TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ANTISYMMETRIC-E8-KERNEL-BRIDGE.txt
Original file SHA-256: 37ddfb2a11aba09a23a6326fd3c929352edca3c8bb575f19778a2e03cce2454c
Inclusive original lines: 1-200
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.

000001 | ANTISYMMETRIC E8 KERNEL BRIDGE IN THE ACTUAL MIXED-SPIN FRAMES
000002 | Independent exact source review, 30 October 2026.
000003 | 
000004 | VERDICT
000005 | 
000006 | The proposed antisymmetrizer is the missing canonical THREE-dimensional
000007 | coefficient map.  With the original Kummer square pairings beta, the actual
000008 | real v117 bilinear form B and the fixed product-one orientation, it is a
000009 | normalized A4 intertwiner
000010 | 
000011 |   A : OffDiag Hom(Fq,Fp) -> Lambda^2 F,
000012 |   A(mu_ab-mu_ba)/sqrt(2)=e_a wedge e_b.                 (1)
000013 | 
000014 | This is the coefficient map.  On the actual fields the typed arrow is
000015 | 
000016 |   id_(K_C) tensor A:
000017 |   K_C tensor OffDiag Hom(Fq,Fp) -> K_C tensor Lambda^2 F.
000018 | 
000019 | Neither A nor Hodge star removes the K_C one-form leg.
000020 | 
000021 | It also intertwines the actual v117 quarter matrix KQ.  Hodge star and B
000022 | then give a canonical coefficient map Lambda^2 F -> F.  Thus every already supplied
000023 | positive six-sector kernel which preserves the antisymmetric subspace has
000024 | the positive compression K3=A K6 A*.  No new metric, projector or relative
000025 | normalization is chosen.
000026 | 
000027 | The spin phase is essential and corrects the phase-free reading of the old
000028 | six-cycle report.  In the natural q/p section frames
000029 | 
000030 |   s*mu_ab=-mu_ba,
000031 |   R*mu_ab=rho^-1 mu_(b+1,a+1),                           (2)
000032 | 
000033 | so the antisymmetric triplet has R^3=+I.  The phase-free U6 shift remains a
000034 | valid orbit-normalized presentation, but its basis has absorbed the partner
000035 | phases in (2).  One may not use that presentation and simultaneously treat
000036 | the raw partner comparison as +transpose.
000037 | 
000038 | This closes the coefficient and positive-kernel arrow into the E8 family
000039 | wedge selected by the Lie bracket.  It does not turn the resulting
000040 | K_C-valued one-form into the scalar finite Yukawa field, select a P1 response,
000041 | or identify the v117 quarter KQ with the native full-E8 quarter clock.
000042 | 
000043 | 
000044 | 1. THE FIRST LOAD-BEARING SPIN SIGN
000045 | 
000046 | Use the original sections
000047 | 
000048 |   psi_q,a=Phi e_q,a/(w+omega^a),
000049 |   psi_p,b=Chi e_p,b/(w-omega^b),
000050 | 
000051 | with Phi Chi=dw.  For s=R^3, w->-w, the fixed square-compatible spin lift is
000052 | 
000053 |   s*Phi=i Chi,  s*Chi=i Phi.                              (3)
000054 | 
000055 | The normalized sign-line transport carries e_q,a to e_p,a and conversely.
000056 | The transformed denominator contributes a further minus:
000057 | 
000058 |   s*psi_q,a=-i psi_p,a,  s*psi_p,b=-i psi_q,b.             (4)
000059 | 
000060 | Ordinary multiplication of these two sections therefore gives the first
000061 | identity in (2).  The same result is visible directly in the exact formula
000062 | 
000063 |   mu_ab=h_q,a h_p,b dw/[(w+omega^a)(w-omega^b)] tensor E_ba:
000064 | 
000065 | s changes dw to -dw and exchanges the two denominators and endpoints.
000066 | 
000067 | For the positive rotation R:w->rho w, R*Phi=lambda Chi,
000068 | R*Chi=lambda Phi and lambda^2=rho.  Since
000069 | 
000070 |   rho w+omega^a=rho(w-omega^(a+1)),
000071 |   rho w-omega^b=rho(w+omega^(b+1)),
000072 | 
000073 | each spin section contributes lambda/rho and their product contributes
000074 | rho^-1.  This proves the second identity in (2).  Cubing it gives
000075 | rho^-3=-1 and the label permutation becomes transpose, hence R^3=-transpose
000076 | on the raw six-space and +I on its antisymmetric half.  Six iterations still
000077 | give I, as the old report correctly states.
000078 | 
000079 | This agrees exactly with the positive U6 orbit frame, but the comparison is
000080 | not phase free.  In the forward orbit
000081 | 
000082 |   (01,21,20,10,12,02)
000083 | 
000084 | put e_j=rho^(-j) mu_(f^j(01)).  Then R e_j=e_(j+1).  Since
000085 | rho^(-3)=-1, the opposite partner has e_(j+3)=-rho^(-j)mu_ba.  Therefore
000086 | 
000087 |   mu_ab-mu_ba = rho^j (e_j+e_(j+3)).
000088 | 
000089 | The raw antisymmetric projector is precisely (I+U6^3)/2 in this positive
000090 | orbit basis.  It is not the U6^3=-1 sheet.  This is the concrete phase
000091 | calibration that the label permutation alone misses.
000092 | 
000093 | 
000094 | 2. THE CANONICAL MAP AND ITS EXACT GROUP ACTION
000095 | 
000096 | Order the source basis as
000097 | 
000098 |   (mu_01,mu_10,mu_12,mu_21,mu_20,mu_02)
000099 | 
000100 | and the target as
000101 | 
000102 |   (e0 wedge e1,e1 wedge e2,e2 wedge e0).
000103 | 
000104 | Then A is the three by six matrix with blocks (1,-1)/sqrt(2).  Exact
000105 | calculation gives
000106 | 
000107 |   A A*=I3,
000108 |   A* A=(I-J)/2,                                            (5)
000109 | 
000110 | where J transposes the two ordered labels.  Thus A is the coisometry onto
000111 | the antisymmetric half, not a dimension match.
000112 | 
000113 | For the actual determinant-one signed permutations c,V4_1,V4_2 and
000114 | 
000115 |   KQ=[[0,1,0],[-1,0,0],[0,0,1]],                          (6)
000116 | 
000117 | the retained real bilinear form identifies F* with F and direct substitution
000118 | gives
000119 | 
000120 |   A (g tensor g)=Lambda^2(g) A.                            (7)
000121 | 
000122 | This includes all of A4 and the actual v117 G-frame quarter coefficient.
000123 | The full spin-product field still carries id_(K_C), its quarter spin lift
000124 | and its endpoint maps.  Equation (7) does not erase those factors or assert
000125 | that the full coupled quarter is the naked KQ action.  In particular it
000126 | explains the old countertest: the quarter obstruction is entirely on the
000127 | symmetric O wedge F triplet.  Its antisymmetric Lambda^2 F half has zero
000128 | residual.
000129 | 
000130 | With e0 e1 e2=1, Hodge sends
000131 | 
000132 |   e0 wedge e1 -> e2,
000133 |   e1 wedge e2 -> e0,
000134 |   e2 wedge e0 -> e1.
000135 | 
000136 | Because all matrices in (6)-(7) are B-orthogonal with determinant one,
000137 | W=*B A satisfies W(g tensor g)=gW.  The product-one normalization fixes the
000138 | orientation.  Reversing it multiplies W by -1 and leaves W K6 W* unchanged.
000139 | 
000140 | The R action retains its source spin grade:
000141 | 
000142 |   A Rraw=omega Lambda^2(c) A,
000143 |   W Rraw=omega c W.                                       (8)
000144 | 
000145 | The scalar omega is forced by rho^-1 and the exchange sign; it is not a
000146 | freely chosen phase.  Tensoring with the existing inverse cubic coefficient
000147 | line would remove it, but the checked original closed Yukawa consumer has
000148 | not supplied that one-sided coefficient reader.  Equation (8) is therefore
000149 | the correctly typed result on the mixed-spin field, while plain c on a
000150 | scalar family vector would be a further contraction.
000151 | 
000152 | 
000153 | 3. POSITIVE THREE-DIMENSIONAL TRANSPORT
000154 | 
000155 | For the bare six-cycle source put
000156 | 
000157 |   D_y=y I-delta Rraw,
000158 |   H6=D_y* D_y+epsilon^2 I.                                (9)
000159 | 
000160 | The antisymmetric projector in (5) commutes with Rraw.  Therefore this
000161 | subspace reduces H6 and every positive Borel function of H6.  Its exact
000162 | three-dimensional Gram is
000163 | 
000164 |   H3=W H6 W*
000165 |     =(y^2+delta^2+epsilon^2)I
000166 |       -y delta(omega c+bar(omega)c*).                      (10)
000167 | 
000168 | Its eigenvalues are
000169 | 
000170 |   (y-delta)^2+epsilon^2                    multiplicity 1,
000171 |   y^2+y delta+delta^2+epsilon^2            multiplicity 2. (11)
000172 | 
000173 | They are positive for the regulated source.  Consequently
000174 | 
000175 |   K3=W H6^(-1/2) W*=H3^(-1/2)                            (12)
000176 | 
000177 | for the original positive inverse-square-root kernel, and the analogous
000178 | identity holds for the Green inverse.  The checker verifies the latter
000179 | exactly at rational y=2/3, delta=1/2, epsilon=1/7 and verifies (10)-(11)
000180 | symbolically in the matrix entries.  This is a canonical restriction of
000181 | the already given source, not a new three-state kernel.
000182 | 
000183 | KQ does not commute with the selected c-oriented H3.  Equation (7) gives
000184 | the correct statement: quarter transport sends H3 to the conjugate source
000185 | KQ H3 KQ*, so the construction is covariant.  Fixed quarter invariance
000186 | would be an additional, false strengthening.
000187 | 
000188 | 
000189 | 4. E8 AND PHYSICAL SCOPE
000190 | 
000191 | The original E8 bracket on two same-chirality branches uses
000192 | 
000193 |   Sym^2(16)->10,  Lambda^2(4)->6.
000194 | 
000195 | For the pulled q/p triplets its family part is exactly (1); the symmetric
000196 | ordered-pair half is killed.  The previous full-six objection remains valid
000197 | for a complex six-dimensional target, but it does not obstruct this canonical
000198 | antisymmetric three-dimensional bracket channel.  No O-family zero mode is
000199 | needed for this three-channel map.
000200 |