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Original ledger classification: Exact [E] characters/point-counts/census + [C] CM-theory reading and named resources (deterministic integer computations; no floats beyond the census prefilter)
Recorded statement: 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…
Statement, assumptions and sources
Paper context: TFPT — Introduction and reading guide
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QGEO.LANDSCAPE.01
Original ledger classification: Exact [E] characters/point-counts/census + [C] CM-theory reading and named resources (deterministic integer computations; no floats beyond the census prefilter)
Original claim text
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.]
Original classification detail
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 as recorded
QGEO.COVER.01; QGEO.DICT.01; QGEO.EMBED.01; v597; v598; v600
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- Architecture, E8 and the electromagnetic fixed point
- Standard Model and flavor structure
- E8 audit and bootstrap
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- Front matter / supportE8 audit and bootstrap
- =150 2026-08-01 (XXVIII) (the periods round lands ( v611 promoted): the analytic layer of the cover is CLASSICAL — all five basic period integrals are exact Beta values with Gamma denominators on the [formula in source] grid ( [formula in source] ); the [formula in source] rotation acts on periods with the exact characters [formula in source] ; [formula in source] exactly; and the cross-factor period ratio admits no low-degree integer relation (PSLQ null) — the two CM factors are period-independent, as the v610 splitting demands)Research changelog; TFPT — Introduction and reading guide
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
- =150 2026-08-01 (XXVII) (the curve landscape lands ( v610 promoted): the Jacobian SPLITS into a [formula in source] -CM abelian surface — the field of deck AND marks together — times the [formula in source] -CM elliptic quotient [formula in source] , point-count-confirmed ( [formula in source] exactly for all ten tested [formula in source] mod [formula in source] ); the integer-spectrum census honestly relativizes single-char hits (7894 integer entries — single spectra are cheap; the anchor class is the most frequent nontrivial family at [formula in source] ; the compiler's distinction is the joint structure); two new canonical period resources named)Research changelog; TFPT — Introduction and reading guide
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
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- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
Registered verification modules
- v610_curve_landscape.pyRegistered verification module · core
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QGEO.LANDSCAPE.01: die Kurven-Landschafts-Runde — WAS DIE KURVE SONST NOCH TRÄGT: die Jacobische ZERFÄLLT (Differential-Charaktere der Ordnungen (12,12,6)) in eine abelsche Fläche mit CM durch Q(ζ₁₂) = Q(i,√3) — den Körper von Deck UND Marks zusammen — und die elliptische Kurve E: y³ = u²−1 mit CM durch Q(ω); PUNKTZÄHLUNGS-BESTÄTIGUNG: a_p(C) = a_p(E) EXAKT für alle 10 getesteten p ≢ 1 mod 12 (die ζ₁₂-Fläche dort supersingulär), Flächen-Spuren nur bei p ≡ 1 mod 12; a_p = 0 für p ≡ 2 mod 3; GANZZAHL-SPEKTREN-CENSUS mit ehrlicher Relativierung: Einzelspektren sind BILLIG (7894 Einträge), die ANCHOR-KLASSE ist die häufigste nichttriviale Familie (286 = Maximum), Nicht-Compiler-Spektren existieren ({1,1,3}: 254) — die Compiler-Auszeichnung ist die GELENK-Struktur (v600), nicht Einzel-Chars; zwei neue kanonische Objekte benannt (Perioden = Beta-Werte) (9 checks, ~1 s)
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