T91 establishes the reduced Q5 two-channel interference law after the \(5\to3\to2\) reduction. The surviving reduced state space is \(\mathbb{C}^2\), with crossing channels \(P\) and \(Q\), and transport generated by \(A=i\sigma_y\), satisfying \(A^2=-I\). Continuous reduced evolution is given by\[\psi(t)=e^{tA}\psi(0),\]with\[e^{tA}=\begin{pmatrix}\cos t & \sin t\\-\sin t & \cos t\end{pmatrix}.\] For the initial relative-phase state\[\psi(0)=\frac1{\sqrt2}\begin{pmatrix}1\\e^{i\phi}\end{pmatrix},\]The evolved channel probabilities are\[\Pr(P)=\frac12\left(1+\sin(2t)\cos\phi\right),\]and\[\Pr(Q)=\frac12\left(1-\sin(2t)\cos\phi\right).\]The observable channel imbalance is therefore\[\Pr(P)-\Pr(Q)=\sin(2t)\cos\phi.\] The theorem shows that reduced Q5 transport naturally produces a two-channel interference structure on the reduced crossing plane. The transport amplitude is controlled by accumulated phase \(t=k\varepsilon\), while the coherence factor is controlled by the relative phase \(\phi\). Phase averaging removes the interference contribution:\[\langle\cos\phi\rangle=0\quad\Longrightarrow\quad\langle\Pr(P)\rangle=\langle\Pr(Q)\rangle=\frac12.\]Loss of coherent phase relation, therefore restores equal channel probabilities. T91 converts the earlier transport-generator chain into an explicit observable probability law. Earlier theorems established\[A^2=-I,\qquadT^k\to e^{tA},\qquad\mu_k=\sin(k\varepsilon),\]while T91 derives the resulting two-channel interference structure generated by the same reduced transport algebra. The discrete form follows from\[t=k\varepsilon,\qquad\varepsilon=\frac1{320},\]giving\[\Pr(P)=\frac12\left(1+\sin\left(\frac{2k}{320}\right)\cos\phi\right).\]The fringe contrast oscillates with transport step number \(k\), with the first maximal contrast near\[k\approx80\pi\approx251,\]matching the near-integer phase-maximization structure identified previously in T84. Classification:Solid: matrix exponential, probability law, normalization, and phase averaging.Conditional: identification of the reduced two-channel Q5 state space and the generator chain \(A=i\sigma_y\) from T37 and T80-T89.Not claimed: direct experimental observability.
T91 establishes the reduced Q5 two-channel interference law after the (5\to3\to2) reduction. The surviving reduced state space is (\mathbb{C}^2), with crossing channels (P) and (Q), and transport generated by (A=i\sigma_y), satisfying (A^2=-I). Continuous reduced evolution is given by[\psi(t)=e^{tA}\psi(0),]with[e^{tA}=\begin{pmatrix}\cos t & \sin t\-\sin t & \cos t\end{pmatrix}.] For the initial relative-phase state[\psi(0)=\frac1{\sqrt2}\begin{pmatrix}1\e^{i\phi}\end{pmatrix},]The evolved channel probabilities are[\Pr(P)=\frac12\left(1+\sin(2t)\cos\phi\right),]and[\Pr(Q)=\frac12\left(1-\sin(2t)\cos\phi\right).]The observable channel imbalance is therefore[\Pr(P)-\Pr(Q)=\sin(2t)\cos\phi.] The theorem shows that reduced Q5 transport naturally produces a two-channel interference structure on the reduced crossing plane. The transport amplitude is controlled by accumulated phase (t=k\varepsilon), while the coherence factor is controlled by the relative phase (\phi). Phase averaging removes the interference contribution:[\langle\cos\phi\rangle=0\quad\Longrightarrow\quad\langle\Pr(P)\rangle=\langle\Pr(Q)\rangle=\frac12.]Loss of coherent phase relation, therefore restores equal channel probabilities. T91 converts the earlier transport-generator chain into an explicit observable probability law. Earlier theorems established[A^2=-I,\qquadT^k\to e^{tA},\qquad\mu_k=\sin(k\varepsilon),]while T91 derives the resulting two-channel interference structure generated by the same reduced transport algebra. The discrete form follows from[t=k\varepsilon,\qquad\varepsilon=\frac1{320},]giving[\Pr(P)=\frac12\left(1+\sin\left(\frac{2k}{320}\right)\cos\phi\right).]The fringe contrast oscillates with transport step number (k), with the first maximal contrast near[k\approx80\pi\approx251,]matching the near-integer phase-maximization structure identified previously in T84. Classification:Solid: matrix exponential, probability law, normalization, and phase averaging.Conditional: identification of the reduced two-channel Q5 state space and the generator chain (A=i\sigma_y) from T37 and T80-T89.Not claimed: direct experimental observability.
T92 establishes that the squared-amplitude probability rule in the reduced Q5 crossing sector is not postulated but follows from unitarity, norm preservation, and additive refinement consistency within the reduced two-channel architecture. The theorem works in the reduced crossing sectorHPQ=C2,obtained after the 5→3→2 reduction. The reduced transport generator is\[A = i\sigma_y=\begin{pmatrix}0 & 1 \\-1 & 0\end{pmatrix},\qquadA^\dagger = -A,\qquadA^2 = -I.\] Continuous transport is generated byU(t)=etA. The proof first establishes anti-Hermiticity:A†=−A. From this,U(t)†=etA†=e−tA=U(t)−1,So the transport operator is unitary:U(t)†U(t)=I. Unitarity then forces norm preservation:∥U(t)ψ∥2=∥ψ∥2. For normalized statesψ=P∣P⟩+Q∣Q⟩,∣P∣2+∣Q∣2=1,The theorem introduces an additive refinement consistency lemma. If a channel decomposes orthogonally,∣P∣2=∣P1∣2+∣P2∣2,Then the probability must satisfyPr(P)=Pr(P1)+Pr(P2). Assuming probability depends only on channel norm weight,Pr(P)=f(∣P∣2),The refinement condition forces the Cauchy functional equation\[f(x+y)=f(x)+f(y),\qquadf(1)=1.\] Monotonicity and nonnegativity force continuity on [0,1], and the unique continuous additive solution isf(x)=x. Therefore, the unique norm-preserving probability assignment in the reduced crossing sector isPr(P)=∣P∣2,Pr(Q)=∣Q∣2. T92 therefore upgrades the earlier reduced-sector Born interpretation from heuristic consistency to a mathematically constrained uniqueness statement within the two-channel architecture. Corollary 92.1 establishes channel resolution into the basis states∣P⟩=(1,0)T,∣Q⟩=(0,1)T,with probabilities determined by squared amplitudes. Corollary 92.2 shows that the Born assignment is preserved under Q5 transport because∣P(t)∣2+∣Q(t)∣2=1for all t. Corollary 92.3 gives the architectural interpretation of measurement. Measurement is interpreted not as an external collapse force but as forced resolution into one admissible barrier-coupled channel. The barrier selects admissible channels, while the amplitude distribution determines the resolution probability:Pr(channel C)=∣⟨C∣ψ⟩∣2. The theorem remains carefully bounded. It does not claim:- a derivation of the Born rule for arbitrary Hilbert spaces,- a complete solution to the quantum measurement problem,- or a derivation independent of the 5→3→2 reduction chain. The solid core is the reduced-sector statement:\[A^\dagger=-A\RightarrowU(t)\ \text{unitary}\Rightarrow\text{norm preservation}\Rightarrow\Pr(P)=|P|^2.\] T92 therefore provides the foundational probability justification underlying the interference law of T91 and strengthens the reduced Q5 transport interpretation by deriving the squared-amplitude rule from unitary reduced transport plus additive refinement consistency rather than taking it as a postulate.
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.
T84 establishes the discrete sampling law for continuous Q5 phase transport. Starting from the reduced phase transport operator\[T_{\mathrm{phase}}=I+\epsilon A+O(\epsilon^2),\qquadA^2=-I,\qquad\epsilon=\frac1{160},\]The theorem defines the discrete observable phase sequenceμk=L(Tphasek),where L is the normalized phase-readout functional satisfying\[L(I)=0,\qquadL(A)=1.\] Using the T80 continuous-limit result\[T_{\mathrm{phase}}^k\toe^{k\epsilon A},\]together withekϵA=Icos(kϵ)+Asin(kϵ),The theorem provesμk=sin(kϵ)+O(ϵ). Substitutingϵ=1601,gives the discrete sampled phase lawμk=sin(160k)+O(1601). T84 is structurally important because it converts the continuous phase-flow lawL(etA)=sintfrom T83 into an indexed observable transport sequencek↦μk. The theorem, therefore, establishes the sampled observable structure inherited from the continuous Q5 transport flow. Several structural consequences follow immediately. Pass-through occurs exactly whensin(kϵ)=0. Maximal pass-through occurs nearkϵ=2π+nπ,Although exact maxima are generally inaccessible because the discrete sampling lattice does not usually intersect those irrational phase locations exactly. For small accumulated transport,k≪160,The small-angle approximation givesμk≈160k,so early-stage phase readout grows linearly with transport depth. Together with T80-T83, the transport/readout chain now becomes\[T_{\mathrm{raw}}\toT_{\mathrm{phase}}\toT_{\mathrm{phase}}^k\toe^{k\epsilon A}\to\mu_k=\sin(k\epsilon).\] Status: solid for the sampled phase lawμk=sin(kϵ)+O(ϵ)given the T80 continuous limit and bounded accumulated phase t=kϵ; solid for the small-step linear regimeμk≈k/160;conditional for higher-order corrections inherited from the unresolved higher-order structure ofTphase.
T83 establishes the phase pass-through condition for reduced Q5 transport. Starting from the continuous phase-flow limit established in T80,\[T_{\mathrm{phase}}^k\longrightarrowe^{tA}=I\cos t + A\sin t,\qquadt=k\varepsilon,\]withA2=−I,the theorem applies the normalized phase-readout functional L, satisfying\[L(I)=0,\qquadL(A)=1.\] It follows thatL(etA)=L(Icost+Asint)=sint. Thus, observable phase transmission occurs exactly whensint=0. At\[t=n\pi,\qquadn\in\mathbb{Z},\]the accumulated transport satisfiesetA=±I,so the transport is diagonal/scalar in the phase readout and no phase passes through. At\[t=\tfrac{\pi}{2}+n\pi,\qquadn\in\mathbb{Z},\]The accumulated transport satisfiesetA=±A,So the readout is purely rotational and phase pass-through is maximal. T83 is structurally important because it separates diagonal retention from observable phase transmission:etA=Icost+Asint.The Icost branch retains the carrier state but is invisible to L, while the Asint branch carries the observable phase coefficient. Together with T80-T82, this completes the reduced phase-flow readout arc:\[T_{\mathrm{raw}}\toT_{\mathrm{phase}}\toe^{tA}\toL(e^{tA})=\sin t.\] Status: solid for the identityL(etA)=sintgiven the T80 continuous limit and the T26-T28 normalization of L; solid for the pass-through conditionsint=0;conditional on the reduced phase-transport formTphase=I+εA+O(ε2)and on the T81-T82 projection-invariance bridge from raw barrier transport.
T104 computes the explicit observable interference correction generated by Q5 leakage-return transport. T101 established the admissible structure of a hidden-half correction term but left the correction function unspecified. T102 derived the leakage-return operator C=L†BL, and T103 identified the leakage generator B=iAL from the explicit Q5 leakage adjacency. T104 completes the chain by directly computing the resulting interference correction. The central structural result is that first-order mixing vanishes identically: L†BL=0. This occurs because the leakage generator transports amplitude through the helical chain e3↔e4↔e5, while the observable return map extracts only the boundary modes e3 and e5. A single application of B moves amplitude into the middle gate-null mode e4, preventing any first-order return contribution. Observable mixing therefore begins only at second order. The first nontrivial return operator is C2=L†B2L=−801(I−iσy), which is Hermitian and generates the leading observable correction. Applying the T100 renormalization procedure yields normalized probability shifts Δ+=−160sinϕsin(4θ),Δ−=+160sinϕsin(4θ), with Δ++Δ−=0. The resulting observable probabilities are P+obs=cos2θ−2η2160sinϕsin(4θ)+O(η3),P−obs=sin2θ+2η2160sinϕsin(4θ)+O(η3). The correction is therefore second order in the leakage coupling and possesses a characteristic angular envelope sin(4θ). This replaces the earlier minimal placeholder structure of T101 and introduces a richer node pattern, with nulls at θ=0,4π,2π, together with maxima at θ=8π,83π. The phase dependence is proportional to sinϕ, arising from the e±iπ/4 phases contained in the leakage generator AL. This shifts the correction into quadrature relative to the leading Born interference fringe. T104 establishes that the Q5 leakage-return mechanism produces a fully explicit interference correction that is antisymmetric between output channels, conserves total probability, vanishes at the required endpoints, and emerges naturally from the two-step transport structure of the leakage chain. The theorem replaces the phenomenological correction function of T101 with a direct Q5-derived prediction and identifies sinϕsin(4θ) as the characteristic observable signature of leakage-return transport.
T93 extends the reduced Q5 interference framework of T91–T92 to an enlarged leakage-state space Hext=C2⊕HL, where C2 is the coherent crossing sector and HL collects non-crossing leakage modes. The theorem distinguishes between visible coherent weight C(t)=∥ΠPQΨ(t)∥2, leakage weight D(t)=∥(I−ΠPQ)Ψ(t)∥2, and gate-admission weight G(t)=∥ΠYΨ(t)∥2, where ΠY projects onto the Y-admissible crossing mode. A key result is that visible coherence and gate alignment are distinct quantities. A state may retain nonzero amplitude in the observable P/Q sector while failing admission into the Y-crossing mode. This leads to a geometric definition of decoherence: DY(t)=1−∥ΠYΨ(t)∥2. Decoherence is therefore interpreted as loss of overlap with the admissible crossing channel rather than destruction of phase information. The theorem further establishes that interference visibility is governed by the gate-admitted component of the state: V(t)=∥ΠYΨ(t)∥, yielding the extended interference law Pr(P)=21(1+V(t)sin(2t)cosϕ). Perfect visibility corresponds to complete gate alignment, while reduced visibility reflects either leakage into non-crossing modes or misalignment with the crossing sector. An extended generator G=(AK−K†B) is introduced to couple the coherent crossing sector to the leakage sector. Under the skew-adjoint conditions \[A^{\dagger}=-A,\qquadB^{\dagger}=-B,\] the evolution operator etG preserves total norm, implying C(t)+D(t)=1. Leakage therefore represents redistribution of phase weight rather than irreversible loss. The theorem provides a geometric interpretation of decoherence within the Q5 framework: phase information may remain globally conserved while becoming inaccessible to coherent crossing dynamics. In this picture, coherence is determined not merely by the presence of phase amplitude, but by its alignment with the unique Y-admissible transport mode.