feat: Algebraic setup and matrix coefficients for Gelfond-Schneider theorem

[mkaratarakis]

This PR introduces the foundational algebraic setup and coefficient bounds required for the proof of the Gelfond-Schneider Theorem (Hilbert's Seventh Problem), which establishes that for algebraic numbers $\alpha \neq 0, 1$ and irrational algebraic $\beta$, the number $\alpha^\beta$ is transcendental.

Following the contradiction argument presented in Loo-Keng Hua's "Introduction to Number Theory" (Chapter 17.9, 490 -493) Gelfond's Proof), this file constructs the common number field $K$ and sets up the scaled algebraic integers for the auxiliary linear system. This system will later be solved via Siegel's lemma (Lemma 8.2, 490, Hua).

This PR is essentially the first half of page 490 in the book.

The proof proceeds by assuming $\gamma = \alpha^\beta$ is algebraic, meaning $\alpha$, $\beta$, and $\gamma$ all lie in an algebraic field $K$ of degree $h$.

1. Bundles the common number field $K$, the complex embeddings, and the algebraic preimages $\alpha', \beta', \gamma'$, alongside the core hypotheses of the theorem.

2. Defines the field degree $h = [K : \mathbb{Q}]$, and bounds $m = 2h + 2$ and $n = q^2 / (2m)$ to control the dimensions of the linear system.

3. Defines $c_1$, a natural number chosen so that $c_1 \alpha$, $c_1 \beta$, and $c_1 \gamma$ are algebraic integers in $K$. We prove numerous bounding and integrality lemmas (isIntegral_c₁α, isInt_β_bound, etc.).

4. Formalizes the core algebraic coefficients $(a+b\beta)^k \alpha^{al} \gamma^{bl}$ which appear in the evaluation of the derivatives of the auxiliary function $R(x)$.

5. By scaling the system by $c_1^{n-1+2mq}$, we successfully restrict the entries of our linear system matrix A entirely to the ring of integers $\mathcal{O}_K$, preparing it for the application of Siegel's lemma.

6. House Bounds (c₂) : Establishes the foundational base integer $c_2$ to bound the absolute values of the conjugates (houses) of our coefficients.

---

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PR summary 486b45d5d1

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.NumberTheory.Transcendental.GelfondSchneider.MainAlg (new file)	2917

---

Declarations diff

+ A
+ IsIntegral.Cast
+ IsIntegral.Nat
+ IsIntegral_assoc
+ Setup
+ a
+ alpha'_beta'_gamma'_ne_zero
+ alpha'_ne_one
+ alpha_gamma_pow_beta_ne_zero
+ b
+ b_sum_ne_zero
+ beta'_ne_zero
+ beta_ne_zero
+ bound_c₁β
+ c_coeffs
+ c_coeffs0
+ c_coeffs_neq_zero
+ c₀
+ c₀Coeff
+ c₀Coeff_ne_zero
+ c₁
+ c₁IsInt
+ c₁_ne_zero
+ c₁α_ne_zero
+ c₁γ_ne_zero
+ c₂
+ exists_common_field_of_isAlgebraic
+ exists_int_smul_isIntegral
+ h
+ house_bound_c₁α
+ isInt_β_bound
+ isInt_β_bound_low
+ isIntegral_c₁_pow_smul_add_smul_pow
+ isIntegral_c₁_pow_smul_pow
+ isIntegral_c₁_pow_smul_α'_pow
+ isIntegral_c₁_pow_smul_α'_pow'
+ isIntegral_c₁_pow_smul_γ'_pow
+ isIntegral_c₁_pow_smul_γ'_pow'
+ isIntegral_c₁α
+ isIntegral_c₁β
+ isIntegral_c₁γ
+ isNumberField_adjoin_of_isAlgebraic
+ k
+ l
+ log_α_ne_zero
+ m
+ n
+ one_le_abs_c₁
+ one_le_c₁
+ one_le_c₂
+ one_le_house_c₁γ
+ one_le_m
+ one_le_n
+ systemCoeffs
+ zsmul_mul_mul_distrib
+ β'_ne_zero
+ ρ

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for scripts/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).