TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-partner-original-action-20261015/PROOF.txt
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000065 | 3. THE ACTUAL TIME ACTION, WITHOUT A NEW HAMILTONIAN
000066 | 
000067 | For any supplied Hermitian one-particle matrix h on E, its exterior action is
000068 | 
000069 |     H_h=dGamma(h)=sum_(jk) h_jk epsilon_j iota_k.
000070 | 
000071 | Directly from CAR,
000072 | 
000073 |     [H_h,epsilon_i epsilon_a]
000074 |        =sum_j h_ji epsilon_j epsilon_a + sum_j h_ja epsilon_i epsilon_j.
000075 | 
000076 | If h commutes with the full 3+2 group, Schur's lemma gives h=a I3 direct-sum b I2.
000077 | Then [H_h,A]=(a+b)A and H_h(A1)=(a+b)A1. Zero energy requires a+b=0.
000078 | For h=Y, the value is 1/6, not zero. If one ALSO imposes tr h=3a+2b=0,
000079 | zero energy of this pair implies a=b=0. These are identities for this supplied
000080 | one-particle family; general invariant carrier operators need not be dGamma(h).
000081 | Neither a time h nor H_phys=dGamma(h) follows from the slot count or from Y.
000082 | 
000083 | Now use the actual previously stipulated affine E8 time, not H_h:
000084 | 
000085 |     K_alpha=L0+delta_alpha.H0,
000086 |     delta_(alpha,j)=(j+alpha)/8-1/2, j=0,...,7,
000087 |     alpha=1/4 or 3/4.
000088 | 
000089 | The existing exact Weyl matrix in fold-results.json gives
000090 | 
000091 |     M delta_alpha=(-1/2,0,alpha/4,-1/2,0,(alpha-1)/4,-1/4,-1/4).
000092 | 
000093 | Hence the energies of the six normalized root-current states in the order
000094 | (e1+e4,e1+e5,e2+e4,e2+e5,e3+e4,e3+e5) are
000095 | 
000096 |     (0,1/2,1/2,1,1/2+alpha/4,1+alpha/4).
000097 | 
000098 | For alpha=1/4 the last two are 9/16,17/16; for alpha=3/4 they are 11/16,19/16.
000099 | The current-mode law is [K_alpha,J_(beta,n)]=(-n+(M delta_alpha).beta)J_(beta,n).
000100 | The three fixed SM simple-root zero modes for e1-e2,e2-e3,e4-e5 have respective
000101 | commutator coefficients -1/2,-alpha/4,-1/2, all nonzero. Thus this transported K
000102 | does not commute with that fixed full color/weak action. Quarter phases are kept.
000103 | 
000104 | A useful necessary condition follows without assuming any one-particle H:
000105 | if a selfadjoint H commutes with the full connected S(U3 x U2) representation,
000106 | its kernel is invariant. A nonzero zero vector in an irreducible (3,2) sector
000107 | generates all six zero vectors under that group. Together with an invariant
000108 | vacuum in a different singlet sector, dim ker H >= 7. The known two-dimensional
000109 | K_alpha kernel is therefore incompatible with this additional fixed unbroken
000110 | symmetry/time identification. This is NOT a universal gauge-theory exclusion:
000111 | a background, time-dependent covariant symmetry, or Gauss projection changes
000112 | the premise. A Gauss projection cannot simultaneously retain a lone colored
000113 | root state as a physical singlet partner. A different time frame must transport
000114 | its connection term and the actual observables, not silently erase delta.