TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/ATLAS-KMS-LOCALITY-AND-SOURCE-WINDING.txt
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000075 | 4. Exact thermal discriminator for the existing global odd-spinor Phi
000076 | 
000077 | Keep the existing ONE-chiral-source fold, p in Z8=D8 union v,
000078 |   H_source=8[Nosc+p.p/2+delta.p], Q=sum p, P=(-1)^Q.
000079 | The already proved Phi is identity on D8 and maps odd v to c by rho(p)=P8p+r,
000080 | with the given oscillator permutation. It preserves H_source,Q and vacuum.
000081 | Write P8 for the coordinate permutation; it is distinct from parity P.
000082 | Its target charges q are integral on D8 and half-integral on c. Therefore
000083 | for EACH of the eight fixed named Cartans,
000084 |   exp(2pi i N_k^old)=I,
000085 |   Phi* exp(2pi i N_k^target) Phi=P.                  (1)
000086 | This is a bounded FULL-word observable. Q=sum N_k is preserved nevertheless;
000087 | total charge cannot replace individual Cartan actions.
000088 | 
000089 | There is an even shorter FULL VACUUM-WORD discriminator, needing no thermal
000090 | preparation. Let U_k be the respective2pi Cartan operator and use the actual
000091 | given charge-one SOURCE winding W1,unitary on the full source charge ladder.
000092 | Since P W1=-W1 P, the same named four-operation word satisfies
000093 |   W1_old* U_k^old W1_old (U_k^old)*=I,
000094 |   Phi*[(W1_target)* U_k^target W1_target (U_k^target)*]Phi=-I. (1a)
000095 | Thus its vacuum expectation is+1 versus-1. Phi preserves the vacuum VECTOR,
000096 | not this word with the unchanged named Cartans. This gives a precise
000097 | source-current/winding comparison required BEFORE invoking KMS uniqueness.
000098 | It does not identify the SOURCE winding with the unknown primitive P1 map.
000099 | Fixed c-number normal-ordering shifts multiply U_k by scalar phases; those
000100 | phases cancel in(1a). They cannot repair its opposite four-word signs.
000101 | 
000102 | Let Z_e=Tr_even exp(-beta H_source), Z_o=Tr_odd exp(-beta H_source).
000103 | Both are finite and strictly positive at every finite beta>0. Phi preserves
000104 | the unflavoured partition function Z_e+Z_o at ALL temperatures. But(1) gives
000105 |   <exp(2pi i N_k^old)>_beta=1,
000106 |   <exp(2pi i N_k^target)>_(Phi beta Phi*)
000107 |      =(Z_e-Z_o)/(Z_e+Z_o) !=1.                       (2)
000108 | Hence equal H,Q,vacuum and every unflavoured Gibbs partition function do
000109 | not establish the unchanged full physical Cartan/field-state dictionary.
000110 | KMS is valid in each specified model; it cannot identify the two named
000111 | observable operators. This is an exact consequence of the already known
000112 | charge-periodicity obstruction, applied to the alleged thermal rescue.
000113 | 
000114 | The actual chiral source spectrum gives, with alpha1/4or3/4,
000115 |   Z=prod_(m>=0)(1+e^-beta(m+alpha))(1+e^-beta(m+1-alpha)),
000116 |   S=Tr(P e^-beta H_source)
000117 |    =prod_(m>=0)(1-e^-beta(m+alpha))(1-e^-beta(m+1-alpha)).
000118 | Their convergence follows from the source heat trace. Each factor is
000119 | positive and sum of its deviations finite, so 0<S/Z<1 at every beta>0.
000120 | Then Z_e=(Z+S)/2,Z_o=(Z-S)/2. Both quarters have the same products after
000121 | permuting the particle/hole towers. No new lattice sector is inserted.
000122 | 
000123 | 5. Both edges and the proposed global-parity repair
000124 | 
000125 | Retain both actual limiting edges and their Hamiltonians. If ONLY total
000126 | fermion parity is projected even, left-odd/right-odd states remain. On this
000127 | space the left target Cartan holonomy in(1) is P_L, with expectation
000128 |   (Z_Le Z_Re-Z_Lo Z_Ro)/(Z_Le Z_Re+Z_Lo Z_Ro) !=1.   (3)
000129 | For the actual equal edge spectra set r=S/Z; this is 2r/(1+r^2)<1.
000130 | The discrepancy is (1-r)^2/(1+r^2), strictly positive at finite beta.
000131 | Both-edge thermal correlations must therefore be kept. Total-even GSO
000132 | does NOT erase the witness by eliminating the whole left odd sector.
000133 | 
000134 | Projecting LEFT parity even instead WOULD erase(2), but removes the given
000135 | charge-one SOURCE winding as an internal operator. The actual original
000136 | chiral W1 satisfies [Q,W1]=W1, hence
000137 |   P W1=-W1 P, Pi_e W1 Pi_e=0.                        (4)
000138 | This does not say a paired left/right pump is removed by TOTAL parity:
000139 | paired odd transitions may survive, precisely as(3) retains odd/odd states.
000140 | A new boundary/gauge sector could retain W1 as an inter-sector operation;
000141 | its locality, state and products would have to be derived, not inferred.
000142 | The SOURCE circle is still not identified with the primitive P1 core
000143 | circle. No division of spectral flow by8or16, no primitive-origin claim.
000144 | 
000145 | 6. Decision and exact remaining physical structure
000146 | 
000147 | The candidate 'use KMS/local modular uniqueness to turn the known global
000148 | Phi into the missing microscopic local spinor map' is excluded under the
000149 | stated unchanged observable/time comparison. Its abstract theorem already
000150 | needs B and its correct local modular dynamics does not supply B. Its
000151 | thermal original-Cartan responses fail(2), including the total-even two-edge
000152 | repair(3); suppressing all left odd states loses SOURCE W1 by(4).
000153 | 
000154 | A successful route must derive an actual local charged algebra/field map
000155 | with its physical Cartan action, state, original time and primitive winding
000156 | comparison. A P1-derived different net/Gauss/statistics dictionary is not
000157 | excluded. The local KMS theorem is useful AFTER that construction; it can
000158 | transport its state/time data then. This is not a new independent coupling
000159 | to choose, and not an all-P1 or all-spinor impossibility theorem.
000160 | 
000161 | No FE-GEN/ALG-EXH or full E8 origin. No verification/paper/ledger promotion.
000162 | Independent read-only formula review is recorded separately. The numerical
000163 | thermal products/finite source controls corroborate the written identities;
000164 | the all-beta and continuum statements are analytic, not extrapolations.