TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-joint-source-origin-audit-20261016/PROOF.txt
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Inclusive original lines: 52-82
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000052 | 3. First attempted bridge: determinant class -> positive carrier rank
000053 | 
000054 | Let S2 denote _archive/tfpt-45/source_extracts/02_carrier_source.tex.
000055 | Its determinant-class theorem starts at1894 and already uses
000056 |   X=-2P3+3P2 (1908; repeated1938)
000057 | and the explicitly named weak rank-two sector (1936-1944).
000058 | Its compact-index theorem then uses L2=det E2=O(1), and the rank-two corollary
000059 | (2075-2095) infers dim E_plus=2 from H^0(P1,O(1))=C^2.
000060 | This is conditional reuse of the closed branch, not an independent upstream
000061 | derivation of that branch. The later carrier-rigidity proof calls this corollary.
000062 | 
000063 | There is also a concrete object-type gap. The dimension of the space of global
000064 | sections of a determinant LINE is not the fiber rank of the underlying bundle:
000065 |   F_r=O(1) direct_sum O(0)^(r-1)
000066 | has det F_r=O(1) for every r>=1, but rank F_r=r and h0(F_r)=r+1.
000067 | At fixed rank2 and determinant degree1, O(1)+O(0) has (h0,h1)=(3,0),
000068 | whereas O(3)+O(-2) has (h0,h1)=(4,1). Thus determinant degree does not fix
000069 | even the kernel of an arbitrary bundle Dolbeault operator at fixed rank.
000070 | These are elementary bundle controls, not alternative physical TFPT worlds.
000071 | 
000072 | The source must supply the actual bosonic operator, its bundle and its
000073 | equivariant identification with the desired internal carrier. The numerical
000074 | identity h0(O(1))=2 cannot substitute for this map. Likewise a geometric
000075 | rotation action on sections cannot silently be identified with weak gauge action.
000076 | The stated analytic index must retain its operator and normalization: chi(O(1))=2;
000077 | counting one doublet is not automatically that same Fredholm index.
000078 | 
000079 | The Lean HiggsIndexCertificate explicitly assumes
000080 |   toForms : Eplus equiv_linear DegreeOneBinaryForms.
000081 | Its dimension theorem is valid under this premise. This audit does not refute it
000082 | and does not claim a new Lean build.