TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SIGN-LINE-COVER-DICTIONARY.txt
Original file SHA-256: 555f75a2d3c67347fa5f8e3c4f7f46e79d94cb4172176c84010224585dc842e7
Inclusive original lines: 181-276
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 | 
000182 | It is another spherical triangle cover, but the order-four and order-two
000183 | fibers are exchanged.  The original ODE fixes order four at t=0 and order two
000184 | at t=infinity, so this packet has the wrong marked clock.
000185 | 
000186 | The previously proposed antipodal packet
000187 | 
000188 |     L_a^anti=O_C(p_a-q_a),
000189 |     u_a^2=(w-omega^a)/(w+omega^a)                                 (20)
000190 | 
000191 | is mathematically valid and unramified over C, but its reduced V4 cover of the
000192 | w-sphere branches at all six p_a and q_a.  Over t this gives
000193 | 
000194 |     (4,3,4).                                                       (21)
000195 | 
000196 | Riemann--Hurwitz at degree 24 gives genus three.  It is not the spherical
000197 | (4,3,2) cover of the v117 ODE.  Matching its local A4 character dimensions or
000198 | choosing a favorable lift of the quarter map cannot repair this global
000199 | passport mismatch.
000200 | 
000201 | This comparison is the smallest decisive selection test among the three
000202 | natural C3-stable two-torsion packets.
000203 | 
000204 | 
000205 | 6. COMPARISON WITH THE OLD MU_12 QUARTER COVER
000206 | 
000207 | Let
000208 | 
000209 |     f(z)=(z^2-1)/(z^2+1),
000210 |     D: H^12=f(z),
000211 |     g=H^3,
000212 |     w=H^4.                                                        (22)
000213 | 
000214 | Then D maps to C and to the old elliptic square curve:
000215 | 
000216 |     D -> C,   (z,H) |-> (w=H^4, eta=z(1-w^3)),
000217 |     D -> E,   (z,H) |-> (z, y=H^6(z^2+1)),
000218 |     E: y^2=z^4-1.                                                 (23)
000219 | 
000220 | This cyclic root fiber product does not contain the sign lines (1).  Over C it
000221 | is H^4=w.  The divisor of w has two simple zeros and two simple poles, so the
000222 | relative C4 cover is totally ramified at all four points.  Its unique quadratic
000223 | intermediate is also ramified there.  Every L_a^q is unramified.  Hence none of
000224 | the three selected lines can be that cyclic quadratic intermediate.
000225 | 
000226 | The quarter lift
000227 | 
000228 |     (z,H) |-> (i z,i/H)                                           (24)
000229 | 
000230 | still proves w->w^(-1) and g->-i g^(-1).  It transports the old unordered
000231 | quarter pair and the genus-two geometry.  It does not itself supply the V4
000232 | coefficient packet or a charged-field intertwiner.
000233 | 
000234 | 
000235 | 7. CONSEQUENCE AND REMAINING BOUNDARY
000236 | 
000237 | The old statement that the genus-two curve needs an additional sign line
000238 | system is now closed at coefficient level.  The exact system is the negative
000239 | orbit packet (1), selected by the original order-four fiber at t=0.  Its
000240 | fiber product is literally the A4 kernel of the already proved S4 triangle
000241 | source.
000242 | 
000243 | The remaining source problem is narrower: lift this exact coefficient
000244 | identification to the original charged fields, state, local support, and full
000245 | response functional.  The point hexagon and the mu_12 cover do not perform
000246 | that lift automatically.
000247 | 
000248 | 
000249 | 8. REPRODUCTION
000250 | 
000251 | [OMITTED: non-scientific conversation or execution metadata]
000252 | 
000253 |     python3 -B -OO research-20261026-real-boundary/check_sign_line_cover.py
000254 | 
000255 | Expected verdict:
000256 | 
000257 |     PASS_EXACT_Q_PACKET_V117_COEFFICIENT_LIFT
000258 | 
000259 | The script writes:
000260 | 
000261 |     research-20261026-real-boundary/sign-line-cover-results.json
000262 | 
000263 | It checks the Kummer product relations, cyclic transport, rationality of Z_q,
000264 | genus five of Y_q, the quotient coordinate t, all three competing passports,
000265 | their Riemann--Hurwitz genera, the oriented v117 matrix orders, and the
000266 | ramification obstruction for the old mu_12 cover.
000267 | 
000268 | 
000269 | SOURCE FILES
000270 | 
000271 | [OMITTED: non-scientific conversation or execution metadata]
000272 | [OMITTED: non-scientific conversation or execution metadata]
000273 | [OMITTED: non-scientific conversation or execution metadata]
000274 | [OMITTED: non-scientific conversation or execution metadata]
000275 | [OMITTED: non-scientific conversation or execution metadata]
000276 | [OMITTED: non-scientific conversation or execution metadata]