TFPT scientific source excerpt Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/PROOF.txt Original file SHA-256: adb8dac5389d79f7e1374d491ef848c5cc30d8851f6891c03a668c36f5807769 Inclusive original lines: 1-88 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 | ORIGINAL SEAM, SPIN LIFT AND E8 DEFECT: JOINT SYNTHESIS 000002 | ====================================================== 000003 | 25 October 2026 000004 | 000005 | Research question 000006 | ----------------- 000007 | Can the older original principles supply the spin and quarter data and an 000008 | actual operator connection to E8, without a new fitted source or an illegal 000009 | local odd-sector identification? 000010 | 000011 | Result: two mathematical connections close in their stated scope. A4 fixes 000012 | the local NS spin double loop and its unordered spin-reference quarter pair. 000013 | Additional gauge holonomy is separate connection data. The retained CAR 000014 | winding has a complete finite-energy extension as an E8 sector-changing 000015 | defect in its actual D8 frame. Its apparent remaining two-sign ambiguity is 000016 | an intertwiner-basis gauge. Neither result proves the full interacting TFPT 000017 | source, its physical OS-time identification, or its EM response contract. 000018 | 000019 | 1. The original order-four lift already has a consequence 000020 | -------------------------------------------------------- 000021 | fullv21 A4 says that the lift is of the SAME geometric Moebius Z2 000022 | identification and has order four. Read that condition literally. On the 000023 | universal collar write tau(s,t)=(s+2pi,-t). The orientable double corresponds 000024 | to tau^2. For its Pin lift r, 000025 | 000026 | r^2=-I, Hol_spin(double core)=-1. (1) 000027 | 000028 | Thus its pulled-back spinors are antiperiodic in the flat nonrotating 000029 | frame. This is the local bounding NS condition. The two Pin-minus choices 000030 | r and -r have the same square. A flat sign twist on the BASE also has square 000031 | one and does not change the double-core condition. 000032 | 000033 | The earlier tentative R counterexample changed the spin twist on the COVER, 000034 | preserved only thinner A2/A3 data and violated A4. It is not a countermodel 000035 | to the full older data. ORIGINAL-SPIN-AND-SEAM-DICTIONARY.txt records this 000036 | correction and the exact original passages. 000037 | 000038 | This does not identify the geometric r with the finite E8 complex structure 000039 | J, the Calderon involution, OS reflection, or many-body fermion parity. 000040 | The local spin conclusion uses a genuine lift of the same tangent 000041 | identification. An independent internal mu4 operator would not imply (1). 000042 | 000043 | 2. Exact quarter decomposition and the primitive winding 000044 | ------------------------------------------------------- 000045 | On the longitudinal double-core SPIN reference channel, without an 000046 | additional flat U1 connection, put 000047 | 000048 | H_cov=L2_NS(S1), B_cov=-i partial_theta, 000049 | f(theta+2pi)=-f(theta), u_cov=M_exp(i theta). 000050 | 000051 | The sign is the explicit orientation repair of fullv21's printed i partial 000052 | with eigenvalues n+a. The full Pinor monodromy on a collar is c J_perp, 000053 | where J_perp includes the transverse reflection. For c=+/-i its two 000054 | eigenspaces give the same unordered quarter pair. No transverse boundary 000055 | condition from a separate model is imported into TFPT. 000056 | An additional flat gauge line changes the total quasiperiodic phase. Its 000057 | holonomy is extra connection data, not another Pin structure or a 000058 | consequence of A4. Thus (1) fixes the spin contribution; it does not fix 000059 | every total charged Wilson phase in an interacting background. 000060 | 000061 | In the periodic chart on K=L2_per(S1), the exact Fourier map is 000062 | 000063 | W(g0,g1)(theta) 000064 | =exp(i theta/2)g0(2theta)+exp(3i theta/2)g1(2theta). 000065 | 000066 | With normalized circle measure it is unitary, and maps the corresponding 000067 | H1 domains. Define B_a=-i partial_x+a on the periodic chart. Then 000068 | 000069 | W* B_cov W=2(B_1/4 direct-sum B_3/4), 000070 | W* exp(i pi B_cov) W=diag(i,-i), 000071 | W* u_cov W=[[0,u_base],[I,0]], 000072 | W* u_cov^2 W=diag(u_base,u_base). (2) 000073 | 000074 | Consequently SF(u_cov)=1 and SF(u_cov^2)=2=1+1. The compressed positive 000075 | Toeplitz operators have indices -1 and -2 respectively. A single quarter 000076 | channel is not invariant under u_cov. 000077 | 000078 | This is the missing distinction between a primitive cover winding and one 000079 | block of its square. It prevents doubling the primitive SF by silently 000080 | replacing the original source by one quarter component. On a fixed 000081 | descended Pin lift the scalar u_cov changes c to -c; it is a matching 000082 | transport, not an observable endomorphism of a fixed descended sector. 000083 | 000084 | The direct minimal EM determinant from the previous contract needs only 000085 | 0