TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-HIGGS-INNER-SCALE-PROOF.txt
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000001 | ORIGINAL HIGGS PRODUCER: CENTRAL SCALE, AFFINE SOURCE AND NORMAL INDEX TYPE
000002 | 31 October2026. R034/R035. Scoped continuation of R026-R033.
000003 | No new Y,physical field,geometry,reference,action term or ledger promotion.
000004 | 
000005 | 1. Exact source question and original objects
000006 | The metrology001 owner lines15-17 fixes Dgeo=Dref+Drel=D. Lines105-126
000007 | define f=e^(-sigma/2), Ksigma=fDgeo f+Brel and the SAME relative trace.
000008 | Lines42-44 name Brel in the reorganized primitive seed bundle package.
000009 | They do not there give its actual matrix/symbol H component. This report
000010 | does not turn that uncomputed component into a corpus-wide absence claim.
000011 | The orientation002 master lines71-105 also names Brel in Aseed and its
000012 | bosonic indices. The carrier007 source lines95-139 instead defines an
000013 | explicit B_Ea=sigma_normal^mu nabla_mu^(a)+Phi_a on the NORMAL S2.
000014 | It identifies the retained light field with a weak doublet. Retain that
000015 | stated Higgs result, rather than deriving a different Higgs/Y family.
000016 | 
000017 | These are three distinct objects: normal index operator; light FIELD;
000018 | and its actual fermion COUPLING. R026 already supplies the last arrow
000019 | from the original wedge tensor and the actual full96 real one-form.
000020 | No equality Brel=B_E2 or Brel=oneform follows from the shared letter B.
000021 | 
000022 | 2. Inner fluctuation and central scaling commute EXACTLY
000023 | Let D be the actual given self-adjoint anchor, and use a common invariant
000024 | core for D,a,b and f. Assume f is positive real bounded with bounded inverse,
000025 | [f,a]=[f,b]=0 for the algebra factors used, and JfJ^-1=f. For smooth compact
000026 | base charts these are satisfied by scalar f=e^(-sigma/2) in the given product
000027 | realization. The identity itself does not require a newly chosen spin4 chart.
000028 | Hold the same a,b fixed as original operation arguments. Then
000029 |  [fDf,b]=fD(bf)-f(bD)f=f[D,b]f.                     (1)
000030 | Thus for A=sum a[D,b],
000031 |  A_sigma=sum a[fDf,b]=fAf,
000032 |  JA_sigma J^-1=f(JAJ^-1)f,
000033 |  (fDf)+A_sigma+epsilon JA_sigma J^-1
000034 |        =f(D+A+epsilon JAJ^-1)f.                     (2)
000035 | No [D,f] term is omitted in(1). There IS a derivative-of-f term in fDf;
000036 | it cancels from this commutator because b commutes with f. It is retained
000037 | in the full operator. Central here means the stated commutations, not
000038 | [D,f]=0. A differential operator on a separate normal factor can commute
000039 | with f if f depends only on base coordinates; do not assert otherwise.
000040 | 
000041 | If bounded self-adjoint A is used and f preserves the prescribed Sobolev
000042 | domain, the identity extends to the realization. The domain for fDf is
000043 | f^-1 Dom(D); equality to an independently chosen APS domain is not automatic.
000044 | Without bounded invertibility/global regularity only the common-core/local
000045 | identity is claimed. No continuum determinant or measure theorem follows.
000046 | 
000047 | 3. Execute(2) on the WHOLE ORIGINAL96 Higgs producer
000048 | Use the already proved original chart
000049 |  q(H)=[[bar h2,h1],[-bar h1,h2]], H=(h1,h2),
000050 |  Phi(H)=L(H)+DM,
000051 |  Hstar=(0,1), L(Hstar)=DY.
000052 | The current original oneform and its selfadjoint/JF completion give L(H).
000053 | For the seed Dstar=Dkin+gamma5(DY+DM) the CHANGE is
000054 |  V(H)=gamma5[L(H)-DY]=gamma5 L(H-Hstar),             (3)
000055 | not gamma5 Phi(H). Majorana is not inserted a second time. Therefore
000056 |  Dstar+V(H)=D_H=Dkin+gamma5[L(H)+DM].
000057 | On the same centrally scaled anchor, the actual inner change is fV(H)f.
000058 | Writing q=e^-sigma=f² gives
000059 |  K_in(H)=fDstar f+q V(H)=f D_H f.                   (4)
000060 | This is the full supplied Higgs inner-fluctuation family, not B0 truncation.
000061 | The ordinary fixed-connection spin4 realization of R031 is still conditional.
000062 | If the original Brel is THIS actual inner change, R033 applies to its full
000063 | sum by(4). Its Higgs/Einstein comparison cannot be bypassed by merely saying
000064 | that a B summand had previously been left out. Identifying the original
000065 | unresolved Brel with(3) requires its original producer; it is not inferred.
000066 | 
000067 | 4. An unscaled outside change has an EXACT AFFINE field dictionary
000068 | For comparison ONLY, evaluate the same given change outside the sandwich:
000069 |  K_out(H)=fDstar f+V(H).
000070 | For every smooth sigma and whole H, put
000071 |  h=T_sigma H=Hstar+q^-1(H-Hstar),
000072 |  H=q h+(1-q)Hstar.                                 (5)
000073 | Since L is real-linear and q positive scalar,
000074 |  q L(h-Hstar)=L(H-Hstar), hence
000075 |  K_out(H)=K_in(T_sigma H)                           (6)
000076 | as full operators in the SAME fixed spin frame/domain. No derivative of H
000077 | is in D_H itself. Derivatives of h DO enter its square, automatically with
000078 |  dh=q^-1 dH+q^-1(H-Hstar)d sigma
000079 | in this fixed chart. If the original reference section Hstar is not constant
000080 | under the chosen covariant derivative, differentiate(5) including D Hstar.
000081 | Never omit these terms when comparing kinetic actions. Real-valued sigma
000082 | does not mix the four existing Y matrices or add a Majorana coupling.
000083 | 
000084 | At the SAME named H the operators differ in general. In particular
000085 |  T_sigma(0)=(1-q^-1)Hstar, not0.
000086 | Consequently expansion about H=0 must be compared with the transported
000087 | nonzero h background. Resetting that background to0 is a changed problem.
000088 | This is an affine change of the total-H coordinate around its original
000089 | seed; it is not the purely multiplicative Hhat=qH conformal map of R033.
000090 | The two maps answer different comparisons and must not be conflated.
000091 | 
000092 | 5. Whole finite source, absolute Z0 and every higher word
000093 | Work with the existing finite regulated H fiber, sigma and all other fields
000094 | held conditional as in R024/R026. No new boson measure is proposed. With a
000095 | flat four-real-coordinate measure at each of N sites, dH=prod_x q_x^4 dh.
000096 | Let W include the ACTUAL K_out dependent determinant/Pfaffian, reference,
000097 | eta and all other insertions. Let S_B denote all separately displayed H
000098 | dependence. All additional H dependence in W must also be composed with
000099 | the inverse(5); do not assume it depends only on K if the original does not.
000100 | For the case W=W[K_out(H)] the exact change-of-variables statement is
000101 |  Z_out[j]=int dH W[K_out(H)] exp(-S_B[H]+sum j.H)
000102 |  =Q exp(sum(1-q)j.Hstar)
000103 |     int dh W[K_in(h)]
000104 |           exp(-S_B[qh+(1-q)Hstar]+sum qj.h),          (7)
000105 |  Q=prod_x q_x^4, jhat_x=q_x j_x.
000106 | This is valid coefficientwise with existing external Grassmann sources
000107 | when the same finite integrals exist. The fermion fields have not been
000108 | changed; equality(6) is in their same original frame. Every extra boson/
000109 | fermion vertex must be pulled back in(7), not discarded or duplicated.
000110 | 
000111 | For an actual non-flat measure use its PUSHFORWARD by T_sigma instead.
000112 | Q alone is not a general measure law. Infinite field limits and regularized
000113 | Jacobian/anomaly questions need their actual original limit prescription.