TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-HIGGS-FIBER-PROOF.txt
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000001 | ORIGINAL HIGGS FIBER: WHOLE96 ORBIT, DETERMINANT JET AND RESIDUE MATCH
000002 | 31 October2026. Results R028/R029. Established scoped algebra; source-match
000003 | failure of specific printed formulas. No physical completion or ledger change.
000004 | 
000005 | 1. Question and original data
000006 | Continue F1.M6 at its actual source consumer. Keep the original four supplied
000007 | 3x3 matrices Yu,Yd,Ye,Ynu and symmetric MR, weak quaternion q(H), whole96
000008 | particle/antiparticle and chirality slots, and the original source/state maps.
000009 | Do not optimize Yukawas, change primitive profile, add Gamma_H to a master,
000010 | or identify a bare Hessian with the full connected H covariance.
000011 | Original v252 full_D and rep supply the actual coefficients. R026's288 images
000012 | already identify the complete real Higgs one-form. That result is reused.
000013 | The new question is whether all constant H directions require a search, and
000014 | whether the displayed geometric producer actually matches the displayed H
000015 | potential before it is used as the joint H integration weight.
000016 | 
000017 | 2. Entire constant H orbit on the actual original coefficient
000018 | Use real coordinates H=(x1+i x2,x3+i x4), u=sum x_a^2 and r=sqrt(u).
000019 | There is no1/sqrt2 factor in these coordinates. The original quaternion is
000020 |  q(H)=[[bar h2,h1],[-bar h1,h2]].
000021 | It obeys q* q=u I2 and det q=u. For r>0, U=q/r lies in SU2.
000022 | Let V(U) act by U on EACH original weak LEFT doublet, identity on all
000023 | right singlets, and complex-conjugate actions on the antiparticle slots.
000024 | Then, with DY=full_D(Yu,Yd,Ye,Ynu,0), DM=full_D(0,0,0,0,MR),
000025 |  Phi(H)=V(U) [r DY+DM] V(U)*.                          (1)
000026 | Proof is the actual LR block multiplication U(r diag(Yup,Ydown)) and its
000027 | adjoint. No commutation with Yup or Ydown is used. DM is between right
000028 | neutrino singlets and remains unchanged. V commutes with original grading
000029 | and JF. At H=0, Phi=DM, so no singular polar coordinate is needed.
000030 | All72 real Y coefficient directions and12 real symmetric MR directions
000031 | are checked against the unmodified original builder at an integral q with
000032 | r=5 using W=rV: W*W=25I and W Phi_rad W*=25 Phi(H), exact integer arithmetic.
000033 | The generic q identities and block proof establish ALL H and ALL matrices;
000034 | one quaternion replay is a constructor check, not the generic proof.
000035 | This is a weak LEFT gauge rotation, not a custodial right symmetry. Unequal
000036 | Yu,Yd or Ynu,Ye do not destroy (1). A contrary Gemini proposal was rejected.
000037 | 
000038 | 3. Full determinant reduction, with Majorana and all multiplicities
000039 | For z>0 put P(z,u)=det96(zI+Phi(H)^2). In the original neutrino chiral
000040 | ordering plus=(nuL,antiNuR),minus=(nuR,antiNuL), the whole12 block is
000041 |  N=[[0,B],[B*,0]], B=[[r Ynu,0],[MR*,r Ynu^T]], B a6x6 matrix.              (2)
000042 | Here * is adjoint; bar is entrywise conjugation. Consequently
000043 |  P=det3(z+u Yu*Yu)^12 det3(z+u Yd*Yd)^12
000044 |    det3(z+u Ye*Ye)^4 det6(z+B*B)^2.                    (3)
000045 | For each charged species/color the L/R pair and conjugate pair give fourth
000046 | power; three colors give twelfth power. Neutrinos give the squared6x6
000047 | determinant. The partition accounts for all96 slots without discarding a
000048 | Majorana leg. These are INTERNAL spectral multiplicities, not permission
000049 | to integrate96 independent physical Grassmann fermions again.
000050 | 
000051 | The neutrino6 determinant can itself be evaluated with two3x3 matrices:
000052 |  A=z+u bar(Ynu) Ynu^T,
000053 |  E=z+u Ynu*Ynu+MR MR*
000054 |      -u MR Ynu^T A^-1 bar(Ynu) MR*,
000055 |  det6(z+B*B)=det3(A) det3(E).                           (4)
000056 | A>0 for z>0 even at singular Y. Thus (4) is exact and stable as a Schur
000057 | identity. At z=0 use the polynomial continuation rather than division by
000058 | singular A. All higher constant-field spectral derivatives follow from
000059 | (3)-(4); no scan over orientations is needed. For one family this is
000060 |  det6_reduced=(z+u |Ynu|^2)^2+z |MR|^2.
000061 | The checker uses exact dense Gaussian-integer controls with all original
000062 | slots, not physical fitted coefficient values.
000063 | 
000064 | 4. First nontrivial spectral Higgs coefficient, entire matrix class
000065 | Let F_z(u)=log P(z,u). For complex SYMMETRIC MR,
000066 |  F_z'(0)=4/z Tr(3Yu*Yu+3Yd*Yd+Ye*Ye)
000067 |          +4 Tr[(z+MR MR*)^-1 Ynu*Ynu].                (5)
000068 | This is nonnegative, strictly positive whenever a supplied Yukawa is nonzero.
000069 | The trace of a product of two positive matrices is nonnegative; no simultaneous
000070 | diagonalization of Ynu and MR is assumed.
000071 | 
000072 | Proof retaining the noncommuting Majorana cross term: write
000073 |  Phi(r)^2+z=K0+rQ+r^2 DY^2,
000074 |  K0=z+DM^2, Q={DM,DY}. Then
000075 |  F_z'(0)=Tr(K0^-1 DY^2)-1/2 Tr(K0^-1 Q K0^-1 Q).        (6)
000076 | Dropping the second term is wrong. Equivalently differentiating (4) yields
000077 |  d_u logdet6|0=Tr[(z+MR MR*)^-1 Ynu*Ynu]
000078 |              +Tr[(z+MR*MR)^-1 Ynu^T bar(Ynu)].
000079 | Symmetry of MR makes the two traces equal by transposition, giving (5)
000080 | with the square in (3). The direct full96 inverse calculation agrees exactly
000081 | with (5) at two z controls; coefficients97072/213 and75984/487 belong only
000082 | to the diagnostic Gaussian-integer matrices.
000083 | 
000084 | On the retained constant, scalar-connection product circle h=p gamma-i gammaPhi,
000085 | p=n+alpha, alpha=1/4 or3/4, h^2=p^2+Phi^2. Hence the derivative of the
000086 | RELATIVE full circle logdet(h^2+mu^2), at mu^2>0 and constant H=0, is
000087 |  4 s_alpha(mu^2) Tr(3Yu*Yu+3Yd*Yd+Ye*Ye)
000088 |  +4 Tr[s_alpha(mu^2+MR MR*) Ynu*Ynu],                 (7)
000089 |  s_alpha(v)=pi/sqrt(v) sinh(2pi sqrt(v))
000090 |               /[cosh(2pi sqrt(v))-cos(2pi alpha)].
000091 | Functional calculus defines the last matrix function. The standard paired
000092 | partial-fraction expansion is s_alpha(v)=sum_n[(n+alpha)^2+v]^-1.
000093 | Terms are O(n^-2); the differentiated relative determinant is trace class
000094 | in this1D circle setting. Thus (7) includes ALL modes, not a finite cutoff
000095 | substituted for a limit. It is positive with a nonzero Y. This is the given
000096 | circle/product source coefficient, not an evaluation of the full4D master,
000097 | not a new AP/APS domain selection and not an infinite4D positivity claim.
000098 | No claim equating the primitive Sigma eta to an eta of this circle is made.
000099 | 
000100 | 5. Whole constant-field Hessian and the preserved directional source
000101 | For a scalar invariant F(u), the complete four-real-coordinate Hessian is
000102 |  d_a d_b F=2F'(u) delta_ab+4F''(u) x_a x_b.             (8)
000103 | Three tangent eigenvalues equal2F'; radial eigenvalue is2F'+4uF''.
000104 | The symbolic check fixes coordinate factors. Vanishing tangents at a nonzero
000105 | stationary invariant background do not prove physical massless particles:
000106 | the gauge quotient, kinetic terms and source must still be treated.
000107 | In the given product chart (1) also conjugates h and every entry of the
000108 | full two-face Cauchy projector, as V commutes with gamma. Field vectors
000109 | and adjoints transform with it. In a fixed frame the whole source is NOT
000110 | replaced by an invariant determinant. At fixed nontrivial weak connection,
000111 | transform A too; for varying U(x), retain V*partial V. Existing nonzero
000112 | weak-connection/mixing finding remains valid. Commutation of Padm with V,
000113 | transport of reference and APS domain must be established before extending
000114 | (1) to an original globally projected spectral functional.
000115 | 
000116 | 6. Source types before H integration
000117 | The printed v42 §027 master has only xi=(alpha,chi_geo,delta_ph,rho_vac).
000118 | It contains -logZrel, logdet, eta, U6 and vacuum terms. It is not a displayed
000119 | H-field source functional or displayed bare H action. rho_vac is NOT H's VEV.
000120 | The formula alone does not establish M=-W[0] or M=Legendre Gamma. The correct
000121 | field source remains original §035 with actual local H arguments and measure.
000122 | R026's Z_H[J-j|A] integrates H FIRST in the affine Yukawa case. The sign,
000123 | counting and full determinant factor in (3) are not by themselves an
000124 | identification with an integrated fermion contribution to M.
000125 | Thus neither (5) nor (7) may simply be added to Gamma_grav or used to call
000126 | an inverse bare Hessian the full G_H. The relative Z and original bound
000127 | reference can contribute to the same H coefficient. They must be retained.
000128 |