TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/native-arc-charge-source-interface-20261026/BILOCAL-RECONSTRUCTION.txt
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000001 | UR.NATIVE.ARCCHARGE.BILOCAL.01
000002 | 2026-10-26. Reconstruct the actual neutral transfer fields from original current.
000003 | 
000004 | 1. Question, changed premise and relation to the previous result
000005 | The previous exact obstruction applies to the additive scalar density/stress
000006 | net on eight separate proper arcs. It does not exclude current operators
000007 | whose containing connected interval crosses the intervening gaps.
000008 | Test whether the actual neutral CAR transfer product has an exact original
000009 | current expression before adding a new field-dependent interaction.
000010 | Success: same matrix elements, quarter phase and original normal ordering.
000011 | Abort: a mismatch at the first finite-energy matrix element or a hidden change
000012 | of vacuum, current normalization or source support.
000013 | This continues the existing source-algebra route; it is not a new toy dynamics.
000014 | 
000015 | 2. Exact original conventions
000016 | Keep the original single-circle CAR a_r, r integer, with occupied sea r<0.
000017 | Original J_m=sum_r :a_r* a_(r+m): has [J_m,J_n]=m delta_(m+n,0).
000018 | The relative Q=0 sector has the actual original partition/Maya realization
000019 | r_i=lambda_i-i, J_-m=p_m, J_m=m partial_(p_m).
000020 | Define the analytically point-split coefficient field
000021 |  B(z,w)=sum_(r,s) a_r* a_s z^r w^(-s).
000022 | Take0<|w|<|z|<1 on fixed branches for the analytic calculation.
000023 | For the physical cylinder endpoints z=exp(-i theta_x),w=exp(-i theta_y),
000024 | the original quarter field product has the additional factor(z/w)^alpha,
000025 | alpha=1/4 or3/4. The8fold normalization1/8 remains in eta_r* eta_s.
000026 | This follows directly from z^(r+alpha) w^(-(s+alpha))
000027 | =(z/w)^alpha z^r w^(-s). The checker only tests consistency with this
000028 | analytically derived factor; it does not independently infer the twist.
000029 | The boundary values are field distributions; analytic coordinates are not
000030 | extra physical positions or a newly chosen time.
000031 | 
000032 | 3. Reconstructed bilocal identity
000033 | On the relative charge0 finite-energy matrix elements,
000034 |  B(z,w) = [w/(z-w)]
000035 |    exp(sum_(m>0) (z^m-w^m) J_-m/m)
000036 |    exp(sum_(m>0) (w^(-m)-z^(-m)) J_m/m).                 (1)
000037 | There is no fitted coefficient. The prefactor is the infinite original sea
000038 | contraction sum_(r<0)(z/w)^r=w/(z-w), not a finite occupied-sea surrogate.
000039 | On a fixed chargeq sector, the prefactor acquires(z/w)^q. The physical
000040 | quarter factor must additionally be retained; Q=0 does not set alpha=0.
000041 | 
000042 | Proof of (1): Direct CAR gives [J_m,B]=(z^m-w^m)B for m nonzero.
000043 | Using the displayed CCR, the two ordered exponentials obey exactly these
000044 | same commutators, both positive and negative m. Their vacuum matrix element
000045 | is1, fixing the scalar from the full sea contraction. Moving arbitrary
000046 | positive and negative current words across either side now gives identical
000047 | matrix elements on all finite-energy Heisenberg descendants. Those states
000048 | are the full original Q=0 sector by the existing exact Maya/Schur map.
000049 | Thus (1) is an all-degree matrix-element identity, not inferred from a small
000050 | numerical sample. Radial damping controls the vacuum creation series;
000051 | annihilation-mode series terminate on every finite-energy input.
000052 | Separate closed-operator boundary/point-split limits are not supplied by
000053 | finite matrices alone. Existing native limits are retained, not redefined.
000054 | The same source time is respected: an a_r* a_s term has relative energy r-s,
000055 | which equals (z partial_z+w partial_w) acting on its coefficient. In the
000056 | existing K_alpha normalization divide this relative source energy by8.
000057 | No modular time or new rate is introduced.
000058 | 
000059 | 4. Exact independent finite-energy checks
000060 | The checker imports the ACTUAL original infinite-Maya add_ribbons current
000061 | constructor, then builds the two graded current exponentials in (1).
000062 | Independently it computes CAR matrix entries by occupied-mode replacement
000063 | signs, summing the unchanged infinite diagonal sea analytically.
000064 | It checks every matrix element on all degrees<=4(12 states) and<=8(67 states)
000065 | at two rational point pairs,0<w<z<1. Both original quarter factors are checked;
000066 | omitting them is a nonzero negative control. All comparisons are exact.
000067 | Truncation here selects inspected input/output matrix elements. It is not
000068 | an executed physical projection or a finite-dimensional CCR representation.
000069 | Because annihilators act first and creators only increase degree, this
000070 | ordered calculation retains every intermediate state contributing to the
000071 | chosen matrix elements. No equivalent claim holds for multiplying arbitrary
000072 | separately compressed bilocal matrices.
000073 | Vacuum Wick determinants with1,2,4 and8 neutral pairs match the
000074 | Cauchy determinant/product identity, including its signs.
000075 | These are scoped checks of (1), not a new empirical result.
000076 | 
000077 | 5. Spatial algebra statement and its exact limitation
000078 | For a CONNECTED proper containing interval I, the complex fermion net has
000079 |  F(I)^U(1)=B(I), the level1 scalar current net on its neutral cyclic sector.
000080 | This is the established bosonization net theorem: Bischoff/Tanimoto,
000081 |  arXiv:1111.1671v3, Proposition3.8 and Section3.3,
000082 |  https://arxiv.org/html/1111.1671 .
000083 | The original quarter fields differ from the NS coefficient fields by the
000084 | fixed endpoint phase. Transport this specified field/domain convention;
000085 | do not replace their original geometric stress or state by an NS-vacuum
000086 | claim. On connected intervals the gauge-fixed algebra identification is
000087 | unchanged; the original current subtraction can add its fixed scalar term.
000088 | Therefore bounded neutral products a*(f_r)a(g_s) with supports in different
000089 | arcs of E lie in B(I) for a connected I containing E. They need not lie in
000090 |  B_add(E)=join_r B(I_r).
000091 | Indeed one smooth phase constant separately near each closed arc fixes
000092 | B_add(E) while multiplying the interarc product by a nontrivial phase.
000093 | On the original charge0 cyclic Hilbert space, locality also gives
000094 |  F(E)^U(1) restricted toQ0 subset B(E')',                 (2)
000095 | where B(E') is generated on the complementary intervals. No equality in(2)
000096 | or full classification of intermediate nets is claimed.
000097 | A current construction can thus use connecting gaps or the appropriate
000098 | relative commutant. It cannot silently relabel that construction as an
000099 | operation in the additive8arc net. Smooth Weyl strings approaching endpoint
000100 | vertices need the original renormalization; (1) is not a finite smooth
000101 | unitary pulse or an automatically executable physical intervention.
000102 | 
000103 | 6. Reconstructing the actual native fields, including W8
000104 | Use the already fixed U_alpha,V=M^t O U_+ and A_a=U_alpha diag(V[:,a])U_alpha*.
000105 | Insert (1), WITH its original endpoint phases, into each term eta_r* A_a[r,s]
000106 | eta_s of the EXISTING point-split native current. Retain precisely the old
000107 | reference contraction and the single conformal b/z shift.
000108 | This is an operator-distribution substitution at separated endpoints.
000109 | It reconstructs all four positive native currents and hence every native
000110 | quartic,erasure current polynomial and the existing neutral W8 completion
000111 | by their unchanged products/derivatives/counterterms. The17term W8 ansatz
000112 | is reused,not claimed newly selected. No charged E8 spinor extension or new
000113 | linearj.F coupling is needed for this algebraic composite construction.
000114 | All eight original source arcs remain the actual fermionic support; the
000115 | larger connected interval is a current-net representation of the same
000116 | neutral product, not a replacement for the support of the full CAR field.
000117 | 
000118 | 7. Corrected physical target
000119 | A direct F_a->delta_a D_rel rule would be needed to assert a PARTICULAR
000120 | new primitive field-dependent interaction or to identify a proposed
000121 | spectral-action variation with that field. It is not necessary merely to
000122 | construct an already existing neutral composite observable. Requiring a
000123 | new elementary coupling for each such composite would add an unnecessary
000124 | condition to the original source route.
000125 | For that route the actual outstanding object is the original P1 current
000126 | source image, with its full generating algebra,reference state,time,
000127 | reflection,regional support and admissibility. A unital normal state-preserving
000128 | star homomorphism of nets gives an isometry onto its cyclic image. Once the transported
000129 | interarc products and native point-split limits are admitted there, the
000130 | same construction gives their images; it does not select them by a fit.
000131 | A single local Weyl pair alone does not supply this map. Nor does an
000132 | abstract dual completion prove that P1 physically admits those operators.
000133 | Relative commutants transported by a unitary onto the cyclic image are
000134 | commutants WITHIN that image,not automatically full ambient P1 commutants.
000135 | 
000136 | 8. Original-source audit and remaining origin
000137 | The original operational seed definition already TAKES the quasilocal
000138 | observable net,time,reflection,state,seam class and collar as input.
000139 | Its C_seed functor maps this full seed TO the boundary tuple. The primitive
000140 | relative spectral action by itself does not state the reverse generating
000141 | current-net construction. The physical net is restricted to an invariant
000142 | admissible subspace,not defined by compressing arbitrary global operators.
000143 | Thus this continuation closes the neutral transfer reconstruction within
000144 | the existing source. It does not invert C_seed or derive the source state
000145 | from c3,E8 dimensions or a computer metaphor. Physical interactions,event
000146 | selection and source-to-P1 alignment retain their original contracts.
000147 | 
000148 | References: vonDelft/Schoeller,cond-mat/9805275v3,Sections6,9 and AppendixG
000149 | for constructive bosonization and point splitting; Bischoff/Tanimoto above
000150 | for the operator-algebraic connected-interval statement.
000151 | Originals: joint-native-algebra-selection-20261016/
000152 |  ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-GRADING.txt,33-161;
000153 | original-admissible-source-closure-20261025/
000154 |  QUARTER-STRESS-CURRENT-RECOVERY.txt;
000155 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/
000156 |  sections/002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex,103-143;
000157 |  sections/001-section-primitive-core-and-carrier-interface-handoff.tex,302-343.
000158 | 
000159 | Verdict: EXACT_SCOPED_NEUTRAL_CURRENT_VERTEX_RECONSTRUCTION;
000160 | PHYSICAL_P1_CURRENT_SOURCE_IMAGE_OPEN. Experiment only,no ledger promotion.