TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/AFFINE-DEFECT-INDEPENDENT-REVIEW.txt
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Inclusive original lines: 339-490
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000339 | 7. Original source phase audit and exact gauge resolution
000340 | ---------------------------------------------------------
000341 | 
000342 | This section corrects the provisional physical reading in lines 65--77 and
000343 | 240--303 above.  The two Ramond signs are distinct after one fixes a complete
000344 | four-sector intertwining representative.  The reviewed originals do not fix
000345 | that complete representative.  With every field transported, the two signs
000346 | are isomorphic descriptions of the same two already fixed subtheories.
000347 | 
000348 | Write every charge uniquely as
000349 | 
000350 |   p=n+c0 s,        n in Z^8, c0 in {0,1},
000351 | 
000352 | and let Q(n)=sum_j n_j.  On the four sectors define the diagonal unitary
000353 | 
000354 |   U_tau |n+c0 s> = tau(n+c0 s)|n+c0 s>,
000355 |   tau(n+c0 s)=(-1)^(c0 Q(n)).                             (19)
000356 | 
000357 | Its sector signs in the order (0,v,s,c) are (+,+,+,-).  Hence U_tau is the
000358 | identity on both complete supplied local subextensions:
000359 | 
000360 |   H_0+H_v, the ordered-CAR NS theory;
000361 |   H_0+H_s, the even E8 lattice VOA.                        (20)
000362 | 
000363 | In particular it fixes the old W1 input data on integer charges, every E8
000364 | state, the E8 vacuum, and the spinor state e^s itself.
000365 | 
000366 | Let epsilon_* be the common four-sector cocycle chosen in
000367 | SAME-SOURCE-AFFINE-DEFECT.txt.  The basis change (19) gives
000368 | 
000369 |   epsilon_*'(p,q)
000370 |     =tau(p) tau(q) tau(p+q)^(-1) epsilon_*(p,q).            (21)
000371 | 
000372 | Equation (20) implies that (21) restricts EXACTLY to the same ordered-CAR
000373 | cocycle on Z^8 and to the same fixed E8 cocycle on D8 union(D8+s).  It changes
000374 | only the cross-sector intertwiner coefficients.  Thus those two restrictions
000375 | do not determine the full four-sector cocycle representative.
000376 | 
000377 | For the affine operator of the same-source construction,
000378 | 
000379 |   W_kappa |n+c0 s>
000380 |     =eta_R(n) kappa^c0 |Rn+e0+c0 s>,
000381 |   kappa in {+i,-i},                                       (22)
000382 | 
000383 | one has Q(Rn+e0)=Q(n)+1 and therefore
000384 | 
000385 |   tau(Rn+e0+c0 s)/tau(n+c0 s)=(-1)^c0,
000386 |   U_tau W_kappa U_tau^(-1)=W_(-kappa).                    (23)
000387 | 
000388 | This is the explicit equivalence that is invisible on both original local
000389 | subtheories.
000390 | 
000391 | The decisive mixed word can also be written exactly.  Put X=E_s for the
000392 | ground-state spinor charge shift in the chosen common charge algebra.  Its
000393 | relations are
000394 | 
000395 |   X E_e0 X^(-1)=i E_e0,
000396 |   Rhat X Rhat^(-1)=kappa X,
000397 |   W_kappa=E_e0 Rhat.
000398 | 
000399 | They give
000400 | 
000401 |   W_kappa X W_kappa^(-1) X^(-1)=(-i kappa) I.             (24)
000402 | 
000403 | The adjoint ordering W_kappa^* X W_kappa X^* has the same sign because the
000404 | right hand side is +1 or -1.  Thus, with the COMPLETE operator X held fixed,
000405 | kappa=+i and kappa=-i are distinguished by +I and -I.  Rephasing X by one
000406 | constant cannot alter (24).
000407 | 
000408 | This does not yet make (24) an original observable.  Under the full basis
000409 | transport (19), the spinor shift becomes
000410 | 
000411 |   X' = U_tau X U_tau^(-1),
000412 |   X'|n+c0 s> = (-1)^Q(n) X|n+c0 s>.                       (25)
000413 | 
000414 | It agrees with X on the E8 vacuum representation 0+s and changes sign only
000415 | on its action between v and c.  Conjugating every factor shows
000416 | 
000417 |   M(W_(-kappa),X')=U_tau M(W_kappa,X)U_tau^(-1).           (26)
000418 | 
000419 | Hence the scalar in (24) is unchanged under simultaneous transport.  The
000420 | apparent sign change occurs only if W is changed while the previously
000421 | unfixed v/c action of X is frozen by hand.  This proves the exact distinction:
000422 | 
000423 |   * fixing a full marked AIA, including X on v+c, makes the sign measurable;
000424 |   * fixing only the supplied CAR and E8 subtheories leaves the two signs
000425 |     related by (19)--(26), so there are not yet two physical sources.
000426 | 
000427 | For an actual vertex field Y_s(z), rather than the unitary charge translation
000428 | X on the algebraic core, the displayed star word additionally requires a
000429 | common invariant domain or smearing.  Its one-dimensional ground-intertwiner
000430 | cocycle is (24); bounded local-net status does not follow from this algebraic
000431 | calculation.
000432 | 
000433 | 8. Why the existing clocks and spin lift do not add the missing mark
000434 | -------------------------------------------------------------------
000435 | 
000436 | The targeted original comparison gives four separate domain statements.
000437 | 
000438 |   1. E-SERIES-SOURCE-RECONSTRUCTION, THEOREMS Sections 6--9, fixes the
000439 |      normalized E8 cocycle, all E8 vacuum fields and their naturality.  Its
000440 |      Hilbert space is V_E8=H_0+H_s.  It does not define the action of those
000441 |      spinor fields on H_v+H_c, and it contains no source W1.
000442 | 
000443 |   2. E-SERIES-JOINT-SOURCE, THEOREMS Sections 3--5, transports the marked
000444 |      E8 lattice VOA and its compiler clocks.  The C12 relative Hom theorem is
000445 |      the rank-four Phi12 comparison H -> A^7 H.  Neither construction contains
000446 |      the four D8-module AIA or the affine CAR winding.  Their character carry
000447 |      therefore cannot evaluate (24).
000448 | 
000449 |   3. The native mu4 clock lift acts on the E8 spinor extension, while kappa in
000450 |      (22) lifts the source channel permutation R on the four-sector parent.
000451 |      Identifying these two lifts would require the very marked source map under
000452 |      review.  Equal order and compatible squares do not supply that map.
000453 | 
000454 |   4. The original Moebius A4 spin lift fixes local bounding NS because r and
000455 |      -r have the same square r^2=-I.  ORIGINAL-SPIN-AND-SEAM-DICTIONARY,
000456 |      Section 11.2, proves that both signs induce the same NS structure on the
000457 |      orientable double.  Sections 11.6--11.7 explicitly leave the global
000458 |      quadratic refinement, Arf defect network and marked local-net
000459 |      intertwiner open.  It therefore supplies no independent v/c action of
000460 |      X for (24).
000461 | 
000462 | There is also direct precedent in SEAM-JOINT-WARD-STATE, Section 21.  After
000463 | the even E8 cocycle is fixed, both signs in the extension of the source
000464 | permutation to X preserve all displayed relations, and the original even
000465 | source does not select one.  That theorem concerns W4 on the even sector,
000466 | so it is not itself a W1 calculation, but it confirms that the even cocycle
000467 | does not secretly contain the missing mixed mark.
000468 | 
000469 | Corrected verdict
000470 | -----------------
000471 | 
000472 | PROVED: the same-source four-sector affine extension exists, and a chosen
000473 | common cocycle representative gives the exact mixed scalar (24).
000474 | 
000475 | PROVED: the two kappa signs are conjugate by the explicit sector basis change
000476 | (19) when all cross-sector intertwiners are transported.  This conjugacy fixes
000477 | both original subtheories pointwise.
000478 | 
000479 | NOT FOUND IN THE REVIEWED ORIGINALS: an independently fixed action of the E8
000480 | spinor field on v+c, or an independently fixed defect junction, whose
000481 | normalization is held constant while kappa is varied.  Without that extra
000482 | mark, (24) does not select a physical sign.
000483 | 
000484 | The smallest decisive future source test is now precise: exhibit one original
000485 | mixed operator Y_s^(v<->c), with its cocycle and adjoint fixed independently of
000486 | W1, then evaluate W1^* Y_s W1 Y_s^* on one common finite-energy core.  If no
000487 | such independently marked operator is supplied, changing kappa together with
000488 | (25) is only a change of intertwiner basis and must not be promoted to a
000489 | physical gap.
000490 |