TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/ADMISSIBLE-ACTION-DERIVATION-20261103.txt
Original file SHA-256: d3b7ea73d59c1c671af0fec4a4f9b1048dfc297eb26a932fe18fda2af03b972d
Inclusive original lines: 1-130
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000001 | R112. DERIVING THE ADMISSIBLE ACTION: COVARIANT OPERATOR AND FERMION VARIATION
000002 | 3 November 2026. Actual original operators; no auxiliary physical model.
000003 | Entry plan103, Atlas79704e29c6da26ac16ba909d.
000004 | Verdict: PARTIAL_ACTION_DERIVATION_WITH_FORCED_PS_COVARIANT_TERM.
000005 | 
000006 | The complete primitive physical S_adm is NOT derived here. What is derived
000007 | is a mandatory nonlinear correction of the existing PS source branch,
000008 | its actual whole96 block and spectral-action response, and an inverse
000009 | variation theorem specifying exactly when the fermionic action follows.
000010 | The original spectral-action formula and the general CCvS fluctuation
000011 | theorem are reused; their application to the actual96 coefficients is
000012 | evaluated. No primitive physics/TOE/ledger promotion.
000013 | 
000014 | 1. ORIGINAL INPUTS AND THE DERIVATION QUESTION
000015 | 
000016 | Keep the actual finite v252 full_D, J_F=Kswap times conjugation, grading,
000017 | all four original Yukawa matrices and symmetric Majorana M_R. P1/P2,
000018 | E8, clocks, flavor and alpha remain. The original PS branch is conditional;
000019 | the SM first-order branch is not replaced by a newly gauged PS theory.
000020 | Use only the actual existing right quaternion subgroup of rep_PS here.
000021 | No full SU4 representation theorem is inferred from this restricted test.
000022 | 
000023 | The original v251/v252 establish failure of the first-order condition in
000024 | this PS Majorana direction. v254 describes linear represented oneforms
000025 | and their support. A source-compatible completion must respect the actual
000026 | gauge lift U=uJuJ^-1 on the original operator, not only its scalar labels.
000027 | 
000028 | 2. A QUADRATIC TERM FORCED BY THE ACTUAL GAUGE ORBIT
000029 | 
000030 | For an actual unitary algebra element u, put hatu=J u J^-1 and U=u hatu.
000031 | Order-zero gives [u,hatu]=0. The finite original KO sign is JD=DJ.
000032 | The linear terms of the original inner-fluctuation prescription are
000033 |  A1=u[D,u*]=u D u*-D,
000034 |  A1tilde=hatu[D,hatu*]=hatu D hatu*-D.
000035 | Exact multiplication now gives
000036 |  U D U* = D+A1+A1tilde+A2,
000037 |  A2=hatu[A1,hatu*]=hatu A1 hatu*-A1.                 (1)
000038 | 
000039 | Equation(1) fixes the correction uniquely ON THIS GAUGE ORBIT, with
000040 | coefficient one. No coupling has been fitted. It is the cross term of
000041 | the same two-sided transformation. Gauge covariance alone does not choose
000042 | all gauge-invariant off-orbit additions or the entire primitive dynamics.
000043 | The general canonical inner-fluctuation construction without first-order
000044 | has the analogous A2=sum hat(a_i)[A1,hat(b_i)], for the same original
000045 | normalized perturbation. This is the known CCvS theorem, not a new claim
000046 | that P1 forces a PS realization.
000047 | 
000048 | 3. EXPLICIT ENTIRE96 MAJORANA BLOCK
000049 | 
000050 | On the actual original unitary subgroup u=rep_PS(I2,qR,I3), let
000051 |  qR=[[alpha,beta],[-bar(beta),bar(alpha)]],
000052 |  |alpha|^2+|beta|^2=1, e_nu=(1,0)^T,
000053 |  n=qR e_nu=(alpha,-bar(beta))^T, w=n-e_nu.
000054 | In the reordered original right-lepton basis the particle/antiparticle
000055 | Majorana block is T=P_nu tensor M_R, P_nu=e_nu e_nu^T.
000056 | 
000057 | The mandatory correction is EXACTLY
000058 |  A2_(p,anti)=(w w^T) tensor M_R,
000059 |  A2_(anti,p)=A2_(p,anti)*,                            (2)
000060 | and all other blocks vanish. The four original Dirac-Yukawa blocks have
000061 | zero A2. Nothing outside the Majorana support has been dropped.
000062 | The linear versus correctly transformed Majorana blocks are
000063 |  Tlin=(n e_nu^T+e_nu n^T-P_nu) tensor M_R,
000064 |  Tcorr=(n n^T) tensor M_R=qR P_nu qR^T tensor M_R.     (3)
000065 | Thus A2 introduces no new independent scalar. It supplies the product of
000066 | the existing transformed particle and conjugate-particle components.
000067 | For real qR with cos(theta)=c, sin(theta)=s,
000068 |  ||A2||_HS^2=8(1-c)^2 ||M_R||_HS^2.
000069 | 
000070 | 4. EFFECT ON THE ACTUAL SPECTRAL ACTION
000071 | 
000072 | For any scalar unitarily invariant original spectral functional F,
000073 | unitary covariance gives
000074 |  F(Dcorr)=F(U D U*)=F(D).                              (4)
000075 | Both the actual reference operator and eta/domain data must travel under
000076 | the SAME U, together with the original selector and deck/insertion marks.
000077 | A restricted trace requires its actual selector to be preserved or
000078 | transported as P->U P U*. A deck-defined relative D also carries its actual
000079 | iota. The finite orbit alone does not prove U admissible on every marked
000080 | primitive source. Under this same-source transport, the whole relative
000081 | spectral action is invariant. The correction to its numerator is exactly
000082 |  Delta S_B=Tr f((Dlin+A2)/chi)-Tr f(Dlin/chi)
000083 |           =Tr f(D/chi)-Tr f(Dlin/chi).                (5)
000084 | Use the original squared/unsquared argument and chosen source version;
000085 | profile traces and residue/finite-part versions are not identified.
000086 | At constant bounded finite gauge U, all valid original spectral functional
000087 | versions obey(4). Local transformations also carry their original
000088 | derivative connection and domains.
000089 | 
000090 | For arbitrary ORIGINAL Yukawas and symmetric M_R, the full96 trace obeys
000091 |  Tr Dlin^2-Tr D^2=4|alpha-1|^2 c_R,
000092 |  c_R=Tr(M_R* M_R).                                   (6)
000093 | This follows from ||Tlin||_HS^2-||T||_HS^2
000094 | =2|alpha-1|^2||M_R||_HS^2; its adjoint block doubles the trace.
000095 | 
000096 | For the already existing original quaternion(0,1) of v254.ps_basis,
000097 |  qR=[[0,1],[-1,0]],
000098 |  Tlin=[[-1,-1],[-1,0]] tensor M_R,
000099 |  A2_(p,anti)=[[1,1],[1,1]] tensor M_R,
000100 |  Tcorr=[[0,0],[0,1]] tensor M_R.
000101 | In the SAME full96 operator,
000102 |  Tr Dlin^4-Tr D^4=12d_R+16e_e,
000103 |  d_R=Tr[(M_R M_R*)^2],
000104 |  e_e=Tr[(Y_e*Y_e)(M_R M_R*)].                         (7)
000105 | For M_R!=0 this is strictly positive. To derive(7), conjugate both
000106 | operators back by U*: all original Yukawas are restored. Only Tlin changes
000107 | to [[0,1],[1,-1]] tensor M_R. Its2x2 Gram has diagonal(1,2),square trace7.
000108 | The exact block formula is
000109 |  Tr D^4=2Tr S^4+8Tr(S^2 T T*)+2Tr[(T T*)^2]
000110 | for the original chirality-changing S and right-right Majorana support.
000111 | The disjoint supports kill the remaining mixed trace. Substitution gives
000112 | 2(7-1)d_R+8(2)e_e, proving(7) without selecting physical Y or M_R.
000113 | 
000114 | The original local potential coefficients multiply exactly these traces.
000115 | Thus omitting A2 would introduce spurious changes of the Majorana and
000116 | charged-lepton/Higgs contributions under a mere change of gauge frame.
000117 | Equations(6)-(7) concern the NUMERATOR invariants. If the reference has
000118 | the same contributions, some relative defects may cancel. No nonzero
000119 | defect of every fixed relative cutoff is asserted without its actual
000120 | reference. Correct common conjugation restores the entire relative
000121 | spectral functional/action for all original profiles, not only two
000122 | coefficients. This alone does not identify the entire physical S/Z source.
000123 | 
000124 | With actual transformed marks O_a the exact decorated spectral identity is
000125 |  Tr f(Dcorr+sum J_a O_a)
000126 |   =Tr f(D+sum J_a U* O_a U).                          (8)
000127 | Consequently all corresponding marked spectral derivatives travel together.
000128 | This is not a claim that these spectral derivatives are already the
000129 | complete interacting Schwinger generating functional.
000130 |