TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/automatic_identification/E8-ANCHOR-REVIEW.txt
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000005 | 1. Candidate and provenance
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000014 | 
000015 | The existing original source distinction is already explicit in
000016 | seam-joint-conjugation-20261015/PROOF.txt:189-216: the 10+6 real vector
000017 | fermions have spin sectors with weights5/8+3/8=1. The (16,4) fields in
000018 | this E8 current realization are weight1 currents, not16 additional
000019 | elementary chiral CAR species. This is an old result, reused here.
000020 | Thus the current OPE alone is not a demonstrated four-dimensional
000021 | fermion Yukawa functional. Its state, spacetime and field map still matter.
000022 | 
000023 | 2. What is correct algebraically
000024 | 
000025 | Invariance and symmetry of kappa give B_h(y,x)=-B_h(x,y).
000026 | Under the given Spin10 x SU4 action, in each fixed spinor-weight channel
000027 | the family part is a skew4x4 form omega: the10 spinor bilinear beta is
000028 | symmetric while6=Lambda2(4) is antisymmetric. The same conclusion holds
000029 | under linear superposition in h. No large numerical E8 reconstruction is
000030 | needed to test the next identification.
000031 | 
000032 | For the proposed ordinary local bilinear of the SAME collection of left
000033 | four-dimensional Weyl fields, the Lorentz epsilon is antisymmetric.
000034 | Consequently a skew internal coefficient B gives B tensor epsilon symmetric.
000035 | For Grassmann fields psi_i, psi^T M psi uses only (M-M^T)/2, so this bare
000036 | candidate vanishes. Equivalently the internal mass coefficient must be
000037 | symmetric. This excludes this precise identification, not every possible
000038 | Dirac pairing, derivative vertex, sector field map or interacting theory.
000039 | The distinction is also consistent with the primary SO10 Yukawa literature:
000040 | Bajc and Smirnov, arXiv:1605.07955, abstract and section2, distinguishes
000041 | symmetric10/126 Yukawas from antisymmetric120 Yukawas. No coefficients or
000042 | physical masses are imported from that paper.
000043 | 
000044 | 3. Complete anchor-only polynomial class, not a random sample
000045 | 
000046 | Normalize the single anchor/triplet marking on4=1+3 as
000047 |   X=diag(3,-1,-1,-1), X^2=2X+3I.
000048 | Every polynomial of this SAME X reduces exactly to aI+bX. Consider all
000049 |   M=F(X) omega G(X), omega^T=-omega,
000050 | and their linear combinations, retaining one such bracket form per term.
000051 | Let F=aI+bX,G=cI+dX. The physical symmetric part in this fixed channel is
000052 |   Sym(M)=(ad-bc)(omega X-X omega)/2.
000053 | The identity and X omega X terms vanish on symmetrization. Because X is
000054 | scalar on the triplet, EVERY such Sym(M) has the form
000055 |   A=[[0,v^T],[v,0_3]].
000056 | The physical3x3 triplet mass block is identically zero. In a single4x4
000057 | family channel, rank A<=2. For complex v,
000058 |   A A^dag=diag(||v||^2,v v^dag),
000059 | so the singular values are ||v||,||v||,0,0. For fixed omega the dressing
000060 | image has dimension at most1; as omega ranges over all skew4 forms the
000061 | symmetric image has dimension3. This is not a rank bound on the full64
000062 | spinor-family operator or a theorem about every multi-insertion process.
000063 | 
000064 | All arbitrarily long polynomial dressings of this single mark have been
000065 | covered by the quadratic relation for X. A genetic search in this class
000066 | cannot find a missing full triplet mass matrix. The claim that the anchor
000067 | alone gives four physical masses counted singular values of a coefficient
000068 | before the required Weyl symmetrization and is invalid in this contract.
000069 | 
000070 | 4. Exact decisive test and changed premise
000071 | 
000072 | check_e8_anchor_yukawa.py checks all6 skew basis forms with I/X on both
000073 | legs,24 reduced words, using exact rationals. Result: symmetric image3,
000074 | triplet block0, channel rank<=2. It also verifies the Grassmann symmetry
000075 | and the symbolic real singular-square polynomial; section3 proves the
000076 | complex statement. Independent review is in REVIEW-E8-ANCHOR.txt.
000077 | 
000078 | A distinct-eigenvalue control Z=diag(3,1,-1,-3) produces a6-dimensional
000079 | symmetric image, showing exactly why an operation distinguishing triplet
000080 | directions changes the class. Z is ONLY a negative-scope control, not a
000081 | new original TFPT input and not a derived family connection.
000082 | 
000083 | The next substantive candidate must therefore use an ACTUAL additional
000084 | original family/sector operation, a genuinely different field map or a
000085 | different higher-product functional. Derive that operation and its Weyl,
000086 | charge, product, reflection and time action before fitting masses. The
000087 | existing parallel-irreducible family-connection result is not replaced
000088 | by the diagonal control. The unrestricted original construction remains
000089 | open; no anomaly, source level or physical symmetry is silently changed.