TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/historical-dynamics-source-recovery-20261103/HISTORICAL-QUARTER-SOURCE-AND-HOLONOMY-J.txt
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000064 | 3. COMPLETE BOUND REFERENCE AND GIVEN SOURCE TRANSPORT
000065 | The original R032 SEAM-EVEN charts evaluated in REFERENCE-FIBER-EVALUATION
000066 | give h_A=A gamma_F-i gamma_F Phi, h_ref,S=A gamma_F, h_ref,T=h_A.
000067 | For constant U_X and [A,gamma_F]=0, h_A(-H,-MR)=U_X h_A(H,MR) U_X*.
000068 | The S reference stays fixed; the T reference transforms by the same U_X.
000069 | The mass part of this identity has zero residual on every84 coefficient
000070 | direction. General H_even/H_odd charts retain their already stated pullbacks.
000071 | The full original graphs, coherent kernels and sea factors therefore
000072 | transport on this field image; their actual time parameter is unchanged.
000073 | This does not turn a_R or the external Z_ref[0] into a computed boson weight.
000074 | In particular the actual remaining a_R factor is not set to1.
000075 | 
000076 | For the ordinary product D=D_M+gamma5 tensor Phi, the same constant U_X
000077 | commutes with D_M and conjugates the full D to its transformed-field D.
000078 | On suitable transported selfadjoint domains its spectral functions and
000079 | Calderon/APS projectors conjugate. This is covariance between domains,
000080 | not an unproved assertion that the original fixed domain is invariant.
000081 | At a compatible finite regulator the paired Pfaffian and Berezin measure
000082 | transform inversely. A finite determinant1 does NOT evaluate a global APS
000083 | determinant-line or bordism anomaly. Source/reference/domain phases remain.
000084 | For variable Phi, D^2=D_M^2+Phi^2-gamma5[D_M,Phi] in this convention.
000085 | 
000086 | 4. REUSE THE ACTUAL ORIGINAL HOLONOMY LIFT
000087 | compiler-quartic-holonomy-20260919/PROOF.txt sections3-4 already proves:
000088 |  det R(g)=det g=(-1)^epsilon(g),
000089 |  S(g)=exp(-i pi epsilon/2) Lambda_even R(g),
000090 |  Fplus(g)=exp(+i pi epsilon/4)g,
000091 |  L(g)=[S(g),Fplus(g)] in H_E=(Spin10 x SU4)/<(z,zeta^-1)>,
000092 |  G=[z,1]=[1,zeta], z|16=i, z|10=-1, zeta=iI4.
000093 | The actual product is L(g)L(h)=G^(-epsilon(g)epsilon(h))L(gh).
000094 | The original central extension E=Z4 x G31 uses exactly that cocycle.
000095 | Its kernel in H_E is {(-k,(iI4)^k)}. The orders184320/46080 refer to
000096 | the full extension/effective image respectively, not identical groups.
000097 | Equivalently h=lambda*g, lambda^4 detg=1, belongs to the supplied SU4
000098 | preimage and maps to [Lambda_even R4(h),h], R4(h)=lambda^4 R(g) in SO5.
000099 | No independent phase choice or new theorem is attributed to this reuse.
000100 | 
000101 | Crucially L(r)^2=G^-1 and L(r)^8=1, whereas raw r^2=1. Thus this is
000102 | the original path-holonomy transport, not an honest representation of
000103 | the unchanged original reflection relations on charged sectors.
000104 | The 23Oct equivariant-monodromy theorem retains its exact clock-obstruction
000105 | scope. Fermion-parity language alone does not erase an original lift defect.
000106 | 
000107 | For g=J=iI4, R(J)=I5, S(J)=I16 and Fplus(J)=iI4, hence L(J)=G.
000108 | On the DESCENDING (16,4) sector the phase i comes from the A3 partner.
000109 | An isolated (16,1) is not a representation of the quotient H_E, since
000110 | its supposed trivial kernel acts with i. The actual full decomposition
000111 | is(45,1)+(1,15)+(16,4)+(10,6)+(16bar,4bar).
000112 | Neither rank(A3)=3 nor dim(4)=4 proves a physical three-family restriction.
000113 | The actual family/selector/time insertion map must be retained, with its
000114 | possible complementary paths. No new rank3 projector is introduced here.
000115 | 
000116 | The native quartic code really has rho5(J)=i^4 I5=+I5. Spin10 z acts as
000117 | -I on its vector C5. A DIRECT vector intertwiner obeys T=-T and is zero.
000118 | This does NOT invalidate the existing whole-partner holonomy construction.
000119 | The original code basis was directly evaluated; the character mismatch
000120 | is analytic and independent of numerical rank thresholds.
000121 | 
000122 | 4b. USE THE EXISTING FOUR-CUT FAMILY, WITHOUT CHOOSING A NEW CUT
000123 | The actual original CUTS.txt section8 already constructs B=(16,4) and
000124 | P_t E_(s,u)=delta_(t,u)E_(s,u), Q_t=I-P_t, for all four tetrahedral marks.
000125 | Each Q_t B is literally(16,3), with its D5 and A2 root actions verified
000126 | there, not merely a48-dimensional space. On B the present G is i I64;
000127 | therefore, for EVERY original cut, [G,Q_t]=0 and G|Q_t B=i I48.
000128 | The conjugate block has the conjugate phase. Only IF these48 current
000129 | coordinates are mapped to48 physical left Weyl fields with the same
000130 | specified global symmetry does3*16 give a vanishing Z16 anomaly index.
000131 | Neither that statistical/physical map nor a choice of t or mirror removal
000132 | is proved by these projector identities. Original S4 transports the
000133 | four cuts; one must transport the full source and its chosen marks.
000134 | This is a direct application of the existing projector resolution, not
000135 | an assertion that Q_t is the primitive physical P_adm or that the entire
000136 | holonomy action preserves an individual fixed Q_t.
000137 | Keep SU4_A3 (the partner outside Spin10) separate from SU4_C (the color
000138 | subgroup inside Spin10). The quotient identity [1,zeta]=[z,1] on joint
000139 | representations does not identify these two Lie group factors. The full
000140 | PS compensation in(4) is inside Spin10 and still requires its actual
000141 | primitive field map. No new four-to-three projection is supplied by it.
000142 | 
000143 | 5. WHAT THE PRIMITIVE SELECTOR AND ANOMALY DO NOT FIX HERE
000144 | Original QFT002 defines Pstr=P_seam,+ Psing PTheta and
000145 |  Pgap=1_[eps_adm^2/4,infinity)(Pstr Dadm* Dadm Pstr).
000146 | It supplies commutation of Dadm with iC,Psing,QTheta, not an identification
000147 | of those objects with gamma_F. One cannot infer their commutators with U_X
000148 | or their global covariance by notation. If their actual complete maps
000149 | conjugate, functional calculus also conjugates Pgap; this implication
000150 | does not evaluate those missing actual maps. Charged letters remain