This paper establishes that a single scalar invariant µ = ½Tr(Q⊥R) simultaneously determines phase asymmetry, rotational defect, and interference structure in the reduced Q5 transport system. The central result is that µ links phase asymmetry, rotational defect, and interference source as the unique rotational readout coefficient of the reduced operator sector. The amplitude-square observable form \[ |√P₁ + e^{iφ}√P₂|² \]emerges from the interference structure under conditional uniqueness assumptions rather than being assumed. The solid core establishes: \[ µ ∝ Im(PQ̄) = |P||Q|sin(ΔΩ) \] ; µ is the unique antisymmetric rotational coefficient on the transverse phase plane P⊥; channel and operator definitions of µ coincide under the common-reduction assumption; the interference term µ⟨R⟩ is the sole phase-sensitive contribution to observables; and amplitude-square structure follows from the interference form. Conditional extensions (barrier holonomy, entangling power, effective generator, and emergent time) are explicitly separated from the solid core and stated with their open derivation dependencies. The paper demonstrates that interference and amplitude structure share a single origin in µ, and that the amplitude-square form is derived rather than assumed.
This paper establishes that a single scalar invariant µ = ½Tr(Q⊥R) simultaneously determines phase asymmetry, rotational defect, and interference structure in the reduced Q5 transport system. The central result is that µ links phase asymmetry, rotational defect, and interference source as the unique rotational readout coefficient of the reduced operator sector. The amplitude-square observable form [ |√P₁ + e^{iφ}√P₂|² ]emerges from the interference structure under conditional uniqueness assumptions rather than being assumed. The solid core establishes: [ µ ∝ Im(PQ̄) = |P||Q|sin(ΔΩ) ] ; µ is the unique antisymmetric rotational coefficient on the transverse phase plane P⊥; channel and operator definitions of µ coincide under the common-reduction assumption; the interference term µ⟨R⟩ is the sole phase-sensitive contribution to observables; and amplitude-square structure follows from the interference form. Conditional extensions (barrier holonomy, entangling power, effective generator, and emergent time) are explicitly separated from the solid core and stated with their open derivation dependencies. The paper demonstrates that interference and amplitude structure share a single origin in µ, and that the amplitude-square form is derived rather than assumed.
T61 develops an emergent interference-form representation within the reduced Q5 operator framework. Starting from the reduced expectation structure⟨Q⊥⟩ρ=a+bx+cz+μy,The theorem identifies a matched subclass in which the antisymmetric rotational contribution can be rewritten into a probability-like interference expression involving effective channel weights and a relative phase term. The resulting structure takes the form of a two-channel interference law,Ptot=P1+P2+2P1P2sin(ΔΩ),where the interference contribution arises from the surviving reduced antisymmetric sector generated by the rotational invariant μ. The theorem, therefore, establishes that the reduced defect-sector expectation structure naturally admits quantum-like interference representations under explicit matching conditions. T61 does not derive the Born rule, quantum measurement theory, wavefunction collapse, or universal probabilistic interpretation. The interference form is obtained through parameter matching within a restricted reduced-sector subclass rather than from structural necessity alone. The theorem further clarifies that the effective channel weights P1,P2 are not assumed to be physical probabilities in the full quantum-mechanical sense. Instead, T61 isolates the precise conditions under which the reduced antisymmetric transport structure reproduces a probability-like interference form. Status: solid for the reduced expectation decomposition and interference-form rewriting under the stated matching assumptions; conditional on the reduced-sector parameter correspondence and admissible reduction structure; speculative for any interpretation as a derivation of quantum probability theory or the Born rule.
T59 establishes the equivalence of the channel-interference and barrier-holonomy derivations of the reduced rotational invariant μ within the Q5 transport architecture. The theorem shows that the antisymmetric rotational coefficient obtained from effective-channel interference and the coefficient obtained from barrier-mediated holonomy reduce to the same invariant structure after an admissible projection into the reduced defect sector. The result, therefore, identifies the two constructions not as competing mechanisms, but as equivalent reduced descriptions of the same mediated antisymmetric transport invariant. The theorem is structurally important because it demonstrates consistency between distinct reduction pathways in the Q5 framework. Rather than depending on a particular decomposition picture, the reduced rotational coefficient survives as a common invariant under the admissible reduction pipeline. This strengthens the interpretation of μ as a genuine reduced-sector transport invariant rather than a decomposition artifact. T59 also foreshadows the later residual ecology architecture developed in the TA-series, where mediated transport, projection, and reduced-sector elimination repeatedly produce equivalent effective structures across multiple derivational viewpoints. Status: solid for the equivalence of the reduced antisymmetric invariant under the stated reduction and projection assumptions; conditional on normalization and admissible reduction conventions used in the reduced defect sector; speculative for any claim that the microscopic physical mechanisms are ontologically identical rather than reduced-sector equivalent.
D T58 develops a conditional entangling-power framework arising from irreducible rotational structure in the reduced Q5 transport sector. The theorem analyzes the antisymmetric rotational invariant generated by phase-asymmetric transport amplitudes and studies its consequences under a bipartite reduced-sector decomposition. When the reduced interaction generator contains a non-removable rotational interaction component,Aint=0,The resulting reduced evolution operator becomes non-factorizable under local sector decomposition, implying strictly positive entangling power,ep(U)>0.The theorem, therefore, identifies a structural pathway from irreducible mediated rotational transport to nonlocal reduced-sector coupling behaviour. T58 does not claim that Q5 derives quantum entanglement from first principles, nor that this mechanism is unique. Rather, it establishes that under the stated decomposition and interaction assumptions, the mediated rotational residue generated by asymmetric transport produces nonfactorizable reduced evolution with positive entangling capability. The theorem further clarifies that vanishing rotational structure disables this specific entangling channel without excluding alternative entanglement mechanisms. Status: solid for the nonfactorizable-unitary and positive-entangling-power conclusions under the stated operator assumptions; conditional on the bipartite tensor decomposition and the mapping from reduced rotational structure to interaction generators; speculative for broader physical interpretation or equivalence with quantum entanglement phenomenology.
T63 establishes a conditional uniqueness theorem for the interference and amplitude-square structures developed in T61 and T62. Starting from the reduced two-channel interference form,Ptot=P1+P2+G(P1,P2,ΔΩ),The theorem analyzes the structural constraints imposed by symmetry, continuity, axis-vanishing behaviour, degree-1 homogeneity, and first-harmonic rotational structure generated by the reduced antisymmetric operator algebra. Under these assumptions, the interference contribution is forced into the normal formG(P1,P2,ΔΩ)=2P1P2sin(ΔΩ),yielding the exact amplitude-square representationPtot=P1+eiϕP22.The theorem therefore shows that, within the stated structural class, the Born-compatible amplitude-square form is uniquely selected by the reduced interference geometry together with the harmonic structure induced by the rotational generator satisfying A2=−I. T63 does not derive full quantum mechanics, measurement theory, Hilbert-space axioms, or probabilistic interpretation from first principles. The result is a conditional uniqueness theorem for the algebraic normal form of two-channel interference under the stated assumptions. In particular, the homogeneity assumption is essential: without it, uniqueness does not hold. The theorem further clarifies that the complex amplitude representation is obtained from the interference structure rather than independently postulated. Status: solid for the conditional uniqueness of the amplitude-square normal form under the stated symmetry, homogeneity, and harmonic assumptions; conditional on the separability and structural assumptions defining the interference class; speculative for any claim of a complete derivation of quantum probability theory or the Born rule from Q5 geometry alone.
T60 analyzes the observable consequences of the reduced antisymmetric rotational invariant μ within the Q5 defect-sector operator framework. For a reduced operator of the formQ⊥=aI+bσx+cσz+μR,The theorem shows that expectation values decompose into symmetric contributions together with a distinct antisymmetric rotational quadrature proportional toμTr(ρR).The antisymmetric sector therefore contributes a phase-sensitive observable component that vanishes identically when μ=0 and becomes detectable whenever the reduced rotational invariant survives admissible reduction. The theorem further establishes that states exist which selectively detect the antisymmetric quadrature whenever the rotational contribution is nonzero. T60 is structurally important because it connects the reduced rotational invariant developed in T56-T59 to the observable-sector expectation structure. The theorem does not claim that all interference or observable asymmetry arises from μ. Rather, it identifies μ as the unique antisymmetric contribution associated with the reduced rotational sector of the operator algebra. The result later became conceptually aligned with the selective-accessibility and reduced-sector survival principles formalized in the TA-series residual ecology arc. Status: solid for the reduced-sector expectation decomposition and antisymmetric quadrature structure under the stated operator assumptions; conditional on the admissible reduction framework and reduced-sector normalization conventions; speculative for direct physical observable interpretation beyond the internal operator model.
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.