TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/CONTINUOUS-TIME-NONUNITARY-SECTOR-TEST.txt
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000001 | UR.SEAM.MIXED_CURRENT_ORIGIN.01 -- 23 October 2026
000002 | Continuous-time compactness test for arbitrary nonunitary endpoint limits
000003 | Verdict: PARTIAL physical origin; a stronger source-specific exclusion.
000004 | 
000005 | 1. Precise question and inherited original source
000006 | 
000007 | Can arbitrary renormalized NONUNITARY fields extracted from the actual
000008 | neutral parity pair produce a genuinely new spinor sector, while retaining
000009 | the original full source norm, time and observable comparison? This goes
000010 | beyond the earlier smooth-unitary-return bound. No uniform bound on the
000011 | first energy moment is assumed here.
000012 | 
000013 | Use the original mass1,width8,one-copy QWZ cylinder and fixed quarter r1
000014 | or3, alpha=r/4, full negative filled sea Omega_N, and
000015 |   H_N=N/(2pi)[dGamma(h_N)-Tr(P_N h_N)] >=0.
000016 | Both physical edges and all massive modes remain present. Their original
000017 | full spectral exhaustion is proved in FULL-FINITE-ENERGY-AND-SECTOR-MENU.txt
000018 | sections1-3: for each cutoff E off the limiting spectrum the ENTIRE low
000019 | spectral projection eventually has the limiting finite rank; its occupied
000020 | particle/hole basis converges to the two-edge Fock basis. The limiting H
000021 | has compact resolvent. This is not a claim about all P1 representatives.
000022 | 
000023 | 2. Needed comparison hypotheses, not just a spectrum match
000024 | 
000025 | A. Full low-energy spectral identifications are compatible for increasing
000026 | cutoffs, with limiting two-edge Fock space H_old and finite-rank P(E).
000027 | These follow from the fixed source eigenmode/Fock identifications above.
000028 | A diagonal choice gives isometries J_N of finite source Hilbert spaces
000029 | into H_old, agreeing on every fixed low window for large N. Remaining
000030 | finite high basis vectors can be sent into the unused complement.
000031 | 
000032 | B. For each RETAINED bounded observable A of the old field/observable net,
000033 | its actual microscopic A_N has sup_N||A_N||<infinity and its matrix elements
000034 | on every fixed low-energy window converge to those of pi_old(A).
000035 | In the original CAR source, such matrix elements are fixed finite Wick
000036 | words between the convergent low-mode particle/hole states. Smeared CAR
000037 | generators are uniformly bounded, and their norm-closed algebra extends
000038 | this conclusion. The given fixed fold only reorganizes those fields.
000039 | For D8 currents use the original even-CAR realization; do not silently
000040 | substitute a different action on the old odd sector.
000041 | 
000042 | This is stronger than convergence of the neutral VACUUM correlators alone.
000043 | It is the same FULL-source observable comparison already required when
000044 | transporting arbitrary source words, including odd intermediate states.
000045 | Spectrum alone never supplies B. The known odd-sector Phi refielding is
000046 | not excluded here: it CHANGES this full observable comparison.
000047 | 
000048 | Importantly B concerns the observable operators, NOT the unknown states
000049 | v_N. A disjoint state limit does not automatically violate B. With fixed
000050 | old operators B can hold identically while high-energy states escape;
000051 | in the compact source such an escape must fail D or another hypothesis.
000052 | 
000053 | C. For a fixed field smearing, let v_N=T_N Omega_N. The operators T_N may
000054 | be arbitrary finite-source operators, long histories, coherent sums,
000055 | endpoint dressings, heat-filtered insertions, or N-dependent renormalized
000056 | operators with divergent norms. Require only
000057 |   n_N=||v_N||^2 -> n in(0,infinity).
000058 | No Gaussianity, unitarity or finite word-degree bound on T_N is assumed.
000059 | 
000060 | D. Retain the actual positive time response
000061 |   F_N(t)=<v_N,exp(-t H_N)v_N> -> F(t),  t>0,
000062 | with lim_(t down0) F(t)=F(0)=n.
000063 | This is automatic for a proposed limiting smeared vector with finite norm
000064 | and a strongly continuous positive-energy semigroup IF that semigroup is
000065 | claimed to be the limit of the SAME original response. It does NOT assume
000066 | a finite first moment <H_N>, nor convergence of a derivative at t0.
000067 | A formal separately assigned charged-sector energy does not establish D.
000068 | 
000069 | Alternatively, it suffices to require the actual REAL-time word kernels
000070 | G_N(t)=<v_N,exp(-itH_N)v_N> to converge for t in a neighborhood of0,
000071 | with the limiting G continuous there and G(0)=n. The averaging argument
000072 | below yields the same tightness. Euclidean time is therefore not an extra
000073 | physical input needed for this exclusion.
000074 | 
000075 | 3. Elementary tail bound and theorem
000076 | 
000077 | For every E,t>0, the spectral theorem gives
000078 |   ||1_(E,infinity)(H_N)v_N||^2
000079 |     <= [n_N-F_N(t)]/[1-exp(-tE)].                      (1)
000080 | 
000081 | Given eta>0, continuity at zero lets us choose t>0 with n-F(t)<eta/4.
000082 | For all large N the numerator is then <eta/2. Choose E large enough so
000083 | 1-exp(-tE)>1/2, avoiding the discrete limiting spectrum. Equation(1) gives
000084 | a tail <eta for all sufficiently large N. Hence
000085 |   lim_(E to infinity) limsup_N
000086 |        ||1_(E,infinity)(H_N)v_N||^2 = 0.              (2)
000087 | 
000088 | For the real-time variant, average Re G_N over 0<t<a. Spectral calculus
000089 | gives n_N-a^-1 integral_0^a Re G_N(t)dt
000090 |  =integral[1-sin(a lambda)/(a lambda)]dmu_N(lambda).
000091 | For lambda>E>=2/a, the bracket is >=1/2. Therefore
000092 |   ||1_(E,infinity)(H_N)v_N||^2
000093 |     <=2[n_N-a^-1 integral_0^a Re G_N(t)dt].            (2a)
000094 | Uniform boundedness |G_N(t)|<=n_N permits dominated convergence. Continuity
000095 | of G at0 makes the limiting right side arbitrarily small by taking a small,
000096 | and then E>=2/a. This proves(2) from REAL time as well, with no derivative
000097 | or first-moment hypothesis.
000098 | 
000099 | The finite-window identifications in A now make J_N v_N norm-precompact.
000100 | Proof: choose a low window with tail<eta; its bounded coefficient vectors
000101 | live in one fixed finite-dimensional ball. Take a convergent subsequence,
000102 | then a diagonal subsequence over windows. The uniform tail makes that
000103 | subsequence norm-Cauchy in H_old. It converges to v with ||v||^2=n.
000104 | An arbitrary original sequence need not itself converge; subsequential
000105 | compactness is the exact assertion.
000106 | 
000107 | For a bounded retained A, truncate v_N to u_N=P_N(E)v_N. With
000108 | eta_N=||v_N-u_N||^2, the discarded expectation obeys
000109 |   |<v_N,A_N v_N>-<u_N,A_N u_N>|
000110 |     <= ||A_N||[2 sqrt(n_N eta_N)+eta_N].               (3)
000111 | The finite-block comparison B passes the truncated expectation to the old
000112 | net. Letting E increase in(3) proves on the chosen subsequence
000113 |   <v_N,A_N v_N> -> <v,pi_old(A)v>.                     (4)
000114 | If the full word-state limit exists, every such subsequential expression
000115 | agrees with it. In particular its state is NORMAL in the ORIGINAL global
000116 | representation. For mixed positive source states the same argument uses
000117 | finite-block density matrices and trace-norm tightness instead.
000118 | 
000119 | THEOREM. Under A-D every full retained-observable word-state limit of these
000120 | arbitrary nonunitary insertions lies in the original representation folium.
000121 | It cannot be a representation disjoint from pi_old. A uniform energy
000122 | expectation is unnecessary; norm plus a continuous original time kernel
000123 | is enough.
000124 | 
000125 | This is an elementary spectral compactness argument, not a new general
000126 | theorem of operator-algebra theory. The new use is the joint source/time/
000127 | observable test at the actual TFPT neutral-pair loophole.
000128 | 
000129 | 4. Application to a spinor endpoint and the exact scope
000130 | 
000131 | Under the separately established fixed D8/even-CAR dictionary, the original
000132 | one-chiral-edge NS Fock representation restricts to 1 direct-sum v.
000133 | Neither s nor c is in that global folium. Adding the retained opposite
000134 | physical edge gives multiplicities, not a new left D8 representation.
000135 | Therefore a nonzero charged spinor endpoint state, whose neutral-observable
000136 | representation is s or c, cannot be obtained by T_N satisfying A-D while
000137 | retaining that SAME full D8 observable map. This covers arbitrary
000138 | nonunitary neutral-pair histories and all scalar field renormalizations
000139 | for which the claimed full norms and original time kernels converge.
000140 | 
000141 | The D8 sector statement uses the standard free even-CAR sector theorem
000142 | (Boeckenhauer, arXiv:hep-th/9507047 sections3-6), transported through the
000143 | EXISTING fixed dictionary. It is not a proof that P1 selects that dictionary,
000144 | or that the primitive QWZ source is eight independent complex flavours.
000145 | 
000146 | This extends the previous bounded-mean-energy compactness test and the
000147 | unitary 2pi/eps bound. It does not promote spectrum counting into a
000148 | classification of every possible local scaling net. The crucial EXTRA
000149 | hypothesis is B: unchanged low-window action of the retained observables.
000150 | A local new field representation must actually be compared, not named.