TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/F5-QP-REVERSE-HOM-KERNEL-REVIEW.txt
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Inclusive original lines: 137-200
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000137 |   delta(B)=Pc,q (c*B) Pc,p^-1       for B:Fp -> Fq.    (9)
000138 | Their intermediate factors cancel in the trace. These directions must
000139 | not be interchanged. They include all coefficient signs, not only a naked
000140 | geometrical differential or a scalar native family-center phase.
000141 | 
000142 | Consequently, if an insertion M_chi has character chi under this FULL
000143 | natural action and its tangent/Clifford contraction is covariant, then on
000144 | the given test domain its compression satisfies
000145 |   W_p* M_chi W_q=chi W_p* M_chi W_q.                  (10)
000146 | For the complete singular multiplier this equation is an identity of its
000147 | integrable forms, not an unproved L2 operator identity. For chi!=1 that
000148 | form compression is zero. Averaging the three sheets keeps only
000149 | (M+delta M+delta^2 M)/3. Since J and X_q are equivariant, the whole form
000150 | pairing (8a), or (8) when its stronger domain premise holds, has the same
000151 | selection. This does not require adding a cutoff to the original source.
000152 | 
000153 | The calibration to the ORIGINAL lift is concrete. The normalized f_a and
000154 | beta in the Hom report obey c*f_a=f_(a-1), so the square trivializations
000155 | carry no extra cubic phase. Applying the old R product map twice gives
000156 | exactly this c pullback on both coefficient ends. Its two spin-central
000157 | minus signs cancel. Changing from (L0,L1,L2) to G's actual (L0,L2,L1)
000158 | order only conjugates the coefficient matrices. Thus c=R_V^2 on the
000159 | actual Voff product representation is the natural Hom action in (9),
000160 | including its differential factor. The Pc factors express that action;
000161 | they are NOT an extra second multiplication of its character.
000162 | 
000163 | Hence the rho line has chi=rho^2=omega under the full action. Provided
000164 | the proposed boundary reader really intertwines that SAME action, (10)
000165 | annihilates its insertion in the fully descended source. This is a
000166 | testable condition, not an assumption that every reader has that action.
000167 | The two c orbits on the six off-diagonal products have length three, so
000168 | their invariant space has rank2. The total mu family also has invariant
000169 | components; this does not annihilate the full family, its norm or the
000170 | whole F5 kernel.
000171 | An additional compensating ACTUAL reader/endpoint leg, a marked open arc,
000172 | or a different original sewing consumer changes this test. A reverse Hom
000173 | frame by itself is not such an extra selected character: it is already
000174 | included in (9). No exclusion of all physical rho channels follows.
000175 | 
000176 | 6. Compare the actual original sewing and original output consumer
000177 | Directly read fullv21 introduction:198-205,225-251 and its seam appendix
000178 | 006:346-366,007:1-22. They prescribe FOUR cut pieces C1,C2,Gamma_+,Gamma_-,
000179 | APS on C1,C2, and a unitary phase U_Gamma=e^(i theta) on the seam pair.
000180 | The seam operation is real original data. These passages do not give the
000181 | specific trace formula identifying Gamma_+/- with the two F5 faces and
000182 | the q/p R-related endpoints, including their coefficient/spin frames.
000183 | The appendix's U_Gamma^2=I can mean the WHOLE two-way matching, whose
000184 | return uses the inverse unitary; it does not alone prove a pointwise
000185 | angular reflection or identify the bare multiplier with an involution.
000186 | 
000187 | At fixed ungauged endpoint labels, j* M_mu is supported on the R graph,
000188 | whereas the actual original finite one-form qM is multiplication on the
000189 | diagonal. A nonzero test field in an arc disjoint from its R image detects
000190 | this difference immediately. This excludes their DIRECT equality in those
000191 | labels. It is no general locality obstruction: original seam gluing can
000192 | identify the two arcs, and eliminating the bulk can turn a local variation
000193 | into a nonlocal boundary response. Both effects must use their original
000194 | endpoint/Poisson maps, not a new arbitrary p species or fitted kernel.
000195 | 
000196 | The first concrete equality to check for such a seam realization is
000197 |   T_- psi(theta)=U_Gamma(theta) F_Gamma(theta)
000198 |                     T_+ psi(b_Gamma(theta)),        (11)
000199 | with T_+/- the actual original trace maps. Comparing its b_Gamma and
000200 | F_Gamma with (5)-(6) decides whether (8) is a local insertion on the glued