TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/FOLD-COVARIANCE-TO-NATIVE-FIELD-RESPONSE.txt
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000053 | 2. EXACT COVARIANCE: DIAGONALIZE THE SAME FOLD, DO NOT ADD COPIES
000054 | 
000055 | For the negative-D chiral presentation let
000056 |   f_(n,alpha)(theta)=exp(i(n+alpha)theta)/sqrt(2pi), 0<alpha<1,
000057 |   C_alpha=1_(D_alpha<0), D_alpha=-i partial_theta.
000058 | The already recorded fold is
000059 |   (Ff)_r(x)=f((x+2pi r)/8)/sqrt8, r=0,...,7.
000060 | Write n=8l+k and mu_k=(k+alpha)/8. Then
000061 |   F f_(8l+k,alpha) = U_alpha(:,k) exp(i(l+mu_k)x)/sqrt(2pi),
000062 |   U_alpha(r,k)=exp(2pi i(k+alpha)r/8)/sqrt8.
000063 | U_alpha is unitary: it is a diagonal unitary times the exact 8-point DFT.
000064 | This proves the full spectral diagonalization, not merely a finite check.
000065 | 
000066 | Since 8l+k+alpha<0 iff l<=-1 for every k, in this EIGENCHANNEL chart
000067 |   (U_alpha* F C_alpha F* U_alpha)_(jk)
000068 |       =delta_jk C^-_(mu_k),
000069 |   C^-_mu(s)=sum_(l<=-1) exp(i(l+mu)s)/(2pi).
000070 | In the original ARC chart the off-diagonal correlations are retained by
000071 | U_alpha diag(C^-_mu) U_alpha*. No product of eight independent arc vacua has
000072 | been introduced. A diagonal quasifree covariance factorizes in this actual
000073 | eigenmode decomposition; that is a derived property, not extra copies.
000074 | All finite CAR words follow from this same covariance by the quasifree
000075 | determinant rule. The quotient localization remains a stipulated input.
000076 | 
000077 | For the original positive-filled presentation, the identical calculation
000078 | gives l>=0 in every eigenchannel. This is explicitly verified against
000079 | j-a=n+alpha and j>=1, with alpha=1-a. No negative-sea reinterpretation is
000080 | silently imposed on that original formula.
000081 | 
000082 | 3. THE NORMAL-ORDERING TRANSLATION AND TIME ARE FIXED TOGETHER
000083 | 
000084 | Off coincidence, suppressing only the common factor1/(2pi),
000085 |   C^-_mu(s)=exp(i delta s)/(2i sin(s/2)), delta=mu-1/2.
000086 | Subtract the SAME NS kernel with delta=0. The finite point-split term is
000087 |   lim_(s->0) [exp(i delta s)-1]/[2i sin(s/2)] = delta.
000088 | For the original positive halfline the off-diagonal kernel is the negative
000089 | of this one and the corresponding difference is -delta. Fermion ordering
000090 | and the charge-current sign must be transported with it.
000091 | 
000092 | Let H_sea denote currents normal ordered in the actual sea. In the fixed
000093 | negative-D charge convention the conformal NS currents have the form
000094 |   J_delta(z)=H_sea(z)+delta/z.
000095 | The sign is checked by the actual particle/hole gaps:
000096 |   1/2+delta=mu, 1/2-delta=1-mu.
000097 | It is not a chosen coupling. The counterterm is the difference of two
000098 | specified kernels of the same source. A global periodic gauge equivalence
000099 | is not asserted; the endpoint holonomy is retained.
000100 | 
000101 | On source CAR charge p in Z8 and oscillator number N, the physical excitation energy is
000102 |   H_exc=8[N+|p|^2/2+delta.p].
000103 | The factor8 is inherited from the original fold; the additive conformal
000104 | ground energy |delta|^2/2 is subtracted only when defining the source's
000105 | normal-ordered excitation generator. E8 charges enter only with the separately
000106 | stipulated extension. This does not identify L0 with a
000107 | physical second or delete a holonomy from charged states.
000108 | 
000109 | After existing M and O, put
000110 |   d=M delta, b=U_+^T O^T d,
000111 | where U_+,a=(e_(2a)-i e_(2a+1))/sqrt2, J U_+=i U_+.
000112 | Explicitly,
000113 |   alpha=1/4: b=((7-9i)/32,(1-i)/4,-i/4,(3-3i)/32),
000114 |   alpha=3/4: b=((5-11i)/32,(1-i)/4,-i/4,(1-i)/32).
000115 | Both have ||b||^2=85/256 and ||d||^2/2=85/256.
000116 | 
000117 | 4. QUARTIC AND CHARGED FIELD MAPS, WITH THEIR EXACT SCOPE
000118 | 
000119 | Because the four J-positive currents are bilinearly isotropic, they have no
000120 | mutual self-contractions. Translation therefore gives the literal field law
000121 |   F_i^(d)(z)=:f_i(h_+(z)+b/z):.
000122 | It is a local field identity on the punctured quotient chart with the
000123 | original twist retained. The current derivative/erasure rule commutes with
000124 | this affine substitution; actual f_i and partial_a f_i are not replaced
000125 | by abstract fields of equal dimension.
000126 | 
000127 | For the ALREADY STIPULATED E8 extension this is exactly the restriction of
000128 | Li's Delta construction (arXiv:1004.0864, eq2.4, Proposition2.1):
000129 |   Delta(d,z)=z^(d.h0) exp[sum_(n>0) d.h(n)/(-n) (-z)^(-n)],
000130 |   Y_d(a,z)=Y(Delta(d,z)a,z).
000131 | It extends the current shift to a twisted module and respects the full
000132 | vertex product. Root fields satisfy
000133 |   Y_d(e^beta,z)=z^(d.beta)Y(e^beta,z).
000134 | The conformal modes obey
000135 |   L_n^(d)=L_n+d.h(n)+|d|^2/2 delta_(n,0).
000136 | These are representation consequences of a supplied E8 algebra and d,
000137 | not a derivation of its physical extension from odd CAR fields.
000138 | 
000139 | The twist is sigma_d=exp(2pi i d.h0), of order16 on the actual roots.
000140 | It is the original fold g, not the previously tested g^2: M1/2=e3+e6 is
000141 | an E8 root and nu=M(2mu) satisfies nu-2d=M1 in E8, so sigma_d^2 is that
000142 | order8 g^2. All240 phases are checked. The original internal order4
000143 | character v remains a DIFFERENT named datum.
000144 | 
000145 | Apply the shift exactly ONCE: either shifted fields on the underlying
000146 | vacuum copy with h0 Omega=0, or ordinary fields on a charge-d Heisenberg
000147 | Fock module. Not both. No assertion d in E8 is made. The concrete p=0
000148 | vacuum copy is inherited from the source sea; energy minimization does not
000149 | select it within the extension. In fact e1+e4 has1+d.(e1+e4)=0 for both
000150 | quarter cases, giving another equal-energy root primary, as already known.
000151 | Li's theorem does not supply a new unique physical vacuum.
000152 |