TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ROOT-ACTUAL-PATH-G-METRIC-REVIEW.txt
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000001 | ACTUAL ORIGINAL FUCHS PATH AND C6 KERNEL: ROOT REVIEW
000002 | 30 October 2026. Independent source and operator review.
000003 | 
000004 | 1. The earlier source-type review correctly distinguished two code frames.
000005 | integrate_path_transport returns actual original P,Q and T_raw; the old
000006 | finite_kernel_comparison independently constructs the exact unitary T.
000007 | It did not feed those numerical path matrices into its kernel comparison.
000008 | That code fact is not an additional scientific metric ambiguity.
000009 | 
000010 | 2. The already proved ORIGINAL-FUCHS-ALGEBRAIC-FRAME.txt supplies G(z),
000011 | G'=A G, for exactly the same v117 plus-sign connection. Its unitary right
000012 | deck matrices make H=G^-dagger G^-1 globally single valued and flat.
000013 | Consequently actual parallel transport obeys P^dagger H_y P=H_x. No fit
000014 | to the target monodromy, new scale, state or metric is introduced.
000015 | 
000016 | 3. On the selected real lift at x=1/2, continue to z=1 from below. At
000017 | w=0 the calibrated values are h0=h1=-omega and h2=+omega. The local
000018 | positive loop acts by w->omega w and
000019 |   (h0,h1,h2)->(-h2,h0,-h1).
000020 | It fixes those values and preserves the three Kummer squares and product1.
000021 | Substituting into the three explicit columns gives
000022 |   G -> G C_s,  C_s=[[0,0,1],[-1,0,0],[0,-1,0]].
000023 | Hence G_x^-1 QP G_x=C_s for the actual primitive +1 loop. This is a signed
000024 | monodromy statement, not an inference from its three eigenvalues.
000025 | 
000026 | 4. Put C_pair=diag(G_x,P G_x), H_pair=diag(H_x,H_(1/x)). Then directly
000027 |   C_pair^-1 T_raw C_pair = [[0,C_s],[I,0]],
000028 |   C_pair^dagger H_pair C_pair=I,
000029 |   T_raw^sharp=H_pair^-1 T_raw^dagger H_pair=T_raw^-1.
000030 | The Euclidean raw adjoint remains different. It is the source adjoint
000031 | that the original positive kernel must use.
000032 | 
000033 | 5. The connection to the original exact M is explicit. Let
000034 |   F=(e0,M e0,M^2 e0), S=diag(1,-1,1), B=S F^dagger.
000035 | F is unitary, F^dagger M F=P_c, and C_s=S P_c S. Thus
000036 |   B^dagger C_s B=M, B^dagger B=I.
000037 | Root checked these identities exactly with the stated rational complex M.
000038 | The original source_cycle_frame uses one-based e1 for the same e0.
000039 | With C=C_pair diag(B,B) the raw path is mapped isometrically to the
000040 | already used exact T=[[0,M],[I,0]]. This B is the adjoint of the B in
000041 | FLAVOR-PATH-TRANSPORT-G-METRIC-ADDENDUM.txt; the total map is identical.
000042 | Finally the existing exact J gives J^dagger T^-1 J=U6. Therefore C J
000043 | transports D_y, its metric Gram operator, Green and its positive inverse
000044 | square root from the actual original path to the original U6 consumer.
000045 | Any specified endpoint vectors and P_str must be transported by that
000046 | same map. This does not select new physical endpoint vectors or P_str.
000047 | 
000048 | 6. Root independently evaluated G at0.5 and2 and reused only P,Q from
000049 | the public original integrate_path_transport call. Results:
000050 |   signed monodromy residual                 8.50e-13,
000051 |   P source metric residual                 5.38e-13,
000052 |   Q source metric residual                 2.87e-14,
000053 |   pair frame isometry residual             1.26e-12,
000054 |   actual path conjugacy residual           8.50e-13,
000055 |   T_raw source unitarity residual          5.39e-13.
000056 | The symbolic proof is(2)-(5); these numerics independently replay the
000057 | actual path implementation. Raw Euclidean unitarity defect12.6373 is
000058 | not a contradiction. ROOT-ACTUAL-PATH-G-METRIC-CHECK.json preserves them.
000059 | 
000060 | 7. A useful exact coefficient consequence, independently checked, is
000061 |   C_s^2 E02=E22, tr(C_s^2 kappa E02)=kappa.
000062 | U6^2=diag(C_s^2,C_s^2) in the signed frame. Thus the known nonzero Hardy
000063 | contribution has an explicit even transport word, without identifying
000064 | the two transport fibers with the F5 boundary faces. This is a certified
000065 | contribution, not the trace of an as-yet-unevaluated full matrix. Calling
000066 | it an actual P1 action variation additionally requires the original
000067 | source/observable to insert this word. The algebra does not choose it.
000068 | 
000069 | 8. Closed: actual original halfpaths, their fixed G source metric, signed
000070 | monodromy, exact M/U6 representation and finite positive functional
000071 | calculus now share one explicit isometric map. The earlier metric-step
000072 | wording is superseded by this positive result and the independent addendum.
000073 | Open source comparison: actual full D4 spine/transport words, prescribed
000074 | P_str/endpoints and Hol K_f^str source insertion. One primitive path is
000075 | not automatically the entire consumer. This is not a new universal CAR
000076 | isometry gate on K as a coupling coefficient. Existing whole G/W source
000077 | state, reflection and time remain intact; this finite bridge does not
000078 | derive physical time or the entire original physical source.