TFPT scientific source excerpt Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/LATTICE-AND-SOURCE-FRAME.txt Original file SHA-256: 77e0687412cb79bf1d51d079946484b6e8be337adf8a72a85bf2e6079b045ae7 Inclusive original lines: 1-171 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 | CANONICAL SEAM FERMIONIZATION: LATTICE AND SOURCE FRAME 000002 | ======================================================= 000003 | 000004 | Question 000005 | -------- 000006 | 000007 | The marked compiler lattice is 000008 | 000009 | K=D5+A3 subset E8, 000010 | E8/K= isomorphic to Z4. 000011 | 000012 | Its unique order-two subgroup determines one canonical intermediate even 000013 | lattice. Is that marked intermediate lattice, in the already fixed O/M 000014 | dictionary and without another Weyl rotation, exactly the D8 charge lattice 000015 | of the eight folded components of the one CAR source? 000016 | 000017 | Verdict 000018 | ------- 000019 | 000020 | SCOPED FALSE. 000021 | 000022 | The order-two intermediate of the marked D5+A3 glue is exactly a D8. The 000023 | folded CAR even lattice is also exactly a D8. In the common target frame 000024 | fixed by the existing O and M, however, they are different embedded D8 000025 | sublattices of E8: 000026 | 000027 | canonical marked D8 root set: 112 roots, 000028 | M(folded CAR D8) root set: 112 roots, 000029 | intersection: 48 roots, 000030 | canonical only: 64 roots, 000031 | folded only: 64 roots. 000032 | 000033 | Their Cartan defect characters and their actions of the actual quarter-time 000034 | vector also differ. Equality of Dynkin type, determinant and root count is 000035 | therefore insufficient in the simultaneously fixed O/M marking. 000036 | 000037 | This is not a coordinate-invariant TFPT no-go. A Weyl transformation is a 000038 | legitimate change of coordinates if it transports every marked datum at 000039 | once: the D5+A3 flag, the O frame, the time vector, the source embedding, 000040 | the field words and their cocycle/locality data. What fails here is the 000041 | identity comparison after the two existing dictionaries O and M have both 000042 | been fixed. Rotating only one D8 to align it with the other while leaving 000043 | the remaining data fixed would introduce a new dictionary. No such fitted 000044 | Weyl rotation is made here. 000045 | 000046 | Because this first equality gate fails, the proposed common identification 000047 | of the nonisotropic vector coset, the odd neighbour Z8 and the E8 defect 000048 | sector called v+c is not performed. Those names belong to the discriminant 000049 | group of a specified D8 embedding and cannot be carried across these two 000050 | frames by identity. 000051 | 000052 | Status split 000053 | ------------ 000054 | 000055 | PROVED: 000056 | 000057 | * the unique order-two glue intermediate is the canonical target D8; 000058 | * O maps the original Gaussian E8 roots bijectively to the fixed target 000059 | E8 frame and maps that intermediate to the standard target D8; 000060 | * M is the already supplied orthogonal E8 root automorphism; 000061 | * M maps the folded CAR D8 to a different target D8 with the exact 000062 | 48/64/64 root intersection stated above; 000063 | * the two Z2 characters on charged root fields differ; 000064 | * d_alpha=M delta_alpha transports the actual folded quarter time, and 000065 | its root-phase multiset on M(D8_CAR) differs from its phase multiset on 000066 | the canonical marked D8 for alpha=1/4 and 3/4; 000067 | * p=(+,+,+,+,-,-,-,-)/2 is an E8 spinor root in the standard even-parity 000068 | spinor coset, M p=e_1+e_4 in one-based target notation, and both actual 000069 | quarter shifts give p^2/2+delta.p=0; 000070 | * this is the exact source preimage of the equal-energy target root line 000071 | already recorded in FOLD-COVARIANCE-TO-NATIVE-FIELD-RESPONSE.txt. 000072 | 000073 | SCOPED FALSE: 000074 | 000075 | O(D8_halfway)=M(D8_CAR) 000076 | 000077 | in the existing marked O/M dictionary. 000078 | 000079 | CONDITIONAL ABSTRACT FACT, NOT A CROSS-FRAME IDENTIFICATION: 000080 | 000081 | For any fixed copy of D8, D8*/D8 has one nonisotropic vector class and 000082 | two isotropic spinor classes. Adjoining the vector class gives the odd 000083 | unimodular neighbour Z8; adjoining either spinor class gives an E8 000084 | neighbour. This standard discriminant-form fact does not say which class 000085 | of one embedded D8 is the named class of another embedded D8. 000086 | 000087 | OPEN AND NOT TESTED AFTER THE FAILED GATE: 000088 | 000089 | * a primitive source reason selecting a common Weyl presentation in which 000090 | the D5+A3 flag, O, M, time, source and fields are transported together, 000091 | or selecting a new E8 automorphism which aligns the two D8s while the 000092 | other markings are retained; 000093 | * a common microscopic identification of the CAR odd vector coset with a 000094 | canonical compiler spinor or defect sector; 000095 | * a local odd-sector field construction. The known global re-fielding 000096 | does not supply this locality statement. 000097 | 000098 | 1. The canonical order-two intermediate 000099 | ----------------------------------------- 000100 | 000101 | Use the fixed standard target coordinates in which A3 is the usual D3 root 000102 | lattice on the last three coordinates. Then 000103 | 000104 | K={z in Z8: 000105 | sum_(i=0)^4 z_i is even, 000106 | sum_(i=5)^7 z_i is even}. 000107 | 000108 | Choose the marked Z4 glue generator 000109 | 000110 | g=(1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2). 000111 | 000112 | It has order four modulo K: 2g=(1,...,1) is not in K, whereas 000113 | 4g=(2,...,2) is in K. A cyclic group of order four has one subgroup of 000114 | order two, namely <2g>. Its intermediate lattice is 000115 | 000116 | L_2=K union (K+2g). 000117 | 000118 | For an integral vector z, membership in K means that the first-five and 000119 | last-three coordinate sums are both even. Membership in K+2g means that 000120 | both sums are odd. Their union is therefore 000121 | 000122 | L_2={z in Z8: sum_i z_i is even}=D8. (1) 000123 | 000124 | Equation (1) proves the full lattice identity. It is not inferred from the 000125 | number of roots. 000126 | 000127 | On roots, use doubled coordinates as in the original programs. The marked 000128 | glue degree is 000129 | 000130 | deg(r)=sum_(i=0)^4 r_i mod 4. 000131 | 000132 | The exact E8 root counts in degrees 0,1,2,3 are 000133 | 000134 | 52,64,60,64. 000135 | 000136 | The roots in the order-two subgroup are degrees 0 and 2. Their total is 000137 | 52+60=112, and their set is exactly 000138 | 000139 | R_can={+/-e_i +/-e_j: i