Manuscript XVI-P@p23 · XVI-P.tex · section · 9b0037a7
Counterterm domain and solve
P.5 identity step (p19): the projected one-loop κ 2 ch-identity defect › Counterterm domain and solve · lines 479–509
| assignment | defect pole | coeff. of L | local finite |
|---|---|---|---|
hline Q4S (declared, R2) | 0 | 0 | 0 |
QdS (hat) | 0 | 0 | 0 |
c tilde, h hat | 0 | 0 | 0 |
c tilde, h mixed | 0 | 0 | 0 |
c hat, h mixed | 0 | 0 | 0 |
all tilde | 0 | 0 | 0 |
c hat, h tilde | 0 | 0 | 0 |
Route G over Q(i): rank = 5, dim = 6, dim = 6, cohomology 0, quotient 5 (complement h.h#0, h.h#1, h.h#2, c.cb#0, c.cb#1); rank = 3 (Route C: 3).
Over Q with the re/im split (p14r convention): dim = 3, quotient 8, dim = 5.
Quotient coordinates of the local data: pole ['0', '0', '0', '-3/8', '1/4'], coeff. of L ['0', '0', '0', '3/8', '-1/4'], local ['0', '0', '0', '-2/3', '5/12'].
| mutation | outcome | changes |
|---|---|---|
hline drop_graph_D03 | OBSTRUCTION (nonlocal residual: the L-coeffi | yes |
drop_graph_D01 | OBSTRUCTION (nonlocal residual: the L-coeffi | yes |
flip_phase_D03 | OBSTRUCTION (nonlocal residual: the L-coeffi | yes |
symmetry_factor_D03_x2 | OBSTRUCTION (nonlocal residual: the L-coeffi | yes |
wrong_projection_trace_operator_hh | OBSTRUCTION (nonlocal residual: the L-coeffi | yes |
wrong_projection_gauge_fixed_operator_Kc | OBSTRUCTION (nonlocal residual: the L-coeffi | yes |
naive_4D_D03_evanescent_dropped | OBSTRUCTION (nonlocal residual: the L-coeffi | yes |
naive_4D_D13_Gt_dropped | NONZERO, CANCELLABLE by a counterterm in the | yes |
Counterterm solve: ZERO; x = 0 (no counterterm needed beyond the MS-bar poles); the identity-preserving finite renormalisations are ker(P_ch sigma_0) cap D (dim 8), of which 2 modulo D cap Im sigma_0.
Why zero. each surviving ch row vanishes separately in the projected slot; no counterterm cancellation and no cancellation between rows is involved: (i) hh row (D01+D02+D03+D11+D12): full-row transverse decoupling – D01 and D03 are individually nonzero and cancel against each other in the slot (p17/p18 transverse sum), D11 is transverse by itself, D02 = D12 = 0; (ii) K.c row (D13): D13 reaches the ch slot only through T1 and E1 (T2, T3, E2 are longitudinal, E transverse) and A_T1 = A_E1 = 0 exactly in d; (iii) one-point row: the tadpoles D14a-c vanish under the declared prescription. The defect is therefore zero without any counterterm (solve x = 0).