TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/SOURCE-ZERO-MODE-NORM-INDEPENDENT.txt
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000001 | INDEPENDENT CHECK: ACTUAL ZERO-MODE LIFT AND INVERSE POINT NORM
000002 | 2026-10-29. Bounded analytic review; no new enumeration or source mutation.
000003 | 
000004 | Reviewed: SOURCE-ZERO-MODE-ACTUAL-LIFT-NORM.txt in this directory.
000005 | Original comparisons: SOURCE-SELECTED-THETA-SPIN.txt,
000006 | CLOCK-Q-PACKET-COMPATIBILITY.txt and MARKED-Q-DIVISOR-CENTRAL-TWIST.txt
000007 | in elliptic-moebius-source-bridge-20261026.
000008 | 
000009 | VERDICT
000010 | The corrected comparison is valid as a comparison of one-dimensional
000011 | representations of the group generated by the specified c, Q, deck V4
000012 | and spin-center lifts:
000013 | 
000014 |  Delta=det(bigoplus_a H0(T_a)) isomorphic Nq(S0)^-1,
000015 |  Z=bigotimes_a H0(T_a) isomorphic Bq(S0)^-1.
000016 | 
000017 | Here Z has the ordinary unordered tensor permutation, while Delta and
000018 | Bq include the exterior permutation sign. These are different lines
000019 | with different quarter characters. Equality of their characters proves
000020 | existence of equivariant isomorphisms up to scalar, not a canonical map,
000021 | a physical insertion rule, or equivariance under additional symmetries.
000022 | 
000023 | 1. THE LOCAL FIBER CALIBRATION IS ESSENTIAL
000024 | 
000025 | Write D_a=q_a-q_(a+1), H_a=(w+omega^a)/(w+omega^(a+1)),
000026 | p(a)=-a-1 and xi=exp(-i pi/4). At the Q-fixed q0, with local parameter
000027 | eta, the divisor orders are (+1,0,-1). Q^2 sends eta to -eta.
000028 | Thus the canonical divisor linearization on O(D_a) ALREADY has the
000029 | fiber square (-1,+1,-1) at q0. Meromorphic factors are not these fiber
000030 | values; a nonvanishing local frame requires division by the appropriate
000031 | power of eta from the tautological meromorphic section.
000032 | 
000033 | For the meromorphic Q map g_a=delta_a omega/H_p(a), composition twice
000034 | has meromorphic factor delta_a delta_p(a). Taking
000035 | delta=(1,-1,1) gives meromorphic square +1 in all slots and therefore
000036 | the actual required flat FIBER square (-,+,-). The fixed-line value
000037 | is +1 because H1(q0)=-omega. This is compatible with the original
000038 | rotational coefficient representation, whose Q square is (-,+,-).
000039 | 
000040 | The Kummer-function cover lift epsilon=(1,-1,-1) has a different role.
000041 | Used directly for this meromorphic associated-line map, it contributes
000042 | another (-,+,-), leaving fiber square (+,+,+). It must not replace the
000043 | original coefficient normalizer representation. The earlier compact
000044 | Kummer-cover identities themselves are not refuted by this distinction.
000045 | 
000046 | 2. KERNEL QUARTER AND ITS DETERMINANT
000047 | 
000048 | With the same spin branch and corrected associated-line map,
000049 | 
000050 |  Q*psi_a=v_a psi_p(a),  v_a=delta_a xi omega^(1-a).
000051 | 
000052 | Explicitly v0=xi omega, v1=-xi, v2=xi omega^-1. The exchanged
000053 | 0/2 block has product xi^2=-i; the fixed slot has eigenvalue -xi.
000054 | Consequently
000055 | 
000056 |  Q_kernel^2=-i I3,
000057 |  spec(Q_kernel)={xi,-xi,-xi},
000058 |  product_a v_a=-xi^3,
000059 |  det(Q_kernel)=xi^3.
000060 | 
000061 | The last sign is the transposition p=(02). Thus Q|Z=-xi^3 and
000062 | Q|Delta=xi^3. The nontrivial 2/1 eigenvalue multiplicity cannot become
000063 | the three distinct eigenvalues of the original coefficient U by scalar
000064 | rescaling or conjugation. This corrects the claimed direct intertwiner,
000065 | without giving a general physical obstruction from the closed kernel.
000066 | 
000067 | 3. POINT NORM QUARTER: CHECK DIRECTLY AT q0 AND ITS PAIRED FIBERS
000068 | 
000069 | The chosen spin branch has alpha(q0)=i xi, whose square is +i.
000070 | Q exchanges q1 and q2, and the product of their spin fiber maps is
000071 | beta(q1)=+i. The latter follows from the source-selected S0 lift:
000072 | the meromorphic Phi has simple zeros at all q, beta Phi=-i Phi,
000073 | and eta changes sign, so the beta FIBER phase is +i at every q.
000074 | Therefore, independently of individual meromorphic frame factors,
000075 | 
000076 |  Q|Nq=(i xi)(+i)=-xi,
000077 |  Q|Nq^-1=xi^3.
000078 | 
000079 | On Bq the transposition supplies an additional minus sign, so
000080 | Q|Bq=xi and Q|Bq^-1=-xi^3. This proves exactly the two corrected
000081 | quarter-character comparisons above. Changing the overall spin-lift
000082 | sign changes both compared quarter characters together.
000083 | 
000084 | As a useful check, beta acts on Delta by (-i)^3=+i, and on Nq^-1
000085 | by ((+i)^3)^-1=+i. This beta character alone cannot distinguish the
000086 | previous wrong quarter assignment: the quarter generator and its
000087 | permutation sign must actually be checked.
000088 | 
000089 | 4. CUBIC, DECK AND CENTRAL CHARACTERS
000090 | 
000091 | The actual order-three c lift cycles the three T_a. Its three successive
000092 | line maps multiply to identity because c^3=I. The 3-cycle is even, so
000093 | c|Delta=c|Z=1. On the three point-spin fibers the same order-three
000094 | condition gives product one; hence c|Nq^-1=c|Bq^-1=1. A scalar cubic
000095 | renormalization enters these products through its third power and does
000096 | not change this result.
000097 | 
000098 | Each deck element acts on H0(T_a) by the line character chi_a;
000099 | chi0 chi1 chi2=1. Both kernel-volume characters are therefore trivial
000100 | on deck V4. The point-spin lines come from C, so deck V4 acts trivially
000101 | there as well. The spin central -I acts as -1 on either kernel volume
000102 | and on either inverse three-point norm. Thus all listed generators
000103 | match. Character equality on generators proves equality on every word
000104 | in these specified lifts, including their existing relations.
000105 | 
000106 | 5. PRECISE LIMITS
000107 | 
000108 | No Jac(C)[2] tensor-translation action has been constructed on this
000109 | isolated three-dimensional kernel. Such translations move S0 and alter
000110 | h0; they cannot be imported from the full theta-family module merely
000111 | because the above one-dimensional characters agree. No full N64
000112 | equivariance follows. The reviewed file explicitly keeps these limits.
000113 | 
000114 | The residue legs remain valid linear maps between individual lines.
000115 | Their direct unordered tensor does not automatically supply the corrected
000116 | Delta-to-Nq^-1 equivariant isomorphism: the actual edge reversal and
000117 | fermionic permutation signs still have to occur in a proposed full
000118 | construction. The character proof supplies a nonzero intertwiner up to
000119 | scalar and leaves its canonical normalization and physical norm open.
000120 | 
000121 | Nor does it choose H0 rather than H1, prescribe one insertion, or prove
000122 | that its dual is the original Berezin measure. The full determinant of
000123 | cohomology includes H1 and retains scalar weight h0-h1=0. The corrected
000124 | one-dimensional match is accepted at precisely its stated generated-group
000125 | scope. It does not identify the compact curve operator with the P1/APS
000126 | operator, its domain, real structure or physical state.