> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # ACTM Round 31 ## Two-Observable Kummer Orbit, Generic Fiber Escape, and Quadratic Joint-Degree Growth ### Two-Observable Kummer Orbits, Escape from Exceptional Fibers, and Quadratic Joint-Degree Growth **Research programme:** Algebraic Combinatorial Transcendence Methodology (ACTM) **Authors:** Neo.K; Aletheia (GPT-5.6 Sol) **Institution:** EveMissLab / 一言諾科技有限公司 **Date:** 2026-10-03 **Version:** Round 31 / v0.1 **Document type:** Formal research note / two-observable Kummer-fiber round **Status statement:** This round does not claim the impossibility of real BADT$_2$, nor that an individual interpolation slope has cubic algebraic degree. It proves that, for fixed interpolation size $M$, the first- and second-layer extremal interpolation slopes, considered as a pair of observables, have joint algebraic degree growing at least as $Q^2$. The core mechanism is that the combined BADT Kummer rank is $4$, the two-observable map has generic fiber dimension $2$, and the BADT rational-flattening path hits exceptional higher-dimensional fibers only finitely many times. --- ## Abstract For the first-layer interpolation slope: $$ \alpha_Q = R_M(A_Q,B_Q,C_Q) $$ Round 30 proves: $$ [\mathbb Q(\alpha_Q):\mathbb Q] \gg_M Q. $$ where: $$ A_Q=A^{1/Q}, \quad B_Q=B^{1/Q}, \quad C_Q=C^{1/Q}. $$ The reasons are: - Kummer rank $3$ supplies: $$ \Omega(Q^3) $$ twists; - a scalar rational observable has generic value-fiber dimension $2$; - Within: $$ \mu_Q^3 $$ a bounded-degree fiber has: $$ O_M(Q^2) $$ points. Round 31, for the first time, genuinely uses the second BADT layer as an additional independent observable. Let: $$ D=e^{s^4\lambda}, $$ $$ D_Q=D^{1/Q}, $$ and define the second-layer interpolation slope: $$ \beta_Q = R_M(A_Q,C_Q,D_Q). $$ Define the pair-observable map: $$ \boxed{ \Phi_M: (\mathbb G_m)^4 \dashrightarrow \mathbb A^2 } $$ by: $$ \boxed{ \Phi_M(X,Y,Z,W) = \left( R_M(X,Y,Z), R_M(X,Z,W) \right). } $$ The core theorem of this round: ## Two-Observable Joint-Degree Theorem For fixed: $$ M\ge2, $$ there exist: $$ c_M>0, $$ and: $$ Q_0(M,A,B,C,D), $$ such that for all: $$ Q\ge Q_0, $$ we have: $$ \boxed{ [ \mathbb Q(\alpha_Q,\beta_Q) : \mathbb Q ] \ge c_MQ^2. } $$ Thus the second layer strictly improves the single-layer linear element-degree lower bound: $$ \Omega(Q) $$ to the pair-level bound: $$ \boxed{ \Omega(Q^2). } $$ This is the first time the second BADT layer provides a rigorous gain in the arithmetic complexity of interpolation elements. --- # 1. Universal interpolation rational function Fix: $$ M\ge2, $$ $$ L=M^2. $$ For formal variables: $$ X,Y,Z, $$ let: $$ R_M(X,Y,Z) $$ be the universal extremal interpolation derivative defined in Rounds 29–30: $$ R_M(X,Y,Z) = H_{X,Y,Z}'(1), $$ where: $$ H_{X,Y,Z} $$ interpolates: $$ X^aY^b \mapsto Y^aZ^b $$ for: $$ 0\le a,b0$; 5. its toric translation stabilizer is finite. This round needs only items 2 and 3 to establish two-observable independence. --- # 2. Two observables Define: $$ R_1 = R_M(X,Y,Z), $$ $$ R_2 = R_M(X,Z,W). $$ Note that: $$ R_1 $$ does not depend on: $$ W, $$ while: $$ R_2 $$ does not depend on: $$ Y. $$ and: $$ \frac{\partial R_1}{\partial Y} \not\equiv0, $$ $$ \frac{\partial R_2}{\partial W} \not\equiv0. $$ --- # 3. Jacobian minor Consider: $$ \Phi_M = (R_1,R_2). $$ The Jacobian with respect to the variables: $$ Y,W $$ has the $2\times2$ minor: $$ \det \begin{pmatrix} \partial_YR_1 & \partial_WR_1\\ \partial_YR_2 & \partial_WR_2 \end{pmatrix}. $$ Since: $$ \partial_WR_1=0, $$ $$ \partial_YR_2=0, $$ we have: $$ \boxed{ J_{YW} = (\partial_YR_1) (\partial_WR_2) \not\equiv0. } $$ Thus: $$ \boxed{ \operatorname{rank}d\Phi_M=2 } $$ generically. --- # 4. Algebraic independence By the generic Jacobian rank $2$: $$ \boxed{ R_1, \quad R_2 } $$ in: $$ \mathbb Q(X,Y,Z,W) $$ are algebraically independent. Equivalently: $$ \Phi_M $$ is a dominant rational map: $$ (\mathbb G_m)^4 \dashrightarrow \mathbb A^2. $$ Thus the generic fiber dimension is: $$ \boxed{ 4-2=2. } $$ --- # 5. Domain of regular definition Let: $$ U_M \subset (\mathbb G_m)^4 $$ be the maximal open subset on which: $$ R_1, R_2 $$ both functions are regular. For the positive-real BADT point: $$ p_Q = (A_Q,B_Q,C_Q,D_Q), $$ all first- and second-layer interpolation nodes are distinct. A node collision, for example: $$ A_Q^aB_Q^b = A_Q^{a'}B_Q^{b'} $$ would give: $$ (a-a')+(b-b')s=0, $$ which is excluded by the irrationality of $s$. The same reasoning applies to the second layer. Thus: $$ \boxed{ p_Q\in U_M } $$ for every: $$ Q\ge1. $$ --- # 6. Generic-fiber open set The dominant rational map: $$ \Phi_M $$ restricts to: $$ U_M \to \mathbb A^2. $$ By the generic-fiber dimension theorem / generic flatness, there exists a nonempty Zariski-open subset: $$ \Omega_M \subset \mathbb A^2 $$ such that for: $$ y\in\Omega_M, $$ the fiber: $$ \Phi_M^{-1}(y) $$ has dimension: $$ \boxed{ 2. } $$ Let the exceptional target locus be: $$ E_M = \mathbb A^2 \setminus \Omega_M. $$ Then: $$ \boxed{ E_M } $$ is a proper algebraic subset. --- # 7. Flattening path Define the analytic one-parameter BADT path: $$ p(t) = \left( e^{\lambda t}, e^{s\lambda t}, e^{s^2\lambda t}, e^{s^4\lambda t} \right). $$ Then: $$ p_Q=p(1/Q). $$ --- # 8. Zariski density of the BADT path If a nonzero Laurent polynomial: $$ F = \sum_m c_m X_0^{m_0} X_1^{m_1} X_2^{m_2} X_4^{m_4} $$ vanishes identically on: $$ p(t), $$ then: $$ 0 = F(p(t)) = \sum_m c_m e^{ \lambda t ( m_0+m_1s+m_2s^2+m_4s^4 ) }. $$ Since: $$ 1,s,s^2,s^4 $$ are Q-linearly independent, distinct exponent vectors: $$ m $$ give distinct real frequencies. Distinct exponentials are linearly independent. Thus: $$ c_m=0 $$ for all: $$ m. $$ a contradiction. Therefore: $$ \boxed{ \overline{ \{p(t):t\in\mathbb C\} }^{\mathrm{Zar}} = (\mathbb G_m)^4. } $$ Under the local analytic interpretation, any small interval of nonzero $t$ already suffices for Zariski density. --- # 9. Exceptional fibers cannot recur infinitely along $1/Q$ Suppose: $$ \Phi_M(p_Q) \in E_M $$ for infinitely many: $$ Q. $$ Since: $$ E_M $$ is proper, there exists a nonzero polynomial: $$ H(U,V) $$ vanishing on: $$ E_M. $$ Then: $$ h(t) = H( R_1(p(t)), R_2(p(t)) ) $$ is analytic in a punctured neighborhood of: $$ 0 $$ and the interpolation asymptotics make its singularity at: $$ t=0 $$ removable. If: $$ h(1/Q)=0 $$ infinitely often, with accumulation at: $$ 0, $$ the identity theorem gives: $$ h(t)\equiv0. $$ Thus the rational function: $$ H(R_1,R_2) $$ vanishes on the Zariski-dense BADT path. Hence: $$ H(R_1,R_2)\equiv0. $$ But: $$ R_1,R_2 $$ are algebraically independent. A contradiction. Therefore: ## Exceptional-Fiber Escape Lemma $$ \boxed{ \Phi_M(p_Q)\in\Omega_M } $$ for all sufficiently large: $$ Q. $$ Thus: $$ \boxed{ \dim \Phi_M^{-1} ( \alpha_Q,\beta_Q ) = 2 } $$ eventually. --- # 10. Uniform fiber degree Let: $$ \Gamma_M $$ be the projective / affine closure of the graph of: $$ \Phi_M $$ as displayed above. Since: $$ M $$ is fixed: $$ \Gamma_M $$ has degree: $$ \Delta_M $$ a fixed finite constant. For: $$ y\in\Omega_M, $$ the fiber: $$ \Gamma_{M,y} $$ is the intersection of the graph with a target-coordinate linear slice. By Bézout / degree monotonicity: $$ \boxed{ \deg \Gamma_{M,y} \le \Delta_M. } $$ Projecting the fiber back to the source torus and taking its closure gives a degree controlled by another constant depending only on $M$. Thus there exists: $$ \boxed{ D_M^{\mathrm{fib}} } $$ such that every good fiber has: - dimension $2$; - degree at most: $$ D_M^{\mathrm{fib}}. $$ --- # 11. Algebraic-grid point bound Use the standard degree–dimension Cartesian-grid bound: If: $$ V \subset \mathbb A^n $$ is an algebraic set of dimension: $$ d $$ and degree: $$ \Delta $$ as displayed above, and: $$ S $$ is a finite subset of the ground field, then: $$ \boxed{ |V\cap S^n| \le C(n,\Delta)|S|^d. } $$ For fixed: $$ n=4, $$ and bounded: $$ \Delta, $$ one need only write: $$ O_{\Delta}( |S|^d ). $$ This bound can be proved by Bézout + induction / variety Schwartz–Zippel. Take: $$ S=\mu_Q, $$ $$ |S|=Q. $$ Thus a good two-dimensional fiber has at most: $$ \boxed{ C_MQ^2. } $$ --- # 12. Translation to collision fibers Define: $$ y_Q = (\alpha_Q,\beta_Q) = \Phi_M(p_Q). $$ The pair-value collision set is: $$ \mathcal C_Q = \{ \xi\in\mu_Q^4: \Phi_M(\xi p_Q) = \Phi_M(p_Q) \}. $$ The diagonal multiplication: $$ \xi \mapsto \xi p_Q $$ is a torus automorphism. Thus: $$ \mathcal C_Q $$ is the intersection of a good fiber with a translated torsion grid. Degree / dimension are unchanged. Therefore: $$ \boxed{ |\mathcal C_Q| \le C_MQ^2 } $$ for sufficiently large: $$ Q. $$ --- # 13. Combined Kummer rank Let: $$ K_4 = \mathbb Q(A,B,C,D). $$ Round 29 proves: $$ \langle A,B,C,D\rangle $$ is a torsion-free multiplicative subgroup of rank $4$. The Perucca–Sgobba theorem gives a fixed: $$ C_{\mathrm{Kum}} $$ such that: $$ \boxed{ [ K_4( \zeta_Q, A_Q,B_Q,C_Q,D_Q ) : K_4(\zeta_Q) ] \ge \frac{ Q^4 }{ C_{\mathrm{Kum}} }. } $$ --- # 14. Kummer Galois twists Let: $$ L_Q = K_4( \zeta_Q, A_Q,B_Q,C_Q,D_Q ). $$ The extension: $$ L_Q/K_4(\zeta_Q) $$ is a Kummer Galois extension. Its Galois group: $$ G_Q $$ injects into: $$ \mu_Q^4 $$ by: $$ (A_Q,B_Q,C_Q,D_Q) \mapsto ( \xi_1A_Q, \xi_2B_Q, \xi_3C_Q, \xi_4D_Q ). $$ and: $$ \boxed{ |G_Q| \ge Q^4/C_{\mathrm{Kum}}. } $$ --- # 15. Stabilizer of the observable pair A Galois automorphism: $$ \sigma_\xi $$ fixes: $$ (\alpha_Q,\beta_Q) $$ if and only if: $$ \Phi_M(\xi p_Q) = \Phi_M(p_Q). $$ Thus: $$ \operatorname{Stab}_{G_Q} (\alpha_Q,\beta_Q) \subset \mathcal C_Q. $$ Therefore: $$ \boxed{ | \operatorname{Stab}_{G_Q} (\alpha_Q,\beta_Q) | \le C_MQ^2. } $$ --- # 16. Pair orbit size Orbit–stabilizer gives: $$ | G_Q\cdot (\alpha_Q,\beta_Q) | = \frac{ |G_Q| }{ | \operatorname{Stab}_{G_Q} (\alpha_Q,\beta_Q) | }. $$ Thus: $$ \boxed{ | G_Q\cdot (\alpha_Q,\beta_Q) | \ge \frac{ Q^4/C_{\mathrm{Kum}} }{ C_MQ^2 } = c_MQ^2. } $$ --- # 17. Relative joint field degree The Galois orbit size is: $$ \boxed{ [ K_4( \zeta_Q, \alpha_Q,\beta_Q ) : K_4(\zeta_Q) ] } $$ so: $$ \boxed{ [ K_4( \zeta_Q, \alpha_Q,\beta_Q ) : K_4(\zeta_Q) ] \ge c_MQ^2. } $$ --- # 18. Absolute joint degree Let: $$ E_Q = \mathbb Q(\alpha_Q,\beta_Q). $$ Then: $$ K_4(\zeta_Q,E_Q) = K_4(\zeta_Q,\alpha_Q,\beta_Q). $$ The general field-compositum inequality gives: $$ [ K_4(\zeta_Q,E_Q) : K_4(\zeta_Q) ] \le [E_Q:\mathbb Q]. $$ Therefore: ## Two-Observable Joint-Degree Theorem $$ \boxed{ [ \mathbb Q(\alpha_Q,\beta_Q) : \mathbb Q ] \ge c_MQ^2. } $$ for sufficiently large: $$ Q. $$ --- # 19. Why this is genuinely a second-layer gain The first layer alone: $$ \alpha_Q $$ comes from a Kummer torus of rank $3$ + one scalar observable: $$ 3-1=2 $$ fiber dimensions, giving: $$ \boxed{ \deg\alpha_Q \gg Q. } $$ With the second layer: $$ (\alpha_Q,\beta_Q) $$ comes from a Kummer torus of rank $4$ + two independent observables: $$ 4-2=2 $$ fiber dimensions, giving: $$ \boxed{ \deg(\alpha_Q,\beta_Q) \gg Q^2. } $$ Thus: $$ \boxed{ \text{second layer contributes one full additional power of }Q } $$ in rigorously visible element-level arithmetic complexity. --- # 20. Fiber-dimension accounting principle This round suggests a general pattern: If: - the multiplicative Kummer rank is: $$ r; $$ - the number of algebraically independent scalar observables is: $$ k; $$ - good fibers are: $$ r-k $$ dimensional; then a bounded-degree Cartesian-grid argument naturally gives: $$ \boxed{ \text{joint observable degree} \gtrsim Q^k. } $$ provided the flattening path eventually avoids exceptional fibers. Round 30: $$ (r,k)=(3,1) \Rightarrow Q. $$ Round 31: $$ (r,k)=(4,2) \Rightarrow Q^2. $$ --- # 21. Why exceptional fibers mattered If the pair-value fiber dimension jumps to: $$ 3, $$ torsion-grid collisions can return to: $$ O(Q^3), $$ leaving only: $$ \Omega(Q) $$ as the joint orbit size. Thus one cannot rely solely on the generic-dimension heuristic from algebraic independence. The genuine additional result of this round is: $$ \boxed{ \text{BADT flattening path eventually escapes the exceptional fiber locus}. } $$ This makes the: $$ Q^2 $$ bound a theorem, not a generic heuristic. --- # 22. Why the escape argument works unusually well here The key is that: $$ p_Q=p(1/Q) $$ is not an arbitrary sequence. It comes from: $$ p(t) = ( e^{\lambda t}, e^{s\lambda t}, e^{s^2\lambda t}, e^{s^4\lambda t} ). $$ Since: $$ 1,s,s^2,s^4 $$ are Q-linearly independent, this analytic curve is Zariski dense. Also: $$ 1/Q $$ accumulates at: $$ 0. $$ Thus, if any fixed algebraic exceptional condition is hit for infinitely many $Q$, the analytic identity theorem turns it into an identity along the whole exponential curve, which Zariski density then excludes. --- # 23. Relation to torsion-point counting literature Schefer 2025 gives general sharp growth upper bounds for torsion points on fixed torus subvarieties and power savings away from maximal torsion-coset behavior. The method of Round 31 is different: - fibers vary with: $$ Q $$ the parameter displayed above; - no power saving is needed; - only the good fiber dimension: $$ 2 $$ and a uniform degree are needed; - then use the generic algebraic-grid: $$ O(Q^2) $$ count. Thus it avoids the varying-fiber torsion-coset obstacle of Round 30. If one wishes in future to improve: $$ Q^2 $$ further, a finer theory of torsion-coset sparsity is still needed. --- # 24. Pair approximation remains nonexact Round 29 has proved: $$ \boxed{ |\beta_Q-\alpha_Q^2| \asymp Q^{-(M^2-1)}. } $$ Thus, even though: $$ (\alpha_Q,\beta_Q) $$ has joint degree at least: $$ Q^2, $$ they are still not exact points of the parabola: $$ V=U^2. $$ Therefore: $$ \boxed{ \text{joint degree growth alone does not close BADT}. } $$ --- # 25. Height status The interpolation-height argument of Round 28 also applies to the second layer: $$ h(\alpha_Q)=O_M(1), $$ $$ h(\beta_Q)=O_M(1). $$ Thus: $$ \boxed{ h(\alpha_Q,\beta_Q)=O_M(1) } $$ for fixed: $$ M. $$ We therefore now have: $$ \boxed{ \text{bounded height} + \text{joint degree }\Omega(Q^2) + \text{distance to parabola }Q^{-(M^2-1)}. } $$ these three quantities simultaneously. --- # 26. A new arithmetic-geometric interface Define the defect: $$ \delta_Q = \beta_Q-\alpha_Q^2. $$ Then: $$ \boxed{ |\delta_Q| \asymp Q^{-(M^2-1)}. } $$ and: $$ h(\delta_Q)=O_M(1). $$ But: $$ \delta_Q $$ has currently unknown individual algebraic degree. The joint degree of the pair: $$ \Omega(Q^2) $$ does not automatically give: $$ \deg\delta_Q \gg Q^2. $$ because subtraction / squaring may greatly enlarge the stabilizer. Thus the new natural observable is: $$ \boxed{ \Psi_M = R_M(X,Z,W) - R_M(X,Y,Z)^2. } $$ Its Kummer orbit / fiber geometry can be studied separately. --- # 27. Defect-Degree Problem A candidate question for Round 32: **Defect-Degree Problem (DDP)** For: $$ \delta_Q = \Psi_M(p_Q), $$ can one prove: $$ \boxed{ [\mathbb Q(\delta_Q):\mathbb Q] \gg Q^\theta } $$ for some: $$ \theta>1? $$ The generic fiber of a scalar observable of rank $4$ gives only: $$ \theta=1. $$ To exceed $1$, one must use: - the special structure of $\Psi_M$; - the pair's joint degree; - the relation between the pair fiber and the defect fiber; - torsion-coset sparsity. --- # 28. Double-scaling alternative Another Round 32 route is not to improve the defect degree, but to let the interpolation order: $$ r=M^2-1 $$ grow with: $$ Q $$ the parameter displayed above. Since: $$ |\delta_Q| \sim Q^{-r}, $$ if one can control: - height growth in $M$; - upper degree growth in $Q,M$; - leading constants; then a Liouville / norm lower bound may collide with high-order analytic approximation. This requires moving from the fixed-$M$ regime into: $$ \boxed{ M=M(Q) } $$ double scaling. --- # 29. Round 31 Assessment **PASS — The two BADT interpolation slopes form algebraically independent observables; the flattening path eventually avoids exceptional higher-dimensional fibers; combined rank-4 Kummer theory yields quadratic joint-degree growth** This round completes: 1. Definition of: $$ \Phi_M(X,Y,Z,W) = ( R_M(X,Y,Z), R_M(X,Z,W) ). $$ 2. Using the genuine variable dependence of Round 30 to establish a nonzero Jacobian minor. 3. A proof that the two observables are algebraically independent. 4. The generic fiber dimension: $$ 2. $$ 5. Definition of the exceptional higher-dimensional target locus. 6. A proof that the one-parameter BADT flattening path is Zariski dense in: $$ (\mathbb G_m)^4. $$ 7. Using analytic accumulation at: $$ t=0 $$ to exclude infinitely recurring exceptional fibers. 8. The exact fiber dimension for sufficiently large $Q$: $$ 2. $$ 9. Construction of a uniform fiber-degree bound. 10. Use of the algebraic-grid / variety Schwartz–Zippel bound: $$ O_M(Q^2). $$ 11. Application of the Perucca–Sgobba Kummer lower bound of rank $4$: $$ \Omega(Q^4). $$ 12. Using orbit–stabilizer to obtain: $$ \boxed{ [ K_4(\zeta_Q,\alpha_Q,\beta_Q) : K_4(\zeta_Q) ] \gg_M Q^2. } $$ 13. The resulting absolute bound: $$ \boxed{ [ \mathbb Q(\alpha_Q,\beta_Q) : \mathbb Q ] \gg_M Q^2. } $$ 14. Construction of the fiber-dimension accounting principle for Kummer rank / observable count. 15. Identification of the first rigorous element-level gain from the second layer: $$ Q\to Q^2. $$ 16. Pinpointing the next defect-degree / double-scaling problem. --- # 30. The Formal Entry Point for Round 32 It is proposed that Round 32 assess two lines simultaneously, ultimately retaining the stronger one. ## R32-A: The Defect Observable's Kummer Orbit Study: $$ \Psi_M(X,Y,Z,W) = R_M(X,Z,W) - R_M(X,Y,Z)^2. $$ First analyse: - the toric stabilizer; - the generic fiber; - special factorization; - the torsion-grid collision count. If defect fibers are unusually sparse, one can improve: $$ \deg\delta_Q. $$ --- ## R32-B: Joint-to-Defect Descent It is known that: $$ \deg(\alpha_Q,\beta_Q)\gg Q^2. $$ Study the extension tower: $$ \mathbb Q(\delta_Q) \subset \mathbb Q(\alpha_Q,\beta_Q), $$ where: $$ \beta_Q = \alpha_Q^2+\delta_Q. $$ Thus: $$ \mathbb Q(\alpha_Q,\beta_Q) = \mathbb Q(\alpha_Q,\delta_Q). $$ If one can upper-bound: $$ [ \mathbb Q(\alpha_Q,\delta_Q) : \mathbb Q(\delta_Q) ] $$ by: $$ O(Q^\rho) $$ with: $$ \rho<2, $$ then part of the joint degree can be passed to the defect. --- ## R32-C: Double Scaling Let: $$ M=M(Q)\to\infty. $$ Balance: $$ |\delta_Q| \approx Q^{-M^2} $$ against algebraic lower bounds depending on: - degree; - height; - $M$. The objective is to find whether there is a growth window: $$ M(Q) $$ in which analytic decay overwhelms arithmetic complexity. --- # 31. One-Sentence Conclusion > **Round 31 proves that the second layer genuinely provides a quantifiable element-level arithmetic gain for the first time. The universal interpolation slopes $R_M(X,Y,Z)$ and $R_M(X,Z,W)$ are algebraically independent, so the pair map has two-dimensional generic fibers; the BADT rational-flattening path is also Zariski dense because $1,s,s^2,s^4$ are linearly independent, so it can hit higher-dimensional exceptional fibers only finitely many times. For large $Q$, the pair-collision fiber is a bounded-degree two-dimensional variety, and on $\mu_Q^4$ it has only $O(Q^2)$ points, while the combined Kummer orbit has $\Omega(Q^4)$ twists. Thus the joint algebraic degree of $(\alpha_Q,\beta_Q)$ is at least $\Omega(Q^2)$.** The current state: $$ \boxed{ h(\alpha_Q),h(\beta_Q)=O_M(1), } $$ $$ \boxed{ \deg(\alpha_Q,\beta_Q)\gg_M Q^2, } $$ $$ \boxed{ |\beta_Q-\alpha_Q^2| \asymp Q^{-(M^2-1)}. } $$ The genuine question for the next round is whether this $Q^2$ joint complexity can be passed to the tiny defect, or whether allowing $M$ to grow with $Q$ can produce a collision with algebraic lower bounds. --- # References 1. Antonella Perucca and Pietro Sgobba, **Kummer theory for number fields and the reductions of algebraic numbers**, *International Journal of Number Theory* 15 (2019), 1617–1633. DOI: 10.1142/S179304211950091X. Their main theorem states that for a torsion-free rank-$r$ subgroup $G\subset K^\times$, $$ \frac{ n^r }{ [K(\zeta_n,\sqrt[n]{G}):K(\zeta_n)] } $$ is bounded independently of $n$. 2. Antonella Perucca, **The degree of non-Galois Kummer extensions of number fields**, *Rivista di Matematica della Università di Parma* 11 (2020), 301–313. 3. Standard generic-fiber dimension theorem and generic flatness for dominant morphisms of algebraic varieties. 4. Standard Bézout degree inequalities and degree–dimension Cartesian-grid bounds / variety Schwartz–Zippel estimates. 5. Gerold Schefer, **Counting torsion points on subvarieties of the algebraic torus**, *Acta Arithmetica* 218 (2025), 297–336. DOI: 10.4064/aa221025-14-1. Provides sharp torsion-point growth bounds for fixed torus subvarieties and power-saving bounds away from maximal torsion-coset behavior. --- **End of ACTM Round 31**