TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/automated_source_search_20261029/PRIOR-CONDITIONAL-REFLECTION-NOTE-20261028.txt
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000001 | REFLECTION, HALF KINK AND THE ACTUAL FOLDED SOURCE
000002 | Corrected independent source and type review, 2026-10-28
000003 | 
000004 | Question
000005 | ========
000006 | 
000007 | Does the already present Pin and theta reflection transport the established
000008 | half winding sector rho_s and its bounded Weyl transporters on the actual
000009 | folded quarter source?  The test must retain the quarter holonomy, the
000010 | original positive winding convention, and the original K_alpha time.
000011 | 
000012 | 
000013 | Corrected verdict
000014 | =================
000015 | 
000016 | The geometric quarter switch, sector charge, and K_alpha time are mutually
000017 | compatible.  The existing sources do not yet prove that their OS word map
000018 | acts on the bounded current Weyls.  That missing typed intertwiner prevents
000019 | an unconditional reflection formula for T_IJ.  Two corrections and this
000020 | remaining gap are essential.
000021 | 
000022 | First, reflection of the eight fold fibres is not the bare permutation
000023 | r -> -r mod 8.  Crossing the quasi periodic seam contributes the quarter
000024 | phase.  With P the bare arc reversal and
000025 | 
000026 |   S_alpha = diag(1,e^(-2 pi i alpha),...,e^(-2 pi i alpha)),              (1)
000027 | 
000028 | the exact linear quarter switch is S_alpha P, not P.
000029 | 
000030 | Second, the actual v519 checker supplies an anti linear, order reversing
000031 | rule on the tested normal ordered disjoint words.  It is not
000032 | 
000033 |   A -> Rhat A^* Rhat^-1                                  (false),         (2)
000034 | 
000035 | when Rhat is antiunitary.  The map in (2) is linear.  The scalar test already
000036 | separates them:
000037 | 
000038 |   Theta_OS(i I) = -i I,
000039 |   Rhat (i I)^* Rhat^-1 = +i I.                            (3)
000040 | 
000041 | For a single disjoint even Majorana word, the phase and reversal sign in the
000042 | v519 rule cancel.  This does not prove an operator algebra identity on the
000043 | completed even CAR algebra and does not justify exponentiating to W(phi).
000044 | The candidate current relation that would close the gap is
000045 | 
000046 |   Theta_OS(J(phi))=J(phi o r),
000047 |   Theta_OS(W(phi))=W(-phi o r).                         (4, conditional)
000048 | 
000049 | Finally, reflection reverses the winding orientation.  For the standard
000050 | positive oriented kink profiles,
000051 | 
000052 |   b_IJ^+ o r = -b_(rI,rJ)^+.                              (6)
000053 | 
000054 | If the missing current relation (4) is supplied, the established smooth
000055 | transporter obeys
000056 | 
000057 |   T_IJ^+ = W(-b_IJ^+ o F),
000058 |   Theta_OS(T_IJ^+) = T_(rI,rJ)^+.                     (7, conditional)
000059 | 
000060 | There is no adjoint in (7).  An adjoint appears only if one keeps the raw
000061 | negative oriented reflected profile b_raw=b_IJ^+ o r instead of converting
000062 | back to the standard positive orientation:
000063 | 
000064 |   Theta_OS(T_IJ^+) = W(+b_raw o F)
000065 |                     = W(-b_raw o F)^*.                (8, conditional)
000066 | 
000067 | The exact sector and time dictionary remains
000068 | 
000069 |   alpha -> 1-alpha,
000070 |   q     -> -q,
000071 |   s     -> -s,
000072 |   I     -> rI,
000073 |   t     -> -t.                                            (9)
000074 | 
000075 | Here [s]=[-s] in the D8 discriminant group, while s and -s remain distinct
000076 | oriented representatives before passing to the even observable net.
000077 | 
000078 | Separately, the local normality conclusion survives unchanged.  It applies
000079 | to every quotient interval I whose closure is compactly contained in the
000080 | complement of the cut mark.  On such I the actual folded covariance is a
000081 | smooth local U(8) gauge transform of eight NS covariances.  No additional
000082 | local normality assumption is needed there.  This supplies no global vacuum,
000083 | time, theta, or primitive P1 identification.
000084 | 
000085 | 
000086 | 1. The three reflection roles
000087 | =============================
000088 | 
000089 | The original sources contain three typed operations.
000090 | 
000091 | 1. The linear Green reflection on the paired orientation cover is
000092 | 
000093 |      R_G = sigma_1 R_x,
000094 |      R_G P_i R_G = P_-i.                                  (10)
000095 | 
000096 |    It exchanges the two Pin lifts.  The paired cover is essential for the
000097 |    positive full NS Green form.  A single fixed quarter fails its temporal
000098 |    zero mode test.
000099 | 
000100 | 2. The antiunitary one particle symmetry