Within the kernel transport framework of T17 - T31, establishes that the scalar-channel component of the round-trip transport operator T(a) = X_{d→n}X_{n→d} decomposes as: P_sc·T(a)·P_sc = P_sc·X_{d→n}·P_sc·X_{n→d}·P_sc + P_sc·X_{d→n}·P⊥·X_{n→d}·P_sc The first term is transport remaining entirely within the scalar channel. The second term is support-mediated transport, trajectories that exit the scalar channel, propagate through the complementary sector P⊥, and return. The decomposition is an algebraic identity requiring only P_sc + P⊥ = I and linearity: no further assumptions. Whenever the mixed term is nonzero, the scalar return weight is not intrinsic to the scalar channel alone. This provides the structural mechanism underlying the coefficient β(a) in T31: β(a) depends on support-mediated transport whenever the mixed term is nonzero. Status: Decomposition identity solid: algebraic, no conditions. Nonvanishing of the mixed term conditional on assumption (ii), transport maps not block-diagonal with respect to P_sc ⊕ P⊥ removes the obstruction but does not guarantee nonvanishing; explicit verification from T17 kernel geometry remains open. β(a) dependence inherits T31 diagonalizability assumption. All results inherit T16/T17/T20 conditionality. Dependencies: T14, T15, T16, T17, T18, T19, T20, T26, T27, T28, T29, T30, T31.
Within the kernel transport framework of T17 - T31, establishes that the scalar-channel component of the round-trip transport operator T(a) = X_{d→n}X_{n→d} decomposes as: P_sc·T(a)·P_sc = P_sc·X_{d→n}·P_sc·X_{n→d}·P_sc + P_sc·X_{d→n}·P⊥·X_{n→d}·P_sc The first term is transport remaining entirely within the scalar channel. The second term is support-mediated transport, trajectories that exit the scalar channel, propagate through the complementary sector P⊥, and return. The decomposition is an algebraic identity requiring only P_sc + P⊥ = I and linearity: no further assumptions. Whenever the mixed term is nonzero, the scalar return weight is not intrinsic to the scalar channel alone. This provides the structural mechanism underlying the coefficient β(a) in T31: β(a) depends on support-mediated transport whenever the mixed term is nonzero. Status: Decomposition identity solid: algebraic, no conditions. Nonvanishing of the mixed term conditional on assumption (ii), transport maps not block-diagonal with respect to P_sc ⊕ P⊥ removes the obstruction but does not guarantee nonvanishing; explicit verification from T17 kernel geometry remains open. β(a) dependence inherits T31 diagonalizability assumption. All results inherit T16/T17/T20 conditionality. Dependencies: T14, T15, T16, T17, T18, T19, T20, T26, T27, T28, T29, T30, T31.
Within the T17 - T30 kernel transport framework, two realizations of a triadic kernel sharing the same spectral carrier and commutator source are shown to produce distinct scalar outputs when their transport segments are rearranged. The native kernel K = [A, T, B] and its mirrored anti-automorphic realization J(K) = [B, T^rev, A] share the common carrier χ₀(η,p) = (1/p)sin(ηp) − η·cos(ηp) and the same commutator source. The mirrored scalar branch satisfies: Ξ_J(η,p,a) = −β(a)·χ₀(η,p) where β(a) is the scalar-channel coefficient of the round-trip transport closure X_{d→n}X_{n→d} = α(a)P⊥ + β(a)P_sc. Three further results are established: the scalar-visible transport residue is β(a) − α(a); the complementary-sector closure defect is Δ⊥(a) = (α(a) − 1)P⊥; and this defect is generically nonzero whenever transport-realization change is not confined to the scalar channel. The theorem is purely internal. No physical or Hilbert-space identification is asserted. The interpretive remark notes structural analogy to path-dependent amplitude formation in quantum systems: analogy only. Status: Carrier invariance, transport coefficient isolation, round-trip closure readout, and complementary-sector defect formula are solid by direct algebraic argument. Conditional on assumption (iv), the decomposition of the round-trip transport operator is assumed, not derived; explicit computation of α(a) and β(a) from the T17 kernel architecture remains open. Conditional on assumption (iii), transport non-neutrality assumed; structural derivation from T17 geometry is open. All results inherit T16/T17/T20 conditionality. Dependencies: T14, T15, T16, T17, T18, T19, T20, T26, T27, T28, T29, T30.
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 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 derives the ambient Schur landing of complement-sector feedback and establishes its complete observable classification. The complement block of admissible form H_cc = aI + bJ (J² = I) is invertible for a² ≠ b² with H_cc⁻¹ = (aI − bJ)/(a² − b²). The Schur correction ΔH = H_rc·H_cc⁻¹·H_cr decomposes into: ΔH = (a/(a²−b²))·Δ₀ − (b/(a²−b²))·Δ_J where Δ₀ = H_rc·H_cr and Δ_J = H_rc·J·H_cr. Five lemmas establish the landing structure. The helicity-twisted term satisfies P_sc·Δ_J·P_sc = 0, J is odd under scalar projection, so the twisted feedback contributes nothing to the scalar channel. The transverse projection of Δ_J lands in span{I, σ_z, σ_x}, J anticommutes with R = iσ_y, so the twisted component has no R-component and lies entirely in the symmetric transverse sector. A critical bridge identifies Δ_J as the operator-level origin of the T30 spectral deformation: both the Schur denominator (a² − b²) and the T30 discriminant (p² − 4a²) arise as discriminants of two-level systems with identical admissible commutant structure. Four observable consequences: scalar channel protected (μ invariant); helicity-twisted correction annihilated by extraction functional L before readout; T30 spectral deformation identified as Schur correction; sole observable manifestation is spectral redistribution between channels at p and ω = √(p² − 4a²). Two independent protection mechanisms operate: local null-class annihilation (scalar observables) and global mirror-orbit cancellation via T38 (embedded observables). Status: Commutant inversion, Schur decomposition, scalar nullity given J odd under scalar projection, and transverse symmetric landing given anticommutation of J with R, all solid by direct computation. T30 spectral bridge structural, identification of denominator with p² − 4a² is a leading-order structural identification, not a derived equality. Scalar invariance and λ_eff = 1 solid given lemmas and T38. Orbit preservation solid via T34–T35. Admissible form H_cc = aI + bJ is a structural assumption; derivation from the full ambient T12/T16 block structure is the primary remaining open step. Dependencies: T12, T16, T26, T29, T30, T34, T35, T37, T38, T54.
This Theorem extracts the phase-bearing 2-plane from the T23–T25 ambient structure and identifies the scalar phase defect Δφ(τ) with the rotational coefficient of the induced collapsed operator on that plane. Two supporting lemmas are established. Lemma 26.A performs the branch-axis decomposition: the T22 triplet T₃ splits canonically into the branch-selected zero-weight axis span{u₃} and the transverse phase plane P⊥ = span{u₂, u₄}. Lemma 26.B establishes the readout kernel: every Hermitian operator on P⊥ decomposes uniquely in the real basis {I, σ_z, σ_x, R}, and the U(1)-holonomy readout ρ_phase annihilates I, σ_z, and σ_x, retaining only the coefficient μ of the rotation generator R. The main theorem establishes that the collapsed branch object C_A induces an operator Q⊥ on P⊥, the holonomy readout factors through P⊥, and (after fixing the T15 normalization convention) Δφ(τ) = μ exactly. The coefficients α, β, γ drop out of the readout regardless of their values. Status: Parts (1)–(3) solid by direct structural argument. Part (4) inherits the conditionality of T20; the identification of Δφ(τ) as the scalar shadow of the collapsed kernel is a load-bearing assumption carried forward from T20. Dependencies: T15, T18, T19, T20, T23, T24, T25.