TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/v16-operator-pullback-20261014/FOLD-AUDIT.md
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Inclusive original lines: 125-245
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000125 | ## 4. The actual quarter-holonomy source is a separate exact test
000126 | 
000127 | The source's charged modes in `source-localized-spinor` are
000128 | exp(i(j-a)theta), with a=1/4 or3/4. Their full-circle phases are ±i.
000129 | Those physical cases are alpha=1/4 and3/4, up to orientation convention,
000130 | not alpha=0 merely because the geometric spin lift was called periodic.
000131 | More precisely, the stated source modes have phase exp(-2pi i a),
000132 | whereas this fold uses exp(+2pi i alpha). In the same orientation,
000133 | alpha=-a modulo1: a=1/4 maps to alpha=3/4 and a=3/4 to alpha=1/4.
000134 | Changing the representative of alpha requires the corresponding mode
000135 | reindexing. This phase dictionary must be fixed before transporting H_a;
000136 | the finite clock character alone is not the source energy generator.
000137 | 
000138 | The relevant primary topological statement is conditional on the Pin
000139 | structure: Kaidi, Parra-Martinez and Tachikawa, arXiv:1911.11780v2,
000140 | section2.2.2, give periodic Möbius-boundary fermions for Pin+ and
000141 | antiperiodic ones for Pin-. This supports the two spin cases; it does not
000142 | select Pin+ from P1/P2 or remove a separate charged Wilson holonomy.
000143 | 
000144 | On the proposed folded E8 action, the exact orders of g² are:
000145 | 
000146 | | Full-circle phase alpha | g² order on E8 | Fixed-root type | Central abelian rank |
000147 | |---|---:|---|---:|
000148 | | 0 | 4 | A3+D5 | 0 |
000149 | | 1/2 | 4 | D6+A1 | 1 |
000150 | | 1/4 or3/4 | 8 | D4+A1+A1 | 2 |
000151 | 
000152 | The extra abelian directions are necessary because a torus element fixes
000153 | all eight Cartan directions. For the antiperiodic deck, the fixed roots
000154 | are E7+A1; for the quarter-phase deck they are D6+A1 with one central
000155 | U1. These are computations of the stipulated fold action on the already
000156 | chosen E8 lattice, not a proof of an E8 source extension.
000157 | 
000158 | Thus the proposed geometric g⁴ is not automatically the actual charged
000159 | double-cover involution. Multiplying g by a compensating gauge phase can
000160 | change the full-circle holonomy and the charged clock. It requires a
000161 | derived source dictionary, not deletion of the Wilson phase. The existing
000162 | physical generator H_a=L0+(a-1/2)Q explicitly retains that distinction.
000163 | 
000164 | Spin-lift changes mu->mu+e1 leave g² and g⁴ root actions unchanged, because
000165 | 2e1 and4e1 belong to E8. They can affect g itself. Therefore lift-dependent
000166 | root types for g must not be attributed to g² without specifying the power.
000167 | 
000168 | ### 4a. The source charge has an explicit location in the TFPT dictionary
000169 | 
000170 | The same matrix gives more than a failed period test. Let b=(1,...,1)
000171 | be the total folded fermion-charge vector. Direct multiplication gives
000172 | 
000173 |     M b=2 r_*, r_*=e3+e6 in E8.
000174 | 
000175 | Here e3 is a D5 carrier-plane direction and e6 an A3/D3 family-plane
000176 | direction in the target t dictionary; r_* is a mixed integer root.
000177 | For every source boundary phase alpha, nu_alpha=nu_0+(alpha/4)b, hence
000178 | 
000179 |     M nu_alpha=t+lambda+(alpha/2)r_*, lambda in E8.
000180 | 
000181 | On every target root current of charge p the source g² action is therefore
000182 | 
000183 |     exp(2pi i t.p) exp(pi i alpha r_*.p).
000184 | 
000185 | This identifies the inherited source U1 generator as Q'=2(H3+H6) within
000186 | the stipulated folded E8 extension. It does not identify elementary TFPT
000187 | fermions with the original seam fermions or construct an actual field map.
000188 | The 240-root phase equality is checked for all four boundary phases.
000189 | 
000190 | For the two actual quarter phases the displacement is respectively
000191 | (e3+e6)/8 and3(e3+e6)/8. A candidate compensation must have exactly the
000192 | inverse character on actual local charged fields. The formula specifies
000193 | what must be derived; it does not authorize deleting the phase or prove
000194 | that the original source contains that compensating gauge action. It also
000195 | does not identify a finite clock automorphism with physical time without
000196 | the separately transported Hamiltonian.
000197 | 
000198 | ### 4b. The transported charge selects a lift within the original packet
000199 | 
000200 | This charge also joins the fold to the original E/F connection formula.
000201 | In the fixed target-plane dictionary put
000202 | 
000203 |     Q_E=2 diag(0,0,1,0,0), tr Q_E=2.
000204 | 
000205 | Q'=2(H3+H6) gives family-plane charges (2,0,0). In the original formula
000206 | their charges are (tr Q_E) a_i with a_i=m_i+m4+1. Hence
000207 | 
000208 |     a=(1,0,0), sum m_j=-2
000209 |     => m4=(sum a_i-1)/2=0
000210 |     => m=(0,-1,-1,0), beta=(1/2,-1/2,-1/2,1/2).
000211 | 
000212 | This is the unique lift in this ordered-plane and charge convention.
000213 | The original packet formula with these m and Q_E, constructed independently
000214 | of the Clifford inverse, returns exactly
000215 | 
000216 |     Omega_V=-2 L_4,5-2 L_10,11
000217 | 
000218 | in the interleaved realification convention. These are the carrier third
000219 | and family first plane rotations; Omega_V=i Q' in that convention. The
000220 | skewadjoint connection/action sign is separate from the Hermitian torus
000221 | charge convention. No residual is discarded. The actual quarter phase
000222 | then multiplies this same generator through the character in section4a.
000223 | 
000224 | The choice of the E8 extension and this field/plane dictionary remains
000225 | upstream. The lift is selected from charge matching within that dictionary,
000226 | not from primitive P1/P2 alone. A simultaneous reversal of carrier and
000227 | family charge leaves their ratios a_i and the same lift unchanged. A
000228 | different relative family-plane orientation is a different dictionary;
000229 | it must not be introduced merely by replacing alpha with -alpha.
000230 | 
000231 | ## 5. How this joins the ongoing origin derivation
000232 | 
000233 | The positive sequence now has two explicit maps: a source-preserving
000234 | pushforward to the quotient's rank8 complex/rank16 real operator, and an
000235 | exact periodic E8 triality map to the TFPT mark. The connection inverse in
000236 | PROOF.md supplies the local Spin16/vector action dictionary as well.
000237 | These are connected constructions, not independent numerical evidence.
000238 | 
000239 | The physical identification still needs the original source to select that
000240 | quotient/localization and its actual Pin/gauge lift, and to realize the E8
000241 | extension required by the triality step. Folding derives V_fold; it does
000242 | not directly give Xi onto the differently fermionized TFPT V16.
000243 | The root calculation starts from E8, so it cannot by itself derive E8
000244 | from the primitive seam. No complete origin/T1–T8 solution is claimed.
000245 |