Phase Pass-Through Condition for Reduced Q5 Transport

Craig Edwin Holdway

Published May 25, 2026• 0 citations

Abstract

T83 establishes the phase pass-through condition for reduced Q5 transport. Starting from the continuous phase-flow limit established in T80,\[T_{\mathrm{phase}}^k\longrightarrowe^{tA}=I\cos t + A\sin t,\qquadt=k\varepsilon,\]with\[A^2=-I,\]the theorem applies the normalized phase-readout functional $L$, satisfying\[L(I)=0,\qquadL(A)=1.\] It follows that\[L(e^{tA})=L(I\cos t + A\sin t)=\sin t.\] Thus, observable phase transmission occurs exactly when\[\sin t\neq 0.\] At\[t=n\pi,\qquadn\in\mathbb{Z},\]the accumulated transport satisfies\[e^{tA}=\pm I,\]so the transport is diagonal/scalar in the phase readout and no phase passes through. At\[t=\tfrac{\pi}{2}+n\pi,\qquadn\in\mathbb{Z},\]The accumulated transport satisfies\[e^{tA}=\pm A,\]So the readout is purely rotational and phase pass-through is maximal. T83 is structurally important because it separates diagonal retention from observable phase transmission:\[e^{tA}=I\cos t + A\sin t.\]The $I\cos t$ branch retains the carrier state but is invisible to $L$, while the $A\sin t$ branch carries the observable phase coefficient. Together with T80-T82, this completes the reduced phase-flow readout arc:\[T_{\mathrm{raw}}\toT_{\mathrm{phase}}\toe^{tA}\toL(e^{tA})=\sin t.\] Status: solid for the identity\[L(e^{tA})=\sin t\]given the T80 continuous limit and the T26-T28 normalization of $L$; solid for the pass-through condition\[\sin t\neq 0;\]conditional on the reduced phase-transport form\[T_{\mathrm{phase}}=I+\varepsilon A+O(\varepsilon^2)\]and on the T81-T82 projection-invariance bridge from raw barrier transport.

DOI: 10.5281/zenodo.20401412

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Reduced Q5 Interference Law

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T91 establishes the reduced Q5 two-channel interference law after the $5\to3\to2$ reduction. The surviving reduced state space is $\mathbb{C}^2$, with crossing channels $P$ and $Q$, and transport generated by $A=i\sigma_y$, satisfying $A^2=-I$. Continuous reduced evolution is given by\[\psi(t)=e^{tA}\psi(0),\]with\[e^{tA}=\begin{pmatrix}\cos t & \sin t\\-\sin t & \cos t\end{pmatrix}.\] For the initial relative-phase state\[\psi(0)=\frac1{\sqrt2}\begin{pmatrix}1\\e^{i\phi}\end{pmatrix},\]The evolved channel probabilities are\[\Pr(P)=\frac12\left(1+\sin(2t)\cos\phi\right),\]and\[\Pr(Q)=\frac12\left(1-\sin(2t)\cos\phi\right).\]The observable channel imbalance is therefore\[\Pr(P)-\Pr(Q)=\sin(2t)\cos\phi.\] The theorem shows that reduced Q5 transport naturally produces a two-channel interference structure on the reduced crossing plane. The transport amplitude is controlled by accumulated phase $t=k\varepsilon$, while the coherence factor is controlled by the relative phase $\phi$. Phase averaging removes the interference contribution:\[\langle\cos\phi\rangle=0\quad\Longrightarrow\quad\langle\Pr(P)\rangle=\langle\Pr(Q)\rangle=\frac12.\]Loss of coherent phase relation, therefore restores equal channel probabilities. T91 converts the earlier transport-generator chain into an explicit observable probability law. Earlier theorems established\[A^2=-I,\qquadT^k\to e^{tA},\qquad\mu_k=\sin(k\varepsilon),\]while T91 derives the resulting two-channel interference structure generated by the same reduced transport algebra. The discrete form follows from\[t=k\varepsilon,\qquad\varepsilon=\frac1{320},\]giving\[\Pr(P)=\frac12\left(1+\sin\left(\frac{2k}{320}\right)\cos\phi\right).\]The fringe contrast oscillates with transport step number $k$, with the first maximal contrast near\[k\approx80\pi\approx251,\]matching the near-integer phase-maximization structure identified previously in T84. Classification:Solid: matrix exponential, probability law, normalization, and phase averaging.Conditional: identification of the reduced two-channel Q5 state space and the generator chain $A=i\sigma_y$ from T37 and T80-T89.Not claimed: direct experimental observability.

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Coarse Spinor Cycling and Fine Phase Accumulation

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Phase Pass-Through Condition for Reduced Q5 Transport

Craig Edwin Holdway

Published May 25, 2026• 0 citations

Publisher

Abstract

T83 establishes the phase pass-through condition for reduced Q5 transport. Starting from the continuous phase-flow limit established in T80,[T_{\mathrm{phase}}^k\longrightarrowe^{tA}=I\cos t + A\sin t,\qquadt=k\varepsilon,]with[A^2=-I,]the theorem applies the normalized phase-readout functional (L), satisfying[L(I)=0,\qquadL(A)=1.] It follows that[L(e^{tA})=L(I\cos t + A\sin t)=\sin t.] Thus, observable phase transmission occurs exactly when[\sin t\neq 0.] At[t=n\pi,\qquadn\in\mathbb{Z},]the accumulated transport satisfies[e^{tA}=\pm I,]so the transport is diagonal/scalar in the phase readout and no phase passes through. At[t=\tfrac{\pi}{2}+n\pi,\qquadn\in\mathbb{Z},]The accumulated transport satisfies[e^{tA}=\pm A,]So the readout is purely rotational and phase pass-through is maximal. T83 is structurally important because it separates diagonal retention from observable phase transmission:[e^{tA}=I\cos t + A\sin t.]The (I\cos t) branch retains the carrier state but is invisible to (L), while the (A\sin t) branch carries the observable phase coefficient. Together with T80-T82, this completes the reduced phase-flow readout arc:[T_{\mathrm{raw}}\toT_{\mathrm{phase}}\toe^{tA}\toL(e^{tA})=\sin t.] Status: solid for the identity[L(e^{tA})=\sin t]given the T80 continuous limit and the T26-T28 normalization of (L); solid for the pass-through condition[\sin t\neq 0;]conditional on the reduced phase-transport form[T_{\mathrm{phase}}=I+\varepsilon A+O(\varepsilon^2)]and on the T81-T82 projection-invariance bridge from raw barrier transport.

DOI: 10.5281/zenodo.20401412

Related papers

Discrete Sampling of Continuous Q5 Phase Transport

Craig Edwin Holdway

T84 establishes the discrete sampling law for continuous Q5 phase transport. Starting from the reduced phase transport operator\[T_{\mathrm{phase}}=I+\epsilon A+O(\epsilon^2),\qquadA^2=-I,\qquad\epsilon=\frac1{160},\]The theorem defines the discrete observable phase sequence $\mu_k=L(T_{\mathrm{phase}}^k),$ where $L$ is the normalized phase-readout functional satisfying\[L(I)=0,\qquadL(A)=1.\] Using the T80 continuous-limit result\[T_{\mathrm{phase}}^k\toe^{k\epsilon A},\]together with $e^{k\epsilon A}=I\cos(k\epsilon)+A\sin(k\epsilon),$ The theorem proves $\mu_k=\sin(k\epsilon)+O(\epsilon).$ Substituting $\epsilon=\frac1{160},$ gives the discrete sampled phase law $\mu_k=\sin\left(\frac{k}{160}\right)+O\left(\frac1{160}\right).$ T84 is structurally important because it converts the continuous phase-flow law $L(e^{tA})=\sin t$ from T83 into an indexed observable transport sequence $k\mapsto \mu_k.$ The theorem, therefore, establishes the sampled observable structure inherited from the continuous Q5 transport flow. Several structural consequences follow immediately. Pass-through occurs exactly when $\sin(k\epsilon)\neq0.$ Maximal pass-through occurs near $k\epsilon=\frac{\pi}{2}+n\pi,$ Although exact maxima are generally inaccessible because the discrete sampling lattice does not usually intersect those irrational phase locations exactly. For small accumulated transport, $k\ll160,$ The small-angle approximation gives $\mu_k\approx\frac{k}{160},$ so early-stage phase readout grows linearly with transport depth. Together with T80-T83, the transport/readout chain now becomes\[T_{\mathrm{raw}}\toT_{\mathrm{phase}}\toT_{\mathrm{phase}}^k\toe^{k\epsilon A}\to\mu_k=\sin(k\epsilon).\] Status: solid for the sampled phase law $\mu_k=\sin(k\epsilon)+O(\epsilon)$ given the T80 continuous limit and bounded accumulated phase $t=k\epsilon$ ; solid for the small-step linear regime $\mu_k\approx k/160;$ conditional for higher-order corrections inherited from the unresolved higher-order structure of $T_{\mathrm{phase}}.$ T84 establishes the discrete sampling law for continuous Q5 phase transport. Starting from the reduced phase transport operator\[T_{\mathrm{phase}}=I+\epsilon A+O(\epsilon^2),\qquadA^2=-I,\qquad\epsilon=\frac1{160},\]The theorem defines the discrete observable phase sequence\[\mu_k=L(T_{\mathrm{phase}}^k),\]where $L$ is the normalized phase-readout functional satisfying\[L(I)=0,\qquadL(A)=1.\] Using the T80 continuous-limit result\[T_{\mathrm{phase}}^k\toe^{k\epsilon A},\]together with\[e^{k\epsilon A}=I\cos(k\epsilon)+A\sin(k\epsilon),\]The theorem proves\[\mu_k=\sin(k\epsilon)+O(\epsilon).\] Substituting\[\epsilon=\frac1{160},\]gives the discrete sampled phase law\[\mu_k=\sin\left(\frac{k}{160}\right)+O\left(\frac1{160}\right).\] T84 is structurally important because it converts the continuous phase-flow law\[L(e^{tA})=\sin t\]from T83 into an indexed observable transport sequence\[k\mapsto \mu_k.\] The theorem, therefore, establishes the sampled observable structure inherited from the continuous Q5 transport flow. Several structural consequences follow immediately. Pass-through occurs exactly when\[\sin(k\epsilon)\neq0.\] Maximal pass-through occurs near\[k\epsilon=\frac{\pi}{2}+n\pi,\]Although exact maxima are generally inaccessible because the discrete sampling lattice does not usually intersect those irrational phase locations exactly. For small accumulated transport,\[k\ll160,\]The small-angle approximation gives\[\mu_k\approx\frac{k}{160},\]so early-stage phase readout grows linearly with transport depth. Together with T80-T83, the transport/readout chain now becomes\[T_{\mathrm{raw}}\toT_{\mathrm{phase}}\toT_{\mathrm{phase}}^k\toe^{k\epsilon A}\to\mu_k=\sin(k\epsilon).\] Status: solid for the sampled phase law\[\mu_k=\sin(k\epsilon)+O(\epsilon)\]given the T80 continuous limit and bounded accumulated phase $t=k\epsilon$; solid for the small-step linear regime\[\mu_k\approx k/160;\]conditional for higher-order corrections inherited from the unresolved higher-order structure of\[T_{\mathrm{phase}}.\]

May 25, 20260 citations

Observable Invariance under Non-Rotational Transport Dressing

Craig Edwin Holdway

T82 establishes observable invariance under non-rotational transport dressing within the reduced Q5 phase-transport sector. Starting from the raw barrier transport operator $T_{\mathrm{raw}}=I+\epsilon A+\mathcal{S}_{\mathrm{nr}}+O(\epsilon^2),\qquad\epsilon=\tfrac{1}{160},$ with\[A=i\sigma_y,\qquadA^2=-I,\]The theorem proves that the T26-T29 phase-extraction functional satisfies $L(T_{\mathrm{raw}})=\epsilon+O(\epsilon^2)=L(T_{\mathrm{phase}})+O(\epsilon^2),$ where $T_{\mathrm{phase}}=I+\epsilon A+O(\epsilon^2).$ The non-rotational dressing $\mathcal{S}_{\mathrm{nr}}\in\operatorname{span}\{I,\sigma_x,\sigma_z\}$ is annihilated by the extraction functional: $L(\mathcal{S}_{\mathrm{nr}})=0.$ Thus, the observable phase coefficient is determined entirely by the rotational generator $A$ and is invariant under scalar, grading, or symmetric transverse dressing at first order in $\epsilon$ . The theorem is structurally important because it completes the T80-T82 transport arc: $T_{\mathrm{raw}}\xrightarrow{\mathcal{P}_{\mathrm{phase}}}T_{\mathrm{phase}}\xrightarrow{\times k,\ \epsilon\to0}e^{tA}\xrightarrow{L}\sin t.$ T80 established the discrete-to-continuous phase-flow limit\[T_{\mathrm{phase}}^k\toe^{tA}=I\cos t + A\sin t.\] T81 established that the canonical T26-T29 projection kills the non-rotational dressing: $\mathcal{P}_{\mathrm{phase}}(T_{\mathrm{raw}})=T_{\mathrm{phase}}.$ T82 now proves the observable consequence: the measurable phase readout is invariant under the dressing sector and depends only on the rotational transport component. The theorem therefore replaces any remaining implicit “raw purity” interpretation with observable-sector equivalence under canonical extraction. The non-rotational dressing changes the representation of the raw transport operator but not the observable phase coefficient. The gauge-redundancy interpretation introduced in the theorem is structurally important: $T_{\mathrm{raw}}=N_\epsilon \cdot T_{\mathrm{phase}},$ with $N_\epsilon=I+\mathcal{S}_{\mathrm{nr}},$ defines a family of raw transport operators projecting to the same observable phase sector, analogous to gauge-equivalent representations sharing the same observable curvature structure. Status: solid for the first-order observable invariance $L(T_{\mathrm{raw}})=L(T_{\mathrm{phase}})+O(\epsilon^2)$ under the T26-T29 null-class structure; conditional on the assumption that the full dressing sector remains entirely within $\operatorname{span}\{I,\sigma_x,\sigma_z\},$ and for higher-order corrections beyond first order in $\epsilon$ .T82 establishes observable invariance under non-rotational transport dressing within the reduced Q5 phase-transport sector. Starting from the raw barrier transport operator\[T_{\mathrm{raw}}=I+\epsilon A+\mathcal{S}_{\mathrm{nr}}+O(\epsilon^2),\qquad\epsilon=\tfrac{1}{160},\]with\[A=i\sigma_y,\qquadA^2=-I,\]The theorem proves that the T26-T29 phase-extraction functional satisfies\[L(T_{\mathrm{raw}})=\epsilon+O(\epsilon^2)=L(T_{\mathrm{phase}})+O(\epsilon^2),\]where\[T_{\mathrm{phase}}=I+\epsilon A+O(\epsilon^2).\] The non-rotational dressing\[\mathcal{S}_{\mathrm{nr}}\in\operatorname{span}\{I,\sigma_x,\sigma_z\}\]is annihilated by the extraction functional:\[L(\mathcal{S}_{\mathrm{nr}})=0.\] Thus, the observable phase coefficient is determined entirely by the rotational generator $A$ and is invariant under scalar, grading, or symmetric transverse dressing at first order in $\epsilon$. The theorem is structurally important because it completes the T80-T82 transport arc:\[T_{\mathrm{raw}}\xrightarrow{\mathcal{P}_{\mathrm{phase}}}T_{\mathrm{phase}}\xrightarrow{\times k,\ \epsilon\to0}e^{tA}\xrightarrow{L}\sin t.\] T80 established the discrete-to-continuous phase-flow limit\[T_{\mathrm{phase}}^k\toe^{tA}=I\cos t + A\sin t.\] T81 established that the canonical T26-T29 projection kills the non-rotational dressing:\[\mathcal{P}_{\mathrm{phase}}(T_{\mathrm{raw}})=T_{\mathrm{phase}}.\] T82 now proves the observable consequence: the measurable phase readout is invariant under the dressing sector and depends only on the rotational transport component. The theorem therefore replaces any remaining implicit “raw purity” interpretation with observable-sector equivalence under canonical extraction. The non-rotational dressing changes the representation of the raw transport operator but not the observable phase coefficient. The gauge-redundancy interpretation introduced in the theorem is structurally important:\[T_{\mathrm{raw}}=N_\epsilon \cdot T_{\mathrm{phase}},\]with\[N_\epsilon=I+\mathcal{S}_{\mathrm{nr}},\]defines a family of raw transport operators projecting to the same observable phase sector, analogous to gauge-equivalent representations sharing the same observable curvature structure. Status: solid for the first-order observable invariance\[L(T_{\mathrm{raw}})=L(T_{\mathrm{phase}})+O(\epsilon^2)\]under the T26-T29 null-class structure; conditional on the assumption that the full dressing sector remains entirely within\[\operatorname{span}\{I,\sigma_x,\sigma_z\},\]and for higher-order corrections beyond first order in $\epsilon$.

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Discrete-to-Continuous Phase Transport Theorem

Craig Edwin Holdway

T80 establishes the discrete-to-continuous phase transport bridge for the reduced defect-sector transport operator. Starting from the near-identity transport form\[T_{\mathrm{phase}}=I+\epsilon A+O(\epsilon^2),\qquadA^2=-I,\]the theorem proves that iterated discrete transport converges to continuous exponential phase evolution:\[T_{\mathrm{phase}}^k\longrightarrowe^{tA}=I\cos t + A\sin t,\qquadt=k\epsilon.\] The theorem is structurally important because it provides the canonical analytic mechanism connecting discrete barrier iteration to continuous sinusoidal phase flow within the Q5 reduced transport sector. T80 strengthens the T64-T65 emergent-generator arc by showing that repeated discrete transport naturally generates a continuous unitary phase evolution governed by the same rotational generator responsible for the coarse $\mathbb{Z}_8$ barrier cycle. The decomposition $e^{tA}=I\cos t + A\sin t$ separates diagonal retention structure from rotational pass-through structure and links directly to the extraction and admissibility framework developed in T26-T29. Status: solid for the convergence\[T_{\mathrm{phase}}^k\toe^{tA}\]under the stated assumptions and for the closed-form identity $e^{tA}=I\cos t + A\sin t;$ conditional for the identification of the full raw barrier transport operator with the isolated phase transport sector prior to the canonical projection result deferred to T81.T80 establishes the discrete-to-continuous phase transport bridge for the reduced defect-sector transport operator. Starting from the near-identity transport form\[T_{\mathrm{phase}}=I+\epsilon A+O(\epsilon^2),\qquadA^2=-I,\]the theorem proves that iterated discrete transport converges to continuous exponential phase evolution:\[T_{\mathrm{phase}}^k\longrightarrowe^{tA}=I\cos t + A\sin t,\qquadt=k\epsilon.\] The theorem is structurally important because it provides the canonical analytic mechanism connecting discrete barrier iteration to continuous sinusoidal phase flow within the Q5 reduced transport sector. T80 strengthens the T64-T65 emergent-generator arc by showing that repeated discrete transport naturally generates a continuous unitary phase evolution governed by the same rotational generator responsible for the coarse $\mathbb{Z}_8$ barrier cycle. The decomposition\[e^{tA}=I\cos t + A\sin t\]separates diagonal retention structure from rotational pass-through structure and links directly to the extraction and admissibility framework developed in T26-T29. Status: solid for the convergence\[T_{\mathrm{phase}}^k\toe^{tA}\]under the stated assumptions and for the closed-form identity\[e^{tA}=I\cos t + A\sin t;\]conditional for the identification of the full raw barrier transport operator with the isolated phase transport sector prior to the canonical projection result deferred to T81.

May 25, 20260 citations

Canonical Projection of Raw Barrier Transport onto Reduced Phase Transport

Craig Edwin Holdway

T81 establishes the canonical projection of the raw Q5 barrier transport operator onto the reduced phase transport sector. The theorem begins from the first-order raw transport expansion $T_{\mathrm{raw}}=I+\tfrac{1}{160}A+\mathcal{S}_{\mathrm{nr}}+O(\epsilon^2),$ where\[A=i\sigma_y,\qquadA^2=-I,\]and $\mathcal{S}_{\mathrm{nr}}$ is symmetric/scalar non-rotational dressing arising from the paired transport structure identified in T48. T81 proves that raw purity is false: $\mathcal{S}_{\mathrm{nr}} \neq 0$ in general. The correct structural statement is instead that the canonical phase-readout projection induced by the T26-T29 extraction pipeline kills the non-rotational dressing: $\mathcal{P}_{\mathrm{phase}}(T_{\mathrm{raw}})=I+\tfrac{1}{160}A+O(\epsilon^2)=T_{\mathrm{phase}}.$ Equivalently, $T_{\mathrm{raw}}=N_\epsilon \cdot T_{\mathrm{phase}}+O(\epsilon^2),$ where $N_\epsilon=I+\mathcal{S}_{\mathrm{nr}}$ is projection-null symmetric/scalar dressing invisible to the phase-readout sector. The theorem is structurally important because it closes the open gap identified in T80. The continuous phase-flow result\[T_{\mathrm{phase}}^k\toe^{tA}\]no longer depends on manually removing unwanted operator dressing by hand. The T26-T29 extraction architecture canonically projects onto the rotational transport sector from first principles. T81 also marks a conceptual shift in the framework: observable structure is not identified with the full raw operator but with the projection-defined observable sector of a richer transport architecture. The theorem therefore replaces earlier implicit “raw purity” assumptions with observable-sector equivalence under canonical projection. The non-rotational dressing lies in the null-class operator sector $\operatorname{span}\{I,\sigma_z,\sigma_x\},$ while the rotational component $\tfrac{1}{160}A$ survives the projection uniquely. Status: solid for the projection-null structure of the non-rotational dressing under the T26-T29 extraction pipeline and for the canonical reduction $\mathcal{P}_{\mathrm{phase}}(T_{\mathrm{raw}})=T_{\mathrm{phase}};$ conditional for the near-identity crossing model inherited from T48-T49 and for extension beyond the reduced defect-sector transport plane.T81 establishes the canonical projection of the raw Q5 barrier transport operator onto the reduced phase transport sector. The theorem begins from the first-order raw transport expansion\[T_{\mathrm{raw}}=I+\tfrac{1}{160}A+\mathcal{S}_{\mathrm{nr}}+O(\epsilon^2),\]where\[A=i\sigma_y,\qquadA^2=-I,\]and\[\mathcal{S}_{\mathrm{nr}}\]is symmetric/scalar non-rotational dressing arising from the paired transport structure identified in T48. T81 proves that raw purity is false:\[\mathcal{S}_{\mathrm{nr}} \neq 0\]in general. The correct structural statement is instead that the canonical phase-readout projection induced by the T26-T29 extraction pipeline kills the non-rotational dressing:\[\mathcal{P}_{\mathrm{phase}}(T_{\mathrm{raw}})=I+\tfrac{1}{160}A+O(\epsilon^2)=T_{\mathrm{phase}}.\] Equivalently,\[T_{\mathrm{raw}}=N_\epsilon \cdot T_{\mathrm{phase}}+O(\epsilon^2),\]where\[N_\epsilon=I+\mathcal{S}_{\mathrm{nr}}\]is projection-null symmetric/scalar dressing invisible to the phase-readout sector. The theorem is structurally important because it closes the open gap identified in T80. The continuous phase-flow result\[T_{\mathrm{phase}}^k\toe^{tA}\]no longer depends on manually removing unwanted operator dressing by hand. The T26-T29 extraction architecture canonically projects onto the rotational transport sector from first principles. T81 also marks a conceptual shift in the framework: observable structure is not identified with the full raw operator but with the projection-defined observable sector of a richer transport architecture. The theorem therefore replaces earlier implicit “raw purity” assumptions with observable-sector equivalence under canonical projection. The non-rotational dressing lies in the null-class operator sector\[\operatorname{span}\{I,\sigma_z,\sigma_x\},\]while the rotational component\[\tfrac{1}{160}A\]survives the projection uniquely. Status: solid for the projection-null structure of the non-rotational dressing under the T26-T29 extraction pipeline and for the canonical reduction\[\mathcal{P}_{\mathrm{phase}}(T_{\mathrm{raw}})=T_{\mathrm{phase}};\]conditional for the near-identity crossing model inherited from T48-T49 and for extension beyond the reduced defect-sector transport plane.

May 25, 20260 citations

Reduced Q5 Interference Law

Craig Edwin Holdway

T91 establishes the reduced Q5 two-channel interference law after the $5\to3\to2$ reduction. The surviving reduced state space is $\mathbb{C}^2$ , with crossing channels $P$ and $Q$ , and transport generated by $A=i\sigma_y$ , satisfying $A^2=-I$ . Continuous reduced evolution is given by $\psi(t)=e^{tA}\psi(0),$ with $e^{tA}=\begin{pmatrix}\cos t & \sin t\\-\sin t & \cos t\end{pmatrix}.$ For the initial relative-phase state $\psi(0)=\frac1{\sqrt2}\begin{pmatrix}1\\e^{i\phi}\end{pmatrix},$ The evolved channel probabilities are $\Pr(P)=\frac12\left(1+\sin(2t)\cos\phi\right),$ and $\Pr(Q)=\frac12\left(1-\sin(2t)\cos\phi\right).$ The observable channel imbalance is therefore $\Pr(P)-\Pr(Q)=\sin(2t)\cos\phi.$ The theorem shows that reduced Q5 transport naturally produces a two-channel interference structure on the reduced crossing plane. The transport amplitude is controlled by accumulated phase $t=k\varepsilon$ , while the coherence factor is controlled by the relative phase $\phi$ . Phase averaging removes the interference contribution: $\langle\cos\phi\rangle=0\quad\Longrightarrow\quad\langle\Pr(P)\rangle=\langle\Pr(Q)\rangle=\frac12.$ Loss of coherent phase relation, therefore restores equal channel probabilities. T91 converts the earlier transport-generator chain into an explicit observable probability law. Earlier theorems established\[A^2=-I,\qquadT^k\to e^{tA},\qquad\mu_k=\sin(k\varepsilon),\]while T91 derives the resulting two-channel interference structure generated by the same reduced transport algebra. The discrete form follows from $t=k\varepsilon,\qquad\varepsilon=\frac1{320},$ giving $\Pr(P)=\frac12\left(1+\sin\left(\frac{2k}{320}\right)\cos\phi\right).$ The fringe contrast oscillates with transport step number $k$ , with the first maximal contrast near $k\approx80\pi\approx251,$ matching the near-integer phase-maximization structure identified previously in T84. Classification:Solid: matrix exponential, probability law, normalization, and phase averaging.Conditional: identification of the reduced two-channel Q5 state space and the generator chain $A=i\sigma_y$ from T37 and T80-T89.Not claimed: direct experimental observability.T91 establishes the reduced Q5 two-channel interference law after the $5\to3\to2$ reduction. The surviving reduced state space is $\mathbb{C}^2$, with crossing channels $P$ and $Q$, and transport generated by $A=i\sigma_y$, satisfying $A^2=-I$. Continuous reduced evolution is given by\[\psi(t)=e^{tA}\psi(0),\]with\[e^{tA}=\begin{pmatrix}\cos t & \sin t\\-\sin t & \cos t\end{pmatrix}.\] For the initial relative-phase state\[\psi(0)=\frac1{\sqrt2}\begin{pmatrix}1\\e^{i\phi}\end{pmatrix},\]The evolved channel probabilities are\[\Pr(P)=\frac12\left(1+\sin(2t)\cos\phi\right),\]and\[\Pr(Q)=\frac12\left(1-\sin(2t)\cos\phi\right).\]The observable channel imbalance is therefore\[\Pr(P)-\Pr(Q)=\sin(2t)\cos\phi.\] The theorem shows that reduced Q5 transport naturally produces a two-channel interference structure on the reduced crossing plane. The transport amplitude is controlled by accumulated phase $t=k\varepsilon$, while the coherence factor is controlled by the relative phase $\phi$. Phase averaging removes the interference contribution:\[\langle\cos\phi\rangle=0\quad\Longrightarrow\quad\langle\Pr(P)\rangle=\langle\Pr(Q)\rangle=\frac12.\]Loss of coherent phase relation, therefore restores equal channel probabilities. T91 converts the earlier transport-generator chain into an explicit observable probability law. Earlier theorems established\[A^2=-I,\qquadT^k\to e^{tA},\qquad\mu_k=\sin(k\varepsilon),\]while T91 derives the resulting two-channel interference structure generated by the same reduced transport algebra. The discrete form follows from\[t=k\varepsilon,\qquad\varepsilon=\frac1{320},\]giving\[\Pr(P)=\frac12\left(1+\sin\left(\frac{2k}{320}\right)\cos\phi\right).\]The fringe contrast oscillates with transport step number $k$, with the first maximal contrast near\[k\approx80\pi\approx251,\]matching the near-integer phase-maximization structure identified previously in T84. Classification:Solid: matrix exponential, probability law, normalization, and phase averaging.Conditional: identification of the reduced two-channel Q5 state space and the generator chain $A=i\sigma_y$ from T37 and T80-T89.Not claimed: direct experimental observability.

May 25, 20260 citations

Coarse Spinor Cycling and Fine Phase Accumulation

Craig Edwin Holdway

T85 establishes the dual-scale generator structure underlying coarse spinor cycling and fine phase accumulation in the Q5 transport framework. Starting from the rotational generator $A^2=-I,$ The theorem defines the coarse barrier-cycle operator $B=e^{(\pi/4)A}.$ Using the exponential identity $e^{\theta A}=I\cos\theta + A\sin\theta,$ The theorem proves\[B^4=-I,\qquadB^8=I.\] Thus, the coarse barrier operator exhibits a $\mathbb Z_8$ spinor cycle with sign reversal at the half-cycle. Separately, the reduced near-identity phase transport operator $T_{\mathrm{phase}}=I+\frac1{160}A+O\left(\frac1{160^2}\right)$ satisfies, by T80,\[T_{\mathrm{phase}}^k\toe^{(k/160)A}.\] The theorem, therefore establishes that the same generator $A$ supports two compatible transport structures: $B^n=e^{(n\pi/4)A}$ and\[T_{\mathrm{phase}}^k\toe^{(k/160)A}.\] The first defines coarse $\mathbb Z_8$ spinor cycling at step size $\pi/4,$ while the second defines fine accumulated phase transport at the step size $1/160.$ T85 is structurally important because it resolves the apparent tension between coarse barrier cycling and fine transport accumulation. The theorem shows that these are not competing clocks and do not contradict one another. They are structurally distinct operators generated by the same rotational algebra. The coarse barrier cycle governs finite crossing symmetry: $B^8=I.$ The fine transport operator governs the gradual accumulated phase evolution: $\mu_k=\sin(k/160)+O(1/160).$ The theorem therefore unifies the coarse spinor structure from T47-T49/T68 with the continuous transport structure developed in T80-T84. A key consequence is that coarse $\mathbb Z_8$ cycling does not reset fine accumulated transport. Eight fine transport steps accumulate only $8/160=1/20,$ far from a full $2\pi$ phase cycle, so fine accumulation persists independently of the coarse barrier period. The theorem also clarifies that both structures arise from the same Lie-generator algebra: $A^2=-I.$ The coexistence of coarse cycling and fine transport is therefore a structural consequence of the common generator rather than a coincidence of notation. Status: solid for\[B^4=-I,\qquadB^8=I,\]and for the dual exponential structures generated by $A$ ; solid for the compatibility of coarse $\mathbb Z_8$ cycling and fine $1/160$ phase accumulation given the T80 continuous-limit theorem; conditional for the physical identification of $B$ and $T_{\mathrm{phase}}$ with the Q5 barrier and reduced transport operators through the T47-T49 and T80-T82 chains.T85 establishes the dual-scale generator structure underlying coarse spinor cycling and fine phase accumulation in the Q5 transport framework. Starting from the rotational generator\[A^2=-I,\]The theorem defines the coarse barrier-cycle operator\[B=e^{(\pi/4)A}.\] Using the exponential identity\[e^{\theta A}=I\cos\theta + A\sin\theta,\]The theorem proves\[B^4=-I,\qquadB^8=I.\] Thus, the coarse barrier operator exhibits a $\mathbb Z_8$ spinor cycle with sign reversal at the half-cycle. Separately, the reduced near-identity phase transport operator\[T_{\mathrm{phase}}=I+\frac1{160}A+O\left(\frac1{160^2}\right)\]satisfies, by T80,\[T_{\mathrm{phase}}^k\toe^{(k/160)A}.\] The theorem, therefore establishes that the same generator $A$ supports two compatible transport structures:\[B^n=e^{(n\pi/4)A}\]and\[T_{\mathrm{phase}}^k\toe^{(k/160)A}.\] The first defines coarse $\mathbb Z_8$ spinor cycling at step size\[\pi/4,\]while the second defines fine accumulated phase transport at the step size\[1/160.\] T85 is structurally important because it resolves the apparent tension between coarse barrier cycling and fine transport accumulation. The theorem shows that these are not competing clocks and do not contradict one another. They are structurally distinct operators generated by the same rotational algebra. The coarse barrier cycle governs finite crossing symmetry:\[B^8=I.\] The fine transport operator governs the gradual accumulated phase evolution:\[\mu_k=\sin(k/160)+O(1/160).\] The theorem therefore unifies the coarse spinor structure from T47-T49/T68 with the continuous transport structure developed in T80-T84. A key consequence is that coarse $\mathbb Z_8$ cycling does not reset fine accumulated transport. Eight fine transport steps accumulate only\[8/160=1/20,\]far from a full $2\pi$ phase cycle, so fine accumulation persists independently of the coarse barrier period. The theorem also clarifies that both structures arise from the same Lie-generator algebra:\[A^2=-I.\]The coexistence of coarse cycling and fine transport is therefore a structural consequence of the common generator rather than a coincidence of notation. Status: solid for\[B^4=-I,\qquadB^8=I,\]and for the dual exponential structures generated by $A$; solid for the compatibility of coarse $\mathbb Z_8$ cycling and fine $1/160$ phase accumulation given the T80 continuous-limit theorem; conditional for the physical identification of $B$ and $T_{\mathrm{phase}}$ with the Q5 barrier and reduced transport operators through the T47-T49 and T80-T82 chains.

May 25, 20260 citations

\mu_k=L(T_{\mathrm{phase}}^k),

where

L

is the normalized phase-readout functional satisfying\[L(I)=0,\qquadL(A)=1.\] Using the T80 continuous-limit result\[T_{\mathrm{phase}}^k\toe^{k\epsilon A},\]together with

e^{k\epsilon A}=I\cos(k\epsilon)+A\sin(k\epsilon),

The theorem proves

\mu_k=\sin(k\epsilon)+O(\epsilon).

Substituting

\epsilon=\frac1{160},

gives the discrete sampled phase law

\mu_k=\sin\left(\frac{k}{160}\right)+O\left(\frac1{160}\right).

T84 is structurally important because it converts the continuous phase-flow law

L(e^{tA})=\sin t

from T83 into an indexed observable transport sequence

k\mapsto \mu_k.

The theorem, therefore, establishes the sampled observable structure inherited from the continuous Q5 transport flow. Several structural consequences follow immediately. Pass-through occurs exactly when

\sin(k\epsilon)\neq0.

Maximal pass-through occurs near

k\epsilon=\frac{\pi}{2}+n\pi,

Although exact maxima are generally inaccessible because the discrete sampling lattice does not usually intersect those irrational phase locations exactly. For small accumulated transport,

k\ll160,

The small-angle approximation gives

\mu_k\approx\frac{k}{160},

so early-stage phase readout grows linearly with transport depth. Together with T80-T83, the transport/readout chain now becomes\[T_{\mathrm{raw}}\toT_{\mathrm{phase}}\toT_{\mathrm{phase}}^k\toe^{k\epsilon A}\to\mu_k=\sin(k\epsilon).\] Status: solid for the sampled phase law

\mu_k=\sin(k\epsilon)+O(\epsilon)

given the T80 continuous limit and bounded accumulated phase

t=k\epsilon

; solid for the small-step linear regime

\mu_k\approx k/160;

conditional for higher-order corrections inherited from the unresolved higher-order structure of

T_{\mathrm{phase}}.

T82 establishes observable invariance under non-rotational transport dressing within the reduced Q5 phase-transport sector. Starting from the raw barrier transport operator

T_{\mathrm{raw}}=I+\epsilon A+\mathcal{S}_{\mathrm{nr}}+O(\epsilon^2),\qquad\epsilon=\tfrac{1}{160},

with\[A=i\sigma_y,\qquadA^2=-I,\]The theorem proves that the T26-T29 phase-extraction functional satisfies

L(T_{\mathrm{raw}})=\epsilon+O(\epsilon^2)=L(T_{\mathrm{phase}})+O(\epsilon^2),

where

T_{\mathrm{phase}}=I+\epsilon A+O(\epsilon^2).

The non-rotational dressing

\mathcal{S}_{\mathrm{nr}}\in\operatorname{span}\{I,\sigma_x,\sigma_z\}

is annihilated by the extraction functional:

L(\mathcal{S}_{\mathrm{nr}})=0.

Thus, the observable phase coefficient is determined entirely by the rotational generator

A

and is invariant under scalar, grading, or symmetric transverse dressing at first order in

\epsilon

. The theorem is structurally important because it completes the T80-T82 transport arc:

T_{\mathrm{raw}}\xrightarrow{\mathcal{P}_{\mathrm{phase}}}T_{\mathrm{phase}}\xrightarrow{\times k,\ \epsilon\to0}e^{tA}\xrightarrow{L}\sin t.

T80 established the discrete-to-continuous phase-flow limit\[T_{\mathrm{phase}}^k\toe^{tA}=I\cos t + A\sin t.\] T81 established that the canonical T26-T29 projection kills the non-rotational dressing:

\mathcal{P}_{\mathrm{phase}}(T_{\mathrm{raw}})=T_{\mathrm{phase}}.

T82 now proves the observable consequence: the measurable phase readout is invariant under the dressing sector and depends only on the rotational transport component. The theorem therefore replaces any remaining implicit “raw purity” interpretation with observable-sector equivalence under canonical extraction. The non-rotational dressing changes the representation of the raw transport operator but not the observable phase coefficient. The gauge-redundancy interpretation introduced in the theorem is structurally important:

T_{\mathrm{raw}}=N_\epsilon \cdot T_{\mathrm{phase}},

with

N_\epsilon=I+\mathcal{S}_{\mathrm{nr}},

defines a family of raw transport operators projecting to the same observable phase sector, analogous to gauge-equivalent representations sharing the same observable curvature structure. Status: solid for the first-order observable invariance

L(T_{\mathrm{raw}})=L(T_{\mathrm{phase}})+O(\epsilon^2)

under the T26-T29 null-class structure; conditional on the assumption that the full dressing sector remains entirely within

\operatorname{span}\{I,\sigma_x,\sigma_z\},

and for higher-order corrections beyond first order in

\epsilon

\mathbb{Z}_8

barrier cycle. The decomposition

e^{tA}=I\cos t + A\sin t

separates diagonal retention structure from rotational pass-through structure and links directly to the extraction and admissibility framework developed in T26-T29. Status: solid for the convergence\[T_{\mathrm{phase}}^k\toe^{tA}\]under the stated assumptions and for the closed-form identity

e^{tA}=I\cos t + A\sin t;

conditional for the identification of the full raw barrier transport operator with the isolated phase transport sector prior to the canonical projection result deferred to T81.

T81 establishes the canonical projection of the raw Q5 barrier transport operator onto the reduced phase transport sector. The theorem begins from the first-order raw transport expansion

T_{\mathrm{raw}}=I+\tfrac{1}{160}A+\mathcal{S}_{\mathrm{nr}}+O(\epsilon^2),

where\[A=i\sigma_y,\qquadA^2=-I,\]and

\mathcal{S}_{\mathrm{nr}}

is symmetric/scalar non-rotational dressing arising from the paired transport structure identified in T48. T81 proves that raw purity is false:

\mathcal{S}_{\mathrm{nr}} \neq 0

in general. The correct structural statement is instead that the canonical phase-readout projection induced by the T26-T29 extraction pipeline kills the non-rotational dressing:

\mathcal{P}_{\mathrm{phase}}(T_{\mathrm{raw}})=I+\tfrac{1}{160}A+O(\epsilon^2)=T_{\mathrm{phase}}.

Equivalently,

T_{\mathrm{raw}}=N_\epsilon \cdot T_{\mathrm{phase}}+O(\epsilon^2),

where

N_\epsilon=I+\mathcal{S}_{\mathrm{nr}}

is projection-null symmetric/scalar dressing invisible to the phase-readout sector. The theorem is structurally important because it closes the open gap identified in T80. The continuous phase-flow result\[T_{\mathrm{phase}}^k\toe^{tA}\]no longer depends on manually removing unwanted operator dressing by hand. The T26-T29 extraction architecture canonically projects onto the rotational transport sector from first principles. T81 also marks a conceptual shift in the framework: observable structure is not identified with the full raw operator but with the projection-defined observable sector of a richer transport architecture. The theorem therefore replaces earlier implicit “raw purity” assumptions with observable-sector equivalence under canonical projection. The non-rotational dressing lies in the null-class operator sector

\operatorname{span}\{I,\sigma_z,\sigma_x\},

while the rotational component

\tfrac{1}{160}A

survives the projection uniquely. Status: solid for the projection-null structure of the non-rotational dressing under the T26-T29 extraction pipeline and for the canonical reduction

\mathcal{P}_{\mathrm{phase}}(T_{\mathrm{raw}})=T_{\mathrm{phase}};

conditional for the near-identity crossing model inherited from T48-T49 and for extension beyond the reduced defect-sector transport plane.

T91 establishes the reduced Q5 two-channel interference law after the

5\to3\to2

reduction. The surviving reduced state space is

\mathbb{C}^2

, with crossing channels

P

and

Q

, and transport generated by

A=i\sigma_y

, satisfying

A^2=-I

. Continuous reduced evolution is given by

\psi(t)=e^{tA}\psi(0),

with

e^{tA}=\begin{pmatrix}\cos t & \sin t\\-\sin t & \cos t\end{pmatrix}.

For the initial relative-phase state

\psi(0)=\frac1{\sqrt2}\begin{pmatrix}1\\e^{i\phi}\end{pmatrix},

The evolved channel probabilities are

\Pr(P)=\frac12\left(1+\sin(2t)\cos\phi\right),

and

\Pr(Q)=\frac12\left(1-\sin(2t)\cos\phi\right).

The observable channel imbalance is therefore

\Pr(P)-\Pr(Q)=\sin(2t)\cos\phi.

The theorem shows that reduced Q5 transport naturally produces a two-channel interference structure on the reduced crossing plane. The transport amplitude is controlled by accumulated phase

t=k\varepsilon

, while the coherence factor is controlled by the relative phase

\phi

. Phase averaging removes the interference contribution:

\langle\cos\phi\rangle=0\quad\Longrightarrow\quad\langle\Pr(P)\rangle=\langle\Pr(Q)\rangle=\frac12.

Loss of coherent phase relation, therefore restores equal channel probabilities. T91 converts the earlier transport-generator chain into an explicit observable probability law. Earlier theorems established\[A^2=-I,\qquadT^k\to e^{tA},\qquad\mu_k=\sin(k\varepsilon),\]while T91 derives the resulting two-channel interference structure generated by the same reduced transport algebra. The discrete form follows from

t=k\varepsilon,\qquad\varepsilon=\frac1{320},

giving

\Pr(P)=\frac12\left(1+\sin\left(\frac{2k}{320}\right)\cos\phi\right).

The fringe contrast oscillates with transport step number

k

, with the first maximal contrast near

k\approx80\pi\approx251,

matching the near-integer phase-maximization structure identified previously in T84. Classification:Solid: matrix exponential, probability law, normalization, and phase averaging.Conditional: identification of the reduced two-channel Q5 state space and the generator chain

A=i\sigma_y

from T37 and T80-T89.Not claimed: direct experimental observability.

T85 establishes the dual-scale generator structure underlying coarse spinor cycling and fine phase accumulation in the Q5 transport framework. Starting from the rotational generator

A^2=-I,

The theorem defines the coarse barrier-cycle operator

B=e^{(\pi/4)A}.

Using the exponential identity

e^{\theta A}=I\cos\theta + A\sin\theta,

The theorem proves\[B^4=-I,\qquadB^8=I.\] Thus, the coarse barrier operator exhibits a

\mathbb Z_8

spinor cycle with sign reversal at the half-cycle. Separately, the reduced near-identity phase transport operator

T_{\mathrm{phase}}=I+\frac1{160}A+O\left(\frac1{160^2}\right)

satisfies, by T80,\[T_{\mathrm{phase}}^k\toe^{(k/160)A}.\] The theorem, therefore establishes that the same generator

A

supports two compatible transport structures:

B^n=e^{(n\pi/4)A}

and\[T_{\mathrm{phase}}^k\toe^{(k/160)A}.\] The first defines coarse

\mathbb Z_8

spinor cycling at step size

\pi/4,

while the second defines fine accumulated phase transport at the step size

1/160.

T85 is structurally important because it resolves the apparent tension between coarse barrier cycling and fine transport accumulation. The theorem shows that these are not competing clocks and do not contradict one another. They are structurally distinct operators generated by the same rotational algebra. The coarse barrier cycle governs finite crossing symmetry:

B^8=I.

The fine transport operator governs the gradual accumulated phase evolution:

\mu_k=\sin(k/160)+O(1/160).

The theorem therefore unifies the coarse spinor structure from T47-T49/T68 with the continuous transport structure developed in T80-T84. A key consequence is that coarse

\mathbb Z_8

cycling does not reset fine accumulated transport. Eight fine transport steps accumulate only

8/160=1/20,

far from a full

2\pi

phase cycle, so fine accumulation persists independently of the coarse barrier period. The theorem also clarifies that both structures arise from the same Lie-generator algebra:

A^2=-I.

The coexistence of coarse cycling and fine transport is therefore a structural consequence of the common generator rather than a coincidence of notation. Status: solid for\[B^4=-I,\qquadB^8=I,\]and for the dual exponential structures generated by

A

; solid for the compatibility of coarse

\mathbb Z_8

cycling and fine

1/160

phase accumulation given the T80 continuous-limit theorem; conditional for the physical identification of

B

and

T_{\mathrm{phase}}

with the Q5 barrier and reduced transport operators through the T47-T49 and T80-T82 chains.