← ACTM: Algebraic Combinatorial Transcendence Methodology / ACTM R04 · Structural classification of a hypothetical counterexample

ACTM: Algebraic Combinatorial Transcendence Methodology ACTM R04 · Structural classification of a hypothetical counterexample Neo.K / Aletheia

Structural Classification of a Hypothetical MID = 2 State: Projective Orbit Avoidance, Rank Splitting, and an Exact Three-Term Power Chain

Assuming a Four-Exponentials counterexample (a 'MID = 2' state): a, b and e^{g(a)}, e^{g(b)} (g ∈ PGL₂(Q)) are transcendental, trdeg Q(λ, a, b) ≥ 2, and the state is Type P (b a rational Möbius image of a; normal form λ, tλ, t²λ ∈ L_alg with t³λ ∉ L_alg, trdeg 2) or Type B (1, a, b, ab linearly independent over Q̄).

R04 is a structural classification of a hypothetical counterexample, entirely conditional on its existence. The boxed '2^(π³) must be transcendental' in §24 is conditional on 2^π and 2^(π²) both being algebraic; excerpted alone it reads like a new unconditional result (the classical form: at least one of 2^π, 2^(π²), 2^(π³) is transcendental). After R05 only Type P is pursued; Type B is never revisited.

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PreviousACTM R03 · The one-dimensional gap of Four Exponentials · Exponential Matrices, Multiplier Intersections, and the One-Dimensional Gap of Four Exponentials

NextACTM R08 · The binary-tower survivor · Difference-Saturation Graphs, Three-Term Progression Certificates, and the Binary-Tower Survivor

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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