TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/charged_source_alternatives_20261027/MODULAR-SEWING-INDEPENDENT-REVIEW.txt Original file SHA-256: 71e46a6b7a60fc1a837bee0225912afd086dbf0bbdd5b98001499e37662cac20 Inclusive original lines: 1-200 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | INDEPENDENT REVIEW: CHARGE-RESOLVED MODULAR SEWING 000002 | 27 October 2026 000003 | 000004 | VERDICT 000005 | 000006 | The proposed finite modular statement is correct. Inside the already proved 000007 | class of connected local unitary same-conformal extensions of 000008 | 000009 | U = V_D5 tensor V_A3, 000010 | 000011 | a one-dimensional chiral vacuum block closed under cycle exchange selects 000012 | exactly the two Lagrangian supports Hplus and Hminus. Both extensions are 000013 | V_E8. The criterion therefore selects the unmarked E8 endpoint without 000014 | assuming occurrence of the particular local module M_(1,1). 000015 | 000016 | It does not select the fixed-U marked embedding Hplus. Hminus passes the same 000017 | modular test. The two become equivalent only after allowing the A3 diagram 000018 | reversal already identified in SIMPLIFICATION.txt. Thus the modular criterion 000019 | is a less marked source test for the abstract E8 VOA, but not a weaker form of 000020 | holomorphy and not a replacement for the source marking when the labelled 000021 | D5 x A3 embedding matters. 000022 | 000023 | 000024 | 1. EXACT FINITE CALCULATION 000025 | 000026 | Let A=C4 x C4 and 000027 | 000028 | q(a,b) = (5a^2+3b^2)/8 mod 1, 000029 | B((a,b),(c,d)) = (5ac+3bd)/4 mod 1. 000030 | 000031 | In the charge-resolved conformal-block basis e_x, the finite Fourier matrix is 000032 | 000033 | S_(x,y) = |A|^(-1/2) exp(-2 pi i B(x,y)) 000034 | = (1/4) exp(-2 pi i (5ac+3bd)/4). 000035 | 000036 | For v_H=sum_(h in H)e_h, character orthogonality gives 000037 | 000038 | S v_H = (|H|/4) v_(H perp). (1) 000039 | 000040 | This is an identity in the abstract 16-dimensional modular representation. It 000041 | does not rely on numerical values of specialized characters. 000042 | 000043 | The four extension supports from SIMPLIFICATION.txt give: 000044 | 000045 | H0 |H|=1 |Hperp|=16 S v_H=(1/4)v_A, 000046 | H2 |H|=2 |Hperp|=8 S v_H=(1/2)v_Hperp, 000047 | Hplus |H|=4 Hperp=H S v_H=v_H, 000048 | Hminus |H|=4 Hperp=H S v_H=v_H. 000049 | 000050 | Hence S fixes the vacuum vector exactly iff H is Lagrangian. If the premise is 000051 | only invariance of its line, the conclusion is unchanged: equality of supports 000052 | forces H=Hperp, hence |H|^2=|A|=16 and the scalar in (1) is one. 000053 | 000054 | The relative T matrix is exp(2 pi i q). It fixes every v_H because all four 000055 | supports are isotropic. For actual VOA characters there is also the common 000056 | central-charge factor exp(-2 pi i c/24); at c=8 the vacuum line is T-stable with 000057 | that common projective phase. It should not be described as pointwise T-fixed 000058 | unless this standard scalar has first been removed. 000059 | 000060 | The exact checker reproduces these statements, verifies S* S=I, S^2=charge 000061 | conjugation, and provides the rank-16 paired control. One focused replay was 000062 | byte-identical to the stored result: 000063 | 000064 | check_modular_sewing.py 000065 | SHA256 9fec3ac832649d79b13fe5d84f198f15e8b1a8a570aab1be6aa3185f59049d87 000066 | 000067 | modular-sewing.json 000068 | SHA256 7c6b99699f1eed64a966a93eca7c4ed4adb988f59baf030cd64f6652a34c7769 000069 | 000070 | 000071 | 2. WHAT THE PREMISE MEANS 000072 | 000073 | For an extension algebra B with support H, v_H is the restriction of the 000074 | extended vacuum character to U. Requiring its span to be closed under the 000075 | torus S move says that the extension has no other chiral sector into which its 000076 | vacuum character can transform. In this pointed category this is precisely 000077 | the Lagrangian, or holomorphic, condition. 000078 | 000079 | Therefore "a single chiral vacuum line" is substantial global input. It is not 000080 | ordinary sewing consistency of an arbitrary rational theory. Non-holomorphic 000081 | extensions have consistent multi-component modular representations, and the 000082 | vacuum component is allowed to mix with the others. 000083 | 000084 | The criterion is also not derivable from a scalar doubled partition function. 000085 | The diagonal matrix M=I obeys 000086 | 000087 | S^* M S=M, T^* M T=M 000088 | 000089 | for the whole 16-sector theory. Its rank is 16, so this invariance supplies no 000090 | rank-one chiral vacuum line. Likewise tau=i is a fixed point of S; equality of 000091 | one specialized numerical value at tau=i does not establish (1) as a vector 000092 | identity. 000093 | 000094 | Charge resolution is load-bearing for the full support and the marked 000095 | embedding. The leading vacuum coefficient of an unrefined S-transformed 000096 | character still gives the necessary equation |H|/4=1, because every nonvacuum 000097 | U module has positive lowest weight. Thus it detects the required cardinality 000098 | four inside the already classified four-extension list. Coincident unrefined 000099 | q-series can nevertheless hide the rest of the support. The safe object for 000100 | the complete statement is the labelled modular tensor-category basis e_x, or 000101 | Jacobi characters with independent charge variables. Scalar specialized 000102 | characters cannot distinguish Hplus from Hminus and need not retain the full 000103 | 16-dimensional action faithfully. 000104 | 000105 | 000106 | 3. COMPARISON WITH THE MARKED-GENERATOR THEOREM 000107 | 000108 | The two hypotheses are not logically equivalent on the classified set: 000109 | 000110 | Hom_U(M_(1,1),W) nonzero holds only for Hplus; 000111 | one chiral S-stable vacuum line holds for Hplus and Hminus. 000112 | 000113 | Thus modular sewing is strictly less marked. It is sufficient for the 000114 | unlabelled VOA isomorphism class V_E8, because Hplus and Hminus differ by the A3 000115 | diagram reversal. It is insufficient for the literal fixed-U extension and 000116 | for every later dictionary that distinguishes (1,1) from (1,-1). 000117 | 000118 | Conversely, the new condition is exactly holomorphy within this classified 000119 | extension class, whereas the local marked-generator theorem starts only from 000120 | occurrence of one charged module and proves saturation. Calling the modular 000121 | premise weaker would therefore be wrong. It is less marked as a test for the 000122 | unlabelled endpoint, while making a global statement about all 000123 | superselection sectors. 000124 | 000125 | The local-extension premise remains. The classification used here begins 000126 | with a connected local unitary extension sharing U's conformal vector. A 000127 | formal vector satisfying Sv=v does not by itself construct the extension, 000128 | prove locality, or map the primitive P1 source into it. 000129 | 000130 | 000131 | 4. CIRCULARITY TEST AGAINST THE ORIGINAL SOURCE 000132 | 000133 | The active original text already names holomorphy as the closing condition: 000134 | 000135 | documents/tfpt_research_contracts/sections/005-source.tex:60-67 000136 | 000137 | identifies "mu-index 1, single vacuum sector" as the AQFT face selecting E8. 000138 | Lines 78-96 identify the still load-bearing passage as the step from the RP 000139 | bulk to a holomorphic boundary net. Lines 97-121 reformulate it as det K=1, 000140 | absence of topological degeneracy, or short-range entanglement. 000141 | 000142 | This supports the mathematical relevance of the modular criterion, but it does 000143 | not by itself provide a new primitive derivation. In particular, the same 000144 | file's lines 45-50 already phrase the keystone as equality of the raw seam with 000145 | the holomorphic E8 net at tau=i. Using that named equality, or calculating 000146 | det K from the already selected E8 lattice, to justify the S-fixed line would be 000147 | circular for the present selection question. 000148 | 000149 | The existing local theorem is correctly scoped in 000150 | 000151 | experiments/theory-contracts/e-series-source-reconstruction-20261024/ 000152 | SIMPLIFICATION.txt:8-21, 84-111, 113-147, 162-168. 000153 | 000154 | It proves the marked route after local occurrence of M_(1,1), and explicitly 000155 | states that primitive P1 realization is still an input. The modular route 000156 | removes that particular occurrence premise only if holomorphy is established 000157 | independently upstream. 000158 | 000159 | 000160 | 5. SMALLEST DECISIVE ORIGINAL-SOURCE TEST 000161 | 000162 | Before naming an extension, construct from the primitive source one of the 000163 | following equivalent charge-resolved data: 000164 | 000165 | (a) its torus chiral conformal-block vector 000166 | Z_ch=sum_(x in A) n_x chi_x(tau,z), 000167 | with n_0=1 and n_x nonnegative integers, and prove the source sewing map 000168 | sends Z_ch to the same one-dimensional line under S; 000169 | 000170 | (b) the boundary net's mu-index and prove it equals one; 000171 | 000172 | (c) the torus conformal-block, equivalently topological state, space of the 000173 | associated modular theory and prove it has dimension one, without using 000174 | an E8 K matrix or E8 lattice already chosen downstream. 000175 | 000176 | Item (c) never means the ordinary vacuum-energy eigenspace of a chiral CFT. 000177 | D8 also has a unique ordinary vacuum vector; that fact does not make its 000178 | modular category holomorphic. 000179 | 000180 | Within the established extension classification, any one of these tests gives 000181 | H=Hperp and therefore the abstract E8 extension. To recover the original 000182 | marked embedding as well, one further charge-resolved datum must distinguish 000183 | 000184 | n_(1,1)=1, n_(1,3)=0 000185 | 000186 | from the reversed choice. That last bit is exactly the surviving orientation 000187 | or marking information. It is smaller than constructing the local (16,4) 000188 | generator explicitly, but it cannot be obtained from an unrefined character or 000189 | from modular invariance alone. 000190 | 000191 | 000192 | FINAL ASSESSMENT 000193 | 000194 | The proposed S-vector theorem and the now-written 000195 | CHIRAL-SEWING-SELECTION.txt proof are exact in their stated class. They give a 000196 | clean global alternative to the local charged-generator test for selecting the 000197 | unmarked E8 endpoint. Their real source premise is holomorphy, not generic 000198 | modular invariance. The rank-16 paired example is only a logical control; it 000199 | is not presented as a full P1 countermodel. The next decisive step is therefore an upstream, 000200 | charge-resolved proof that the primitive boundary source has a one-dimensional