TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/native-arc-charge-source-interface-20261026/ELLIPTIC-CURRENT-SOURCE.txt
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000065 | For any finite ordered list use the ORIGINAL central commutator
000066 |  c(f,g)=sum_(m>0)m[conj(f_m)g_m-f_m conj(g_m)],
000067 |  omega(W(f1)...W(fk))=exp[-1/2 sum_(i<j)c(fi,fj)]
000068 |                       times omega(W(sum_i fi)).         (5)
000069 | The covariance,normalization,charge factor and phase in(3)-(5) follow from
000070 | the existing geometry and free source. They are not a physical intervention
000071 | schedule or an identification with exp(-S_prim) for arbitrary operator
000072 | variations. Coherent state interventions and gauge orbits of D remain distinct.
000073 | 
000074 | 5. Independent derivation of the neutral current norms
000075 | Direct fermionic Wick contraction with(1) yields,for every integer m>0,
000076 |  omega(J_m J_-m)=sum_r c_r(1-c_(r+m)),
000077 |  omega(J_-m J_m)=sum_r c_(r+m)(1-c_r).
000078 | Put y=exp(bm). Each term obeys
000079 |  c_r(1-c_(r+m))=(c_r-c_(r+m))/(1-y^-1),
000080 |  c_(r+m)(1-c_r)=(c_r-c_(r+m))/(y-1).
000081 | The infinite difference sum telescopes to m,since c_(-infinity)=1 and
000082 | c_(+infinity)=0. Therefore,without a regulator inference,
000083 |  omega(J_m J_-m)=m/(1-exp(-bm)),
000084 |  omega(J_-m J_m)=m/(exp(bm)-1).                          (6)
000085 | Their difference is the unchanged level1 central term m. This both fixes
000086 | the current algebra and verifies the coth covariance in(4) independently.
000087 | 
000088 | 6. Neutral field words from the same current functional
000089 | The previous exact bilocal reconstruction on each chargeq sector carries
000090 | (z/w)^(q+alpha) times the original sea prefactor and current exponentials.
000091 | Average this SAME operator identity with rho_alpha using(3)-(4).
000092 | For the grouped neutral word
000093 |  psi*(z1)...psi*(zn) psi(wn)...psi(w1),
000094 | put R=product_i zi/wi, A_m=sum_i(zi^m-wi^m),
000095 | B_m=sum_i(wi^-m-zi^-m), and C0(z,w)=w/(z-w).
000096 | Its current expression is
000097 |  R^alpha det[C0(zi,wj)]
000098 |  times [sum_q exp(-bF_alpha(q))R^q / Theta_alpha(0)]
000099 |  times exp[sum_(m>0) A_m B_m/(m(exp(bm)-1))].             (7)
000100 | The current commutators of the grouped word,its exact vacuum Cauchy
000101 | determinant and cyclicity prove the charge-sector operator identity,as in
000102 | the preceding bilocal contract. Taking the thermal trace proves(7).
000103 | The same expression equals det[K_alpha(zi,wj)],where
000104 |  K_alpha(z,w)=(z/w)^alpha sum_r (z/w)^r c_r.
000105 | For this analytic formula keep consistent branches and
000106 |  1<|zi/wj|<exp(b), with separated radially ordered endpoints.
000107 | Physical boundary values retain the original i0 and smearing prescriptions.
000108 | The grouped word order is essential. Alternating bilocal words require
000109 | their actual Wick order,including I-C contractions where appropriate.
000110 | Equations(4)-(5) independently cover arbitrary ordered WEYL words.
000111 | Since c_r-1_(r<0) decays exponentially,its spatial kernel is smooth with
000112 | all derivatives. The ultraviolet singular part and the old native point-
000113 | split subtraction are unchanged. Thus the actual transported native
000114 | currents,quartics and neutral W8 can be evaluated in this source with their
000115 | old operator definitions and single conformal shift. Their expectations
000116 | are not silently equated with their earlier Hardy values. Existing energy
000117 | bounds,domains and point-split limits remain the assumptions for unbounded
000118 | local composite realizations;finite determinant samples do not replace them.
000119 | 
000120 | 7. Decisive comparison with the original Hardy source
000121 | On relative charge0 the theta factor in(4) drops out,but thermal oscillators
000122 | remain. In particular the EXACT neutral current norm is
000123 |  omega_(ell,Q0)(J_-1 J_1)=1/(exp(4pi)-1)
000124 |                         =0.00000348735451780811656...>0. (8)
000125 | The original Hardy state has J1 Omega=0 and hence norm0 for this SAME
000126 | current and time dictionary. Removing global charge fluctuations does not
000127 | repair the state mismatch. This is not inferred from numerical regulators.
000128 | It is also a strictly local current distinction:choose nonzero real smooth
000129 | f with support in any proper interval and mean0,for example a derivative
000130 | of a compactly supported bump. Then
000131 |  omega_(ell,Q0)(W(f))/omega_Hardy(W(f))
000132 |    =exp[-sum_(m>0) m|f_m|²/(exp(bm)-1)]<1.               (9)
000133 | Consequently the existing source-current-preserving,unchanged-state
000134 | identification with Hardy fails already for the neutral current subnet.
000135 | No exclusion of other P1 states,differently derived current/time maps or
000136 | admissible subalgebras follows. Purifying the marginal preserves its current
000137 | expectations and therefore does not turn(8) into0.
000138 | 
000139 | 8. What the checker establishes
000140 | Exact symbolic Fermi telescoping identities and cover/quarter scaling.
000141 | Exact charge-resolved Jacobi partition coefficients through tdegree80 for
000142 | both quarters,177 energy/charge monomials each,with t=exp(-b/4).
000143 | Direct Fermi sums and charge partition comparison at140 digit precision.
000144 | Independent finite fermion determinants for real profile Weyls and ordered
000145 | pairs at cuts16/32:errors about6e-12 down to2e-15. Their limits are not
000146 | proved by two cutoffs;the analytic bosonization proof supplies(4)-(5).
000147 | Grouped Wick determinants with1,2,4,8 pairs match(7) to about1e-126 or
000148 | better. The infinite Hardy sea is summed analytically;only exponentially
000149 | decaying thermal corrections are cut in those diagnostics.
000150 | Wrong b=2pi at fixed quarter generator and omitted Weyl central phases are
000151 | detected. Normal and-OO replay preserve all RuntimeError checks.
000152 | 
000153 | 9. Remaining physical origin,not a new Gaussian assertion
000154 | This closes the free same-E kernel -> current functional -> neutral native
000155 | field evaluation route in the explicitly retained Gaussian premise.
000156 | The original primitive operational Seed already includes A_loc,time,Theta,
000157 | state,seam and collar;its C_seed map points from these data to boundary data.
000158 | The relative spectral action is not automatically the Weyl functional(4).
000159 | To identify the present source with original P1,derive that actual measure
000160 | and insertion map,with proper regional admissibility and closed domains.