← ACTM: Algebraic Combinatorial Transcendence Methodology / ACTM R31 · Joint degree of two observables by Kummer theory

ACTM: Algebraic Combinatorial Transcendence Methodology ACTM R31 · Joint degree of two observables by Kummer theory Neo.K / Aletheia

Two-Observable Kummer Orbits, Escape from Exceptional Fibers, and Quadratic Joint-Degree Growth

The interpolation slopes of the two BADT layers are algebraically independent observables; the flattening path eventually avoids exceptional higher-dimensional fibers; combined rank-4 Kummer theory gives [Q(α_Q, β_Q) : Q] ≥ c_M·Q² (for Q ≥ Q₀, a non-effective threshold) — the second layer's first rigorous arithmetic gain.

R31 is the second layer's one rigorous arithmetic gain: the joint degree of the two observables grows at least like Q². The non-effective threshold Q₀ is disclosed; it depends on an external Kummer-theory theorem (Perucca–Sgobba) that this site has not re-checked.

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This is a full-text translation of the Chinese source, which remains authoritative for its research claims. Translation is not independent mathematical verification and does not upgrade the original evidence level.

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