TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/e-series-quarter-clock-20261023/PROOF.txt
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Inclusive original lines: 177-230
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000177 | 8. Intrinsic square curves, covers and the precise descent theorem
000178 | 
000179 | Each Gaussian root line ell=C alpha has exactly four roots. The function
000180 |   f_ell(v)=(v/alpha)^4-1
000181 | is independent of the root representative alpha, since its replacements
000182 | are i^k alpha. Thus ell with its point at infinity carries an intrinsic
000183 | square four-mark set and the cover w^2=f_ell(v). This construction is
000184 | functorial for actual lattice isometries commuting with J. It gives60
000185 | embedded charts of the SAME marked modulus, not60 different moduli.
000186 | The60 original reflections act transitively on these lines, verified
000187 | from their actual permutations. Therefore the unmarked(E8,J) data do
000188 | not equivariantly select one chart. Retaining the full family is valid
000189 | and needs no added unique-chart axiom.
000190 | 
000191 | Now separately SUPPLY the existing base model X=CP1 minus mu4 and its
000192 | rho(gamma_i)=J representation. Its C2 parity character has a connected
000193 | degree2 cover Y:y^2=x^4-1, with the four branch points removed. The
000194 | compactification has genus1 by Riemann-Hurwitz and j1728 from harmonic
000195 | cross ratio. Its restricted fibre holonomy is{I,-I}. Every CHOSEN real
000196 | E7 or E6 root subsystem is invariant under that remaining holonomy.
000197 | On X the chosen E7 descends as the unordered pair of exchanged slices;
000198 | Y orders the pair, and the common D6 already descends individually.
000199 | This repairs any inference that lack of a fixed J-sector forbids a
000200 | transported reduction. It does NOT make real rank7 into a complex subspace.
000201 | 
000202 | The removed points matter. A peripheral loop upstairs projects to
000203 | gamma_i^2 and has fibre holonomy-I; this flat bundle does not extend
000204 | across them as an ordinary flat bundle on the compact elliptic curve.
000205 | The full kernel cover z^4=x^4-1 has compact genus3 and trivial restricted
000206 | holonomy. For w^n=x^4-1,
000207 |   g=((n-1)*3-gcd(n,4)+1)/2,
000208 | so n=2,3,4 gives1,3,3. Infinity ramifies for n=3, not n=2 or4.
000209 | The existing TFPT w^3=x^4-1 cover explicitly retains infinity as an
000210 | extra branch point in v617 and its original TeX account. It is not the
000211 | degree4 kernel cover just because both have genus3.
000212 | 
000213 | The algebraic deck is d(x,y)=(x,-y). The two lifts of base x->ix are
000214 | R_pm(x,y)=(ix,+/-y), with square(-x,y), not d. Likewise, an odd path
000215 | lift representing fibre J must not be equated to constant deck action:
000216 | canonical deck equivariance on a pullback bundle squares to identity,
000217 | whereas J^2=-I is compensated by the restricted loop holonomy.
000218 | 
000219 | The parity cover alone admits eight C4 monodromy lifts on the ordered
000220 | four-punctured base: gamma_i->J^(1+2b_i), with sum b_i even. Requiring
000221 | cyclic invariance of the four peripheral exponents leaves the all-J
000222 | and all-J^-1 choices. Thus the cover does not by itself force the given
000223 | quarter orientation. A loop enclosing all four finite marks has trivial
000224 | monodromy in both the parity and full given C4 representation. Under the
000225 | uniform mu3 monodromy its image is instead omega^4=omega. Any claim that
000226 | these describe the SAME original closed seam must specify and intertwine
000227 | its actual loop and peripheral markings. A cover degree or genus alone
000228 | cannot provide that identification.
000229 | 
000230 | 9. E7 as a discriminator for normalization prescriptions