TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/REVERSE-HOM-COMPOSED-Q-HODGE-REVIEW.txt
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000001 | ACTUAL REVERSE HOM COMPOSITION AND THE q BILINEAR HODGE MAP
000002 | Independent bounded source/operator review, 30 October 2026.
000003 | No new kernel, state, added line, reader normalization or Atlas mutation.
000004 | 
000005 | VERDICT
000006 | The actual B=G_q J* does cancel the intermediate p deck signs in B mu.
000007 | The resulting whole operator has the correct q input and output transitions.
000008 | On its already smoothing same-face diagonal, the existing q bilinear form
000009 | and oriented volume give a global skew/Hodge coefficient without appending
000010 | any line. This is a positive connection supplied by the actual source.
000011 | It is a differential-valued boundary response, not automatically a finite
000012 | 3 by3 Yukawa operator or an originally selected P1 insertion.
000013 | 
000014 | 1. Use the actual kernel and retain both endpoint fibers
000015 | Reuse F5-QP-REVERSE-HOM-KERNEL-REVIEW.txt:19-74 and the actual
000016 | RHO-DUAL-COEFFICIENT-HOM-REVIEW.txt:55-152. Write beta_q for the original
000017 | complexified real bilinear form on Fq, to distinguish it from the KERNEL B.
000018 | The normalized flat frames have beta_q=I, their product gives the fixed
000019 | oriented volume, and the actual V4 signs S_q,S_p have determinant one.
000020 | 
000021 | mu(v):Fq_v -> K_v tensor Fp_v is an original spin-product insertion.
000022 | The given reverse kernel has B(u,v):Fp_v -> Fq_u with its actual boundary
000023 | half-density and normal-spin factors. Thus the composition has kernel
000024 |   T(u,v)=B(u,v) mu(v)
000025 | with family type Hom(Fq_v,Fq_u), and the displayed K_v insertion weight.
000026 | After its GIVEN tangent/Clifford contraction this is the actual integral
000027 | composition B M_mu, or its already proved integrable form realization.
000028 | Before that contraction it remains differential-valued; no scalarization
000029 | has happened merely by multiplying family matrices.
000030 | 
000031 | For each joint Kummerdeck element, the actual transition laws are
000032 |   mu(v) -> S_p(v) mu(v) S_q(v)^-1,
000033 |   B(u,v) -> S_q(u) B(u,v) S_p(v)^-1,
000034 | so
000035 |   T(u,v) -> S_q(u) T(u,v) S_q(v)^-1.                 (1)
000036 | This cancels the p leg exactly on all actual components. It proves the
000037 | correct global q endomorphism-operator bundle type; components need not
000038 | be individually invariant scalar functions. The adjoint source slots and
000039 | all normal components remain the ones already supplied by T_faces.
000040 | 
000041 | 2. Where skew_beta and Hodge are actually pointwise defined
000042 | A bilocal kernel is not a single-fiber End(Fq_u) matrix. Its bilinear
000043 | OPERATOR transpose exchanges u,v and uses the two beta maps, together
000044 | with the actual differential/density transpose. Raw family transposition
000045 | at fixed unmatched endpoints is not that operation.
000046 | 
000047 | For the two already smoothing same-face blocks there is a canonical
000048 | well-defined diagonal density:
000049 |   t(u)=B^(ii)(u,u) mu(u), i=0,1,
000050 |   B^(00)=B^(11)=X_q j*, X_q=(2cosh(ell h_q))^-1,
000051 |   j=i gamma_F F_R R_x.                              (2)
000052 | Here (X_q j*)(u,u)=X_q(u,R^-1 u) j0(u)*, with the original
000053 | spin/half-Jacobian in j0. R_x has not been replaced by a free fiber matrix.
000054 | Then t(u) has family type K_u tensor End(Fq_u). In a beta-orthonormal
000055 | frame define
000056 |   a(u)=(t(u)-t(u)^t_beta)/2,
000057 |   v_k(u)=(1/2) sum_ij epsilon_kij a_ij(u).           (3)
000058 | Because beta and the existing oriented volume are invariant, (1) gives
000059 |   a -> S_q a S_q^-1, v -> S_q v.
000060 | Thus (3) is a GLOBAL K-valued Fq coefficient, with any remaining D5,
000061 | normal-spin or charge operators still attached. It uses no extra line
000062 | and no new choice of an inverse cubic character. This is complex bilinear
000063 | skew/Hodge, not the Hermitian anti-selfadjoint projection (t-t*)/2.
000064 | The nonsmoothing other graph blocks do not acquire a diagonal by analogy.
000065 | 
000066 | At the four original ramification marks the actual scalar boundary reader
000067 | has the retained L1 behavior |theta-theta_j|^(-2/3). The smooth X_q j*
000068 | diagonal therefore makes (3) an integrable density/form globally. The
000069 | existing positive-kernel sandwich gives its trace-class form realization;
000070 | raw M_mu X_q j* is still not asserted to be an everywhere-defined L2
000071 | operator. This uses F5-REVERSE-HOM-FORM-TRACE-ADDENDUM.txt, not a cutoff.
000072 | 
000073 | 3. Smallest actual Y-free nonzero test, with its source class retained
000074 | In the ALREADY GIVEN scalar common-quarter collar at Phi=0,
000075 |   h0=A_alpha gamma_F, A_alpha e_n=(n+alpha)e_n,
000076 |   X0=(2cosh(ell A_alpha))^-1 tensor I_F.
000077 | In a regular source frame its actual circle kernel is
000078 |   k_alpha(x,y)=(1/(2pi)) sum_n exp(i n(x-y))
000079 |                             /(2cosh(ell(n+alpha))).  (4)
000080 | It is smooth by exponential decay. The actual same-face reverse kernel
000081 | is k_alpha(x,R_x y) j0(y)*, not a freely selected representation matrix.
000082 | At x=y=0 on the lifted circle, R_x0=0 and k_alpha(0,0)>0. The fixed spin
000083 | factor and the actual q/p label map F_R are invertible. Hence this kernel
000084 | is nonzero on a neighboring regular arc as well. The original quarter
000085 | carry and spin phase in j0 must be retained; they cannot annul this
000086 | operator. This makes no ordinary normal-spin trace assertion.
000087 | 
000088 | In the corresponding local q/p frames let F_R* send p_b to a nonzero
000089 | multiple of q_sigma(b). For any original mu_ba component with
000090 | sigma(b)!=a, (2) has a nonzero matrix-unit contribution proportional to
000091 |   e_sigma(b) e_a^t.
000092 | Its beta skew is nonzero. Each original spin product mu_ba is nonzero
000093 | away from its two prescribed zero points; thus the response map on the
000094 | existing insertion space is not identically zero, even without an input
000095 | Yukawa. No arbitrary B, new species, common state or scalar norm was used.
000096 | This does not say a particular originally selected sum of insertions
000097 | cannot cancel. It also does not trivialize actual Fuchs monodromy: a
000098 | nontrivial family clutch may give different alpha+kappa_a channels, in
000099 | which case the full original covariant X and W must be used instead of
000100 | (4) tensor I. Formula (1) remains valid for that actual whole kernel.
000101 | 
000102 | 4. Precise first remaining identification
000103 | The output (3) is a boundary section/form. X_q(u,Ru) need not be a
000104 | holomorphic coefficient on C. No automatic restriction to H0(K tensorFq),
000105 | constant three-vector, or finite family-mode matrix follows. No trace over
000106 | normal/D5 slots or projection to a physical weak field has been supplied.
000107 | An equivariant B also does not erase a nontrivial TOTAL C3 character: cubic
000108 | source compression, when applicable, must still use the actual full lifts.
000109 | 
000110 | The actual original finite one-form is a_F[D_F,b_F], with its evaluated
000111 | weak block q(H)M and the supplied Yukawa matrices. See original_insertions/
000112 | PROOF.txt:29-49 and MU-CURRENT-DZ-FINITE-INSERTION-REVIEW.txt:65-114.
000113 | At D_F=0 this finite one-form is identically zero, while the existing
000114 | neutral-source response above can be nonzero. Therefore they are not the
000115 | same operation by their types alone. The positive Y-free response can be
000116 | a candidate coefficient for an original generating variation; identifying
000117 | it as primitive D_F or a Yukawa requires that actual source variation or
000118 | sewing consumer to produce it. No new state/time condition is needed for
000119 | the established response, and no new P1 origin is inferred from it.