TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/ORIGINAL-P1-VARIATION-GAP-AND-EVENT-SELECTION.txt
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000037 | 2. Exact gap-normalized action differential
000038 | 
000039 | Put lambda(B)=lambda_1^+(|B|)>0, chi=1/lambda. Work on a regular stratum
000040 | with constant nullity and an isolated signed eigenvalue E0!=0 of
000041 | selfadjoint B whose modulus is the simple smallest positive eigenvalue
000042 | of |B|,
000043 | normalized eigenvector u0, lambda=|E0|. For a differentiable common-domain
000044 | source curve B(eps), V=dot B, differentiating Bu0=E0u0 and normalization
000045 | gives dot E0=<u0,V u0>. Consequently
000046 |  dot lambda=sign(E0)<u0,V u0>, dot chi=-dot lambda/lambda^2.
000047 | This fixes a COVECTOR on supplied variations V. It does not choose V.
000048 | The eigenprojector is independent of the arbitrary phase of u0.
000049 | 
000050 | If the minimum of |B| is degenerate, use its isolated cluster projector P.
000051 | For A=|B|, A dot A+dot A A=B dot B+dot B B on the invertible stratum.
000052 | The right gap derivative is lambda_min(P dot A P), the left derivative
000053 | lambda_max(P dot A P). A mean over the cluster would add a rule. A minimum
000054 | over distinct signed branches is generally only directionally differentiable.
000055 | No hidden choice of a vector is allowed; changing nullity needs separate
000056 | control. Both actual quarter cases below are invertible with simple minimum.
000057 | 
000058 | Write X_rel=lambda D_rel and X_ref=lambda D_ref. With the ORIGINAL declared
000059 | spectral test function f held fixed, the exact regulated first variation is
000060 | 
000061 |  dot S_prim = lambda Tr[f'(X_rel) dot D_rel-f'(X_ref) dot D_ref]
000062 |    +dot lambda Tr[D_rel f'(X_rel)-D_ref f'(X_ref)]
000063 |    +i pi/2 dot(Delta eta_Sigma).
000064 | 
000065 | There is no assumption that D and dot D commute. The formula uses trace
000066 | cyclicity/functional calculus on a finite regulator, or a separately
000067 | justified trace-class differentiable stratum. Moving domains must be
000068 | transported by their actual comparison map; the induced terms stay in
000069 | the displayed operator jets. Eta jumps/crossings require their own terms.
000070 | The original gap contribution is NOT set to zero before specifying V.
000071 | 
000072 | For FIXED geometric R, dot D_rel=(dot D-R dot D R)/2 and
000073 | dot D_ref=(dot D+R dot D R)/2. For varying R, differentiate RDR fully:
000074 | dot R D R+R dot D R+R D dot R. In the product collar, with the normal
000075 | part fixed and fixed geometry, dot D_+=gamma_n dot B. This local identity
000076 | does not determine the entire bulk variation; varying gamma_n/normal
000077 | geometry adds its own terms. These are compatibility identities, not a physical
000078 | choice of that curve or a map from the native five to source variations.
000079 | 
000080 | An event-parameter family j would additionally require an actual map
000081 |  j -> (A_+[j],H_+[j],D_+[j],J[j],Gamma[j],domain[j])
000082 | compatible with the original collar, reflection, state and admissibility.
000083 | Its tangent then inserts dot D and dot B in the formula. The first
000084 | derivative is a scalar response functional of that tangent. It is not
000085 | an autonomous selection of an event, a physical matrix on native5, or
000086 | a probability/measurement law. Even formal stationarity dot S=0 for all
000087 | ACTUALLY admitted tangents does not define a missing source-to-field map.
000088 | 
000089 | Primary mathematical comparison: Greenbaum--Li--Overton,
000090 | arXiv:1903.00785, Theorem1/equations1,3, simple-eigenvalue perturbation.
000091 | The common-domain selfadjoint formula above is also directly derived by
000092 | differentiating the eigenvector equation. Chamseddine--Connes,
000093 | hep-th/9606001 equation1.23 starts with a GIVEN represented algebra/triple
000094 | and describes inner fluctuations thereof; it is not a selection theorem
000095 | for the missing P1 source algebra or vertex. No SM triple is imported.
000096 | https://arxiv.org/pdf/1903.00785
000097 | https://arxiv.org/pdf/hep-th/9606001
000098 | 
000099 | 3. Entire scalar-potential class in the ACTUAL retained quarter source
000100 | 
000101 | Use the already given B_alpha=-i partial_theta+alpha on H1_per(S1),
000102 | alpha=1/4 or3/4 (equivalently transport to the corresponding twisted
000103 | domain). The following is an ALL-PROFILE theorem, not a fitted profile.
000104 | For any smooth real scalar potential v(theta), set vbar=average(v) and
000105 | choose the periodic phi with phi'=vbar-v. Then g=exp(i phi) is unitary,
000106 | preserves H1_per and the original endpoint holonomy, and EXACTLY
000107 |  B_alpha+v=g(B_alpha+vbar)g^*.
000108 | Proof: g(-i partial_theta)g^*=-i partial_theta-phi'.
000109 | The spectrum is therefore {n+alpha+vbar:n in Z}. In the neighborhood of
000110 | vbar=0 the gap is dist(alpha+vbar,Z). At FIXED original holonomy the
000111 | small scalar connection variation has vbar=0 (integer changes are not
000112 | an infinitesimal freedom), so its gap/chi stay unchanged to ALL orders.
000113 | The eigenspace nevertheless moves as g P_gap g^*. For g(eps)=exp(i eps phi),
000114 |  dot P_gap=i[M_phi,P_gap].
000115 | 
000116 | Thus in this actual scalar class dot lambda only reads the constant
000117 | holonomy variation: at alpha1/4 it is+vbar, at alpha3/4 it is-vbar.
000118 | It cannot choose a particular zero-mean local waveform. Those scalar
000119 | operator families can be gauge equivalent; they are NOT exhibited as
000120 | inequivalent minimal P1 source models. This does not
000121 | classify all original matrix, geometry, boundary-domain or charged
000122 | P1 variations. Those may lie outside this class and must be derived.
000123 | 
000124 | Static operator variation and a state intervention must remain distinct.
000125 | The previous coherent W(phi) response acts in a fixed source representation;
000126 | the above family conjugates a one-particle boundary operator. Their
000127 | dictionary is not supplied merely by using the same scalar phi. If the
000128 | ENTIRE original datum, including R, domains, reference and eta, is
000129 | unitarily transported, both relative operators transform covariantly and
000130 | their spectral action is unchanged. If R or reference is physically fixed,
000131 | that conclusion is NOT available; its actual variation is required.
000132 | The nonzero mean-work/coherent-state responses from the previous turn
000133 | therefore do not contradict this fixed-holonomy gap identity.
000134 |