TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/charged_source_alternatives_20261027/CHIRAL-SEWING-SELECTION.txt
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000001 | CHIRAL SEWING AS AN ALTERNATIVE E8 SELECTION TEST
000002 | 27 October 2026
000003 | 
000004 | DECISION
000005 | 
000006 | Within the already classified connected local unitary same-conformal
000007 | extensions of U=V_D5 tensor V_A3, preservation of the single chiral vacuum
000008 | block by exchange of the two torus cycles selects precisely the two marked
000009 | E8 extensions. The original h=(1,1) marking selects the plus embedding once
000010 | that marked field identification is supplied.
000011 | 
000012 | This is a standard finite Fourier consequence of the existing extension
000013 | classification. It supplies an alternative source test; it does not derive
000014 | the cycle-exchange premise from P1 and is not claimed as a new general CFT
000015 | classification theorem. A scalar paired partition function, a pure CAR
000016 | state, or a clock action at the fixed square torus does not provide that
000017 | premise.
000018 | 
000019 | 1. THE EXISTING EXTENSION CLASS
000020 | 
000021 | Retain the exact class of SIMPLIFICATION.txt:
000022 | 
000023 |   U=V_D5 tensor V_A3,
000024 |   A=U-lattice discriminant group=C4 x C4,
000025 |   q(a,b)=(5a^2+3b^2)/8 mod1,
000026 |   b((a,b),(c,d))=(5ac+3bd)/4 mod1.
000027 | 
000028 | W is a connected local unitary VOA extension with the SAME conformal vector.
000029 | The old theorem derives finite support and multiplicity one. Its support
000030 | H is one of exactly four isotropic subgroups:
000031 | 
000032 |   H0={(0,0)},
000033 |   H2={(0,0),(2,2)},
000034 |   Hplus={(a,a):a in C4},
000035 |   Hminus={(a,-a):a in C4}.
000036 | 
000037 | They realize U,D8,E8plus,E8minus. No assumption of a pre-existing marked
000038 | spinor generator is made in this four-way class.
000039 | 
000040 | 2. CYCLE EXCHANGE ON THE ACTUAL DISCRIMINANT DATA
000041 | 
000042 | Use the full labelled character/conformal-block space C[A], or retain Cartan
000043 | chemical potentials if scalar characters are used. This avoids confusing
000044 | conjugate or otherwise equal unrefined characters.
000045 | 
000046 | For Cartan-refined characters, transform the background by z -> z/tau and
000047 | retain the common Jacobi anomaly factor before extracting the finite matrix.
000048 | The lattice modular S action on the labelled blocks is the finite Fourier transform
000049 | 
000050 |   S_(x,y)=|A|^(-1/2) exp(-2 pi i b(x,y))
000051 |          =(1/4) exp(-2 pi i b(x,y)).
000052 | 
000053 | Let v_H=sum_(h in H) e_h be the branching vector of the extension vacuum.
000054 | Character orthogonality gives the exact identity
000055 | 
000056 |   S v_H=(|H|/4) v_(H-perp).                              (1)
000057 | 
000058 | Indeed its x coordinate is (1/4) sum_(h in H) exp(-2 pi i b(x,h)).
000059 | This sum is |H|/4 for x in H-perp and zero otherwise.
000060 | 
000061 | Require that the single chiral vacuum line is preserved: S v_H=lambda v_H.
000062 | Unitarity gives |lambda|=1, while its vacuum coordinate is lambda=|H|/4>0.
000063 | Consequently |H|=4, lambda=1 and H=H-perp. The converse is immediate.
000064 | 
000065 |   support      vacuum coefficient after S     single chiral block fixed
000066 |   H0                         1/4                         no
000067 |   H2                         1/2                         no
000068 |   Hplus                      1                           yes
000069 |   Hminus                     1                           yes
000070 | 
000071 | Thus the cycle-exchange condition is equivalent, within this class, to the
000072 | extension being holomorphic. It selects E8 up to the two labelled embeddings.
000073 | It is not logically weaker than selecting the relevant extension. Its value
000074 | is that global sewing may be the quantity directly specified by a source,
000075 | whereas an individual charged generator may not be directly given.
000076 | 
000077 | Because all four supports are isotropic, the relative twist matrix T0 has
000078 | T0 v_H=v_H. The common gravitational phase exp(-2 pi i c/24), c=8, is retained
000079 | in the physical modular T transformation; it is not set to one.
000080 | 
000081 | For an unrefined character function the same necessary cardinality follows
000082 | from the leading vacuum coefficient after S, since all nonvacuum U modules
000083 | have positive lowest weight. Unrefined functions do not distinguish the two
000084 | E8 embeddings; the labelled formulation retains that distinction explicitly.
000085 | 
000086 | 3. WHY A SCALAR TOTAL ANSWER IS NOT ENOUGH
000087 | 
000088 | Let chi be the column of the sixteen U characters. The paired scalar
000089 | 
000090 |   Z_pair(tau,bar tau)=chi(tau)^* M chi(tau), M=I16
000091 | 
000092 | is invariant under S and T, has nonnegative integral coefficients and vacuum
000093 | multiplicity M00=1. It pairs all sixteen modules. Its pairing matrix has
000094 | rank16, while a factorized single chiral vacuum amplitude would have the
000095 | rank-one form v_H v_H^*.
000096 | 
000097 | The diagonal unitary full CFT is a standard example of this distinction.
000098 | It has both chiralities. It is not a countermodel satisfying all original
000099 | P1 requirements and is not a proposed replacement TFPT source. It disproves
000100 | only the implication 'scalar modular invariant answer implies a single
000101 | holomorphic chiral vacuum block'.
000102 | 
000103 | Likewise the equality of a scalar value before and after tau=i -> -1/i is
000104 | tautological. It is not an intertwining statement on the full labelled block
000105 | space, nor a modular identity for an entire family.
000106 | 
000107 | Purity of a positive CAR state concerns a covariance on a one-particle space.
000108 | Rank one of the pairing matrix M concerns a genus-one conformal-block space.
000109 | No map equating these two notions is furnished by the CAR projector theorem.
000110 | 
000111 | 4. DECISIVE ORIGINAL SOURCE CHECK
000112 | 
000113 | The original spectral-bordism category in tfpt45 specifies tuples
000114 | (A_+,H_+,D_+,J,Gamma), reflection positivity, finite relative action and a
000115 | nonzero seam index. Its morphisms are unitary intertwiners of those data.
000116 | The cited definition supplies no torus conformal-block functor or cycle
000117 | exchange operator on the sixteen U modules.
000118 | 
000119 | The original primitive completion imports the carrier packet and family
000120 | data into a discrete classification. It does not define the chiral vacuum
000121 | branching vector or prove equation (1) describes a physical P1 sewing map.
000122 | The earlier minimality audit already records the missing algebra-fiber
000123 | classification; its absence is reused, not presented as a new discovery.
000124 | 
000125 | The exact new source question is therefore:
000126 | 
000127 |   Does the actual original P1 sewing functional, after its explicit map to
000128 |   the local U theory, define a single chiral vacuum block v_H and transport
000129 |   it to itself under the exchange of the two seam cycles?
000130 | 
000131 | To answer yes one must supply the block map, retained charge marking and
000132 | actual cycle-exchange action. The original source time
000133 | H=8[N+p^2/2+delta.p] must not silently be replaced by conformal L0; its
000134 | chemical background and time normalization require their actual transport.
000135 | A full paired norm, determinant scalar, eta
000136 | phase, fixed-torus quarter symmetry, or pure CAR covariance alone does not
000137 | answer it. If the answer is yes, the existing finite classification closes
000138 | the extension to E8 without a separate search for the (16,4) generator.
000139 | 
000140 | 5. HISTORICAL ROUTE REUSED
000141 | 
000142 | The general holomorphic completion route is already present in the older
000143 | papers. The July20 v490 entry explicitly records mu_pre=4 and theta_v=1,
000144 | while the condensation mu=4 -> 1 is a cited conditional theorem and the
000145 | original seam equivalence remains open. The later celestial/twistor account,
000146 | steps8-10, records the same 16 -> 4 -> 1 index chain and distinguishes
000147 | lattice witnesses from continuum theorems.
000148 | 
000149 | The present calculation is the explicit discriminant-group S-vector form
000150 | of that selection problem. It connects the old conditional completion to
000151 | the newer exhaustive four-extension classification. It is not a newly
000152 | discovered holomorphy route, and no old lattice witness is promoted to the
000153 | actual original P1 conformal-block functor. An ordinary unique CFT vacuum
000154 | also is not a one-dimensional torus conformal-block space.
000155 | 
000156 | 6. REPRODUCIBLE CHECK
000157 | 
000158 | check_modular_sewing.py computes the exact 16x16 S and relative T matrices,
000159 | checks unitarity and S^2=charge conjugation, proves (1) on all four already
000160 | classified supports, and checks the rank16 paired control. It uses exact
000161 | Gaussian rationals and algebraic phases, with explicit exceptions rather
000162 | than Python assert statements.
000163 | 
000164 | The finite computation certifies the finite representation identities. The
000165 | source statement in section4 is an audit of the cited original passages;
000166 | the computation does not prove a physical P1 map.
000167 | 
000168 | SOURCES
000169 | 
000170 | - experiments/theory-contracts/e-series-source-reconstruction-20261024/
000171 |   SIMPLIFICATION.txt, sections1-5: original four-extension classification.
000172 | - Same contract, ORIGINAL-SOURCE-AUDIT.txt, lines1-184: original minimality
000173 |   does not classify the operational source algebra fiber.
000174 | - documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/
000175 |   sections/002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex,
000176 |   lines250-334: actual spectral-bordism objects and unitary morphisms.
000177 | - documents/archive--paper-latex--tfpt-42/sections/
000178 |   027-subsection-primitive-completion-from-the-boundary-kernel.tex,
000179 |   lines1-195: actual primitive kernel and imported discrete carrier steps.
000180 | - Huang,Kirillov,Lepowsky: https://arxiv.org/html/1406.3420,
000181 |   extension/algebra correspondence. Its prerequisites remain explicit.
000182 | - Smith,Lin,Tachikawa,Zheng: https://arxiv.org/html/2303.16917v2,
000183 |   sections3.1-3.2: standard chiral c=8 modular reconstruction. This is
000184 |   background supporting the route, not a proof of P1 chiral sewing.
000185 | 
000186 | - documents/changelog/sections/008-source.tex, lines208-228: original v490
000187 |   spin-parity census and conditional condensation; no origin gate closed.
000188 | - documents/tfpt_3_e8_audit_bootstrap/sections/
000189 |   011-section-the-celestial-and-twistor-continuum-route-label-.tex,
000190 |   lines147-203: existing v501/v504 index and completeness route.