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Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/E0-WHOLE-FINITE-SOURCE-TYPE-REVIEW.txt
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000001 | E0 REFLECTION ON THE WHOLE v252 FINITE SOURCE: TYPE AND COVARIANCE REVIEW
000002 | Independent operator review, 29 October 2026.
000003 | No original source or Atlas record was changed.
000004 | 
000005 | RESULT
000006 | 
000007 | The actual E0 coefficient reflection extends to the entire original
000008 | 96-dimensional v252 finite coefficient and to the complete F5 two-face graph.
000009 | It is a covariance of the already given source family.  For a generic v252
000010 | Yukawa/Majorana coefficient it is not a symmetry of one fixed Phi.
000011 | 
000012 | The existing finite charge J_F does not already contain the E0 flavour
000013 | matrix.  The two antiunitaries instead commute.  In the F5 complex structure
000014 | their correctly typed combination transports the whole operator, its pure
000015 | graph covariance, all charged/conjugate slots and the circle group.  The
000016 | circle parameter is reversed.  No new covariance or state is selected.
000017 | 
000018 | For real constructor controls there is an additional exact commutant unitary
000019 | which turns the family covariance into a same-Phi antiunitary.  This is a
000020 | unitary equivalence of the finite triples,
000021 | but it is not an inner gauge transformation of the displayed algebra A_F.
000022 | Its availability in the original physical source action menu, with the
000023 | native K/clock marking fixed, is not established here.
000024 | 
000025 | 1. ACTUAL E0 MATRIX AND THE 96-DIMENSIONAL EXTENSION
000026 | 
000027 | In the checked horizontal G-column frame the original algebraic Fuchs
000028 | solution gives
000029 | 
000030 |   D_E = Gbase^-1 U_F bar(Gbase) = omega^2 S,
000031 |   omega^2+omega+1=0,  S=diag(1,-1,1).
000032 | 
000033 | Thus D_E bar(D_E)=I.  Let P_E be the actual intrinsic NS reflection
000034 | theta -> pi/2-theta and kappa pointwise OLD complex conjugation.  On v252,
000035 | let Dcal act by D_E on the three generations of every particle field and by
000036 | bar(D_E) on every antiparticle field.  Define
000037 | 
000038 |   K_E = Dcal kappa P_E.
000039 | 
000040 | The particle/antiparticle distinction is essential.  Directly from the
000041 | blocks one obtains
000042 | 
000043 |   K_E^2=I,  [K_E,gamma_F]=0,  [K_E,J_F]=0,
000044 |   J_F=K_swap kappa.
000045 | 
000046 | The last equality uses
000047 | 
000048 |   Dcal=diag(D_E,bar(D_E)),
000049 |   K_swap bar(Dcal)=Dcal K_swap.
000050 | 
000051 | Therefore E0 is not inserted by composing two unrelated copies of complex
000052 | conjugation.  Literal composition of two antiunitaries would be OLD linear
000053 | and would answer a different KO/type question.  Here K_E and J_F remain two
000054 | commuting antiunitaries on the same actual 96-dimensional space.
000055 | 
000056 | 2. THE FULL v252 PHI AND THE MAJORANA TRANSPOSE
000057 | 
000058 | Write the actual constructor exactly as
000059 | 
000060 |   Phi(Y_f,MR)=[[S_Y,T_MR],[T_MR^*,bar(S_Y)]],
000061 | 
000062 | where T_MR occupies the particle-to-antiparticle nu_R block and MR is
000063 | symmetric.  Then
000064 | 
000065 |   K_E Phi(Y_f,MR) K_E^-1 = Phi(Y'_f,MR') =: Phi^E,
000066 |   Y'_f = D_E bar(Y_f) D_E^* = S bar(Y_f) S,
000067 |   MR'  = D_E bar(MR) D_E^T = omega S bar(MR) S.       (1)
000068 | 
000069 | The transpose in the Majorana formula is forced because its second leg lies
000070 | in the antiparticle block.  Replacing D_E^T by D_E^* would be a type error and
000071 | would lose the phase omega.  Symmetry of MR implies symmetry of MR', so the
000072 | transformed matrix is again an input of the original full_D constructor.
000073 | Since K_E commutes with J_F, the transformed Phi^E still obeys the original
000074 | KO6 reality relation J_F Phi^E=Phi^E J_F.
000075 | 
000076 | For generic complex v252 controls, Phi^E is not Phi.  A targeted replay using
000077 | the actual full_D, grading and Kswap with seed 2904 Gaussian-integer controls
000078 | gave
000079 | 
000080 |   ||K_E Phi K_E^-1-Phi^E|| < 4e-15,
000081 |   ||K_E Phi K_E^-1-Phi|| = 54.0053.
000082 | 
000083 | The nonzero second number is only a concrete generic counterexample; equation
000084 | (1) is the exact result.  For real diagonal Yukawas, Y'_f=Y_f.  The actual
000085 | finite-typing control has a real tridiagonal MR, however, so its precise
000086 | image is omega S MR S, not omega MR.  The shorter omega MR formula applies
000087 | only when [S,MR]=0, for example for diagonal MR or MR=0.  Thus the actual
000088 | finite-typing control is a family covariance before both parts of the extra
000089 | unitary in section 5.
000090 | 
000091 | 3. F5 COMPLEX STRUCTURE, C0 AND THE COMPLETE GRAPH
000092 | 
000093 | Use only the already established whole coefficient data
000094 | 
000095 |   Gamma=Z tensor gamma_F,
000096 |   Ical=i0 Gamma,
000097 |   C0=i0 N tensor I_96,
000098 |   B_Phi=Z tensor A-S_normal tensor Phi,
000099 |   A=-i0 partial_theta+a(theta).
000100 | 
000101 | The actual E0 pullback preserves the intrinsic NS A, with its background
000102 | transported if a is not E0 invariant.  Since K_E commutes with Gamma and is
000103 | OLD antilinear,
000104 | 
000105 |   K_E Ical=-Ical K_E,  K_E C0=-C0 K_E.
000106 | 
000107 | Thus K_E is also antilinear for the NEW F5 complex structure, but it has the
000108 | wrong sign with C0 for the oriented selfdual chart.  The existing complex
000109 | structure supplies the correctly typed involution
000110 | 
000111 |   Jcal_E=Ical K_E.
000112 | 
000113 | It satisfies
000114 | 
000115 |   Jcal_E^2=I,
000116 |   [Jcal_E,C0]=0,
000117 |   [Jcal_E,C_faces]=0,
000118 |   Jcal_E B_Phi Jcal_E^-1=B_(Phi^E).                 (2)
000119 | 
000120 | There is no identification here of coefficient-real J_F, the NEW real
000121 | structure C0 and the physical CAR/OS word reflection.  They are three
000122 | different typed operations.  In particular J_F anticommutes with gamma_F
000123 | and is NEW complex linear after extension to Ical, whereas C0 and Jcal_E are
000124 | NEW antilinear.
000125 | 
000126 | Real functional calculus applied to (2) gives the whole two-face identity
000127 | 
000128 |   Jcal_E G(B_Phi) Jcal_E^-1=G(B_(Phi^E)).             (3)
000129 | 
000130 | This includes both diagonal and crossing blocks.  The pure quasifree graph
000131 | state therefore transports to the graph state for Phi^E.  No second state,
000132 | Gaussian fit or selected finite channel is introduced.  Equation (3) becomes
000133 | a symmetry of one graph exactly when Phi^E=Phi, or after an independently
000134 | allowed equivalence which identifies the two backgrounds.
000135 | 
000136 | 4. CHARGED T_FACES CHART AND TIME
000137 | 
000138 | Let
000139 | 
000140 |   E v=(P_F,+ v,i0 P_F,- v),
000141 |   h_Phi=A gamma_F-i0 gamma_F Phi.
000142 | 
000143 | The exact pullback of Jcal_E through E is
000144 | 
000145 |   Jcal_E E=E j_E,  j_E=i0 gamma_F K_E,
000146 |   j_E^2=I,
000147 |   j_E h_Phi j_E^-1=h_(Phi^E).                         (4)
000148 | 
000149 | The same j_E acts on the two charged face entries; the conjugate entries in
000150 | T_faces carry the corresponding conjugate action.  Consequently (4)
000151 | transports the already existing decomposition
000152 | 
000153 |   G(h_Phi) directsum (I-bar(G(h_Phi)))
000154 | 
000155 | to the same complete expression with Phi^E.  This is the concrete
000156 | charged-plus-dual covariance which remains when Phi is not fixed.
000157 | 
000158 | Because j_E is antiunitary, it reverses the unitary circle parameter:
000159 | 
000160 |   j_E exp(i0 t h_Phi) j_E^-1=exp(-i0 t h_(Phi^E)).    (5)
000161 | 
000162 | Likewise Jcal_E exp(t Ical B_Phi) Jcal_E^-1 is the minus-t group for
000163 | B_(Phi^E).  A plus-time equality would silently count the anti-linearity
000164 | wrong.  This is source-family time reversal.  It does not identify circle
000165 | time with a positive radial OS Hamiltonian.
000166 | 
000167 | 5. THE omega MAJORANA PHASE AND THE ACTUAL GAUGE BOUNDARY
000168 | 
000169 | For any phase t define the actual 96-dimensional unitary
000170 | 
000171 |   V_t=diag(t I_48,bar(t) I_48).
000172 | 
000173 | It commutes with the represented A_F, gamma_F and J_F.  It leaves every
000174 | particle Yukawa block unchanged and sends MR to t^2 MR.  Thus t=omega
000175 | removes the common omega in MR', but it does not remove the two S factors.
000176 | V_omega alone identifies Phi^E with Phi only for an S-real background,
000177 | namely S bar(Y_f) S=Y_f and S bar(MR) S=MR.
000178 | 
000179 | To test the actual real finite-typing control, let L_S apply S to generations
000180 | on both particle and antiparticle blocks.  It too commutes with A_F, gamma_F
000181 | and J_F.  Put
000182 | 
000183 |   W_E=V_omega L_S.
000184 | 
000185 | Then the exact general identity is
000186 | 
000187 |   W_E Phi^E W_E^*=Phi(bar(Y_f),bar(MR)).
000188 | 
000189 | In particular every fully real constructor input, including the actual real
000190 | tridiagonal MR in check_finite_typing.py, obeys
000191 | 
000192 |   W_E Phi^E W_E^*=Phi.                                (6)
000193 | 
000194 | Moreover Jcal_E W_E Jcal_E^-1=W_E^*, so
000195 | 
000196 |   Jcal'_E=W_E Jcal_E
000197 | 
000198 | is again an antiunitary involution.  It preserves the full B_Phi, graph and
000199 | charged source for real controls.  If only V_omega is composed, its effective
000200 | coefficient matrix is S rather than omega^2 S and only the S-real subclass is