TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/FUCHS-FRAME-INDEPENDENT-TRACE-REVIEW.txt
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000001 | ACTUAL ALGEBRAIC FUCHS FRAME: INDEPENDENT OPERATOR AND METRIC REVIEW
000002 | 29 October 2026. No original edit, extra source/state or Atlas mutation.
000003 | 
000004 | 1. Direct original-system check
000005 | Read the actual v117 ode_monodromy input, verification/v117_monodromy_weyl_a3.py:
000006 | 78-109, and original September28 section004:205-255. Summing its FOUR original
000007 | residues gives, D=z^4-1, a=2sqrt(2)/3, b=sqrt(5)/3,
000008 |   A(z)=D^-1 [[2z^3,a z^2,0],[a,z^3,bz],[0,bz,z^3]].
000009 | This independently confirms the three equations used by the new checker;
000010 | its p2,p1 definitions are respectively the original third and second rows.
000011 | The remaining residual is the first row, not a fitted replacement ODE.
000012 | 
000013 | The existing check_fuchs_algebraic_frame.py was executed ONCE independently
000014 | with the established source-search Python. Output:
000015 |   d=omega c; all original scaled row residuals=[0,0,0];
000016 |   all three Q-column residual triples=[0,0,0];
000017 |   detG/(z^4-1)=-27sqrt(2)(2omega+1)/10.
000018 | The normalised detPsi=1-z^4 is also independently forced by Abel's formula:
000019 | Tr A=4z^3/(z^4-1), Psi(0)=I. No numerical ODE was used.
000020 | The zero solution printed for the p alternative concerns that specified
000021 | two-coefficient ansatz; it is not alone an exclusion of every p-type source.
000022 | 
000023 | 2. Independent exact G0 metric calculation
000024 | Use the branch at z=0 with w=-1, its two deck translates, and h0h1h2=1.
000025 | Set tau=i omega^2, rho^2=2/[3(omega-1)], sigma=1/(rho tau).
000026 | Then |rho|^2=2/(3sqrt(3)), |sigma|^2=3sqrt(3)/2, and the three G0 columns are
000027 |   (0,-3rho/(2b),-3rho(1+2omega)/2),
000028 |   (0,-sigma omega^2/[b(1-omega)],sigma),
000029 |   (-3tau(1+2omega)/sqrt(10),0,0).
000030 | The two contributions to the lower off-diagonal row Gram entry are
000031 | -3i/(2b) and +3i/(2b), hence cancel EXACTLY. Consequently
000032 |   G0 G0*=diag(27/10,9sqrt(3)/5,3sqrt(3)),
000033 |   H0=(G0 G0*)^-1=diag(10/27,5/[9sqrt(3)],1/[3sqrt(3)]).
000034 | Here * denotes Hermitian adjoint, not a source pullback.
000035 | 
000036 | 3. The actual flat metric, and which frame is horizontal
000037 | The columns of G are horizontal solution sections of the ORIGINAL connection
000038 |   nabla=d-A(z)dz.
000039 | They are not nonvanishing meromorphic divisor frames of each L_a on C.
000040 | On the unramified sign cover their deck changes are signed permutation
000041 | matrices, unitary for the standard column norm. Thus
000042 |   H(z)=G(z)^(-*) G(z)^(-1)
000043 | is single valued and positive away from the actual singular marks. It obeys
000044 |   H=Psi^(-*) H0 Psi^-1, Psi=G G0^-1,
000045 | and is the explicit invariant flat coefficient metric, up to common scale.
000046 | It is generally NOT the Euclidean norm on the printed ODE row coefficients.
000047 | Finite irreducible monodromy makes this a coefficient-metric determination;
000048 | it does not specify a many-body field state or the physical P1 metric.
000049 | 
000050 | 4. The formerly abstract spin/coefficient trace is now an explicit matrix
000051 | Keep the canonical spin pullback from CANONICAL-BRANCHED-SPIN-PULLBACK.txt.
000052 | In holomorphic half-form frames the actual map on a regular lifted arc is
000053 |   W psi(u)=3^(-1/2) sqrt(pi'(u)) G(pi(u))^-1 psi(pi(u)).      (1)
000054 | The G branch and signed column order are the ones just checked. Its column
000055 | deck signs make the result a section of the actual sum L0+L1+L2; this is not
000056 | an identification of a Kummer function value with a flat fiber value.
000057 | The spin factor carries S0 exactly. In boundary half-density frames replace
000058 | sqrt(pi') by the corresponding unit spin phase times sqrt(|dtheta/dt|).
000059 | Then pointwise norm change and the three-sheet integral give
000060 |   ||W psi||_up^2=||psi||_(base,H)^2.
000061 | The image is the C3-equivariant pullback subspace, not the full ordinary
000062 | rank9 pushforward. This replaces an unspecified coefficient gauge in the
000063 | earlier trace note by an actual algebraic matrix from the original ODE.
000064 | 
000065 | 5. Explicit covariant B and a genuine closed domain, with its input named
000066 | Here G is the analytic branch in the ORIGINAL open interior z disk,
000067 | fixed by the same basepoint continuation used above. H1_NS is glued with
000068 | that SAME interior continuation; no independent extension is selected
000069 | between the four marked points. Another global gauge must carry the
000070 | actual signed coefficient continuation and its domain, not only a matrix
000071 | equality on separate arcs. In that fixed NS coefficient frame put
000072 |   nabla_theta=partial_theta-i z A(z), B_H=-i nabla_theta.
000073 | Writing psi=G c gives exactly
000074 |   B_H=G(-i partial_theta)G^-1.
000075 | Thus on the weighted Hilbert space L2_H, the transported ORIGINAL NS domain
000076 |   D_H={G c:c in H1_NS}
000077 | gives a closed selfadjoint operator, by this unitary coefficient map. No
000078 | claim that the printed raw Euclidean H1 domain equals D_H is justified.
000079 | The four detG zeros occur at the ORIGINAL marks; they cannot be discarded
000080 | from the graph domain merely because they have measure zero.
000081 | 
000082 | Combining (1) with this B_H cancels the Fuchs matrix connection in the
000083 | horizontal line frames. The remaining differential consumer is exactly
000084 |   -i/theta'(t) [partial_t-partial_t log s_spin(t)],
000085 | as in BRANCHED-SPIN-FUCHS-TRACE-DOMAIN.txt. The domain is W D_H. Locally
000086 | the original residue eigenvalues0,1/3,2/3 give raw powers u,u^2,u^3; G^-1
000087 | removes the coefficient factors, while the genuine spin zero u remains.
000088 | The multiplier is also exact: W M_z=M_pi(u) W.
000089 | 
000090 | This provides an explicit original-connection covariant B and its closed
000091 | domain WHEN the existing interior NS trace is the source domain being
000092 | transported. It does not prove that this particular covariant derivative
000093 | is the physical P1 B_Sigma, identify outer circles with Gamma_+/Gamma_-,
000094 | or identify it with the previous constant-coefficient F5 operator. The
000095 | actual quantum insertion, charge, reference and normal trace maps are not
000096 | replaced by a metric calculation. No Phi or state was chosen in this review.
000097 | 
000098 | Sources checked: the actual v117 input above; original September28
000099 | section004:205-290, SHA256
000100 | 574058876ebe23d693a9fce061f68da6fdec63419af78243ac9d457d7858f74b;
000101 | check_fuchs_algebraic_frame.py, ROOT-FUCHS-FRAME-CHECK.json;
000102 | the canonical spin and weighted trace notes named above. The completed
000103 | ORIGINAL-FUCHS-ALGEBRAIC-FRAME.txt:1-174 was subsequently read in full;
000104 | its original equations, explicit columns, basepoint, actual signed deck/Q
000105 | actions, stated metric and scientific scope agree with this review.