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000001 | OFFDIAGONAL HOM, THE ORIGINAL K BUNDLE, AND THE ACTUAL E8 PRODUCT TEST
000002 | Independent bounded source review, 30 October 2026.
000003 | No Atlas mutation and no change to a registered report.
000004 | 
000005 | VERDICT
000006 | 
000007 | After the geometrically fixed p-to-q pullback, the six off-diagonal
000008 | components of Hom(Fq,Fp) do admit an explicit normalized A4-equivariant
000009 | unitary bundle isomorphism
000010 | 
000011 |   OffDiag End(F)  ~=  F direct-sum Lambda^2 F
000012 |                     = Lambda^2(O direct-sum F),                 (1)
000013 | 
000014 | where F=L0 direct-sum L1 direct-sum L2 is the triplet of nontrivial sign
000015 | lines. This is more than a dimension match: it uses the old square
000016 | trivializations beta_a, L0 L1 L2=O, the fixed product-one normalization,
000017 | and it intertwines the actual V4 signs and the cubic permutation c.
000018 | 
000019 | Equation (1) is nevertheless NOT the family map induced by bracketing the
000020 | two given triplet fields. Both coefficient legs land in the same
000021 | triplet F inside Kfam=O direct-sum F. Their actual family wedge has image only
000022 | Lambda^2 F, of rank three. The symmetric off-diagonal triplet in (1)
000023 | would have to land in O wedge F and therefore requires one genuine
000024 | singlet input. The selected closed spin source has
000025 | 
000026 |   H^0(C,S0 tensor O)=0,
000027 | 
000028 | so no singlet zero mode can be inserted silently. The full local field
000029 | bundle may contain an O leg, but using it requires an actual original
000030 | field insertion and the matching D5 spinor bilinear. Neither is supplied
000031 | by the six q/p products alone.
000032 | 
000033 | There is a second sharp boundary. The map (1) is A4-equivariant in the
000034 | fixed two-torsion chart, but it is not an SU4 intertwiner and it already
000035 | fails the original quarter generator on its symmetric triplet. Thus it
000036 | identifies the restriction of the family six to the actual A4 sign-line
000037 | system; it does not identify the full native (10,6) field slot.
000038 | 
000039 | 
000040 | 1. INPUTS ACTUALLY RETAINED
000041 | 
000042 | Use the already source-derived data, without adding a p source or a new
000043 | state:
000044 | 
000045 |   C: eta^2=1-w^6,
000046 |   L_a^q=O(q_a-q_(a+1)),  L_a^p=O(p_a-p_(a+1)),
000047 |   (h_a^q)^2=f_a^q,       (h_a^p)^2=f_a^p,
000048 |   product_a h_a^q=product_a h_a^p=1.
000049 | 
000050 | The square trivializations beta_a:L_a^2->O identify each two-torsion line
000051 | with its dual and are isometric in the retained flat metric. The selected
000052 | spin obeys S0^2=K_C. Hence the already proved mixed products are global
000053 | 
000054 |   mu_ab in H^0(K_C tensor Hom(L_a^q,L_b^p)),
000055 | 
000056 | and the six terms a!=b are the geometric U6 product sector. This review
000057 | does not replace their one-form factor by a scalar. It tests only their
000058 | coefficient bundle against the existing family six.
000059 | 
000060 | The E8 source decomposition used below is the original
000061 | 
000062 |   e8=(45,1)+(1,15)+(10,6)+(16,4)+(bar16,bar4),
000063 | 
000064 | with family 6=Lambda^2 4. In the existing K bundle write
000065 | 
000066 |   Kfam=O direct-sum F,  F=L0 direct-sum L1 direct-sum L2.       (2)
000067 | 
000068 | This notation separates the rank-three triplet F from reports that call
000069 | the whole rank-four coefficient bundle Fq.
000070 | 
000071 | 
000072 | 2. THE s AND R COMPARISON HAS NO NEW PHASE
000073 | 
000074 | The normalized Kummer lift already fixes
000075 | 
000076 |   s^*h_a^p=h_a^q,  s^*h_a^q=h_a^p,
000077 |   c^*h_a^(q/p)=h_(a-1)^(q/p).                                 (3)
000078 | 
000079 | Thus s=R^3 pulls the p endpoint back to the corresponding q endpoint in
000080 | the fixed frames. The other inversion representative is
000081 | 
000082 |   R=s c^2,  R^2=c,  R^3=s.
000083 | 
000084 | Its spin lifts satisfy
000085 | 
000086 | [OMITTED: non-scientific conversation or execution metadata]
000087 | 
000088 | Therefore R and s do not define two freely adjustable comparisons. Their
000089 | ratio is the existing c transport, including its fixed spin lift. In a
000090 | two-field product the common central spin signs cancel, while the residual
000091 | c^2 label shift is handled by c-equivariance. Equations (3)-(4) justify
000092 | the following comparison on one q chart. They do not create the missing
000093 | singlet family leg.
000094 | 
000095 | 
000096 | 3. EXPLICIT NORMALIZED A4 BUNDLE ISOMORPHISM
000097 | 
000098 | Choose unit flat frames e_a of L_a and the product-one orientation
000099 | e_0 e_1 e_2=1. Put e_inf=1 in O. For each cyclic triple
000100 | 
000101 |   (a,b,c)=(0,1,2),(1,2,0),(2,0,1),
000102 | 
000103 | write E_ab=e_a tensor e_b^vee and define
000104 | 
000105 |   f_c=e_inf wedge e_c,
000106 |   g_c=e_a wedge e_b,
000107 | 
000108 |   T(E_ab)=(f_c+g_c)/sqrt(2),
000109 |   T(E_ba)=(f_c-g_c)/sqrt(2).                                  (5)
000110 | 
000111 | Both f_c and g_c have line type L_c: beta identifies L_b^vee with
000112 | L_b, and L_a L_b=L_c follows from L0 L1 L2=O. Consequently (5) is a
000113 | typed bundle map, not a map between bare six-dimensional fibers.
000114 | 
000115 | The six displayed target vectors are orthonormal, so T^*T=I. Equivalently,
000116 | 
000117 |   (E_ab+E_ba)/sqrt(2) |-> f_c,
000118 |   (E_ab-E_ba)/sqrt(2) |-> g_c.                                (6)
000119 | 
000120 | The actual V4 sign matrices and c are determinant-one signed
000121 | permutations of (e0,e1,e2). Direct substitution in (5) gives
000122 | 
000123 |   T Ad(g)=Lambda^2(1 direct-sum g) T
000124 | 
000125 | for the two V4 generators and c, hence for A4=V4 semidirect C3. The
000126 | normalizations beta and product h0 h1 h2=1 remove any extra relative
000127 | phase. This proves (1) as a normalized A4 bundle statement.
000128 | 
000129 | An exact six-by-six fiber audit gives T^*T=I and intertwiner residual
000130 | ranks zero for c and both V4 generators. For the quarter matrix in (7)
000131 | the residual has rank three. These ranks are consequences of (5), not a
000132 | numerical fit.
000133 | 
000134 | It also keeps E_ab and E_ba distinct. Holomorphic multiplication of their
000135 | underlying scalar line sections is commutative, but the Hom arrows remain
000136 | separate; their sum and difference in (6) are precisely the two target
000137 | triplets.
000138 | 
000139 | 
000140 | 4. FIRST COUNTERTESTS BEYOND A4
000141 | 
000142 | The original quarter action on the triplet contains the determinant-one
000143 | signed permutation
000144 | 
000145 |   Qe0=-e1,  Qe1=e0,  Qe2=e2.                                 (7)
000146 | 
000147 | Conjugation sends E01+E10 to its negative. Formula (6) sends this vector
000148 | to f2=e_inf wedge e2, which is fixed by Lambda^2(1 direct-sum Q).
000149 | Thus T does not intertwine Q on the symmetric triplet. The antisymmetric
000150 | combination E01-E10 and g2=e0 wedge e1 are both fixed, so the failure is
000151 | exactly the additional O wedge F half, not the old wedge half.
000152 | 
000153 | A generic diagonal SU3 element gives the same obstruction without relying
000154 | on (7). If U=diag(t0,t1,t2), t0 t1 t2=1, then
000155 | 
000156 |   Ad(U)E_ab has weight t_a/t_b,
000157 |   e_inf wedge e_c has weight t_c,
000158 |   e_a wedge e_b has weight t_a t_b=t_c^-1.                    (8)
000159 | 
000160 | These weights agree for the retained order-two signs, but not generically.
000161 | Therefore (1) cannot be promoted to an SU3 or SU4 intertwiner. The full
000162 | SU4 family six remains Lambda^2 Kfam with its original marking; (1) is
000163 | only its A4-restricted sign-line chart.
000164 | 
000165 | 
000166 | 5. THE DECISIVE E8 PRODUCT TEST
000167 | 
000168 | For two same-chirality D5 spinors the relevant E8 bracket uses
000169 | 
000170 |   Sym^2(16) -> 10
000171 | 
000172 | on the carrier factor and
000173 | 
000174 |   Lambda^2(4) -> 6
000175 | 
000176 | on the family factor. The two parities combine to the antisymmetric Lie
000177 | bracket. Now insert the actual q and pulled-back p triplet fields. Their
000178 | family coefficients x,y both lie in F subset Kfam. Hence
000179 | 
000180 |   x wedge y in Lambda^2 F,                                  (9)
000181 | 
000182 | and the family image has dimension at most three. On an off-diagonal
000183 | ordered pair, antisymmetrizing the two family legs keeps
000184 | 
000185 |   E_ab-E_ba |-> e_a wedge e_b
000186 | 
000187 | and kills the symmetric combination. It cannot turn that symmetric
000188 | combination into e_inf wedge e_c. Formula (5) is therefore not
000189 | product-preserving for the given two-triplet bracket.
000190 | 
000191 | To populate the other triplet O wedge F, one input must carry the O
000192 | coefficient and the other an F coefficient. The actual selected closed
000193 | source gives
000194 | 
000195 |   H^0(C,S0 tensor L_a)=1 for a=0,1,2,
000196 |   H^0(C,S0 tensor O)=0.                                     (10)
000197 | 
000198 | Thus the six mixed triplet zero-mode products do not contain the required
000199 | singlet field. One may not manufacture it from the symmetric scalar
000200 | product L_a L_b=L_c: that equality identifies line types, whereas the E8