TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ROOT-FULL-GRAM-RESPONSE-REVIEW.txt
Original file SHA-256: bdf5b023a617154c9c7766c84664cadd9671dd03a37ebf0879a731b93aa7f3e8
Inclusive original lines: 1-49
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000001 | ROOT INDEPENDENT WHOLE GRAM RESPONSE
000002 | 30 October2026. Actual original kernel, declared perturbation only.
000003 | 
000004 | The original mark1 positive cubic M=M0 is unitary of order3. Define
000005 |  Q=(I+M+M²)/3, R=I-Q, y=2/3, g=y-delta,
000006 |  h=y²+delta²+y delta,
000007 |  D=yI-delta M, H=D*D=g²Q+hR.
000008 | For ANY declared perturbation V, its first Gram differential is
000009 |  B=D*V+V*D.
000010 | The ENTIRE inverse-square-root derivative is exactly
000011 |  dK= -Q B Q/(2g³) -R B R/(2h^(3/2))
000012 |      +c(g)(Q B R+R B Q),
000013 |  c(g)=(1/g-1/sqrt(h))/(g²-h).
000014 | This follows by spectral projection, without diagonalizing a fitted
000015 | matrix or discarding cross blocks. It is a real-linear derivative inV.
000016 | 
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000031 | 
000032 | The fixed Q block is Re(kappa)/(3g²)Q EXACTLY. Analytically,
000033 |  c(g)=-1/(3y²g)+O(1),
000034 |  Q B R=-y bar(kappa) QP0(M*-I)R+O(g),
000035 |  norm[QP0(M*-I)R]=sqrt(2/3).
000036 | Therefore at y=2/3 the cross operator norm is
000037 |  |kappa|/(sqrt6 g)+O(1), not a boundedO(1) remainder.
000038 | Even a purely imaginary direction has a divergent1/g response. The
000039 | complex phase cannot be removed by keeping only the resonant Q trace.
000040 | Root's exact projector argument agrees with the agent's numerical
000041 | small-gap limit and full fixed-gap replay.
000042 | 
000043 | These are the original positive kernel's responses to a DECLARED
000044 | operator perturbation. Original EM5 permits gap-stable Pstr-preservingV;
000045 | it does not select this V as the physical source. Keep actualPstr,
000046 | reference, both C6 branches, full adjoint/Clifford/KO legs, and original
000047 | field insertion rule. PositiveC6 block is not physicalPSeam,+, andP0 is
000048 | notPstr orPTheta. No new branch, ellNS, state or physicaltime is chosen.
000049 | Independent output: ROOT-TWO-PROJECTOR-GRAM-RESPONSE-CHECK.json.