TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/D4-RIGIDITY-INDEPENDENT-REVIEW.txt
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Inclusive original lines: 181-244
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000181 | 
000182 |   * X=P1 minus mu4 and its faithful geometric D4 action generated by
000183 |     x -> i x and x -> 1/x;
000184 |   * F=H1(X,C) and its Hodge metric;
000185 |   * a separate flat Hermitian family bundle with determinant trivial
000186 |     connection, hence SU(3) holonomy.
000187 | 
000188 | It does not state that this chosen flat connection descends equivariantly
000189 | through X/D4, nor does it give the reflection linearization b on its fiber.
000190 | The theorem about the D4 invariant puncture configuration is a statement
000191 | about the base. The SU(3) holonomy proposition is a statement about the
000192 | connection. A source level isomorphism joining those two statements is still
000193 | needed.
000194 | 
000195 | The distinction is concrete. verification/v146_moebius_d4.py computes the
000196 | natural D4 action on the cohomology basis. After determinant normalization,
000197 | its rotation has the spectrum (2), but its natural reflection b_nat gives
000198 | 
000199 |   [(b_nat c)^(-1)]^2=I,
000200 | 
000201 | not the regular cubic M. Therefore one cannot silently use the natural H1
000202 | reflection as the b in this rigidity theorem. The desired b is a
000203 | linearization of the nontrivial flat family local system, not merely the
000204 | ordinary action on the underlying cohomology vector space.
000205 | 
000206 | verification/v117_monodromy_weyl_a3.py supplies the exact pair U_v117 and
000207 | M0_v117 and proves its irreducibility and S4 image. Defining b from that
000208 | already known pair proves existence retrospectively, but would not derive the
000209 | pair from the earlier source principles. The new positive derivation becomes
000210 | source complete exactly when the original construction supplies:
000211 | 
000212 |   1. a D4 equivariant structure on the same selected flat Hermitian family
000213 |      connection;
000214 |   2. an honest determinant one linearization with b^2=I and c equal to the
000215 |      marked operator (2), rather than only a projective action;
000216 |   3. the already stated regular cubic peripheral class and stable
000217 |      irreducibility.
000218 | 
000219 | Under these three premises, the calculation above proves that no remaining
000220 | continuous or discrete choice survives and the v117 pair follows.
000221 | 
000222 | 6. Status
000223 | =========
000224 | 
000225 | PROVED IN THE DECLARED CLASS:
000226 | 
000227 |   * exact classification of all determinant one square root branches;
000228 |   * elimination of the trace two branch by positivity of p_1;
000229 |   * elimination of the two complex trace branches by reducibility;
000230 |   * a unique trace zero branch with p=(1/2,1/4,1/4) and a^3=I;
000231 |   * marked unitary equivalence of that branch to the exact v117 pair.
000232 | 
000233 | CONDITIONAL AT THE ORIGINAL SOURCE LEVEL:
000234 | 
000235 |   * existence of the required equivariant linearization of the selected flat
000236 |     family connection over the genuine D4 quotient.
000237 | 
000238 | NOT CLAIMED:
000239 | 
000240 |   * Q equals the modular S path;
000241 |   * base D4 symmetry alone selects a fiber local system;
000242 |   * the natural cohomology reflection already gives the v117 peripheral;
000243 |   * a physical P1 state, time, or observable map follows from this finite
000244 |     monodromy rigidity.