TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/v16-operator-pullback-20261014/FOLD-AUDIT.md
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Inclusive original lines: 15-124
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.

000015 | 
000016 | The change of variables proves that F is unitary from the original L² space
000017 | to rank8 sections on the quotient circle with a cyclic transition. The
000018 | source boundary phase e^(2pi i alpha) remains in the final transition link:
000019 | 
000020 |     Psi_j(2pi)=Psi_(j+1)(0),
000021 |     Psi_7(2pi)=e^(2pi i alpha) Psi_0(0).
000022 | 
000023 | Accordingly, on the actually transported domain,
000024 | 
000025 |     F(-i d/dtheta)F^-1=8(-i d/dx),
000026 |     C_fold=F C_cover F*,
000027 |     H_fold=Gamma(F) H_cover Gamma(F)*.
000028 | 
000029 | The last equation is for the second-quantized excitation Hamiltonian with
000030 | the source's same normal-ordering reference. A general original lattice
000031 | operator can likewise be permuted without changing its spectrum or state.
000032 | The checker verifies the original cyclic lattice shift and boundary link
000033 | for three lengths and phases1,-1,+i,-i. It does not add copies or replace
000034 | the source Hamiltonian by eight separate reference Hamiltonians.
000035 | 
000036 | CAR inner products are preserved; the eight arc subspaces are orthogonal.
000037 | Hence the pushforward fields have independent local CAR components on the
000038 | quotient. The transported state is generally correlated, not eight product
000039 | seas. A quotient interval has eight separated preimage arcs. This is a
000040 | change of localization; it is not a proof that the original TFPT identifies
000041 | those eight locations as one physical point.
000042 | 
000043 | In that quotient localization, each diagonal short-distance kernel has
000044 | the same unit fermion pole, while different arc kernels are smooth away
000045 | from the cyclic cut. The sum of their local stresses has central charge8.
000046 | The original point-local one-channel net still has central charge1.
000047 | There is no contradiction: central charge is tied to the stated net and
000048 | coordinate action, not just the abstract Hilbert space. Calling this a
000049 | mere relabelling that cannot ever have a rank8/c8 local description would
000050 | be too strong; calling it a P1-selected physical rank8 source would also
000051 | be too strong.
000052 | 
000053 | The central charge statement also has a direct source-mode proof. If the
000054 | original modes obey Virasoro at c=1, define
000055 | 
000056 |     Lhat_n=L_(8n)/8+(21/64) delta_n0.
000057 | 
000058 | Their commutator is exactly Virasoro with central charge8. The8 in the
000059 | cubic central term and the21/64 correction both follow from the original
000060 | commutator; they are not fitted. This is the quotient conformal action.
000061 | It is different from the original full point-local action. A source state
000062 | with energy h has Lhat0 weight h/8+21/64; this by itself does not change
000063 | the short-distance dimension of an original field or make it a new local
000064 | primary. In particular a twisted-sector excitation energy must not be
000065 | identified with a field's conformal weight without its actual OPE.
000066 | 
000067 | ## 2. State and time are part of the fold
000068 | 
000069 | The cyclic transition has eigenphases
000070 | 
000071 |     mu_k=(k+alpha)/8, k=0,...,7.
000072 | 
000073 | Mode branch l then has cover energy8l+k+alpha, exactly the original mode
000074 | inventory. The transported vacuum is a twisted-sector state of the
000075 | eight-component description. In the usual free-complex-fermion conformal
000076 | reference its ground weight is
000077 | 
000078 |     h_fold=sum_k (mu_k-1/2)^2/2
000079 |           =21/64+(alpha-1/2)^2/16.
000080 | 
000081 | The values are11/32 for periodic,21/64 for antiperiodic,85/256 for either
000082 | quarter phase. This formula follows from the independent channel twist
000083 | weights; it does not reselect a product vacuum. The factor8 in physical
000084 | time cannot be silently removed. The actual source has its quarter
000085 | holonomy as well as its state; geometric spin periodicity and gauge
000086 | holonomy must be kept separate.
000087 | 
000088 | ## 3. Exact periodic conjugacy to the TFPT glue, not just a centralizer count
000089 | 
000090 | For the stipulated periodic fold take nu=2mu=(0,1/4,...,7/4) and the TFPT
000091 | glue torus vector t=(1/2,1/2,1/2,1/2,1/2,0,0,0).
000092 | Let H be the4-by4 Hadamard matrix with rows
000093 | 
000094 |     ++++ , ++-- , +-+- , +--+.
000095 | 
000096 | Let P reorder coordinates as(0,4,2,6,1,5,3,7), and Q as
000097 | (2,3,4,6,7,0,1,5). Define
000098 | 
000099 |     M=Q diag(H/2,H/2) P.
000100 | 
000101 | M is orthogonal, has determinant+1 and permutes every one of the240 E8
000102 | roots. Moreover
000103 | 
000104 |     M nu-t=(-3/2,-1/2,3/2,-3/2,-1/2,3/2,-1/2,-1/2) in E8.
000105 | 
000106 | Its entries are half-integral and sum to-2, proving the lattice condition.
000107 | Consequently, for every root r,
000108 | 
000109 |     exp(2pi i nu.r)=exp(2pi i t.(M r)).
000110 | 
000111 | This is an explicit exact Weyl conjugacy between the periodic g² root
000112 | action and the original TFPT glue action. No appeal to equality of
000113 | centralizers or an unexecuted Kac-classification check is needed.
000114 | The complete matrix is in fold-results.json.
000115 | 
000116 | The periodic fixed root components are A3 (12 roots) and D5 (40 roots).
000117 | Their old folded-bilinear/spinor counts are respectively4+8 and16+24;
000118 | the deck g⁴ fixes112 roots of typeD8, comprising48 old integer and64
000119 | old spinor roots. This confirms the substantive structure in the paste.
000120 | M preserves E8 but does not preserve the elementary integer fermion
000121 | lattice Z8. Thus it is not a linear change of the original elementary
000122 | fermion species: the re-fermionization must preserve the actual even
000123 | net, extension, state and field-sector maps.
000124 |