TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/e-series-source-reconstruction-20261024/THEOREMS.txt
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Inclusive original lines: 387-448
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000387 | 9. Naturality for the original sign-extension lifts
000388 | 
000389 | Let ghat be any of the original epsilon-preserving lattice-field lifts. Its
000390 | action has the defining form
000391 | 
000392 |   ghat[h_1(-n_1)...h_r(-n_r)e^alpha]
000393 |     =(gh_1)(-n_1)...(gh_r)(-n_r)
000394 |       eta_g(alpha)e^(g alpha),                            (9.1)
000395 | 
000396 | with eta_g(0)=1 and with the original character carry retained under
000397 | composition. Formula (9.1) fixes the vacuum (6.3). Since g is a lattice
000398 | isometry, it fixes the quadratic tensor in (6.4), hence
000399 | 
000400 |   ghat omega=omega,
000401 |   [ghat,L_n]=0 for every n,
000402 |   [ghat,L0]=0.                                             (9.2)
000403 | 
000404 | Every character correction D_chi fixes the neutral Heisenberg sector,
000405 | the vacuum and omega, so (9.2) also holds for every corrected lift. Therefore
000406 | the transition functors and the associated-source construction of
000407 | MATH.SEAM.ESERIES.JOINTSOURCE.05 remain functorial after adjoining the now
000408 | canonical vacuum, conformal vector and L0. No new cocycle or section is
000409 | needed.
000410 | 
000411 | The earlier E6 nonselection also survives. The existing involutive lifts of
000412 | 
000413 |   d=diag(1,1,1,1,1,1,-1,-1)
000414 | 
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000420 | 
000421 | 
000422 | 10. The compiler clock is not conformal time rotation
000423 | 
000424 | On the complexified weight-one Cartan space
000425 | 
000426 |   H_C = {h(-1)1 : h in L tensor C},
000427 | 
000428 | the fixed Gaussian lattice operator Jhat acts through J. Its spectrum is
000429 | 
000430 |   Spec(Jhat|H_C) = {+i with multiplicity 4,
000431 |                     -i with multiplicity 4}.              (10.1)
000432 | 
000433 | By contrast, conformal rotation acts on every homogeneous weight-m space as
000434 | 
000435 |   exp(i theta L0)|V_m = exp(i theta m) I.                 (10.2)
000436 | 
000437 | In particular its restriction to H_C is the single scalar exp(i theta), for
000438 | every theta. Equations (10.1) and (10.2) prove
000439 | 
000440 |   Jhat != exp(i theta L0) for every theta.                 (10.3)
000441 | 
000442 | Both operators preserve conformal grade, and Jhat commutes with L0, but they
000443 | have different actions within weight one. A base or peripheral quarter turn
000444 | must therefore not be identified with conformal time evolution merely from
000445 | its order. This is an exact mathematical distinction, not a physical no-go
000446 | claim.
000447 | 
000448 |