TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/WHOLE-ALPHABET-ORIGINAL-MARKED-JET-REVIEW.txt
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000004 | POSITIVE RESULT
000005 | The complete two-edge answer has an exact original transport expression
000006 | at BOTH endpoints. This strengthens the previously proved single-E02 map.
000007 | It uses the SAME Bstar, M0 and quarter-fixed P0, not new fitted projectors.
000008 | 
000009 | Let alpha=(4Xstar/pi)I02,beta=(4Xstar/pi)I21,
000010 |  F=alpha E02+beta E21, N=-i(alpha E12+beta E20).
000011 | For the already fixed normalized original Gbase frame
000012 |  Bstar=1/sqrt2 [[0,0,-1+i],[1,i,0],[i,1,0]],
000013 | the actual old identities extend to
000014 |  Bstar E02 Bstar*=M0P0,
000015 |  Bstar E21 Bstar*=-P0M0,
000016 |  Fhat=Bstar F Bstar*=alpha M0P0-beta P0M0.                 (1)
000017 | Since D_(y,+)=yI-delta M0, (1) is exactly
000018 |  Fhat=-alpha (partial_delta D)P0+beta P0(partial_delta D).
000019 | Both coefficients, including their relative sign, come from the original
000020 | calibrated periods. No choice of a different endpoint line was made.
000021 | 
000022 | Put W=diag(1,-1,i)P01 and Dsign=diag(1,-1,1). Exactly
000023 |  W^2=-I, Ad(W)F=-N, Ad(W)N=-F,
000024 |  Bstar W Bstar*=i Dsign,
000025 |  Nhat=-Dsign Fhat Dsign.                                  (2)
000026 | These are actual coefficient identities. W has not been declared a new
000027 | physical symmetry, nor has its linear adjoint action been identified
000028 | with the antilinear geometric E0 reflection.
000029 | 
000030 | LINEAR MOMENT INTERTWINER
000031 | For the already established raw spin-product pullback
000032 |  R*mu_ab=rho^-1 mu_(b+1,a+1), rho=exp(i pi/3),
000033 | let nu_j=(R*)^j mu01. The full table gives
000034 |  Reader(nu_j)=F,-N,F,-N,F,-N.
000035 | Therefore Reader R*=Ad(W) Reader on THIS six-dimensional input space.
000036 | Its Fourier images vanish except lambda=+1 and-1, which give3(F-N)
000037 | and3(F+N), respectively. This is a property of this finite moment map.
000038 | The six global twist sectors, full spin lift, quantum source, state and
000039 | time are not identified with its two-dimensional image or involution.
000040 | Original full-deck character obstructions and their recorded scopes stay.
000041 | 
000042 | ACTUAL RESOLVENT TRACE, WITHOUT DISCARDING THE REVERSE RESPONSE
000043 | The original cubic identity M0^3=I gives
000044 |  D^-1=(y^2 I+y delta M0+delta^2 M0^2)/(y^3-delta^3).
000045 | Consequently
000046 |  Tr(D^-1 M0P0)=Tr(D^-1 P0M0)=delta^2/(y^3-delta^3),
000047 |  Tr(D^-1 Fhat)=(alpha-beta)delta^2/(y^3-delta^3),
000048 |  Tr(D^-1 Nhat)=i(alpha-beta)y delta/(y^3-delta^3).          (3)
000049 | The last identity is an additional root check in the SAME fixed M0 block.
000050 | Only the two orderings INSIDE F have equal trace. The full reverse answer
000051 | has the distinct factor i y/delta where delta is nonzero. A rephased or
000052 | discarded N would lose this information.
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