Residual Scattering Operator Theorem

Craig Edwin Holdway

Published May 21, 2026• 0 citations

Abstract

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 $ H_5 = span{B_L, L, M, R, B_R} $, decomposed into a three-channel coherent triadic carrier sector $ H_M = span{L, M, R} $ and a two-channel asymmetric residual-balancing sector $ H_R = span{B_L, B_R} $. The admissible Y-gate split $ ψ_{PQ} = Π_Y ψ_{PQ} + Kψ_{PQ} $ (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 $ ‖DM_in‖ $ and $ ‖CR_in‖ $remain bounded and $ ρ(E) ≤ 1 $. The spectral radius condition classifies three dynamical regimes:$ ρ(E) < 1 $ damps residual mismatch, $ ρ(E) = 1 $gives 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.

DOI: 10.5281/zenodo.20333988

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Inversion-Comparator Residual Scattering Theorem

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Forced Residual Reinjection Block

Craig Edwin Holdway

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 $A_L$.

May 23, 20260 citations

Hamming-Weighted Admissible Deposition Theorem

Craig Edwin Holdway

May 21, 20260 citations

Residual Surface Return in the Extended Q5 Leakage Sector

Craig Edwin Holdway

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 \[\mathcal{H}_{\mathrm{ext}}=\mathbb{C}^{2}\oplus\mathcal{H}_{L},\] with dynamics governed by the coupled generator \[\mathcal{G}=\begin{pmatrix}A & -K^{\dagger} \\K & B\end{pmatrix},\] where $K$ transfers amplitude into the leakage sector, $K^{\dagger}$ returns amplitude from the leakage sector, and $B$ governs internal circulation within $\mathcal{H}_{L}$. The theorem establishes that leakage is redistribution rather than destruction. For an initially coherent crossing state $\psi_{PQ}(0)$, amplitude enters the leakage sector according to \[\lambda(t)=tK\psi_{PQ}(0)+O(t^{2}),\] and subsequently evolves through the internal circulation operator \[e^{tB}.\] The resulting leading-order return contribution to the crossing sector is \[R(t)=K^{\dagger}e^{tB}K\psi_{PQ}(0).\] 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 \[\psi_Y(t)=\Pi_Y \Psi(t).\] The coherence response is governed by the real overlap \[\Re\langle \psi_Y(t),R(t)\rangle.\] Three regimes emerge: \[\Re\langle \psi_Y,R\rangle > 0\] produces reinforcement of coherent crossing, \[\Re\langle \psi_Y,R\rangle < 0\] produces dephasing, and \[\Re\langle \psi_Y,R\rangle = 0\] 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 $e^{tB}$, 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: \[\text{crossing sector}\;\xrightarrow{K}\;\text{leakage sector}\;\xrightarrow{e^{tB}}\;\text{surface circulation}\;\xrightarrow{K^{\dagger}}\;\text{return transport}.\] 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.

Jun 3, 20260 citations

Geometric Block-Forcing Definitions

Craig Edwin Holdway

May 22, 20260 citations

Extended Barrier Transport with Leakage

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Residual Scattering Operator Theorem

Craig Edwin Holdway

Published May 21, 2026• 0 citations

Publisher

Abstract

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 ( H_5 = span{B_L, L, M, R, B_R} ), decomposed into a three-channel coherent triadic carrier sector ( H_M = span{L, M, R} ) and a two-channel asymmetric residual-balancing sector ( H_R = span{B_L, B_R} ). The admissible Y-gate split ( ψ_{PQ} = Π_Y ψ_{PQ} + Kψ_{PQ} ) (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 ( ‖DM_in‖ ) and ( ‖CR_in‖ )remain bounded and ( ρ(E) ≤ 1 ). The spectral radius condition classifies three dynamical regimes:( ρ(E) < 1 ) damps residual mismatch, ( ρ(E) = 1 )gives 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.

DOI: 10.5281/zenodo.20333988

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Inversion-Comparator Residual Scattering Theorem

Craig Edwin Holdway

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 $ψ_out = D_Q M_Q Π_Y R D_N M_N ψ_full$ , acting on states in $H_{5×5}$ . Every reduced transport state decomposes into an admissible crossing component $Π_Y ψ_{PQ}$ and a complementary residual component $Kψ_{PQ}$ . From T88/T89, symmetric dressing contributions cancel under inversion conjugacy $(S_n + S_d = 0)$ , 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 $A_L = (1/√80)·[phase matrix]$ (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 $δP = κsin(2θ)Δ_Y(t)cosφ$ , 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.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 $ ψ_out = D_Q M_Q Π_Y R D_N M_N ψ_full $, acting on states in $ H_{5×5} $. Every reduced transport state decomposes into an admissible crossing component $ Π_Y ψ_{PQ} $and a complementary residual component $ Kψ_{PQ} $. From T88/T89, symmetric dressing contributions cancel under inversion conjugacy $ (S_n + S_d = 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 $ A_L = (1/√80)·[phase matrix] $ (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)Δ_Y(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.

May 21, 20260 citations

Forced Residual Reinjection Block

Craig Edwin Holdway

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 $A_L$.

May 23, 20260 citations

Hamming-Weighted Admissible Deposition Theorem

Craig Edwin Holdway

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 $D_Y: H_M → C[V(Q5)]$ assigns to each admitted transport state an amplitude profile d_v on the 32 vertices. The total deposition defect $ε_Y(ψ) = (Σ_v |d_v − d_v^{adm}|²)^{1/2}$ 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 $δ_C(ψ) = min_φ ‖U_C ψ − e^{iφ}ψ‖$ measures global recursive closure consistency, with the minimization over φ factoring out trivial global phase. The main theorem establishes the defect bound $δ_C(ψ) ≥ κ_C ε_Y(ψ)$ 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 $κ_C > 0$ and is explicit. Consequences: if $ε_Y(ψ) = 0$ (Hamming-compatible deposition), the state remains in the coherent carrier sector with $δ_C = 0. If ε_Y(ψ) > 0$ , recursive closure is defective and the non-admissible component is forced into the residual sector through $D: H_M → H_R$ . Three corollaries establish the identification of D as the defect extraction operator (with $‖Dψ_M‖ ≥ κ_C ε_Y(ψ_M))$ ; 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 $d^{adm}$ is structurally characterized but not yet computed vertex-by-vertex; its explicit realization is a flagged future proof obligation.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 $ D_Y: H_M → C[V(Q5)] $ assigns to each admitted transport state an amplitude profile d_v on the 32 vertices. The total deposition defect $ ε_Y(ψ) = (Σ_v |d_v − d_v^{adm}|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 $ δ_C(ψ) = min_φ ‖U_C ψ − e^{iφ}ψ‖ $ measures global recursive closure consistency, with the minimization over φ factoring out trivial global phase. The main theorem establishes the defect bound $ δ_C(ψ) ≥ κ_C ε_Y(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 $ κ_C > 0 $ and is explicit. Consequences: if $ ε_Y(ψ) = 0 $ (Hamming-compatible deposition), the state remains in the coherent carrier sector with $ δ_C = 0. If ε_Y(ψ) > 0 $, recursive closure is defective and the non-admissible component is forced into the residual sector through $ D: H_M → H_R $. Three corollaries establish the identification of D as the defect extraction operator (with $ ‖Dψ_M‖ ≥ κ_C ε_Y(ψ_M)) $; 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 $ d^{adm} $ is structurally characterized but not yet computed vertex-by-vertex; its explicit realization is a flagged future proof obligation.

May 21, 20260 citations

Residual Surface Return in the Extended Q5 Leakage Sector

Craig Edwin Holdway

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 $\mathcal{H}_{\mathrm{ext}}=\mathbb{C}^{2}\oplus\mathcal{H}_{L},$ with dynamics governed by the coupled generator $\mathcal{G}=\begin{pmatrix}A & -K^{\dagger} \\K & B\end{pmatrix},$ where $K$ transfers amplitude into the leakage sector, $K^{\dagger}$ returns amplitude from the leakage sector, and $B$ governs internal circulation within $\mathcal{H}_{L}$ . The theorem establishes that leakage is redistribution rather than destruction. For an initially coherent crossing state $\psi_{PQ}(0)$ , amplitude enters the leakage sector according to $\lambda(t)=tK\psi_{PQ}(0)+O(t^{2}),$ and subsequently evolves through the internal circulation operator $e^{tB}.$ The resulting leading-order return contribution to the crossing sector is $R(t)=K^{\dagger}e^{tB}K\psi_{PQ}(0).$ 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 $\psi_Y(t)=\Pi_Y \Psi(t).$ The coherence response is governed by the real overlap $\Re\langle \psi_Y(t),R(t)\rangle.$ Three regimes emerge: $\Re\langle \psi_Y,R\rangle > 0$ produces reinforcement of coherent crossing, $\Re\langle \psi_Y,R\rangle < 0$ produces dephasing, and $\Re\langle \psi_Y,R\rangle = 0$ 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 $e^{tB}$ , 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: $\text{crossing sector}\;\xrightarrow{K}\;\text{leakage sector}\;\xrightarrow{e^{tB}}\;\text{surface circulation}\;\xrightarrow{K^{\dagger}}\;\text{return transport}.$ 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.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 \[\mathcal{H}_{\mathrm{ext}}=\mathbb{C}^{2}\oplus\mathcal{H}_{L},\] with dynamics governed by the coupled generator \[\mathcal{G}=\begin{pmatrix}A & -K^{\dagger} \\K & B\end{pmatrix},\] where $K$ transfers amplitude into the leakage sector, $K^{\dagger}$ returns amplitude from the leakage sector, and $B$ governs internal circulation within $\mathcal{H}_{L}$. The theorem establishes that leakage is redistribution rather than destruction. For an initially coherent crossing state $\psi_{PQ}(0)$, amplitude enters the leakage sector according to \[\lambda(t)=tK\psi_{PQ}(0)+O(t^{2}),\] and subsequently evolves through the internal circulation operator \[e^{tB}.\] The resulting leading-order return contribution to the crossing sector is \[R(t)=K^{\dagger}e^{tB}K\psi_{PQ}(0).\] 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 \[\psi_Y(t)=\Pi_Y \Psi(t).\] The coherence response is governed by the real overlap \[\Re\langle \psi_Y(t),R(t)\rangle.\] Three regimes emerge: \[\Re\langle \psi_Y,R\rangle > 0\] produces reinforcement of coherent crossing, \[\Re\langle \psi_Y,R\rangle < 0\] produces dephasing, and \[\Re\langle \psi_Y,R\rangle = 0\] 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 $e^{tB}$, 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: \[\text{crossing sector}\;\xrightarrow{K}\;\text{leakage sector}\;\xrightarrow{e^{tB}}\;\text{surface circulation}\;\xrightarrow{K^{\dagger}}\;\text{return transport}.\] 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.

Jun 3, 20260 citations

Geometric Block-Forcing Definitions

Craig Edwin Holdway

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.

May 22, 20260 citations

Extended Barrier Transport with Leakage

Craig Edwin Holdway

T98 extends the coherent-crossing transport pipeline of T97 into a complete two-branch barrier architecture. Rather than tracking only the gate-admitted component, the theorem follows both the admitted and rejected portions of the reduced crossing state, yielding the minimal extended transport model for barrier interaction. Beginning with an incoming fibre state $\psi_{\mathrm{full}}\in\mathcal{H}_{5\times5},$ The native-side transport and reduction operators produce the reduced crossing state $\psi_{PQ}=\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}\in\mathbb{C}^{2}.$ Using the gate-admission projector $\Pi_Y$ and leakage complement $K=I-\Pi_Y,$ Introduced in T96, the reduced state decomposes orthogonally into $\psi_{PQ}=\Pi_Y\psi_{PQ}+K\psi_{PQ}.$ The first component satisfies the $Y$ -gate condition and is eligible for coherent crossing. The second component fails gate admission and enters the leakage branch. The theorem derives the complete extended transport operator $\mathcal{T}_{\mathrm{ext}}=\begin{pmatrix}D_QM_Q\,\Pi_Y\,\mathcal{R}\,D_NM_N\\[0.4em]K\,\mathcal{R}\,D_NM_N\end{pmatrix},$ which simultaneously produces both outputs of barrier interaction. Applied to an incoming fibre state, the result is $\mathcal{T}_{\mathrm{ext}}\psi_{\mathrm{full}}=\begin{pmatrix}D_QM_Q\,\Pi_Y\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}\\[0.4em]K\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}\end{pmatrix}.$ The upper component is the coherent crossing branch propagating through the barrier into the dual-side sector. The lower component is the leakage branch consisting of phase content that reaches the reduced crossing sector but fails the $Y$ -gate condition. A key consequence is that generic barrier interaction is neither complete transmission nor complete rejection. Most states decompose into both branches simultaneously: $\psi_{PQ}=\Pi_Y\psi_{PQ}+K\psi_{PQ}.$ The barrier, therefore, acts as a filter rather than a binary gate. Coherent crossing and leakage are complementary outcomes of the same reduced crossing state. The theorem further establishes the complete-crossing and complete-leakage limits. A state crosses cleanly if and only if $K\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}=0,$ equivalently, $\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}\in\operatorname{span}\{|Y\rangle\}.$ Conversely, a state exhibits complete leakage if and only if $\Pi_Y\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}=0,$ so that no coherent crossing component survives. Using the orthogonal decomposition established in T96, the theorem preserves the norm split $C_Y+D_Y=1,$ where $C_Y=\|\Pi_Y\psi_{PQ}\|^2$ is the gate-admission weight and $D_Y=\|K\psi_{PQ}\|^2$ is the leakage weight. A major consequence of the theorem is that the leakage branch becomes the explicit source of the residual return dynamics developed in T94. Substituting the reduced crossing state into the return formula yields $R(t)=K^\dagger e^{tB}K\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}},$ providing a complete operator pathway from incoming fibre state to leakage circulation and subsequent return. T98 therefore completes the transport architecture initiated in T97. The barrier is revealed as a two-output filter that separates an incoming reduced state into an admitted crossing component and a rejected leakage component. Together with T93–T97, the theorem provides the first complete operator-level description of barrier transport, leakage generation, and residual return within the Q5 framework.T98 extends the coherent-crossing transport pipeline of T97 into a complete two-branch barrier architecture. Rather than tracking only the gate-admitted component, the theorem follows both the admitted and rejected portions of the reduced crossing state, yielding the minimal extended transport model for barrier interaction. Beginning with an incoming fibre state \[\psi_{\mathrm{full}}\in\mathcal{H}_{5\times5},\] The native-side transport and reduction operators produce the reduced crossing state \[\psi_{PQ}=\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}\in\mathbb{C}^{2}.\] Using the gate-admission projector $\Pi_Y$ and leakage complement \[K=I-\Pi_Y,\] Introduced in T96, the reduced state decomposes orthogonally into \[\psi_{PQ}=\Pi_Y\psi_{PQ}+K\psi_{PQ}.\] The first component satisfies the $Y$-gate condition and is eligible for coherent crossing. The second component fails gate admission and enters the leakage branch. The theorem derives the complete extended transport operator \[\mathcal{T}_{\mathrm{ext}}=\begin{pmatrix}D_QM_Q\,\Pi_Y\,\mathcal{R}\,D_NM_N\\[0.4em]K\,\mathcal{R}\,D_NM_N\end{pmatrix},\] which simultaneously produces both outputs of barrier interaction. Applied to an incoming fibre state, the result is \[\mathcal{T}_{\mathrm{ext}}\psi_{\mathrm{full}}=\begin{pmatrix}D_QM_Q\,\Pi_Y\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}\\[0.4em]K\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}\end{pmatrix}.\] The upper component is the coherent crossing branch propagating through the barrier into the dual-side sector. The lower component is the leakage branch consisting of phase content that reaches the reduced crossing sector but fails the $Y$-gate condition. A key consequence is that generic barrier interaction is neither complete transmission nor complete rejection. Most states decompose into both branches simultaneously: \[\psi_{PQ}=\Pi_Y\psi_{PQ}+K\psi_{PQ}.\] The barrier, therefore, acts as a filter rather than a binary gate. Coherent crossing and leakage are complementary outcomes of the same reduced crossing state. The theorem further establishes the complete-crossing and complete-leakage limits. A state crosses cleanly if and only if \[K\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}=0,\] equivalently, \[\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}\in\operatorname{span}\{|Y\rangle\}.\] Conversely, a state exhibits complete leakage if and only if \[\Pi_Y\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}=0,\] so that no coherent crossing component survives. Using the orthogonal decomposition established in T96, the theorem preserves the norm split \[C_Y+D_Y=1,\] where \[C_Y=\|\Pi_Y\psi_{PQ}\|^2\] is the gate-admission weight and \[D_Y=\|K\psi_{PQ}\|^2\] is the leakage weight. A major consequence of the theorem is that the leakage branch becomes the explicit source of the residual return dynamics developed in T94. Substituting the reduced crossing state into the return formula yields \[R(t)=K^\dagger e^{tB}K\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}},\] providing a complete operator pathway from incoming fibre state to leakage circulation and subsequent return. T98 therefore completes the transport architecture initiated in T97. The barrier is revealed as a two-output filter that separates an incoming reduced state into an admitted crossing component and a rejected leakage component. Together with T93–T97, the theorem provides the first complete operator-level description of barrier transport, leakage generation, and residual return within the Q5 framework.

Jun 3, 20260 citations

ψ_out = D_Q M_Q Π_Y R D_N M_N ψ_full

, acting on states in

H_{5×5}

. Every reduced transport state decomposes into an admissible crossing component

Π_Y ψ_{PQ}

and a complementary residual component

Kψ_{PQ}

. From T88/T89, symmetric dressing contributions cancel under inversion conjugacy

(S_n + S_d = 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

A_L = (1/√80)·[phase matrix]

(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)Δ_Y(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.

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

D_Y: H_M → C[V(Q5)]

assigns to each admitted transport state an amplitude profile d_v on the 32 vertices. The total deposition defect

ε_Y(ψ) = (Σ_v |d_v − d_v^{adm}|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

δ_C(ψ) = min_φ ‖U_C ψ − e^{iφ}ψ‖

measures global recursive closure consistency, with the minimization over φ factoring out trivial global phase. The main theorem establishes the defect bound

δ_C(ψ) ≥ κ_C ε_Y(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

κ_C > 0

and is explicit. Consequences: if

ε_Y(ψ) = 0

(Hamming-compatible deposition), the state remains in the coherent carrier sector with

δ_C = 0. If ε_Y(ψ) > 0

, recursive closure is defective and the non-admissible component is forced into the residual sector through

D: H_M → H_R

. Three corollaries establish the identification of D as the defect extraction operator (with

‖Dψ_M‖ ≥ κ_C ε_Y(ψ_M))

; 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

d^{adm}

is structurally characterized but not yet computed vertex-by-vertex; its explicit realization is a flagged future proof obligation.

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

\mathcal{H}_{\mathrm{ext}}=\mathbb{C}^{2}\oplus\mathcal{H}_{L},

with dynamics governed by the coupled generator

\mathcal{G}=\begin{pmatrix}A & -K^{\dagger} \\K & B\end{pmatrix},

where

K

transfers amplitude into the leakage sector,

K^{\dagger}

returns amplitude from the leakage sector, and

B

governs internal circulation within

\mathcal{H}_{L}

. The theorem establishes that leakage is redistribution rather than destruction. For an initially coherent crossing state

\psi_{PQ}(0)

, amplitude enters the leakage sector according to

\lambda(t)=tK\psi_{PQ}(0)+O(t^{2}),

and subsequently evolves through the internal circulation operator

e^{tB}.

The resulting leading-order return contribution to the crossing sector is

R(t)=K^{\dagger}e^{tB}K\psi_{PQ}(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

\psi_Y(t)=\Pi_Y \Psi(t).

The coherence response is governed by the real overlap

\Re\langle \psi_Y(t),R(t)\rangle.

Three regimes emerge:

\Re\langle \psi_Y,R\rangle > 0

produces reinforcement of coherent crossing,

\Re\langle \psi_Y,R\rangle < 0

produces dephasing, and

\Re\langle \psi_Y,R\rangle = 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

e^{tB}

, 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:

\text{crossing sector}\;\xrightarrow{K}\;\text{leakage sector}\;\xrightarrow{e^{tB}}\;\text{surface circulation}\;\xrightarrow{K^{\dagger}}\;\text{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.

(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.

\psi_{\mathrm{full}}\in\mathcal{H}_{5\times5},

The native-side transport and reduction operators produce the reduced crossing state

\psi_{PQ}=\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}\in\mathbb{C}^{2}.

Using the gate-admission projector

\Pi_Y

and leakage complement

K=I-\Pi_Y,

Introduced in T96, the reduced state decomposes orthogonally into

\psi_{PQ}=\Pi_Y\psi_{PQ}+K\psi_{PQ}.

The first component satisfies the

Y

-gate condition and is eligible for coherent crossing. The second component fails gate admission and enters the leakage branch. The theorem derives the complete extended transport operator

\mathcal{T}_{\mathrm{ext}}=\begin{pmatrix}D_QM_Q\,\Pi_Y\,\mathcal{R}\,D_NM_N\\[0.4em]K\,\mathcal{R}\,D_NM_N\end{pmatrix},

which simultaneously produces both outputs of barrier interaction. Applied to an incoming fibre state, the result is

\mathcal{T}_{\mathrm{ext}}\psi_{\mathrm{full}}=\begin{pmatrix}D_QM_Q\,\Pi_Y\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}\\[0.4em]K\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}\end{pmatrix}.

The upper component is the coherent crossing branch propagating through the barrier into the dual-side sector. The lower component is the leakage branch consisting of phase content that reaches the reduced crossing sector but fails the

Y

-gate condition. A key consequence is that generic barrier interaction is neither complete transmission nor complete rejection. Most states decompose into both branches simultaneously:

\psi_{PQ}=\Pi_Y\psi_{PQ}+K\psi_{PQ}.

The barrier, therefore, acts as a filter rather than a binary gate. Coherent crossing and leakage are complementary outcomes of the same reduced crossing state. The theorem further establishes the complete-crossing and complete-leakage limits. A state crosses cleanly if and only if

K\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}=0,

equivalently,

\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}\in\operatorname{span}\{|Y\rangle\}.

Conversely, a state exhibits complete leakage if and only if

\Pi_Y\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}}=0,

so that no coherent crossing component survives. Using the orthogonal decomposition established in T96, the theorem preserves the norm split

C_Y+D_Y=1,

where

C_Y=\|\Pi_Y\psi_{PQ}\|^2

is the gate-admission weight and

D_Y=\|K\psi_{PQ}\|^2

is the leakage weight. A major consequence of the theorem is that the leakage branch becomes the explicit source of the residual return dynamics developed in T94. Substituting the reduced crossing state into the return formula yields

R(t)=K^\dagger e^{tB}K\,\mathcal{R}\,D_NM_N\,\psi_{\mathrm{full}},

providing a complete operator pathway from incoming fibre state to leakage circulation and subsequent return. T98 therefore completes the transport architecture initiated in T97. The barrier is revealed as a two-output filter that separates an incoming reduced state into an admitted crossing component and a rejected leakage component. Together with T93–T97, the theorem provides the first complete operator-level description of barrier transport, leakage generation, and residual return within the Q5 framework.