TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/INTRINSIC-EVENT-GEOMETRY-AND-COMPOSITION.txt
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000310 | 7. Clock-free event structure and precisely missing marking information
000311 | 
000312 | The six derived rays give all15 oriented differences d_pq=n_p-n_q. Let B
000313 | be the incidence matrix of the complete six-point graph and D_edge=N B.
000314 | Exactly
000315 | 
000316 |   D_edge D_edge*=6I5, spec(B*B)=6^5,0^10,
000317 |   d_pq+d_qr+d_rp=0.
000318 | 
000319 | The ten-dimensional kernel is the cycle relation space. Positive relative
000320 | normalization of all edges yields15 RANK1 EFFECTS
000321 | 
000322 |   E_pq=d_pq d_pq*/6=(1/3)P_root,pq, sum E_pq=I5.
000323 | 
000324 | They are not idempotent: E_pq²=E_pq/3. The normalized A5 root projectors
000325 | sum to3I5. All identities are on the response support, not H_E8 identity.
000326 | This geometry is independent of the chosen orientation of each edge.
000327 | 
000328 | Grant one marked vertex q. Its five star responses form an invertible D_q.
000329 | Their relative normalization gives the orthonormal vectors
000330 | 
000331 |   u_qj=n_j-(sqrt6-1)n_q/5, j!=q.
000332 | 
000333 | Thus there is a canonical UNORDERED five-projector family conditional on
000334 | q. Choosing two projectors gives10 different rank2 marks. Across the six
000335 | possible q there are EXACTLY60 distinct rank2 marks; the checker constructs
000336 | all before consulting any original clock/P2. The old P2 is obtained when
000337 | the additional two-subset is the two fixed labels left by the old sigma.
000338 | The inverse orientation of that three-cycle is immaterial to this P2.
000339 | Therefore the full clock is stronger data than the marking requires:
000340 | q plus an unordered two-subset suffices.
000341 | 
000342 | This entire clock TYPE can itself be found from event relations, without
000343 | inserting the old sigma. In the actual logical S6 event quotient, of the
000344 | 225 ORDERED transposition pairs,15 equal pairs give identity,90 distinct
000345 | commuting pairs give double transpositions, and120 noncommuting pairs give
000346 | exactly40 distinct three-cycles. Each three-cycle fixes three vertices.
000347 | Retaining one of those fixed vertices q leaves three moved and two other
000348 | fixed labels. The120 oriented (three-cycle,q) choices produce precisely
000349 | the SAME60 polar marks (cycle reversal gives the same mark). Thus the
000350 | original logical clock CLASS and polar-mark ORBIT have a common event-
000351 | relation origin, without selecting a member or assuming3+2 as criterion.
000352 | This is a statement about the actual five-dimensional S6 quotient; native
000353 | four-dimensional lifts/central phases and full source clock/erasure action
000354 | are not fixed by the quotient word alone. No rule choosing noncommuting
000355 | rather than commuting event histories is imposed or physically derived.
000356 | 
000357 | A single canonical source-covariant projector must be fixed by every
000358 | automorphism preserving its input data. The unmarked native S6 action on
000359 | L is irreducible, hence invariant projectors have rank0 or5. With q
000360 | marked, its S5 stabilizer gives L=1+4, hence ranks0,1,4,5 only. There is
000361 | no invariant rank2 projector. This argument applies also to a nonlinear
000362 | UNIQUE covariant selector from these symmetric data: its output still
000363 | must be invariant. It does not prohibit an equivariant orbit of candidates,
000364 | an additional distinguished anisotropic source field/state, spontaneous
000365 | branching, or a P1 seam datum selecting a clock. None is silently imposed.
000366 |