Your Task: Your goal is to convert the following high-quality proof sketches for a theorem and its corollary into a single, formal, rigorous proof suitable for publication in a peer-reviewed mathematics journal. You must fill in all logical gaps, define all terms precisely, and pay special attention to the justification of non-trivial claims.
Provided Conjecture and Proof Sketches
(This is the verified material you are to formalize)
Theorem (Main Conjecture): Let V_n be a sequence of finite-dimensional complex vector spaces with m_n = \dim(V_n) \to \infty. Let X_n \subset V_n be a sequence of closed algebraic subvarieties satisfying three conditions for fixed integers N, d > 0 and a fixed real \epsilon > 0:
 * X_n has at most N irreducible components.
 * Each component is parametrizable by polynomials of degree at most d.
 * X_n is \epsilon-dense in the unit ball B_n \subset V_n.
If such a sequence of varieties exists, and k_n is the dimension of X_n, then it must be the case that:
\limsup_{n\to\infty} \frac{k_n}{m_n} = 1
Proof Sketch for the Theorem:
 * Lower Bound on Covering Number: The minimum number of \epsilon-balls needed to cover the unit ball B_n in an m_n-dimensional space, denoted M(B_n, \epsilon), is bounded below by M(B_n, \epsilon) \ge (C/\epsilon)^{m_n} for a universal constant C.
 * Upper Bound on Covering Number: The \epsilon-tube around the variety, T_n, can be covered by a number of \epsilon-balls, M(T_n, \epsilon), bounded above by M(T_n, \epsilon) \le N \cdot (C_1(d)/\epsilon)^{k_n}. This relies on the Lipschitz continuity of the degree-d polynomial parametrizations.
 * Comparison: The \epsilon-density condition implies B_n \subseteq T_n, so M(B_n, \epsilon) \le M(T_n, \epsilon). Comparing the bounds forces the codimension c_n = m_n - k_n to be bounded above by a constant C_{\text{max}}(N, d, \epsilon) that is independent of n.
 * Limit Conclusion: Since the codimension is bounded and m_n \to \infty, it follows that \lim_{n\to\infty} k_n/m_n = 1.
Corollary (Weaker Conjecture) and its Proof Sketch
Corollary: Under the same hypotheses as the Theorem, there exists a constant C > 0 such that \limsup_{n\to\infty} \frac{k_n}{m_n} \ge C.
Proof Sketch for the Corollary:
 * The main theorem establishes that \lim_{n\to\infty} \frac{k_n}{m_n} = 1.
 * By the definition of a limit, this means for any \delta > 0, the sequence k_n/m_n is eventually greater than 1 - \delta.
 * Choosing \delta = 0.5, we see the sequence is eventually greater than 0.5. Thus, we can take C=0.5.
 * Since the sequence is eventually bounded below by a positive constant, its limsup is also bounded below by that constant.
Instructions for Formalization
Your formal proof should adhere to the following requirements:
 * Structure: Present the argument formally with "Theorem," "Proof," and "Corollary" environments. Define all terms, such as the covering number (metric entropy), norms, and properties of algebraic varieties, with mathematical precision.
 * Address the "Weakest Link": Your primary task is to provide a rigorous justification for the critical assumption identified in Step 2 of the main proof sketch. Formally prove or cite a theorem to establish that the Lipschitz constant L of the polynomial parametrizations (which determines the constant C_1(d)) can be bounded uniformly, depending on the degree d but independent of the ambient dimension m_n. You should explicitly reference and explain the role of the Yomdin-Gromov theorem (or related results on metric entropy of algebraic sets) in providing this justification.
 * Manage Constants: Be explicit about what each constant (C, C_1(d), etc.) depends on throughout the proof.
 * Clarity and Detail: Unpack the logic in Step 3 of the main proof. Show the algebraic manipulations that lead from the inequality of covering numbers to the conclusion that the codimension is uniformly bounded.
 * Formalize the Corollary: After the proof of the main theorem, formally state the weaker conjecture as a "Corollary" and provide its short, rigorous proof based on the sketch provided.
 * Concluding Remarks: End the proof of the main theorem with a concluding paragraph that summarizes the result and its significance regarding the limits of low-complexity algebraic approximation in high dimensions.