TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/SOURCE-ZERO-MODE-ACTUAL-LIFT-NORM.txt
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000001 | ACTUAL FLAT-FIBER LIFT: CORRECTION AND KERNEL DETERMINANT LINE
000002 | Independent analytic correction, 2026-10-29.
000003 | 
000004 | Status
000005 | This supersedes the unregistered working file
000006 | SOURCE-ZERO-MODE-NORM-LINE-REVIEW.txt.
000007 | Registered earlier files remain physically unchanged. Their explicit
000008 | divisors, sections, squares and differential transports remain valid.
000009 | The meromorphic Q-phase calibration and the claimed original simultaneous
000010 | kernel/coefficient intertwiner require the correction below.
000011 | 
000012 | 1. The first wrong identification
000013 | The previous phase calculation used the Kummer-function lift signs
000014 | epsilon=(1,-1,-1) directly as the meromorphic line-map signs.
000015 | The functions h_a have zeros/poles at q0; they are NOT nonvanishing flat
000016 | fiber frames there. The induced cover automorphism alone is not the
000017 | additional v117 coefficient-representation normalizer action.
000018 | 
000019 | The actual original coefficient system is regular at z=0.
000020 | Its source identity Psi(iz)=U Psi(z) U^-1 and the printed F comparison
000021 | give the quarter action U on its fiber there, up to the already declared
000022 | pullback/inverse convention. On the q0 sheet it fixes q0 and has
000023 |   A_q^2=diag(-1,1,-1)
000024 | in the stated character-line ordering. This square is unchanged on inversion.
000025 | The fixed L1 line has Q eigenvalue +1.
000026 | 
000027 | Source anchors:
000028 |   original Sept28 section004:203-212,255,263-290;
000029 |   CLOCK-Q-PACKET-COMPATIBILITY.txt:125-170.
000030 | These concern the actual coefficient lift. Meromorphic Kummer functions
000031 | alone must not replace it.
000032 | 
000033 | 2. Local divisor-frame test
000034 | Use eta as the local parameter at q0 and h=Q^2:eta->-eta.
000035 | For D_a=q_a-q_(a+1), its orders at q0 are
000036 |   d=(+1,0,-1).
000037 | A nonvanishing local frame of O(D_a) is eta^(-d_a).
000038 | Even the canonical divisor identification under h therefore has fiber
000039 | factors (-1)^(d_a)=(-1,+1,-1).
000040 | 
000041 | A square-compatible meromorphic Q map to L_p, p=-a-1, has the form
000042 |   g_a=delta_a omega/H_p, delta_a=+1 or -1,
000043 | since Q*H_a=omega^2/H_p.
000044 | Its twice-composed MEROMORPHIC factor is delta_a delta_p.
000045 | To retain the actual fiber square (-,+,-), that factor must be +1 in
000046 | every slot: delta0 delta2=+1. The fixed-line phase at q0 additionally
000047 | requires delta1=-1, because H1(q0)=-omega.
000048 | One compatible choice is
000049 |   delta=(1,-1,1).
000050 | The alternative with both exchanged signs reversed gives the same square
000051 | and determinant data; the source orientation/frame must be tracked.
000052 | 
000053 | By contrast epsilon=(1,-1,-1) gives meromorphic square (-,+,-).
000054 | Combining it with the unavoidable local divisor-frame factors gives
000055 | FIBER square (+,+,+). That was the double-counted sign in the earlier
000056 | kernel-quarter comparison.
000057 | 
000058 | 3. Correct action on the explicit zero sections
000059 | Keep every previous f_a, mu0, H_a and spin square unchanged:
000060 |   f_a=1/(w+omega^a), mu0=(w^3+1)dw/eta,
000061 |   psi_a^2=(w+omega^(a+2))dw/eta.
000062 | For the SAME spin lift xi/w, xi=exp(-i pi/4), the corrected factors are
000063 |   v'_a=delta_a xi omega^(1-a), Q*psi_a=v'_a psi_(-a-1).
000064 | Thus
000065 |   (Q_kernel)^2=-i I3,
000066 |   spec Q_kernel={xi,-xi,-xi}, (Q_kernel)^4=-I3.
000067 | This is also forced locally: on T0,T2 the section is nonzero at q0,
000068 | so beta=+i on S0 times -1 on L gives -i; on T1 the section has a simple
000069 | eta zero, which contributes the remaining minus sign.
000070 | 
000071 | No scalar/conjugacy can turn this Q into the old U with three DISTINCT
000072 | eigenvalues. The previous explicit joint matrix identity remains correct
000073 | for its POSTULATED meromorphic lift, but that lift is not the old flat
000074 | coefficient lift. It is not an original-source intertwiner.
000075 | This is a scoped identification correction, not an all-source no-go.
000076 | 
000077 | 4. Correct tensor and fermionic determinant characters
000078 | Let
000079 |   Z=unordered tensor-product_a H0(T_a),
000080 |   Delta=det(direct-sum_a H0(T_a)),
000081 |   N_q=unordered tensor-product_a S0|q_a,
000082 |   B_q=det(direct-sum_a S0|q_a).
000083 | Now product_a v'_a=-xi^3. The index permutation is a transposition, so
000084 |   Q|Z=-xi^3, Q|Delta=+xi^3.
000085 | For any compatible order-three c lift the cyclic product is1, hence
000086 |   c|Z=c|Delta=1.
000087 | The actual S0 point-fiber factors are unchanged:
000088 |   alpha_a=i xi omega^(-a), Q|N_q=-xi.
000089 | Consequently
000090 |   Q|N_q^-1=xi^3, Q|B_q^-1=-xi^3;
000091 |   c|N_q^-1=c|B_q^-1=1.
000092 | Therefore the CORRECT comparisons are
000093 |   Delta ~= N_q^-1, Z ~= B_q^-1,
000094 | as simultaneous line representations of these given lifts.
000095 | 
000096 | 5. Which full group is checked
000097 | For the actual deck V4, H0(T_a) has character chi_a and
000098 | chi0 chi1 chi2=1. Its determinant and unordered tensor characters are1.
000099 | The point-norm lines are pulled from C and the deck V4 acts trivially.
000100 | The spin central -I acts by -1 on both sides of either corrected comparison.
000101 | Thus the characters agree on c, Q, deck V4 and the source spin center.
000102 | For the actual group generated by THESE lifts, this is sufficient for
000103 | an equivariant one-dimensional isomorphism, unique up to nonzero scalar.
000104 | It does not fix a functorially canonical isomorphism or its physical norm.
000105 | Jac(C)[2] tensor translations are different: they move S0 with h0=0 to
000106 | T_a with h0=1 and do not act invertibly on this isolated three-line kernel.
000107 | No full N64 or theta-family equivalence is established by this comparison.
000108 | 
000109 | 6. Residue maps and physical source boundary
000110 | The principal-part/Szego leg
000111 |   H0(S0(q_a)) ~= S0|q_a^*
000112 | still exists because h0(S0)=h1(S0)=0, and the one-dimensional H0(T_a)
000113 | includes isomorphically into H0(S0(q_a)). This is a valid linear leg.
000114 | It is not individually equivariant for the actual Q or nontrivial deck
000115 | character. In particular the earlier claim that its unordered tensor
000116 | directly gives the required Q-equivariant N_q^-1 map used the wrong lift.
000117 | A canonical entire determinant/residue construction needs the actual
000118 | edge-reversal and fermionic permutation signs, not just these characters.
000119 | 
000120 | The corrected Delta character supplies the mathematical necessary line
000121 | for ONE kernel-volume insertion. It does not prove that original P1
000122 | selects H0, uses its determinant once, or chooses its dual as a Berezin
000123 | measure. The full determinant of cohomology also contains H1; a uniform
000124 | scalar on the bundle has weight h0-h1=0 and still cancels there.
000125 | Reduced nonzero-mode determinants, saturation, physical state, OS and
000126 | time remain distinct source operations. No physical insertion is derived.
000127 |