TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CLAUDE27-DECK-FIRST-TEST.txt
Original file SHA-256: 1b0866e9de6957e97e2598086bb897af312da0b3f5fb772a17e2c0f6abdd99f9
Inclusive original lines: 1-64
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.

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000009 | 1. Valid geometric statement
000010 | For the actual C:w^3=(z^2-1)/(z^2+1),eta=z(1-w^3),
000011 | tau_C(w,eta)=(-bar(w),bar(eta)) is an antiinvolution, and
000012 | z(tau_C y)=1/bar(z(y)). It sends q_a to p_(-a).
000013 | The base antiinvolution fixes the unit circle pointwise.
000014 | The lifted antiinvolution, however, DOES NOT fix the whole preimage of
000015 | that circle pointwise. A base fixed locus and its entire lifted cable
000016 | are different subsets. This is the first failed proposed identification
000017 | if C itself is treated as the double with Fix(tau_C)=the whole cable seam.
000018 | 
000019 | 2. Smallest exact regular-fiber counterexample
000020 | Take z=exp(i*pi/4). Then w^3=i and eta=sqrt(2).
000021 | The three points of the fiber have
000022 |  w0=(sqrt(3)+i)/2, w1=(-sqrt(3)+i)/2, w2=-i.
000023 | All lie on the actual C above the same ordinary seam point; no vertex
000024 | or singular-domain subtlety is involved. Direct substitution gives
000025 |  tau_C(w0,eta)=(w1,eta), tau_C(w1,eta)=(w0,eta),
000026 |  tau_C(w2,eta)=(w2,eta).
000027 | Thus exactly one point is fixed and two are exchanged. Every alternative
000028 | global cubic-deck lift c^k tau_C acts as another reflection on this
000029 | three-point fiber and also has one fixed point, not three.
000030 | This excludes the literal pointwise whole-cable double identification.
000031 | It does not exclude a sheet-permuting reflection, a different genuine
000032 | geometric cut, or transport between a source and its reflected source.
000033 | 
000034 | 3. What remains valid in the spin computation
000035 | For sphere spin e_plus=sqrt(dz), the anti-pullback covering1/bar(z)
000036 | has J_tau e_plus=+/-i z^-1 e_plus. Its square is+1, not-1, since the
000037 | second application conjugates the coefficient after the base map.
000038 | Choosing the outside frame to be exactly this reflected inside frame
000039 | therefore gives an ordinary degree-one spin transition with constant
000040 | phase+/-i. That is a valid explicitly CHOSEN coordinate/frame comparison.
000041 | It has not identified the actual physical Gamma_+/Gamma_- normal frames
000042 | with these two charts. In particular it does not supply a relative phase
000043 | for the previously given T_faces by itself.
000044 | 
000045 | Likewise pi*|dz|^2 gives a specific flat metric on the open lifted disks.
000046 | Its local Jacobian and NS circle computation can be transported exactly.
000047 | The spin-line identity alone does not select this as the physical P1
000048 | metric, coupling or closed seam domain. A half-form identity cannot
000049 | silently settle the whole given charged source or physical time.
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000060 | 
000061 | Source geometry: original September28 section004:276-290, SHA256
000062 | 574058876ebe23d693a9fce061f68da6fdec63419af78243ac9d457d7858f74b.
000063 | Original Pin and weighted-trace working derivations remain separately
000064 | typed. No equality with CAR C0 or an OS-positive closed source is inferred.