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.
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.
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.
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 shows that the propagated rotational mismatch between the protected and dressed transport channels is governed by the same spectral split (p − ω) that drives the T15 phase defect, read through a distinct propagator-level extraction. The protected channel has generator G₋ = pJ (pure antisymmetric rotor); the dressed channel has G₊ = pJ + 2aσ_z (antisymmetric core plus traceless symmetric deformation). The dressed propagator satisfies G₊² = −ω²I where ω = √(p² − 4a²), shifting the effective transport frequency from p to ω. The normalized propagated mismatch is: Δ̂μ(η) = −i[(p/ω)sin(ηω) − sin(ηp)] = −i(p − ω)Ξ(η,p,a) Five corollaries are established: shared common antisymmetric rotational carrier; generator-level invisibility of the symmetric deformation (Tr(SR) = 0 for symmetric S, antisymmetric R); exact factorization through (p − ω); generic visibility with isolated blind times characterized by p·sin(ηω) = ω·sin(ηp); small-deformation asymptotics of blind times governed by tan(ηp) = ηp; and discrete angular accumulation T_rot^k = cos(kθ)I + sin(kθ)A. T30 does not replace T15. T15 gives the exact phase defect. T30 gives the spectral-propagator interpretation of its rotational manifestation through a different readout of the same core. Status: Corollaries 30.1–30.5 solid by direct algebraic computation. All results are conditional on T15 architecture (identification of p and a) and T16/T17/T20 framework (Δφ(τ) = μ chain). Six-segment kernel expansion of Ξ in terms of [M_n, D_n] remains open. Dependencies: T15, T16, T17, T18, T19, T20, T23, T24, T25, T26, T27, T28, T29.
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 paper closes the primary open step of the Bear Claws Theorem (T50–T52), the explicit derivation of the coupling parameter β from Q5 kernel geometry, in two parts. Part I derives β = a²/4 and h₁ = √(5/2) directly from matrix computation. The recurrence Y³ = 5Y (T46) pins h₁ = √(5/2) via 2h₁² = 5. Direct multiplication gives [M_n,D_n]·Y = ah₁·[[1,0,1],[0,0,0],[−1,0,−1]], zero middle row and column, consistent with |8⟩ as the defect bond mediation vertex. The visible-hidden coupling is the (1,3) entry ah₁. With Tr(Y^T Y) = 4h₁², β = a²h₁²/(4h₁²) = a²/4. The observable phase coefficient satisfies μ = −8βh₁ = −a²√10. Part II (Theorem 53) establishes that μ, β, and a are not free parameters; they are uniquely determined by compatibility between the kernel-derived phase extraction pipeline (T26–T29) and the transport-level rotational residual (T48–T49). The reduced phase plane is two-dimensional, so its antisymmetric subspace is one-dimensional; any antisymmetric operator surviving the pipeline is a scalar multiple of R = A = E_d = iσ_y. Since both constructions reference the same normalized generator with coefficients μ (extraction pipeline) and 1/160 (transport residual), and the sign is fixed by σ_Γ = −1 from T18: μ = −1/160, β = 1/(1280√(5/2)), a² = 1/(160√10) The value 1/160 is not an imposed normalization; it is the unique coefficient consistent with both the ordered kernel holonomy and the reduced transport evolution. The full chain T17 → T29 → T46 → β = a²/4 → μ = −1/160 → T52 is non-circular. Status: h₁ = √(5/2), product structure of [M_n,D_n]Y, (1,3) coupling entry, Tr(Y^T Y) = 4h₁², and β = a²/4 all solid by direct matrix computation. Generator identification R = A = E_d solid, one-dimensionality of antisymmetric sector is linear algebra. μ = −1/160 solid given generator identification. Sign propagation from T18 through T29, structural, full explicit sign-tracking through the projection pipeline, is the natural supporting calculation. Parametrized form of [M_n,D_n] on {9,8,24} must ultimately be derived from T17 kernel generators to confirm the alternating antisymmetric form is forced rather than assumed. Dependencies: T29, T36, T37, T46, T52.
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.