Original ledger status
QGEO.LANDSCAPE.01
QGEO.LANDSCAPE.01
The curve landscape round -- what else the mu3-cover carries (one module, 9 checks, ~1 s; discovery probe curve_landscape_probe.py 9/9, verdict LANDSCAPE-MAPPED). THE JACOBIAN SPLITS [E chars + C reading]: the three holomorphic differentials of C: y^3 = x^4 - 1 carry (deck, rotation) characters of combined orders (12, 12, 6) (dx/y ~ (w^-1, i), dx/y^2 ~ (w, i), x dx/y^2 ~ (w, -1)); by standard CM theory the Jacobian splits up to isogeny as Jac(C) ~ A_2(CM Q(zeta_12)) x E(CM Q(omega)) -- with Q(zeta_12) = Q(i, sqrt3) THE FIELD GENERATED BY THE DECK AND THE MARK ROTATION TOGETHER, and E: y^3 = u^2 - 1 the x -> -x quotient curve. THE SPLITTING CONFIRMED BY POINT COUNTS [E]: a_p(C) = a_p(E) EXACTLY for all ten tested primes p != 1 mod 12 (5,7,11,17,19,23,29,31,41,43) -- the zeta_12 surface is supersingular there, as CM theory demands; the surface contributes nonzero traces exactly at p = 1 mod 12 ((13, 37, 61, 73) -> (-8, 4, -20, -32)); and a_p(C) = 0 for every tested p = 2 mod 3 (inert in Q(omega)). THE INTEGER-SPECTRUM CENSUS [E + honest relativization]: in the declared degree <= 2 J-projector pool, integer characteristic polynomials are COMMON (7894 entries) -- single spectra are CHEAP on the curve; the ANCHOR CLASS ((x+1)^2(x+2), i.e. {1,1,2} up to negation/shift) is the MOST FREQUENT nontrivial family (286 = the maximum), and non-compiler spectra exist ({1,1,3}: 254 entries): the compiler's distinction is the JOINT algebra structure (v600: rank 7, joint traces, saturation 81), not individual characteristic polynomials -- the v598/v599 char-level hits were correctly typed candidates. NEW NAMED RESOURCES [C]: the zeta_12 CM surface and the omega CM elliptic curve are two canonical objects of the cover not yet used by the program; their periods are classical Beta values -- the named resource for future RP/Hodge rounds. GATE.QGEO does not move; no marker changes. [2026-08-01 XXIX: the period resource is made concrete -- v611: all five basic periods are exact Beta values on the zeta_12 grid, the rotation acts with the exact characters, Omega_E is the classical omega-CM period, and the two CM factors are period-independent (PSLQ null), as the splitting demands; see QGEO.PERIODS.01.]
Recorded status: Exact [E] characters/point-counts/census + [C] CM-theory reading and named resources (deterministic integer computations; no floats beyond the census prefilter)
Dependencies, status detail and source
Canonical status: LANDSCAPE-MAPPED: differential characters (12,12,6); Jac ~ A_2(zeta_12) x E(omega) point-count-confirmed (10 exact zero-differences; surface traces only at p = 1 mod 12); anchor class most frequent nontrivial integer family (286); single spectra cheap, joint structure the distinction; periods named; 9/9 checks
Dependencies: QGEO.COVER.01; QGEO.DICT.01; QGEO.EMBED.01; v597; v598; v600
Scripts: v610_curve_landscape.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 628