TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/operator_audit.md
Original file SHA-256: f1849345563b8d81cd3861a5a5eaa96f1355ff7d5d55bdbc00e73ab90b7b64e5
Inclusive original lines: 190-220
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000190 | 1&1-2i&-1&i\\
000191 | 1&-1&-i&-1+2i\\
000192 | -1+2i&-i&-1&1\\
000193 | 1&i&-2-i&-i
000194 | \end{pmatrix}.
000195 | \]
000196 | 
000197 | So the proposed central-phase obstruction is false: honest `F20=C5 semidirect C4` subgroups exist in the native carrier, and every surviving copy is actually equivalent to the standalone cyclotomic carrier.
000198 | 
000199 | The six are also **not six inequivalent physical choices**.  Let
000200 | 
000201 | \[
000202 | H_0=S_{\{0,1\}}\times C_{3,(234)},\qquad
000203 | H=S_{\{0,1\}}\times S_{\{2,3,4\}}.
000204 | \]
000205 | 
000206 | On the **logical `V5` carrier**, both preserve the existing point/weak/color marks and `sigma5`; `H` also centralizes the marked `R`.  Exact conjugation on the six Sylow-five subgroups shows that `H0` acts regularly (orbit six, stabilizer one), while `H` acts transitively (orbit six, stabilizer two).  Therefore the logical marks determine one `F20` **orbit/equivalence class**.  No additional embedding has been chosen.  A literal subgroup and a normalized intertwiner matrix are gauge representatives; the remaining interpretive question is whether the marked stabilizer is quotiented as redundancy or later broken by further physical data.  This quotient-orbit result does not by itself prove that the conjugating lifts commute with the native `V4` representative of `sigma` or with a P1 physical source clock; that stronger simultaneous-lift statement was not used.
000207 | 
000208 | ### Which constraints eliminate the old `S3` sign twin?
000209 | 
000210 | For the actual `(1,21,32)` triangle, restriction gives
000211 | 
000212 | \[
000213 | V_4\downarrow S_3=2\,\mathbf1\oplus\mathbf2,
000214 | \quad V_5\downarrow S_3=3\,\mathbf1\oplus\mathbf2,
000215 | \quad V_{20}\downarrow S_3=7\,\mathbf1\oplus\mathbf{sgn}\oplus6\,\mathbf2.
000216 | \]
000217 | 
000218 | The central sign projector has ranks `(0,0,1)`, so `z=1-2e_sgn` is invisible on `V4,V5` and a nontrivial involution on `V20` (`hidden_sgn_check.py:29-63`).  The actual tests sharpen its status:
000219 | 
000220 | - **Marked `R` does not eliminate it.** `z20` commutes exactly with `R tensor I4`.