TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/FOUR-STARTING-POINTS-STATE-RELATIVE-EQUIVALENCE.txt
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000029 | 2. SOURCES, PREMISES AND COMPARISON CLASS
000030 | 
000031 | Retain original source F C_alpha F*, alpha=1/4 or3/4, not eight new vacua;
000032 | original M and literal O=T S^-1/sqrt2; native E8/J quartic embedding;
000033 | fixed conformal NS point-splitting reference; actual source time/adjoint;
000034 | the stated E8 extension with its cocycle and charged sectors. The physical
000035 | P1 selection of this dictionary remains conditional, as in the prior proof.
000036 | 
000037 | Decisive originals in this contract:
000038 |  ORIGINAL-SEAM-STATE-FLAG-SELECTION.txt sections2--6: actual G31 action,
000039 |  global pair01, four flags, difference of internal character and fold g.
000040 |  FOLD-COVARIANCE-TO-NATIVE-FIELD-RESPONSE.txt sections2--5: same-fold
000041 |  covariance, reference subtraction, actual b, full D0..4 and root twists.
000042 |  CONFORMAL-HISTORY-AND-ERASURE-CARRY.txt: actual erasure20 and h5 carry.
000043 |  ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-GRADING.txt: exact source CAR
000044 |  ordering, point splitting, time, full charge carry; no Hall/locality shortcut.
000045 | Original _archive/tfpt-45/source_extracts/01_boundary_kernel_source.tex
000046 | 480--497 defines operational seed morphisms as unitary intertwiners
000047 | preserving ALL displayed net/time/reflection/state/seam/collar structures.
000048 | Thus fixed-state preservation is an original equivalence requirement,
000049 | not a newly imposed selection rule. Whether these actual native quartic
000050 | fields are the relevant physical subobject still uses the stated dictionary.
000051 | Haisheng Li, arXiv:1004.0864, eq2.4 and Proposition2.1, gives the supplied
000052 | E8 module-twist construction, NOT a physical P1 field-selection rule.
000053 | 
000054 | An equivalence here preserves the given field/coordinate/adjoint/reference
000055 | structure and source state, and carries the distinguished quartic ray q
000056 | to ray q' up to phase, with its intrinsic event geometry. A fixed-root flag
000057 | includes this distinguished ray. Arbitrary derivative mixing, independent
000058 | reference changes, changed preparations, reversed time or an unrelated
000059 | coordinate change are not silently permitted as the same marked datum.
000060 | The source's fixed time/seam structure must be preserved, or explicitly
000061 | transported and then described as a different presentation. A passive
000062 | renaming of a whole object is never a change of only its physical flag.
000063 | The previous negative-D versus positive-filled particle/hole orientation
000064 | qualification remains in force: the NS dictionary is conditional, and
000065 | current signs, charge/time and reference must be transported jointly.
000066 | Ray c1 and plane traces survive the corresponding b->-b/conjugation, but
000067 | this does not eliminate the charged orientation/reference obligation.
000068 | 
000069 | 3. THE FOUR FIELDS AND THEIR COMPLETE NEUTRAL RESPONSE
000070 | 
000071 | Let f_i be the actual normalized quartics with norms(96,8,8,8,1), and N
000072 | the previously INTRINSICALLY reconstructed six-ray matrix, in the existing
000073 | literal comparison chart. NN*=I5, N*N=I6-J6/6. It is not fitted here.
000074 | Use unit ray w_q=sqrt(6/5)N[:,q], and F_w=sum_i(w_i)F_i.
000075 | 
000076 | The actual fold fixes delta_k=(k+alpha)/8-1/2, d=M delta and
000077 |  b=U_+^T O^T d,
000078 |  alpha1/4: b=((7-9i)/32,(1-i)/4,-i/4,(3-3i)/32),
000079 |  alpha3/4: b=((5-11i)/32,(1-i)/4,-i/4,(1-i)/32).
000080 | This b is from the SAME covariance subtraction, not a q-dependent state.
000081 | 
000082 | For the fixed NS reference F_i^d(z)=:f_i(h_+(z)+b/z): and
000083 |  (D_r)_ij=sum_|nu|=r conjugate(partial^nu f_i(b))partial^nu f_j(b)/nu!.
000084 | The FULL separated two-point kernel is
000085 |  K_ij(z,w)=sum_r=0..4 (D_r)_ij /[(z-w)^(2r)z^(4-r)w^(4-r)].
000086 | D0 is the disconnected mean; the connected function is r1..4.
000087 | Define c_r(q)=w_q*D_r w_q. The kernel basis is independent: multiply by
000088 | (zw)^4 to get the powers of t=zw/(z-w)^2. Thus equality of full connected
000089 | functions forces equality of EACH c1..4, not merely a sampled profile.
000090 | 
000091 | For every q2..5 in both quarters:
000092 |  c4=1, c3=85/256, c2=4335/131072.
000093 | But the exact c1 numerators, with COMMON denominator1006632960, are:
000094 | 
000095 |  q                   2           3           4           5
000096 |  alpha1/4       1200580     1300621      982285     1446340
000097 |  alpha3/4       1308100     1521805     1089805     1338820
000098 | 
000099 | All six unordered comparisons are different in each row. c1 is positive,
000100 | invariant under a phase of F_w and under jointly transported unitary charts,
000101 | and unchanged by the correctly joint b->-b/conjugation convention change.
000102 | The first difference along THESE six-ray directions is at sixth order in
000103 | b (D1); D2's fourth-order ray diagonal happens to be equal. D2 is not
000104 | asserted scalar as a whole matrix. Distinct mean magnitudes D0 corroborate
000105 | but are not needed. No largest-c1 or nearest-q rule is adopted.
000106 | In fact, for each fixed quarter and any two of the four normalized rays,
000107 |  K_q_connected(z,w)-K_q'_connected(z,w)
000108 |    =(c1(q)-c1(q'))/[(z-w)^2 z^3 w^3].
000109 | This nonzero separated-point distribution is an entire-kernel witness,
000110 | not a difference obtained by tuning a profile. Appropriate local smears
000111 | can distinguish unequal distributions; no physical instrument is thereby
000112 | postulated or selected.
000113 | 
000114 | If a state-preserving *-isomorphism A sends F_wq to exp(i phi)F_wq', it
000115 | preserves <F_wq#(z)F_wq(w)>_connected. The table contradicts this for
000116 | q!=q'. This argument does NOT depend on Stab_G31(d)=1; it also obstructs
000117 | an arbitrary larger equivalence that respects this actual marked-field
000118 | comparison class. It is not a classification of all P1 gauges.
000119 | 
000120 | A stronger test does not need the distinguished ray to be preserved:
000121 |  lambda_q=tr(P(q,{0,1})D1).
000122 | These four EXACT real algebraic numbers are ALSO pairwise distinct in
000123 | both quarters (all values are retained in results). The central leading
000124 | OPE coefficient D4=I fixes the quartic-space metric. A state-preserving
000125 | linear *-equivalence carrying the whole actual quartic space and Pq to
000126 | Pq' acts unitarily in that metric, must transport D1, and hence must
000127 | preserve this trace. Therefore even forgetting q as an extra label does
000128 | not identify these four planes within that comparison class. This is a
000129 | source-response invariant on a supplied plane, not an executed projection
000130 | or a choice of its physical detector. A transformation abandoning the
000131 | specified quartic field space is not covered by this stronger test.
000132 | 
000133 | For the complete rooted five-field frames let J_q={0,...,5}\{q}, with
000134 | the original sorted labels retained, D_q[:,j]=n_j-n_q and G=I5+J5.
000135 |  U_q=D_q G^-1/2, U_q[:,j]=n_j-(sqrt6-1)n_q/5, U_q*U_q=I.
000136 | Their raw kernels are D_q*D_r D_q; orthonormal kernels U_q*D_r U_q.
000137 | Both retain all cross-correlations. The two columns j0,1 reproduce the
000138 | actual P(q,{0,1}); they are not selected as a detector in this calculation.
000139 | 
000140 | Let g_ia=partial_a f_i be the actual erasure20. Its original vacuum Gram
000141 | is I20, explicitly checked; it is not a charged E8 twenty of equal size.
000142 | For the 25 fields p=(f_i,g_ia), degrees k=(4^5,3^20), compute
000143 |  J_r[A,B]=sum_|nu|=r conjugate(partial^nu p_A(b))partial^nu p_B(b)/nu!.
000144 | Every mixed separated kernel is exactly
000145 |  sum_r J_r[A,B]/[(z-w)^(2r)z^(k_A-r)w^(k_B-r)].
000146 | The new results store ALL five 25x25 coefficient matrices for EACH of the
000147 | four frames and BOTH quarters, via
000148 |  L_q=diag(U_q,U_q tensor I4), J_(q),r=L_q*J_r L_q.
000149 | Indices are (i,a). No independently normalized erasure, reset, executed
000150 | projection or missing cross-block is introduced. These 25 fields are not
000151 | claimed to be the entire E8 algebra; their arbitrary products are covered
000152 | by the next generating formula.
000153 | Here L_q is a PASSIVE quartic-frame change with the derivative/Cartan basis
000154 | held fixed. An ACTIVE native G transport must additionally carry the
000155 | contragredient bar4 erasure action, as supplied by the full Y_d transport
000156 | below. L_q alone is not a physical transversal operation.
000157 |