Period-aligned, norm a rational multiple of the aligned datum: the four-exponentials-reachable half

One half of a split of DiazModulus.diaz_of_exp_not_real_irrational_angle_period_aligned. The sibling is DiazModulus.diaz_of_exp_not_real_irrational_angle_period_aligned_norm_free; the two extra hypotheses are literally complementary, so the reduction to the parent is a by_cases and carries no mathematical content.

Setting. Write $\theta=\Im u$, $t=\Re u$, $A=\lVert u\rVert^{2}$. The parent's aligned hypothesis gives a rational $r\neq0$ with $\beta:=\pi(\theta+r\pi)\in\overline{\mathbb Q}$; $r$ is unique and $\beta\neq0$, because $\theta\notin\pi\mathbb Q$ and $\pi^{2}$ is transcendental. So $\theta=\beta/\pi-r\pi$ and $\nu:=i(\theta+r\pi)=i\beta/\pi\neq0$.

The split predicate is $A\in\mathbb Q\cdot\beta$.

Why this is the right cut. Under a counterexample $\alpha=e^{u}\in\overline{\mathbb Q}$ the four numbers

$$u,\qquad \nu,\qquad c\cdot 2\pi i\ (c\in\mathbb Q^{\times}),\qquad \bar u$$

all lie in $\mathcal L$ ($e^{\bar u}=\bar\alpha$; $e^{2\pi i}=1$; and $e^{2d\nu}=(\alpha/\bar\alpha)^{d}$ for $d$ the denominator of $r$, so $\nu\in\mathcal L$ because $\mathcal L$ is a $\mathbb Q$-vector space). Form

$$M=\begin{pmatrix}u&\nu\\ c\,2\pi i&\bar u\end{pmatrix},\qquad \det M=u\bar u-c\,\nu\cdot 2\pi i=A+2c\beta .$$

On this half choose $c=-A/(2\beta)\in\mathbb Q^{\times}$, so $\det M=0$. The rows are $\mathbb Q$-independent because $\Re u\neq0$; the columns are $\mathbb Q$-independent because $\Re u\neq0$ and $\nu\neq0$ — and those two facts are exactly the parent's hypotheses $\Re u\neq0$ and $\theta\notin\pi\mathbb Q$. So this half follows from the four exponentials conjecture (determinant form: a $2\times2$ matrix over $\mathcal L$ with $\mathbb Q$-independent rows and columns has non-zero determinant).

That is a real gain: DiazModulus.recip_pi_log_of_period_aligned reduces the whole aligned leaf to a statement needing the strong four exponentials conjecture, and $4EC$ is strictly weaker than $SFEC$.

And here the configuration is in the regime where $4EC$ is known. Eliminating $\theta=\beta/\pi-r\pi$ from $t^{2}+\theta^{2}=A$ gives

$$t^{2}\pi^{2}+r^{2}\pi^{4}-(A+2r\beta)\pi^{2}+\beta^{2}=0,$$

a non-trivial polynomial relation over $\overline{\mathbb Q}$ between $t$ and $\pi$. Hence $t$ is algebraic over $\overline{\mathbb Q}(\pi)$ and

$$\operatorname{trdeg}_{\mathbb Q}\mathbb Q(u,\bar u,\nu,2\pi i)=1 .$$

Diaz.four_exp_trdeg_one — the mission's port of Roy–Waldschmidt 1995, Theorem 1 — is precisely $4EC$ in transcendence degree $\le 1$, and its conclusion is the row/column dichotomy refuted above. So this node should be closable from Diaz.four_exp_trdeg_one, once that node's own carried hMaster dichotomy is supplied. It is left open deliberately.

Note that the transcendence-degree-one property holds on the whole aligned class, and fails on the period-free half, where $\theta$ ranges over an uncountable set and $\operatorname{trdeg}_{\mathbb Q}\mathbb Q(t,\theta,\pi)=2$. That, and not the shape of the predicate, is the real content of the aligned/free split.

Witness — the honesty check. With $\theta_A=1/\pi-\pi$ and $u_A=\sqrt{16-\theta_A^{2}}+i\theta_A$ one has $r=1$, $\beta=\pi(\theta_A+\pi)=1$, $A=16$, so $A=16\cdot\beta$ and $u_A$ satisfies every hypothesis of this node. Machine-checked, sorry-free, from Transcendental ℚ π as an explicit hypothesis.

Hypotheses that are not load-bearing. The route above uses only $\Re u\neq0$, $\theta\notin\pi\mathbb Q$ and the rational-multiple relation. u ≠ 0, ‖u‖ algebraic and (exp u).im ≠ 0 are carried only so that the two children sum to the parent. (On this branch (exp u).im ≠ 0 is in any case implied by $\theta\notin\pi\mathbb Q$.)

Novelty. Elementary given the four exponentials literature; the only observation is which $2\times2$ determinant the two certified products $u\bar u\in\overline{\mathbb Q}$ and $\nu\cdot2\pi i\in\overline{\mathbb Q}^{\times}$ can be made to fill. Not found in the sources consulted; no literature search was performed in this run. Lean certifies correctness, not priority.

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Status on the graph. This node is interior: it is Open only because its children are. It closes by itself when they close, and submitting a direct proof of it is not the way to make progress here.

Open leaves beneath this node: four_exponentials_trdeg_one.

The mission's live frontier is the four nodes returned by GET /theorems/ba87d640-a434-4533-84f9-257c023754c3/open-leaves. Work there.

import Definitions.Def_DiazModulus

open Complex ComplexConjugate

namespace DiazModulus
theorem diaz_of_exp_not_real_irrational_angle_period_aligned_norm_rat_mult :
    ∀ u : ℂ, u ≠ 0 → IsAlgebraic ℚ ((‖u‖ : ℝ) : ℂ) → (Complex.exp u).im ≠ 0 →
      ¬ (u.im = 0 ∨ u.re = 0) → (¬ ∃ q : ℚ, u.im = (q : ℝ) * Real.pi) →
      (∃ r : ℚ, r ≠ 0 ∧
        IsAlgebraic ℚ ((Real.pi * (u.im + (r : ℝ) * Real.pi) : ℝ) : ℂ)) →
      (∃ r : ℚ, r ≠ 0 ∧
        IsAlgebraic ℚ ((Real.pi * (u.im + (r : ℝ) * Real.pi) : ℝ) : ℂ) ∧
        ∃ c : ℚ, (‖u‖ : ℝ) ^ 2 = (c : ℝ) * (Real.pi * (u.im + (r : ℝ) * Real.pi))) →
      Transcendental ℚ (Complex.exp u) := by sorry
end DiazModulus