TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-DELTA-JET-READS-THE-SAME-OPEN-WORD.txt
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000001 | ORIGINAL DELTA JET AND THE ACTUAL HARDY OPEN WORD
000002 | 30 October2026. Exact original-source continuation, no new model or Atlaswrite.
000003 | 
000004 | RESULT
000005 | The source-derived common G frame gives the certified contribution
000006 |  kappa M0 P0, P0=(I+U²)/2, kappa!=0.
000007 | The SAME open word occurs in an ORIGINAL differentiable source operation:
000008 | 
000009 |  D_y,+=yI-delta M0,
000010 |  (partial_delta D_y,+)P0=-M0 P0.                           (1)
000011 | 
000012 | Thus kappa M0 P0=-kappa(partial_delta D_y,+)P0 is an exact
000013 | operator identification with the canonical marked leg of the original
000014 | transport jet. No new W=delta D has been chosen for this identity.
000015 | It does not mean that P1 has executed the Hardy insertion in Ftr: kappa
000016 | is complex, the original delta is real, and P0 is a fixed marked endpoint
000017 | rather than a new selected Pstr or projected independent variation.
000018 | 
000019 | 1. Original operation actually used
000020 | 
000021 | EM3=documents/archive--tfpt-45--source_extracts--03_em_flavor_source/sections/
000022 | 003-subsection-discrete-phase-operator-and-transport-grammar.tex:
000023 | 100-125 defines Kstr and the ORIGINAL real-parameter action
000024 |  Ftr(delta)=-sum_(y=1,1/3,2/3) log det(D_y*D_y+eps²),
000025 |  D_y=yI-delta U6.
000026 | Its closed branch eps=0 makes partial_delta Ftr(delta*)=0.
000027 | The already reviewed Gmetric and full marked Bstar dictionary identify
000028 | U6 with diag(M0,-M0), not by merely matching its order.
000029 | Both sign branches are therefore retained below.
000030 | 
000031 | EM3 source ID paper-source:8c374b792c3f7ed5c3c4211f;
000032 | SHA256 00a2cf1b249b77280dbbf6dfb41c91c77e08aa63e0f30b682d1a8a621f3dd7ec.
000033 | Original EM4:353-389 supplies the finite resolvent. The full common marked
000034 | frame and actual P0 endpoint are recorded, unchanged, in
000035 | FULL-MARKED-G-FRAME-AND-ORIGINAL-OPEN-WORD.txt.
000036 | 
000037 | 2. The actual Real/adjoint consumer
000038 | 
000039 | For a real path of complex matrices D(s), the ORIGINAL determinant gives
000040 | 
000041 |  d[-log det(D*D)]/ds=-2 Re Tr(D^-1 dD/ds).                 (2)
000042 | 
000043 | It is not the bare complex trace. Work on the invertible closed cusp
000044 | branch, so det(D_y)=y^6-delta^6 is nonzero at the stated admissible cusps.
000045 | Original M0 is unitary cubic and e0,M0e0,M0²e0 are orthonormal. Thus
000046 |  Tr(M0 P0)=Tr(M0² P0)=0, Tr(P0)=1.
000047 | Using the ORIGINAL cubic resolvents on the two sign branches,
000048 | 
000049 |  Tr[(yI-delta M0)^-1 M0 P0]=delta²/(y³-delta³),
000050 |  Tr[(yI+delta M0)^-1 M0 P0]=delta²/(y³+delta³).             (3)
000051 | 
000052 | These traces are real for the actual real parameters. Their signs in(2)
000053 | are opposite because partial_delta D_+=-M0 while
000054 | partial_delta D_-=+M0. The fixed P0 contribution to the real delta
000055 | response of the whole SIX-dimensional block is therefore
000056 | 
000057 |  j_y=2delta²/(y³-delta³)-2delta²/(y³+delta³)
000058 |      =4delta^5/(y^6-delta^6).                             (4)
000059 | 
000060 | The three canonical family legs have the same trace response and sum to
000061 |  partial_delta F_y=3j_y=12delta^5/(y^6-delta^6).           (5)
000062 | Equations(3)-(5) evaluate an actual summand of the original derivative;
000063 | they do not replace the whole derivative by a selected family trace.
000064 | 
000065 | For a complex scalar c multiplying the same matrix jet, (2) retains
000066 | Re(c times the trace). Since(3) is real, its imaginary component is not
000067 | measured by this bare real-delta derivative. The actual kappa phase and
000068 | Clifford/adjoint legs cannot be discarded by identifying kappa with a
000069 | new real value of delta. Nor is the Hardy source asserted to supply
000070 | both six-fiber branches with exactly this diagonal completion.
000071 | 
000072 | 3. Full original cusp sum: stationary zero and nonzero curvature
000073 | 
000074 | Let a_y=y^6 and z=delta^6. The stated closed root is
000075 |  z*=(794-7sqrt9961)/2187,
000076 |  delta*=0.593277280133224001394066545421...
000077 | The exact original stationarity is
000078 | 
000079 |  sum_y 1/(a_y-z*)=0,
000080 |  partial_delta Ftr(delta*)=0,
000081 |  sum_y j_y(delta*)=0.                                    (6)
000082 | 
000083 | A positive single-cusp or single-word response must NOT be called a
000084 | nonzero original first variation. Both original sign branches and the
000085 | three original cusps have been included in(6).
000086 | Differentiating the SAME original action gives
000087 | 
000088 |  Ftr''=60delta^4 sum_y1/(a_y-delta^6)
000089 |           +72delta^10 sum_y1/(a_y-delta^6)^2.
000090 | Hence at the original stationary point,
000091 | 
000092 |  Ftr''(delta*)=72delta*^10 sum_y1/(a_y-z*)²>0
000093 |              =417.715131947579642817910612310... .         (7)
000094 | 
000095 | This is the already specified transport action's actual curvature, not
000096 | a new fitted stiffness, a generated Yukawa, or a physical mass/time scale.
000097 | The fixed marked leg's summed derivative changes at one third of(7).
000098 | No arbitrary complex/individual-leg source coupling has been added.
000099 | 
000100 | 4. What is now gained and what the first source comparison is
000101 | 
000102 | The previous result gave a genuine path and a typed open word.
000103 | Equation(1) locates that SAME operator in an ORIGINAL source derivative;
000104 | (2)-(7) execute the actual real/adjoint consumer and its original summed
000105 | stationarity. This is stronger than merely choosing Tr(Cs² B02).
000106 | The exact identity does not insert a response coefficient into the action:
000107 | it identifies its operator direction, while the source must still carry
000108 | the actual Hardy coefficient/field/adjoint contraction into the shared
000109 | original functional if this is its intended physical use. An entire Y
000110 | need not come from Hardy, and no new state or CAR quantization is required
000111 | for the already complete given Seed.
000112 | 
000113 | The broader Gamma_ij^min address search was continued in the older
000114 | originals. tfpt-31 section010:74-112 names Gamma_ij and uses the hard
000115 | holonomy formula, without an endpoint/path table. The later EM4:180-197
000116 | adds the canonical D4 spine nomenclature. The research-contract source
000117 | 006:233-241 explicitly records that the v34 diagonal-dressing attempt
000118 | failed and that v118 resolves the LEPTON determinant extraction. The
000119 | actual v118:1-41 preserves its scope: exact spectrum/resolvent constants,
000120 | with the site/leg assignment and quark composition separately recorded.
000121 | This historical determinant rule is a valid original alternative for
000122 | those scalar coefficients; it is not a nine-open-path/source-insertion
000123 | formula and does not undo the now closed whole connection/marked frame.
000124 | No new theta4 spine, endpoint or integral weight was substituted.
000125 | 
000126 | Durable original-jet calculation, seven exact identities:
000127 | workspace outputs/multi-model-source-search-20261029/
000128 | ORIGINAL-DELTA-JET-HARDY-WORD-CHECK.json,
000129 | SHA256 a59994cb7fa343e0eecc42457c52bbb989d4d27602932a0d6419a5d00c53ef89.
000130 | No suite rerun or original code modification.