This Theorem establishes θ = π/2 + O(δ_eff) through a complementary algebraic route to T43: Pauli-basis antisymmetry rather than the null-class pipeline. Two lemmas and two propositions establish the result. The commutator [M,D] = MD − DM is antisymmetric under exchange of operator order by definition, it changes sign under M ↔ D and therefore has no symmetric component. The null-class extraction pipeline suppresses symmetric components: since they do not survive the commutator stage, and Lemma 29.1 eliminates residual symmetric contributions, only antisymmetric components remain on the retained support. Within the Pauli basis {I, σ_x, σ_y, σ_z}, the operators σ_x and σ_z are symmetric under transposition while σ_y is antisymmetric. Therefore, R = −σ_y is the unique antisymmetric generator in the transverse operator space, and at leading order Q⊥ = μR with no σ_x or σ_z content. The Bloch vector is forced to the equatorial plane: θ = π/2. The weighted adjacency operator Y introduces O(δ_eff) off-support leakage, giving h_z = O(δ_eff) and θ = π/2 + O(δ_eff). The two independent derivations of θ = π/2, T43 via null-class pipeline, T45 via Pauli-basis antisymmetry, constitute mutual confirmation of the equatorial forcing. Status: Antisymmetric commutator structure exact, follows from the definition of the commutator alone. Pauli-basis identification of σ_y as a unique antisymmetric generator is exact. Null-class suppression of symmetric components is structurally justified within T26–T29 but not independently derived from explicit kernel generators, same conditionality as T43. The correction term inherits T43 parametric control conditionality. Dependencies: T29, T39, T43, T44.
This Theorem establishes θ = π/2 + O(δ_eff) through a complementary algebraic route to T43: Pauli-basis antisymmetry rather than the null-class pipeline. Two lemmas and two propositions establish the result. The commutator [M,D] = MD − DM is antisymmetric under exchange of operator order by definition, it changes sign under M ↔ D and therefore has no symmetric component. The null-class extraction pipeline suppresses symmetric components: since they do not survive the commutator stage, and Lemma 29.1 eliminates residual symmetric contributions, only antisymmetric components remain on the retained support. Within the Pauli basis {I, σ_x, σ_y, σ_z}, the operators σ_x and σ_z are symmetric under transposition while σ_y is antisymmetric. Therefore, R = −σ_y is the unique antisymmetric generator in the transverse operator space, and at leading order Q⊥ = μR with no σ_x or σ_z content. The Bloch vector is forced to the equatorial plane: θ = π/2. The weighted adjacency operator Y introduces O(δ_eff) off-support leakage, giving h_z = O(δ_eff) and θ = π/2 + O(δ_eff). The two independent derivations of θ = π/2, T43 via null-class pipeline, T45 via Pauli-basis antisymmetry, constitute mutual confirmation of the equatorial forcing. Status: Antisymmetric commutator structure exact, follows from the definition of the commutator alone. Pauli-basis identification of σ_y as a unique antisymmetric generator is exact. Null-class suppression of symmetric components is structurally justified within T26–T29 but not independently derived from explicit kernel generators, same conditionality as T43. The correction term inherits T43 parametric control conditionality. Dependencies: T29, T39, T43, T44.
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 closes the second open assumption of the T39 bridge lemma by deriving the leg-matching identity a = √2‖v_L‖ + O(δ²_eff) from Q5 carrier structure. Two independent Q5-native mechanisms force the leading-order identity. The complement involution ι: x ↦ 1 − x maps the Hamming-weight-1 subspace isometrically onto the Hamming-weight-2 subspace, intertwines the T24 ambient su(2) action by sign-reversing Ĵ_z and exchanging Ĵ_±, and satisfies ι(v_L) = v_R with ‖v_R‖ = ‖v_L‖. Within the spin-1 module V₁ carried by the Hamming-weight-1 sector, the raising operator Ĵ₊ acts with matrix element √2 between adjacent weight states, a standard spin-1 Clebsch-Gordan coefficient. Combining: at leading order a = √2‖v_L‖, where the √2 is the canonical spin-1 matrix element, not a normalization convention. The Y-transport correction is quadratic not linear: since θ = π/2 + O(δ_eff) from T43, the amplitude scales as sin θ = cos(O(δ_eff)) = 1 − O(δ²_eff), giving a = √2‖v_L‖ + O(δ²_eff). The leading-order result is stable to first order in the off-support leakage. Together with T43, both open assumptions in the T39 bridge lemma are now derived results. Status: Complement isometry solid: direct computation on the residual carrier basis. Spin-1 matrix element solid: standard result. Leading-order leg-matching conditional on identification of transport leg amplitude with Ĵ₊ matrix element on V₁ is structurally natural within the T24–T25 framework but not yet derived from explicit kernel-level transport generators. Quadratic correction inherits T43 parametric control conditionality. All results inherit T22–T25 and T39 conditionality. Dependencies: T22, T24, T25, T39, T43.
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 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.
This Theorem shows that the geometric structure of T40 arises canonically from the reduced transport operator itself, not from an external construction. Four results are established. The cyclic reduced physical operator Q⊥(λ) = α(λ)I + β(λ)σ_x + γ(λ)σ_z + μ(λ)R with R = −σ_y defines a canonical two-state Bloch Hamiltonian H_geom(λ) = β(λ)σ_x − μ(λ)σ_y + γ(λ)σ_z with Bloch vector h(λ) = (β(λ), −μ(λ), γ(λ)). The rotational trace functional μ(Q) = ½Tr(QR) selects the coefficient of R = −σ_y and isolates the component generating infinitesimal azimuthal rotations of the Bloch vector, μ(Q)dλ is a one-form proportional to the infinitesimal azimuthal displacement. Under adiabatic transport on a closed cycle with h(λ) ≠ 0, the Berry phase is γ_Berry = −½Ω[ĥ]. The rotational trace one-form satisfies: μ(Q)dλ = ½(1 − cosΘ(λ))∂_λΦ(λ)dλ + df(λ) differing from the Berry connection only by an exact differential, so for any closed cycle ∮μ(Q)dλ = γ_Berry. Under the normalization equating the rotational trace functional with the Berry connection: μ_obs = γ_Berry. The observable phase is not merely realizable as a geometric phase; it is intrinsically the Berry phase of the canonical two-state system determined by the same reduced operator family. Status: Canonical Bloch Hamiltonian construction exact. Rotational trace as azimuthal generator exact via Pauli trace identities. Berry phase formula for two-level systems is a standard result. Connection-level identification is solid, rotational trace one-form differs from Berry connection by exact differential, which vanishes on closed cycles. Identification μ_obs = γ_Berry holds under normalization condition equating rotational trace functional with Berry connection — natural but not yet derived from Q5 first principles. Holonomy-level independence from normalization is addressed in T42. Dependencies: T39, T40.
This Theorem derives the canonical extraction formula for the rotational phase coefficient μ from the terminal retained residue Ω_term. Given the transported collapsed object Q⊥ = P⊥ · Ψ · Π_col(Π_A · Ω_term · Π_A) · Ψ⁻¹ · P⊥ on the transverse phase plane P⊥ = span{u₂, u₄}, three equivalent extraction formulas are established: Primary: μ = ½ Tr(Q⊥ · R), verified by direct matrix computation showing Tr(I·R) = Tr(σ_z·R) = Tr(σ_x·R) = 0 and Tr(R·R) = 2. Antisymmetric off-diagonal: μ = (1/2i)(⟨u₂|Q⊥|u₄⟩ − ⟨u₄|Q⊥|u₂⟩), which holds for all γ since the symmetric contribution cancels. Pinned entry: The antisymmetric part of the (u₂, u₄) entry of the transported collapsed object equals iμ. The identity μ = Δφ(τ) is confirmed within the frozen T20/T26 normalization bridge. The explicit six-segment kernel expansion of μ in terms of minimal kernel generators is identified as the next open frontier. Status: Extraction formula solid by direct matrix computation. Identity μ = Δφ(τ) solid within T20/T26 normalization bridge. Six-segment kernel expansion not yet derived. Dependencies: T15, T20, T23, T26.