout=DQMQΠYRDNMNψfull. This expression provides the first complete operator-level description of barrier transport within the Q5 framework. The theorem begins by introducing the transport operators. The native-side operators MN and DN act on the full 5×5 fibre geometry. The reduction map R:H5×5→C2 implements the dimensional reduction that extracts the surviving P/Q crossing sector. The gate projector ΠY=∣Y⟩⟨Y∣ Introduced in T96 selects the admissible quarter-turn crossing component. Finally, the dual-side operators MQ and DQ propagate the admitted state after barrier crossing. The theorem establishes that the reduced crossing state is ψPQ=RDNMNψfull, and that the admissible crossing component is ψY=ΠYψPQ. Only this projected component is eligible for coherent transport. The complementary component KψPQ=(I−ΠY)ψPQ enters the leakage sector described in T93–T96. The outgoing transported state is therefore obtained by applying the dual-side operators to the admitted component: ψout=DQMQψY. Substitution yields the full transport composition. Several structural consequences follow. A state crosses coherently if and only if ΠYRDNMNψfull=0. Thus, coherent transport requires a nonzero overlap between the reduced crossing state and the admissible Y-mode. States with zero projection onto the Y-mode fail the crossing condition and enter the leakage branch. The theorem further demonstrates that the ordering of operators is not arbitrary. Reduction must occur after native-side processing, while gate selection must occur before dual-side propagation. The gate projector, therefore, occupies the central position within the transport architecture, separating state preparation from post-crossing propagation. A direct connection is established with T96. Since ΠY=∣Y⟩⟨Y∣, The crossing condition becomes ψPQ⊥∣Y⟩. The gate-admission strength is therefore CY(ψPQ)=21∣P−iQ∣2, providing a computable measure of crossing eligibility in terms of the reduced crossing amplitudes. T97 serves as the foundational transport-composition theorem of the Q5 framework. It links the reduced crossing sector, the leakage architecture, and the gate projector into a single operator pipeline, establishing the canonical ordering of processes that govern coherent barrier transport.