TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/D4-CONNECTION-LIFT.txt
Original file SHA-256: d3647a78636a3d9d465c52ff17ed578b4961a097a7256a04638c6e4b2c2ba6a4
Inclusive original lines: 181-259
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000181 | rank nine on the nine variables a,b,c,d,e,f,g,h,j. The proof uses no floating
000182 | point monodromy, fitted fundamental matrix or assumed group order.
000183 | 
000184 | Since even a nonzero solution of (1) is absent, allowing
000185 | 
000186 |   G(z)G(1/z)=xi I
000187 | 
000188 | with a constant scalar xi cannot help. Nor can another reflection Q^m s:
000189 | the exact rotation lift (4) would compose any such lift with U^(-m) to give a
000190 | lift of s, contradicting (12).
000191 | 
000192 | 5. Consequence for the proposed finite bridge
000193 | ==============================================
000194 | 
000195 | The result separates three statements that were previously close in the
000196 | numerics.
000197 | 
000198 | PROVED:
000199 | 
000200 |   * the residue system (3) is exactly C4 equivariant;
000201 |   * infinity is apparent in the v116 formal resonance sense when a13=0;
000202 |   * the specific connection has no rational inversion lift preserving its
000203 |     logarithmic extension;
000204 |   * it therefore does not descend as the genuine D4 connection required by
000205 |     the existing orbifold rigidity theorem.
000206 | 
000207 | EXACT POSITIVE CONTINUATION:
000208 | 
000209 |   * the independently written matrix M0_v117 has finite order three;
000210 |   * <U,M0_v117> is exactly the 24-element S4 representation.
000211 |   * the C4 descent t=z^4 has local orders (4,3,2), hence faithful S4
000212 |     orbifold monodromy;
000213 |   * its marked order-four class is U^(-1), and the unique marked Nielsen
000214 |     class identifies it with (U^(-1),M0_v117) up to constant conjugacy;
000215 |   * the ordinary four-puncture image is A4 of order 12.
000216 | 
000217 | A Belyi pullback through the same marked D4 quotient would carry the deck
000218 | linearization back to a rational solution of (1), so that specific direct
000219 | D4/Belyi route is excluded by (12). An unrelated finite cover or a changed
000220 | connection could evade the obstruction, but it would add precisely the new
000221 | geometric or operator input that this test was asked not to choose.
000222 | 
000223 | The successful C4 proof adds neither: it uses the original covariance (4)
000224 | and the original quotient t=z^4.  Its normalized finite target coordinate is
000225 | J_actual(z)=1-z^(-4), which is C4 invariant and not D4 invariant.  This is
000226 | why the positive C4 result and the negative inversion result coexist.
000227 | 
000228 | The exact lesson is also narrower than “a13=0 fails”. The zero a13 entry
000229 | does its intended job at infinity. It does not imply reflection covariance.
000230 | The remaining nonzero couplings sqrt(2)/6 and sqrt(5)/12 make the inversion
000231 | equations full rank.
000232 | 
000233 | 6. Reproduction and source record
000234 | ==================================
000235 | 
000236 | The exact checks performed were:
000237 | 
000238 |   1. symbolic summation of the four residues to (3);
000239 |   2. symbolic verification of (4) and (5);
000240 |   3. symbolic substitution of (11) into (1), giving rank nine;
000241 |   4. normal and Python -OO replay of v117, both 5 PASS and 0 FAIL.
000242 | 
000243 | Canonical sources read through the Atlas and hash-verified locally:
000244 | 
000245 |   source:80af15193155b4dcf60c
000246 |   verification/v116_resonance_uniqueness.py:1-170
000247 |   SHA256 e9cbeb665cf8c1a2bed3f34281be06aee4c661d784e7f13a5bd8e9be37f577ca.
000248 | 
000249 |   source:830238c9699fad6ae325
000250 |   verification/v117_monodromy_weyl_a3.py:1-206
000251 |   SHA256 55eca6a64144f6e49789f3f1bbe84e256764e7ce92b207044be693f8684d5df5.
000252 | 
000253 | Prior conditional theorem and review reused without modification:
000254 | 
000255 |   research-20261026-real-boundary/D4-ORBIFOLD-RIGIDITY.txt
000256 |   SHA256 2f7ec5e5a3e7a39edc3aa232a47a97f9f3a2b8cf289f67ebfd4be71975bdaf33.
000257 | 
000258 |   research-20261026-real-boundary/D4-RIGIDITY-INDEPENDENT-REVIEW.txt
000259 |   SHA256 97dbe63dfb00bd6ebecd649e48fca2097709686428acb21d4e53fb73f49cfffd.