Discrete Sampling of Continuous Q5 Phase Transport

T84 establishes the discrete sampling law for continuous Q5 phase transport. Starting from the reduced phase transport operator\[Tₚₕₐₛₑ=I+ε A+O(ε^2),^2=-I,ε=1{160},\]The theorem defines the discrete observable phase sequence\[μ_k=L(Tₚₕₐₛₑ^k),\]where \(L\) is the normalized phase-readout functional satisfying\[L(I)=0,(A)=1.\] Using the T80 continuous-limit result\[Tₚₕₐₛₑ^kkε A,\]together with\[ekε A=Icos(kε)+Asin(kε),\]The theorem proves\[μ_k=sin(kε)+O(ε).\] Substituting\[ε=1{160},\]gives the discrete sampled phase law\[μ_k=sin(k/160)+O(1{160}).\] T84 is structurally important because it converts the continuous phase-flow law\[L(etA)=sin t\]from T83 into an indexed observable transport sequence\[k↦ μ_k.\] The theorem, therefore, establishes the sampled observable structure inherited from the continuous Q5 transport flow. Several structural consequences follow immediately. Pass-through occurs exactly when\[sin(kε)≠0.\] Maximal pass-through occurs near\[kε=π/2+nπ,\]Although exact maxima are generally inaccessible because the discrete sampling lattice does not usually intersect those irrational phase locations exactly. For small accumulated transport,\[k160,\]The small-angle approximation gives\[μ_k/160,\]so early-stage phase readout grows linearly with transport depth. Together with T80-T83, the transport/readout chain now becomes\[Trawₚₕₐₛₑₚₕₐₛₑ^kkε A→μ_k=sin(kε).\] Status: solid for the sampled phase law\[μ_k=sin(kε)+O(ε)\]given the T80 continuous limit and bounded accumulated phase \(t=kε\); solid for the small-step linear regime\[μ_k≈ k/160;\]conditional for higher-order corrections inherited from the unresolved higher-order structure of\[Tₚₕₐₛₑ.\]