Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Coherence–Alignment Structure of Reduced Q5 Transport

Craig Edwin Holdway

Published May 3, 2026• 0 citations
Publisher

Abstract

This collects the T26–T49 arc into a unified structural account of coherence in the reduced phase system. No new operators are introduced. Seven results are established. The observable phase is carried entirely on P⊥ = span{u₂, u₄} with generator A = iσ_y = E_d, A² = −I (T26, T37). Any reduced observable state admits Q⊥ = αI + βσ_z + γσ_x + μA with μ = ½Tr(Q⊥A) (T27); scalar, grading, and off-axis components are annihilated by the extraction functional. Coherence magnitude C := |μ| is nonzero if and only if the transported residue has a nonzero projection onto A. By T29, μ = L̃(Π_sub Π_pair Π_ret([M_n,D_n]Y)), coherence is entirely commutator-driven and transport-weighted. Decoherence occurs precisely when the commutator-driven transverse component fails to survive transport, through four mechanisms: detuning imbalance (T11), transport asymmetry (T31–T33), support leakage outside O_q(s) (T35), or boundary-induced inversion (T47–T49). From T48–T49, each boundary crossing produces a fixed rotational increment (T − I)_rot = (1/160)A plus scalar and symmetric components annihilated by the extraction functional. Three corollaries establish: density matrix correspondence |γ| ∝ |μ|; joint-state coherence condition tied to shared commutator survival; and the equivalence C ≠ 0 ≡ survival of the commutator-driven transverse mode under transport. No Lindblad equation or external decay rate is introduced. The density matrix is a representation of Q⊥, not a physical state postulate. Status: Phase-plane structure, four-parameter Hermitian representation, extraction formula, null-class annihilation, coherence identification, and commutator-driven reduction are all solid within the T26–T38 framework. Failure modes structural, each referenced theorem established the mechanism. (T − I)_rot = (1/160)A conditional on T48 near-identity model and T49 barrier symmetry action. All conditionality entirely inherited from T17, T29, T36, T38, T48, T49: no new assumptions introduced. Dependencies: T11, T26, T27, T29, T31, T32, T33, T35, T37, T38, T47, T48, T49.

DOI
10.5281/zenodo.20017336

Related papers

Effective Decoherence Rate Structure

Craig Edwin Holdway

This Theorem identifies the structural form of the effective decoherence rate arising from the irreversible coupling established in T50. The rate is not a free parameter but is determined by three structural inputs from the Q5 transport chain: Γ_eff = γ_loss · G² · [1 + β·A²] · f(d_orbit) · L(δ_detune) Four factors: (1) Base rate γ_loss·G², decay requires irreversible sector coupling; Γ_eff = 0 when γ_loss = 0, regardless of all other factors. (2) Asymmetry term [1 + β·A²], coupling asymmetry A ~ |β_J − β| = |μν'c'b − b'c| from T33; operationally A ~ |g₁ − g₀|/√(g₁² + g₀²) from T8; quadratic dependence consistent with simulation showing decay enhancement scales as (g₁ − g₀)². (3) Orbit depth term f(d_orbit) = 1 − e^{−α(d_orbit − 2)}, saturation form reflecting finite Q5 combinatorial depth; f(2) = 0 at the minimal orbit of T36–T37. (4) Lorentzian detuning suppression L(δ_detune) = 1/(1 + (δ_detune/γ_c)²), detuning suppresses coupling efficiency, not a decay channel. Three structural statements: scrambling is necessary but not sufficient for decoherence; asymmetry amplifies scrambling; irreversible coupling converts scrambling into decay. Three corollaries: zero-loss limit (Γ_eff = 0 when γ_loss = 0); resonance maximum at δ_detune = 0; symmetric coupling limit (A = 0 returns base rate). Status: Asymmetry identification A ~ |μν'c'b − b'c| from T33 solid. f(2) = 0 at minimal orbit solid within T36–T37. Three qualitative boxed statements solid from T50 and factor structure. Overall structural form model-level, functional forms of f and L are natural leading-order models, not uniquely forced. Five structural constants (γ_loss, G, α, β, γ_c) were identified qualitatively; explicit values from the T17 kernel geometry are open. All results inherit T8, T17, T33, T35, T36, T37, T50 conditionality. Dependencies: T8, T11, T29, T33, T35, T36, T37, T50.

•May 3, 2026•0 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.

•May 25, 2026•0 citations

Coherence, Decoherence Rate, and Attractor Structure in Reduced Q5 Transport

Craig Edwin Holdway

We present three theorems that complete the structural account of coherence and decoherence in the reduced transport sector of the Q5 framework. This mathematical system derives operator structures from the combinatorial geometry of the five-dimensional hypercube and its Gray code Hamiltonian cycle. Theorem 50 identifies coherence as the survival of the commutator-driven transverse mode under transport, and establishes a precise structural distinction between reversible phase scrambling and irreversible decoherence. No new operators are introduced; all objects are inherited from the prior theorem chain (T1–T49). Theorem 51 derives the structural form of the effective decoherence rate as a product of four independently motivated factors: a base loss rate, a quadratic coupling asymmetry term, a saturating orbit depth function, and a Lorentzian detuning suppression factor. Each factor is grounded in a specific prior result in the Q5 transport chain. Theorem 52 establishes that the coherence decay rate is the unique globally attracting fixed point of a self-consistent kernel-induced dynamical map. Existence and uniqueness are proved via the intermediate value theorem applied to a strictly decreasing map; global convergence follows from even/odd subsequence monotonicity. The exponential saturation law is derived as the asymptotic reduction of the fixed-point structure in the strong-suppression regime, rather than postulated. The primary remaining open step is the explicit derivation of the coupling parameter β from the commutator [M_n, D_n]Y geometry of Theorem 29. All results are conditional on the admissible linked-pair reduction of Theorem 17 and the retained-pairing assumption of Theorem 36. Epistemic classifications are provided for each result.

•Apr 20, 2026•0 citations

Derivation of the Bloch Polar Angle from Q5 Transport Null-Class Structure

Craig Edwin Holdway

This Theorem closes the open assumption θ = π/2 + O(δ_eff) from the T39 bridge lemma by deriving it from Q5 transport structure. Two results are established. At leading order, the null-class structure of the T28 extraction functional L forces the Bloch vector to the equatorial plane. The functional L annihilates all operators whose transverse projection lies in span{I, σ_x, σ_z}, only the R = −σ_y component survives. At leading order (Y = I on the retained sector), the commutator image [M_n, D_n] after diagonal absorption of X carries no symmetric component; the antisymmetric commutator structure suppresses σ_x and σ_z content, leaving the Bloch vector at θ = π/2 exactly. The residual weighted adjacency operator Y is not retained-support invariant (T29, Proposition 29.4). Its action introduces an O(δ_eff) symmetric correction to the transverse projection, giving θ = π/2 + O(δ_eff), where δ_eff parametrizes the off-support leakage of Y on the retained sector. The assumption θ = π/2 + O(δ_eff) in the T39 bridge lemma is therefore not an external imposition but follows from the null-class architecture of the extraction functional and the antisymmetric structure of the transported commutator. This closes the connection between the Berry phase arc (T39–T42) and the commutator-driven extraction pipeline (T26–T29). Status: Part (1) solid within the T29 null-class framework and T17 ordered kernel architecture. Part (2) conditional on the parametric control assumption on δ_eff — Y-induced off-support leakage is assumed small and expressible as a controlled perturbation. The antisymmetry argument in Lemma 2 is structurally justified but not independently derived from explicit kernel generators. All results inherit T29 conditionality (remainder null-class decomposability, diagonal absorption) and T17 conditionality. Dependencies: T17, T29, T39, T42.

•May 3, 2026•0 citations

Crossing Holonomy Theorem

Craig Edwin Holdway

This Theorem establishes the round-trip transport residual (T − I)_rot = (1/160)A from a second independent derivation route (barrier symmetry B = e^{(π/4)A}Θ acting on an ordered pair of noncommuting generators), confirming and strengthening the T48 result. The barrier symmetry acts by exchange-and-inversion: BPB¹ = −Q, BQB⁻¹ = −P, consistent with T47. One-leg transport is modeled by X_{n→d} = (I + λP)(I + λQ); the return leg by X_{d→n} = (I − λQ)(I − λP). A supporting lemma establishes by explicit second-order expansion that T = X_{d→n}X_{n→d} satisfies: T − I = 2λ²A − 2λ²I + O(λ³) where A = ½[P,Q] is the antisymmetric generator and −2λ²I is an isotropic normalization drift that does not affect relative phase or branch observables. Using P² = Q² = I on the reduced two-state space (supported by T36–T37) and [P,Q] = 2A. Under the one-leg normalization λ² = 1/320 inherited from T8 via T48: (T − I)_rot = (1/160)A That both the T48 near-identity route and the T49 barrier-symmetry route converge on the same residue is structurally significant. Status: First-order cancellation, second-order decomposition, and scalar term analysis solid by direct computation. λ² = 1/320 inherited from T8 via T48, not rederived here. B exchange-and-inversion action on P, Q assumed consistent with T47, explicit derivation from T17 kernel geometry open. P² = Q² = I supported by T36–T37, not independently rederived. Qubit algebra correspondence structural, holds within T36–T37 hypotheses, does not assert the full framework is a quantum theory. Dependencies: T7, T8, T17, T36, T37, T47, T48.

•May 3, 2026•0 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 \(\epsilon\).

•May 16, 2026•0 citations
This collects the T26–T49 arc into a unified structural account of coherence in the reduced phase system. No new operators are introduced. Seven results are established. The observable phase is carried entirely on P⊥ = span{u₂, u₄} with generator A = iσ_y = E_d, A² = −I (T26, T37). Any reduced observable state admits Q⊥ = αI + βσ_z + γσ_x + μA with μ = ½Tr(Q⊥A) (T27); scalar, grading, and off-axis components are annihilated by the extraction functional. Coherence magnitude C := |μ| is nonzero if and only if the transported residue has a nonzero projection onto A. By T29, μ = L̃(Π_sub Π_pair Π_ret([M_n,D_n]Y)), coherence is entirely commutator-driven and transport-weighted. Decoherence occurs precisely when the commutator-driven transverse component fails to survive transport, through four mechanisms: detuning imbalance (T11), transport asymmetry (T31–T33), support leakage outside O_q(s) (T35), or boundary-induced inversion (T47–T49). From T48–T49, each boundary crossing produces a fixed rotational increment (T − I)_rot = (1/160)A plus scalar and symmetric components annihilated by the extraction functional. Three corollaries establish: density matrix correspondence |γ| ∝ |μ|; joint-state coherence condition tied to shared commutator survival; and the equivalence C ≠ 0 ≡ survival of the commutator-driven transverse mode under transport. No Lindblad equation or external decay rate is introduced. The density matrix is a representation of Q⊥, not a physical state postulate. Status: Phase-plane structure, four-parameter Hermitian representation, extraction formula, null-class annihilation, coherence identification, and commutator-driven reduction are all solid within the T26–T38 framework. Failure modes structural, each referenced theorem established the mechanism. (T − I)_rot = (1/160)A conditional on T48 near-identity model and T49 barrier symmetry action. All conditionality entirely inherited from T17, T29, T36, T38, T48, T49: no new assumptions introduced. Dependencies: T11, T26, T27, T29, T31, T32, T33, T35, T37, T38, T47, T48, T49.
This Theorem identifies the structural form of the effective decoherence rate arising from the irreversible coupling established in T50. The rate is not a free parameter but is determined by three structural inputs from the Q5 transport chain: Γ_eff = γ_loss · G² · [1 + β·A²] · f(d_orbit) · L(δ_detune) Four factors: (1) Base rate γ_loss·G², decay requires irreversible sector coupling; Γ_eff = 0 when γ_loss = 0, regardless of all other factors. (2) Asymmetry term [1 + β·A²], coupling asymmetry A ~ |β_J − β| = |μν'c'b − b'c| from T33; operationally A ~ |g₁ − g₀|/√(g₁² + g₀²) from T8; quadratic dependence consistent with simulation showing decay enhancement scales as (g₁ − g₀)². (3) Orbit depth term f(d_orbit) = 1 − e^{−α(d_orbit − 2)}, saturation form reflecting finite Q5 combinatorial depth; f(2) = 0 at the minimal orbit of T36–T37. (4) Lorentzian detuning suppression L(δ_detune) = 1/(1 + (δ_detune/γ_c)²), detuning suppresses coupling efficiency, not a decay channel. Three structural statements: scrambling is necessary but not sufficient for decoherence; asymmetry amplifies scrambling; irreversible coupling converts scrambling into decay. Three corollaries: zero-loss limit (Γ_eff = 0 when γ_loss = 0); resonance maximum at δ_detune = 0; symmetric coupling limit (A = 0 returns base rate). Status: Asymmetry identification A ~ |μν'c'b − b'c| from T33 solid. f(2) = 0 at minimal orbit solid within T36–T37. Three qualitative boxed statements solid from T50 and factor structure. Overall structural form model-level, functional forms of f and L are natural leading-order models, not uniquely forced. Five structural constants (γ_loss, G, α, β, γ_c) were identified qualitatively; explicit values from the T17 kernel geometry are open. All results inherit T8, T17, T33, T35, T36, T37, T50 conditionality. Dependencies: T8, T11, T29, T33, T35, T36, T37, T50.
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 expansionTraw=I+1160A+Snr+O(ϵ2),T_{\mathrm{raw}}=I+\tfrac{1}{160}A+\mathcal{S}_{\mathrm{nr}}+O(\epsilon^2),Traw​=I+1601​A+Snr​+O(ϵ2),where\[A=i\sigma_y,\qquadA^2=-I,\]andSnr\mathcal{S}_{\mathrm{nr}}Snr​is symmetric/scalar non-rotational dressing arising from the paired transport structure identified in T48. T81 proves that raw purity is false:Snr≠0\mathcal{S}_{\mathrm{nr}} \neq 0Snr​=0in 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:Pphase(Traw)=I+1160A+O(ϵ2)=Tphase.\mathcal{P}_{\mathrm{phase}}(T_{\mathrm{raw}})=I+\tfrac{1}{160}A+O(\epsilon^2)=T_{\mathrm{phase}}.Pphase​(Traw​)=I+1601​A+O(ϵ2)=Tphase​. Equivalently,Traw=Nϵ⋅Tphase+O(ϵ2),T_{\mathrm{raw}}=N_\epsilon \cdot T_{\mathrm{phase}}+O(\epsilon^2),Traw​=Nϵ​⋅Tphase​+O(ϵ2),whereNϵ=I+SnrN_\epsilon=I+\mathcal{S}_{\mathrm{nr}}Nϵ​=I+Snr​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 sectorspan⁡{I,σz,σx},\operatorname{span}\{I,\sigma_z,\sigma_x\},span{I,σz​,σx​},while the rotational component1160A\tfrac{1}{160}A1601​Asurvives 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 reductionPphase(Traw)=Tphase;\mathcal{P}_{\mathrm{phase}}(T_{\mathrm{raw}})=T_{\mathrm{phase}};Pphase​(Traw​)=Tphase​;conditional for the near-identity crossing model inherited from T48-T49 and for extension beyond the reduced defect-sector transport plane.
We present three theorems that complete the structural account of coherence and decoherence in the reduced transport sector of the Q5 framework. This mathematical system derives operator structures from the combinatorial geometry of the five-dimensional hypercube and its Gray code Hamiltonian cycle. Theorem 50 identifies coherence as the survival of the commutator-driven transverse mode under transport, and establishes a precise structural distinction between reversible phase scrambling and irreversible decoherence. No new operators are introduced; all objects are inherited from the prior theorem chain (T1–T49). Theorem 51 derives the structural form of the effective decoherence rate as a product of four independently motivated factors: a base loss rate, a quadratic coupling asymmetry term, a saturating orbit depth function, and a Lorentzian detuning suppression factor. Each factor is grounded in a specific prior result in the Q5 transport chain. Theorem 52 establishes that the coherence decay rate is the unique globally attracting fixed point of a self-consistent kernel-induced dynamical map. Existence and uniqueness are proved via the intermediate value theorem applied to a strictly decreasing map; global convergence follows from even/odd subsequence monotonicity. The exponential saturation law is derived as the asymptotic reduction of the fixed-point structure in the strong-suppression regime, rather than postulated. The primary remaining open step is the explicit derivation of the coupling parameter β from the commutator [M_n, D_n]Y geometry of Theorem 29. All results are conditional on the admissible linked-pair reduction of Theorem 17 and the retained-pairing assumption of Theorem 36. Epistemic classifications are provided for each result.
This Theorem closes the open assumption θ = π/2 + O(δ_eff) from the T39 bridge lemma by deriving it from Q5 transport structure. Two results are established. At leading order, the null-class structure of the T28 extraction functional L forces the Bloch vector to the equatorial plane. The functional L annihilates all operators whose transverse projection lies in span{I, σ_x, σ_z}, only the R = −σ_y component survives. At leading order (Y = I on the retained sector), the commutator image [M_n, D_n] after diagonal absorption of X carries no symmetric component; the antisymmetric commutator structure suppresses σ_x and σ_z content, leaving the Bloch vector at θ = π/2 exactly. The residual weighted adjacency operator Y is not retained-support invariant (T29, Proposition 29.4). Its action introduces an O(δ_eff) symmetric correction to the transverse projection, giving θ = π/2 + O(δ_eff), where δ_eff parametrizes the off-support leakage of Y on the retained sector. The assumption θ = π/2 + O(δ_eff) in the T39 bridge lemma is therefore not an external imposition but follows from the null-class architecture of the extraction functional and the antisymmetric structure of the transported commutator. This closes the connection between the Berry phase arc (T39–T42) and the commutator-driven extraction pipeline (T26–T29). Status: Part (1) solid within the T29 null-class framework and T17 ordered kernel architecture. Part (2) conditional on the parametric control assumption on δ_eff — Y-induced off-support leakage is assumed small and expressible as a controlled perturbation. The antisymmetry argument in Lemma 2 is structurally justified but not independently derived from explicit kernel generators. All results inherit T29 conditionality (remainder null-class decomposability, diagonal absorption) and T17 conditionality. Dependencies: T17, T29, T39, T42.
This Theorem establishes the round-trip transport residual (T − I)_rot = (1/160)A from a second independent derivation route (barrier symmetry B = e^{(π/4)A}Θ acting on an ordered pair of noncommuting generators), confirming and strengthening the T48 result. The barrier symmetry acts by exchange-and-inversion: BPB¹ = −Q, BQB⁻¹ = −P, consistent with T47. One-leg transport is modeled by X_{n→d} = (I + λP)(I + λQ); the return leg by X_{d→n} = (I − λQ)(I − λP). A supporting lemma establishes by explicit second-order expansion that T = X_{d→n}X_{n→d} satisfies: T − I = 2λ²A − 2λ²I + O(λ³) where A = ½[P,Q] is the antisymmetric generator and −2λ²I is an isotropic normalization drift that does not affect relative phase or branch observables. Using P² = Q² = I on the reduced two-state space (supported by T36–T37) and [P,Q] = 2A. Under the one-leg normalization λ² = 1/320 inherited from T8 via T48: (T − I)_rot = (1/160)A That both the T48 near-identity route and the T49 barrier-symmetry route converge on the same residue is structurally significant. Status: First-order cancellation, second-order decomposition, and scalar term analysis solid by direct computation. λ² = 1/320 inherited from T8 via T48, not rederived here. B exchange-and-inversion action on P, Q assumed consistent with T47, explicit derivation from T17 kernel geometry open. P² = Q² = I supported by T36–T37, not independently rederived. Qubit algebra correspondence structural, holds within T36–T37 hypotheses, does not assert the full framework is a quantum theory. Dependencies: T7, T8, T17, T36, T37, T47, T48.
T82 establishes observable invariance under non-rotational transport dressing within the reduced Q5 phase-transport sector. Starting from the raw barrier transport operatorTraw=I+ϵA+Snr+O(ϵ2),ϵ=1160,T_{\mathrm{raw}}=I+\epsilon A+\mathcal{S}_{\mathrm{nr}}+O(\epsilon^2),\qquad\epsilon=\tfrac{1}{160},Traw​=I+ϵA+Snr​+O(ϵ2),ϵ=1601​,with\[A=i\sigma_y,\qquadA^2=-I,\]The theorem proves that the T26-T29 phase-extraction functional satisfiesL(Traw)=ϵ+O(ϵ2)=L(Tphase)+O(ϵ2),L(T_{\mathrm{raw}})=\epsilon+O(\epsilon^2)=L(T_{\mathrm{phase}})+O(\epsilon^2),L(Traw​)=ϵ+O(ϵ2)=L(Tphase​)+O(ϵ2),whereTphase=I+ϵA+O(ϵ2).T_{\mathrm{phase}}=I+\epsilon A+O(\epsilon^2).Tphase​=I+ϵA+O(ϵ2). The non-rotational dressingSnr∈span⁡{I,σx,σz}\mathcal{S}_{\mathrm{nr}}\in\operatorname{span}\{I,\sigma_x,\sigma_z\}Snr​∈span{I,σx​,σz​}is annihilated by the extraction functional:L(Snr)=0.L(\mathcal{S}_{\mathrm{nr}})=0.L(Snr​)=0. Thus, the observable phase coefficient is determined entirely by the rotational generator AAA and is invariant under scalar, grading, or symmetric transverse dressing at first order in ϵ\epsilonϵ. The theorem is structurally important because it completes the T80-T82 transport arc:Traw→PphaseTphase→×k, ϵ→0etA→Lsin⁡t.T_{\mathrm{raw}}\xrightarrow{\mathcal{P}_{\mathrm{phase}}}T_{\mathrm{phase}}\xrightarrow{\times k,\ \epsilon\to0}e^{tA}\xrightarrow{L}\sin t.Traw​Pphase​​Tphase​×k, ϵ→0​etAL​sint. 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:Pphase(Traw)=Tphase.\mathcal{P}_{\mathrm{phase}}(T_{\mathrm{raw}})=T_{\mathrm{phase}}.Pphase​(Traw​)=Tphase​. 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:Traw=Nϵ⋅Tphase,T_{\mathrm{raw}}=N_\epsilon \cdot T_{\mathrm{phase}},Traw​=Nϵ​⋅Tphase​,withNϵ=I+Snr,N_\epsilon=I+\mathcal{S}_{\mathrm{nr}},Nϵ​=I+Snr​,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 invarianceL(Traw)=L(Tphase)+O(ϵ2)L(T_{\mathrm{raw}})=L(T_{\mathrm{phase}})+O(\epsilon^2)L(Traw​)=L(Tphase​)+O(ϵ2)under the T26-T29 null-class structure; conditional on the assumption that the full dressing sector remains entirely withinspan⁡{I,σx,σz},\operatorname{span}\{I,\sigma_x,\sigma_z\},span{I,σx​,σz​},and for higher-order corrections beyond first order in ϵ\epsilonϵ.