TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/WHOLE-QUARK-SHIFT-BALANCE-PROOF.txt
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000001 | R084: THE WHOLE QUARK SOURCE, PARTNER CANCELLATION AND SELECTIVE SHIFT
000002 | 2 November 2026
000003 | 
000004 | GOAL AND ORIGINAL CONTRACT
000005 | R083 studied a single R081 Yu singular pair and the full 18-dimensional
000006 | gamma+ quark marginal. The present test retains the entire 36-dimensional
000007 | quark covariance of the same original v252 constructor, including the
000008 | off-diagonal Higgs/partner blocks. We ask whether the conditional common
000009 | scalar seam shift produces the primitive winding on THIS whole source.
000010 | The scalar shift on this same carrier is still an explicit hypothesis; its
000011 | identity with the old P1 seam, the full E8 source and physical readout is
000012 | not assumed. Constants/seed252 are constructor controls, not a physical fit.
000013 | 
000014 | 1. EXACT WHOLE-SPACE FACTOR
000015 | Let h_p=p gamma+V, V=V*, {gamma,V}=0, dim gamma+=dim gamma-=18, p=n+a,
000016 | C_n=expit(2 ell h_p), ell>0, V constant in the given circle background.
000017 | Let U=S tensor I36, S e_n=e_(n+1). The actual R081 even/odd matrix obeys
000018 | these algebraic identities. The 18 nonnegative singular values m_j of V_+-
000019 | give paired eigenvalues +/-E_nj, E_nj=sqrt(p²+m_j²). Thus
000020 |  det_36(I-z C_n)=product_j [1-z+z²/(4 cosh²(ell E_nj))].
000021 | For |z|<1 it is nonzero and tends to (1-z)^18 at both ends n->+/-infty.
000022 | 
000023 | Importantly the FULL C-P_free has off-diagonal V/(2|n|)+O(n^-2), so is
000024 | generically Hilbert Schmidt but NOT trace class when V != 0. Nevertheless
000025 | C_(n-1)-C_n=O(n^-2) in finite-dimensional trace norm. Therefore
000026 |  U C U* - C belongs to S1, and the true relative operator
000027 |  R(z)=(I-z U C U*)(I-z C)^(-1) is I+S1. The product over all finite
000028 | mode blocks is legitimate and telescopes to the ratio of the two endpoint
000029 | determinants. Hence det_F R(z)=1 exactly. The result applies to alpha=1/4
000030 | and 3/4, any fixed ell>0 and fixed finite V of this class. The cancellation
000031 | is of the WHOLE operator: the two marginal determinants may not be
000032 | multiplied in place of the coupled block (nonzero P C Q).
000033 | 
000034 | The source still responds to H: actual native covariance at p=.25 changes
000035 | with the given H by Frobenius norm .296833544922. At the same point, the
000036 | log determinant of the full block differs from the product of its two
000037 | diagonal marginal blocks by .00548972555, so forgetting the partners
000038 | changes the answer. Both are finite controls at ell=.2,z=.3; not physical
000039 | predictions. The native full block, its singular-value spectrum, both
000040 | quarter twists and finite telescoping were checked in 23 explicit tests.
000041 | 
000042 | 2. WHY THE SIMPLE CHIRAL-ONLY FIX DOES NOT WORK
000043 | To retain gamma+ one might be tempted to replace U by
000044 |  U_sel=S tensor P_gamma+ + I tensor P_gamma-.
000045 | This is a diagnostic hypothesis, not the original P_adm. In an actual
000046 | nonzero Higgs singular pair, C_+-(n)=b_n with
000047 |  b_n=m/(2 E_n) tanh(ell E_n) ~ m/(2|n|).
000048 | The +/- block of U_sel C U_sel* - C is
000049 |  S diag(b_n)-diag(b_n).
000050 | Its diagonal is -b_n and sum_n |b_n| diverges. Trace-class operators have
000051 | absolutely summable diagonals in every orthonormal basis, so the FULL
000052 | difference is NOT S1 for m != 0. It IS S2 because b is square summable.
000053 | For |z|<1, R_sel-I=-z(U_sel C U_sel*-C)(I-zC)^-1 cannot be S1 either:
000054 | otherwise multiplying by bounded I-zC would make U_sel C U_sel*-C S1.
000055 | Thus an ordinary Fredholm determinant of the full selected action is NOT
000056 | justified. The projected C_++ marginal has an ordinary determinant, but
000057 | this is a different object from the full action with both coupled sides.
000058 | At m=0 this particular obstruction vanishes. Trace-class estimates at finite
000059 | cutoff do not prove S1: native diagonal lower bounds at N=20,80,320 are
000060 | 1.5164, 2.4496, 3.3924 while HS norms remain bounded .3670,.3897,.3952.
000061 | Eight native tests execute the same R081 original Yu singular value.
000062 | 
000063 | The existing R080 det2 source may provide a correctly typed S2 reader; det2
000064 | is a regularized determinant, not an ordinary one, and its multiplication
000065 | law has an essential trace correction, see Britz et al.
000066 | https://arxiv.org/abs/2007.12834. It does not by itself supply the old
000067 | original normalization, eta, choice of physical selector, or entire source.
000068 | 
000069 | An explicit det2 reader exists mathematically for this DIAGNOSTIC selected
000070 | operator: X=-z(U_sel C U_sel*-C)(I-zC)^-1 is S2 and therefore det2(I+X)
000071 | is defined and approximated by open Fourier compressions in S2. We compute
000072 | the complete 2x2 original R081 singular pair, not just its marginal, using
000073 | det2(I+X_N)=det(I+X_N)exp(-tr X_N). For z=.3, ell=.2, alpha=.25, and the
000074 | same seed252 Yukawa singular value used in R081, the fixed cutoffs N=64,128
000075 | give .97682564645,.97637154583 when the control Higgs mass is .68238816507,
000076 | and .99574841189,.99574841189 in the zero-mass control. These are FINITE
000077 | truncations, not claimed exact limiting values or original TFPT weights.
000078 | Continuity of det2 under S2 convergence guarantees the actual mathematical
000079 | limit exists; the displayed data alone do not provide a rigorous numerical
000080 | error bound for its value. The difference illustrates how a mathematically
000081 | well-defined whole-pair reader can retain H-dependence even while the original
000082 | ordinary relative determinant is not defined. It does not determine the
000083 | physical source, and no free correction is added to the original functional.
000084 | 
000085 | 3. IMPLICATION FOR THE WHOLE THEORY
000086 | Conditional common shift: gamma+ marginal gives (1-z)^-18 (R083), whereas
000087 | the entire 36-dimensional coupled quark reader gives 1 and still has a
000088 | nontrivial Higgs response. A selective shift preserving both partner sides
000089 | is mathematically S2 rather than S1. This makes the sought primitive
000090 | one-winding original source into a specific operator/state/normalization
000091 | matching question, not an assumed 1-vs-18 contradiction in TFPT. The prior
000092 | unperturbed scalar original u=M_z, P_-, C_beta source has the already proved
000093 | factor 1-z; retain its own domain, orientation and full NS cover. One must
000094 | trace its actual image into the Higgs source, including all P/Q fields,
000095 | clock and reflection actions, Z0, relative reference, eta and higher words.
000096 | No claim of local 3+1 geometry, Einstein equations, physical Pfaffian,
000097 | physical alpha, or entire dynamically integrated H measure is made here.
000098 | 
000099 | Executable original-source diagnostics:
000100 |  check_whole_quark_shift_balance.py / WHOLE-QUARK-SHIFT-BALANCE.json (23),
000101 |  check_asymmetric_shift_class.py / ASYMMETRIC-SHIFT-CLASS.json (8).
000102 |  check_regularized_selective_source.py / REGULARIZED-SELECTIVE-SOURCE.json (7).
000103 | Old pair-index theorem and exact previous -18 result are not re-counted.