TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SOURCE-INDUCED-SEAM-TRANSPORT.txt
Original file SHA-256: 1474792f0edfc9f0fc34e354b73cfc8c0a8591d0ccc78312bcf8e57f68058e05
Inclusive original lines: 381-510
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000381 | The outer and inner push offs each choose such a pairing. They are the two
000382 | smooth resolutions (1). Requiring a smooth ambient alpha section merely says
000383 | that all four limiting values equal the one value in alpha at z=0. That trace
000384 | space is one dimensional. It is isotropic but not maximal, so it is not a
000385 | self adjoint graph Dirac vertex condition.
000386 | 
000387 | Keeping both resolutions therefore gives a canonical pair, not a canonical
000388 | single vertex operator. Choosing an arbitrary U(2) scattering matrix would be
000389 | unjustified, and none is introduced here. The remaining decision is whether
000390 | the original cut problem names one face, retains both as a two face Calderon
000391 | space, or supplies a matching operator between them.
000392 | 
000393 | 5. Comparison with the original source
000394 | =======================================
000395 | 
000396 | The oldest explicit source says:
000397 | 
000398 |   D_Gamma=i d/dtheta,
000399 |   U_Gamma(theta)=exp(i theta),
000400 | 
000401 | and separately interpolates twisted domains with eigenvalues
000402 | n+phi/(2pi). It uses the words take and modeled by. It does not derive the
000403 | domain or U_Gamma from the elliptic branch data:
000404 | 
000405 |   documents/archive--paper-latex--tfpt-theory-fullv21/sections/
000406 |   001-section-introduction-the-problem-of-parameters.tex:198-255
000407 |   SHA256 85bb11511be05385a7cadf6071dbede216b12befeb9d6767bc4460748cbec5d7.
000408 | 
000409 | The later v28 seam setup likewise takes the one dimensional tangential
000410 | operator after already supplying a unitary matching operator:
000411 | 
000412 |   documents/archive--paper-latex--tfpt-theory-fullv28/sections/
000413 |   013-section-texorpdfstring-seam-gluing-and-the-eta-invariant.tex:7-17
000414 |   Atlas paper-source:fc0457edc74d21592ac3a36d
000415 |   SHA256 69adc16e3ba4106f164611a1104d42e7fa65de47bba8263f4780ba2ac542ce98.
000416 | 
000417 | v510 explicitly distinguishes the free covering deck from the branched
000418 | pillowcase involution. Its branch control says that the mark creating Z2 has
000419 | four fixed points and restricts as a reflection, whereas the original
000420 | Moebius covering deck is free:
000421 | 
000422 |   verification/v510_seam_bit_freedom.py:1-47,63-93
000423 |   SHA256 5b174a8914b14308621e0be98fc9d44e23b89713400ea5c09e1cae17d3546193.
000424 | 
000425 | That distinction prevents P in (1) from being silently renamed the original
000426 | free seam deck. It is the branch sheet exchange.
000427 | 
000428 | v519 also keeps the roles separate: the deck gives the nonsplit Kramers
000429 | class, while the seam circle reflection furnishes Theta. Its Q/Spin matrix
000430 | uses zeta_8 exactly, consistent with (5) and (7), but it does not provide the
000431 | ramified trace domain:
000432 | 
000433 |   verification/v519_woit_theta_rp_free.py:1-86,97-105
000434 |   SHA256 e86e3a055b58051c7978d8f10ee527948bac758fd4665ac216e18a71cc8c1ee9.
000435 | 
000436 | The preceding exact curve results used here are:
000437 | 
000438 | [OMITTED: non-scientific conversation or execution metadata]
000439 | [OMITTED: non-scientific conversation or execution metadata]
000440 | [OMITTED: non-scientific conversation or execution metadata]
000441 | [OMITTED: non-scientific conversation or execution metadata]
000442 | 
000443 | That original contract fixes alpha and proves the same central sign on the
000444 | twisted two dimensional section space H0(alpha tensor p*O(1)). Equations
000445 | (7a)-(7d) are the separate calculation on the full, untwisted flat alpha
000446 | line. They do not infer the present commutator merely from the finite section
000447 | space.
000448 | 
000449 |   research-20261026-elliptic-source-continuation/
000450 |   ORIGINAL-EQUATORIAL-REAL-LIFTS.txt:332-444
000451 |   SHA256 434eaab2fa67fbf157967a78f4b3f742cb7731d61c9632cdb06f12773c0ba1dd.
000452 | 
000453 |   research-20261026-elliptic-source-continuation/
000454 |   ELLIPTIC-GREEN-RP.txt:1-216
000455 |   SHA256 d8799f04851a9bd539b05b4d013b38eb00de9bec07dddde6438023aaf1facd80.
000456 | 
000457 | The latter supplies a regular intrinsic elliptic Green and Calderon operator
000458 | on the smooth annulus cut. Equation (12) explains why that result does not by
000459 | itself identify the ramified base pullback operator.
000460 | 
000461 | 6. Exact certificate and final classification
000462 | ==============================================
000463 | 
000464 | An inline exact SymPy replay, with no new checker file, returned
000465 | 
000466 |   P^2=I,
000467 |   T_out^4=T_in^4=I,
000468 |   T_out^(-1)T_in=P,
000469 |   M_flat,local^ccw=M_flat,local^cw=P,
000470 |   (M_0,M_1,M_2,M_3)=(P,-P,P,-P),
000471 |   M_0 M_1 M_2 M_3=I,
000472 |   in the character basis M_j=(Z,-Z,Z,-Z),
000473 |   X M_j X^(-1)=M_(j+1),
000474 |   M_tangent^ccw=+iP,
000475 |   M_tangent^cw=-iP=(+iP)^(-1),
000476 |   H_out^4=H_in^4=-I,
000477 |   H_out^(-1)H_in=-iP,
000478 |   zeta_8^2=i, zeta_8^4=-1, zeta_8^8=1,
000479 |   hQz-Qhz=1,
000480 |   U_h U_Q=-U_Q U_h,
000481 |   h_hat^2=-I, Q_hat^4=-I,
000482 |   h_hat Q_hat=-Q_hat h_hat,
000483 |   product_j M_tilde_j=I for both coherent choices c=+i and c=-i.
000484 | 
000485 | It also checked the mode condition exactly for integer and half integer test
000486 | values: the flat product I accepts exactly the integers, while the naive
000487 | moving tangent product -I accepts exactly the half integers.
000488 | 
000489 | Final status:
000490 | 
000491 |   [E] Local sheet continuation, flat alpha transport and tangent Spin phases.
000492 |   [E] Global raw peripherals (P,-P,P,-P), their product relation, reducible
000493 |       character lines and their exchange by Q.
000494 |   [E] Exact Q relation Q_out=P Q_in and the order 8 central lift.
000495 |   [E] Exact global alpha anticommutator h_hat Q_hat=-Q_hat h_hat; Q exchanges
000496 |       the two h_hat quarter eigenspaces.
000497 |   [E] Two self adjoint radial domains, their complete spectra and their
000498 |       unmarked unitary equivalence.
000499 |   [E] The 1/z obstruction for the ramified pullback operator.
000500 |   [E] Smooth ambient continuity is not a self adjoint valence 4 graph domain.
000501 |   [O] Which original face/domain and matching U_Gamma is physical.
000502 |   [O] Identification with the old quarter Hardy domain and its state.
000503 | 
000504 | The constructive gain is precise: the two source induced radial resolutions
000505 | contain no arbitrary local U(2) choice. The original C2 curve and alpha fix
000506 | the pair {I,P}, with Q relating the pair by the local branch deck. Closing the
000507 | source map still requires an original statement selecting or jointly using
000508 | those two faces. Globally, the fixed alpha central extension forbids selecting
000509 | one h_hat eigensector while retaining Q. Spin transport alone cannot supply
000510 | the remaining face or joint domain statement.