TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ROOT-ORIGINAL-JET-AND-FOUR-PATH-REVIEW.txt
Original file SHA-256: a917b2333628aa580666adec61c99f1701823cfe308cc47085bc1188cc565f68
Inclusive original lines: 1-101
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000001 | ROOT REVIEW: ORIGINAL ACTION JET AND FOUR ACTUAL OPEN PATHS
000002 | 30 October2026. Scoped original-source continuation, not full physical closure.
000003 | 
000004 | 1. Actual original action response
000005 | 
000006 | Read original EM3:100-250,250-352 and EM4:164-244. The same original
000007 | D_y=yI-delta U6, both sign branches U6=diag(M0,-M0), and the three cusps
000008 | are used. With P0=(I+U^2)/2, the common marked Gbase dictionary gives
000009 | kappa M0P0=-kappa(partial_delta D_y,+)P0. It matches the direction of an
000010 | actual original derivative rather than an independently chosen test matrix.
000011 | For the original real action, however,
000012 |  d[-logdet(D*D)]=-2 ReTr(D^-1 dD).
000013 | Root independently reduced the two branch traces to
000014 |  delta^2/(y^3-delta^3), delta^2/(y^3+delta^3), with both residuals exactly0.
000015 | The paired marked-leg current is 4delta^5/(y^6-delta^6); three family legs
000016 | sum to 12delta^5/(y^6-delta^6). At the actual original root
000017 |  zstar=(794-7sqrt9961)/2187=delta_star^6,
000018 |  delta_star=0.593277280133224001394066545420926568...,
000019 | the exact three-cusp sum is0. The full first derivative therefore vanishes.
000020 | The same action has positive dimensionless curvature
000021 |  72delta_star^10 sum_y 1/(y^6-zstar)^2
000022 |  =417.715131947579642817910612309596... .
000023 | These exact and high-precision checks are saved in ROOT-ORIGINAL-DELTA-JET-CHECK.json.
000024 | The curvature is not a physical mass, time scale or generated Yukawa. A
000025 | complex Hardy coefficient is not a new freely chosen real delta variation.
000026 | The source's actual coefficient/field insertion has not followed merely
000027 | from matching the operator direction.
000028 | 
000029 | 2. Four actual open classes, not a fitted endpoint matrix
000030 | 
000031 | The fixed original connection is integrated on z=r exp(i(pi/4+k pi/2)),
000032 | k=0,1,2,3, using the existing original_fuchs_connection and _integrate
000033 | implementation. Infinity is regularized by z^(-D), D=diag(2,1,1).
000034 | The algebraic G calculation gives T_k=V^k T0 U^(-k),
000035 |  V=i^(-D)U=diag(-1,1,-1), U=diag(1,i,-i).
000036 | The already fixed bundle metrics are
000037 |  H0=diag(10/27,5/(9sqrt3),1/(3sqrt3)),
000038 |  Hinf=[[50,0,2sqrt10],[0,15,0],[2sqrt10,0,17]]/81.
000039 | The exact report proves T_k*Hinf T_k=H0 and all four original meridians
000040 |  H0^(1/2) T_(k+1)^(-1) T_k H0^(-1/2)=U^(k+1)M0 U^(-k-1).
000041 | Cyclic k is understood, including the last returning edge.
000042 | 
000043 | Root independently integrated ALL FOUR genuine original ODE paths at
000044 | R=1000,2000 using a smooth radial parametrization and extrapolated their
000045 | O(R^-2) endpoint errors. Endpoint quarter residuals were below2.9e-13,
000046 | metric isometry residuals below3.2e-13, all four meridian residuals
000047 | below1.3e-13. This numerical replay is independent of the exact endpoint
000048 | limit; it confirms its actual original-path meaning, not its proof by
000049 | numeric tolerance. ROOT-FOUR-OPEN-PATH-ODE-CHECK.json preserves the output.
000050 | 
000051 | 3. Crucial frame correction before promotion
000052 | 
000053 | An initial unpublished endpoint check multiplied raw T0 by normalized M0.
000054 | That mixes frames and its radical coefficient was wrong. The actual raw
000055 | word is Mraw=Gbase Cs Gbase^-1. Equivalently use
000056 |  T_hat,k=T_k H0^(-1/2)
000057 | with normalized M_k=U^kM0U^(-k). Root caught this mismatch and the final
000058 | agent report/JSON were corrected BEFORE Atlas registration. Correctly,
000059 |  (T_hat,0 M0)10=-3sqrt30/10,
000060 |  sum_k T_hat,k M_k P0=-6sqrt30/5 E10.
000061 | After normalizing the target too, the sum is -2sqrt2 E10. Both agree with
000062 | the independent actual ODE replay. The matched fully raw sum is -4E10.
000063 | A frozen insertion instead has zero sum. These are different consumers.
000064 | The nonzero covariant result is retained; the preliminary mixed-frame
000065 | coefficient must not be reused. No original rule summing four paths is
000066 | asserted by this finite identity.
000067 | 
000068 | 4. Geometry and original consumer scope
000069 | 
000070 | The diagonal graph is a reduced theta4 spine with four punctured faces.
000071 | The original anti-real reflections fix its two full diagonals. For the
000072 | canonical complete hyperbolic metric of the given conformal sphere their
000073 | fixed sets are geodesic. This introduces no fitted physical metric, and
000074 | flat transport depends on path classes, not that auxiliary metric.
000075 | The reduced rank3 valence proof forces two valence4 vertices when rotation
000076 | fixes0,infinity and permutes the four punctured faces. Additional invariance
000077 | under the actual two anti-real reflections fixes the diagonal arcs.
000078 | The printed original EM4 calls its spine D4-geodesic but does not specify
000079 | a base metric there. Holomorphic D4 alone is not silently identified with
000080 | the larger real-reflection symmetry. Thus the four actual path calculations
000081 | are established; equality to the unnamed original spine retains that
000082 | explicit geometric reading. No nine Gamma_ij table is invented.
000083 | 
000084 | [OMITTED: non-scientific conversation or execution metadata]
000085 | [OMITTED: non-scientific conversation or execution metadata]
000086 | [OMITTED: non-scientific conversation or execution metadata]
000087 | [OMITTED: non-scientific conversation or execution metadata]
000088 | [OMITTED: non-scientific conversation or execution metadata]
000089 | [OMITTED: non-scientific conversation or execution metadata]
000090 | [OMITTED: non-scientific conversation or execution metadata]
000091 | 
000092 | 5. Continuing the same original problem
000093 | 
000094 | Use the now explicit four edge groupoid, all four original meridians and
000095 | the actual action jet in the original prescribed Hol_G K / Gamma_ij or
000096 | sewing consumer. Keep all endpoint frames, adjoints, both branches and
000097 | original structural projector. The primitive wedge consumer and whole
000098 | positive field/state/reflection/time transport are already available.
000099 | Do not reopen their derivations or impose Hardy=entireY. The original
000100 | source must decide the actual coefficient/field contraction and packaging.
000101 | No extra state, native-clock phase or physical time rule has been chosen.