TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/COMMON-COVER-COMPOSITUM.txt
Original file SHA-256: c92e9aff9717118a4b78fc4cf582dedf75921aad114a4147d08557445735ac04
Inclusive original lines: 181-292
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 | 
000183 | The map X -> Y_q is the base change of D -> C.  Since Y_q -> C is unramified
000184 | of degree four, each of the four branch points of D -> C has four preimages
000185 | on Y_q.  Thus X -> Y_q is cyclic of degree four and is totally ramified at
000186 | 16 points.  Since g(Y_q)=5, Riemann-Hurwitz gives
000187 | 
000188 |   2g(X)-2
000189 |     =4(2*5-2)+16(4-1)
000190 |     =32+48
000191 |     =80.
000192 | 
000193 | Therefore
000194 | 
000195 |   g(X)=41.                                               (10)
000196 | 
000197 | Equivalently, for X -> C of degree 16, each of the four branch points has
000198 | four points above it with inertia C4.  This gives
000199 | 
000200 |   16(2*2-2)+4*4*(4-1)=32+48=80.
000201 | 
000202 | 6. The actual quarter lift on the compositum
000203 | ==============================================
000204 | 
000205 | On C the existing geometric quarter is
000206 | 
000207 |   Q_C:(z,w,eta) |-> (i z,w^(-1),-i eta/w^3).            (11)
000208 | 
000209 | It has order four and its square is the hyperelliptic involution over w.
000210 | On D it lifts as
000211 | 
000212 |   Q_D:(z,H) |-> (i z,i/H),                              (12)
000213 | 
000214 | so that w=H^4 goes to w^(-1) and g=H^3 goes to -i/g.
000215 | 
000216 | On Y_q the accepted marked lift is represented by
000217 | 
000218 |   Q_Y^* h_a=epsilon_a omega/h_(-a-1),
000219 |   (epsilon_0,epsilon_1,epsilon_2)=(1,-1,-1),             (13)
000220 | 
000221 | up to the already recorded V4 frame change.  It restricts to (11), has order
000222 | four, and on the three coefficient frames has square (+,-,-), up to ordering.
000223 | 
000224 | Because (12) and (13) have the same restriction to C, their pair defines an
000225 | automorphism of the normalized compositum:
000226 | 
000227 |   Q_X=(Q_D,Q_Y):X -> X.                                  (14)
000228 | 
000229 | It covers z->iz.  Both components have fourth power one, so Q_X^4=1; since
000230 | its base action has exact order four, Q_X itself has exact order four.  This
000231 | is one diagonal quarter action on the common field.  Taking two independent
000232 | quarter generators would change the object and is neither required nor
000233 | canonical.
000234 | 
000235 | 7. Why this is not the joint compiler C12 clock
000236 | ================================================
000237 | 
000238 | Let delta be the deck generator of D/P1_z,
000239 | 
000240 |   delta(H)=zeta_12 H.
000241 | 
000242 | The geometric quarter (12) satisfies
000243 | 
000244 |   Q_D delta Q_D^(-1)=delta^(-1).                        (15)
000245 | 
000246 | In particular, for the cubic deck sigma=delta^4,
000247 | 
000248 |   (Q_D sigma)^2=Q_D^2,
000249 | 
000250 | and Q_D sigma still has order four.
000251 | 
000252 | The compiler cover in COVERS.txt is a different root problem,
000253 | 
000254 |   s^12=x^4-1.
000255 | 
000256 | There the plain base quarter Q_12 commutes with Delta_12, and
000257 | 
000258 |   R_12=Q_12 Delta_12^4
000259 | 
000260 | has order twelve, with R_12^4=Delta_12^4 and R_12^9=Q_12.  Equations
000261 | (15) and the commuting compiler relation are inequivalent.  The present
000262 | compositum must not turn Q_X into R_12 or claim that the new C4 factor is a
000263 | second compiler clock.
000264 | 
000265 | What does match is narrower and exact: the common C3 quotient in (6) is the
000266 | already marked w rotation, and the q coefficient packet carries its actual
000267 | C4 covariance.  A total compiler field action would require an additional
000268 | proved map between these two root problems.  No such map is introduced here.
000269 | 
000270 | 8. Scope and executable certificate
000271 | ====================================
000272 | 
000273 | The result supplies the minimal common holomorphic carrier for:
000274 | 
000275 |   * the existing g and H root coordinates;
000276 |   * the three actual v117 flat coefficient lines;
000277 |   * their already proved geometric quarter transport.
000278 | 
000279 | It does not identify charged fields, a CAR state, local support, a seam
000280 | projector, or the full compiler lattice action.  The genus 41 curve is a
000281 | normalization forced by existing fields, not a new physical worldsheet.
000282 | 
000283 | The targeted certificate is:
000284 | 
000285 |   research-20261026-real-boundary/check_common_cover_compositum.py
000286 |   research-20261026-real-boundary/common-cover-compositum-results.json
000287 | 
000288 | It checks the A4/V4 quotient, the order 48 fiber product, the exact CRT
000289 | isomorphism with A4 times C4, both degree calculations, genus 41, recovery of
000290 | H from (w,g), the inversion relation (15), and the order-four versus
000291 | order-twelve clock distinction.  Normal Python and Python -OO give byte
000292 | identical JSON.