> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # ACTM Round 33 ## Exact Quadratic Log-Defect Asymptotics, the Taylor–Liouville Barrier, and the Pivot to Multivariate Auxiliaries ### Exact Quadratic Log-Defect Asymptotics, the Taylor–Liouville Barrier, and the Pivot to Multivariate Auxiliaries **Research program:** 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:** 2026-10-04 **Version:** Round 33 / v0.1 **Document type:** Formal research notes / quadratic-log defect and scalar-small-value barrier round **Status statement:** This round does not claim the impossibility of real BADT$_2$ or a proof of the Four Exponentials Conjecture. It computes exactly the distortion of the Type-P quadratic relation caused by algebraic truncated-log approximants, and proves a structural efficiency barrier for ordinary Taylor truncation combined with scalar Liouville / norm lower bounds: however the truncation order grows, analytic gain per order is only $\log Q$, while Kummer arithmetic cost carries a $Q^3$ factor. Subsequent research is thus forced away from a single scalar defect toward genuine multivariate auxiliary functions, interpolation determinants, and zero estimates. --- ## Abstract The first three Type-P / BADT algebraic logarithms: $$ \lambda, \qquad s\lambda, \qquad s^2\lambda $$ satisfy the exact quadratic relation: $$ \boxed{ (s\lambda)^2 - \lambda(s^2\lambda) = 0. } $$ Round 32 proposed no longer eliminating $\lambda$, but directly approximating these three logarithms with algebraic approximants while retaining the transcendence-degree-$2$ geometry. Fix the positive algebraic: $$ A=e^\lambda, $$ $$ B=e^{s\lambda}, $$ $$ C=e^{s^2\lambda}, $$ and: $$ Q\ge1. $$ Take positive real roots: $$ A_Q=A^{1/Q}, $$ $$ B_Q=B^{1/Q}, $$ $$ C_Q=C^{1/Q}. $$ Define the truncated logarithm: $$ T_m(z) = \sum_{j=1}^{m} (-1)^{j+1} \frac{z^j}{j}. $$ and the scaled approximants: $$ L_{A,m,Q} = Q\,T_m(A_Q-1), $$ $$ L_{B,m,Q} = Q\,T_m(B_Q-1), $$ $$ L_{C,m,Q} = Q\,T_m(C_Q-1). $$ then: $$ L_{A,m,Q}\to\lambda, $$ $$ L_{B,m,Q}\to s\lambda, $$ $$ L_{C,m,Q}\to s^2\lambda. $$ Study the quadratic defect: $$ \boxed{ \Delta_{m,Q}^{\log} = L_{B,m,Q}^2 - L_{A,m,Q}L_{C,m,Q}. } $$ The main asymptotic of this round is: ## Exact Quadratic Log-Defect Theorem For fixed: $$ m\ge1, $$ we have: $$ \boxed{ \Delta_{m,Q}^{\log} = \frac{(-1)^m}{m+1} \lambda^{m+2} s^2 (s^m-1)^2 Q^{-m} + O(Q^{-m-1}). } $$ The leading coefficient is strictly nonzero because: $$ \lambda\neq0, $$ $$ s\neq0, $$ and $s$ is transcendental, so: $$ s^m\neq1. $$ Therefore: $$ \boxed{ |\Delta_{m,Q}^{\log}| \asymp Q^{-m} } $$ for fixed $m$. Thus, ordinary Taylor truncation creates no hidden higher-order cancellation in the Type-P quadratic relation. --- If instead we use unscaled logarithm approximants: $$ \widehat L_{A,m,Q} = T_m(A_Q-1), $$ $$ \widehat L_{B,m,Q} = T_m(B_Q-1), $$ $$ \widehat L_{C,m,Q} = T_m(C_Q-1), $$ then: $$ \widehat L_A \approx \frac{\lambda}{Q}, $$ $$ \widehat L_B \approx \frac{s\lambda}{Q}, $$ $$ \widehat L_C \approx \frac{s^2\lambda}{Q}. $$ Define: $$ \widehat\Delta_{m,Q}^{\log} = \widehat L_{B,m,Q}^2 - \widehat L_{A,m,Q} \widehat L_{C,m,Q}. $$ Since: $$ \widehat\Delta_{m,Q}^{\log} = Q^{-2} \Delta_{m,Q}^{\log}, $$ we have: $$ \boxed{ \widehat\Delta_{m,Q}^{\log} = \frac{(-1)^m}{m+1} \lambda^{m+2} s^2 (s^m-1)^2 Q^{-(m+2)} + O(Q^{-(m+3)}). } $$ This unscaled version is more natural for the arithmetic-height analysis. --- # 1. Individual truncated-log error For fixed: $$ t\in\{1,s,s^2\}, $$ Let: $$ z_{t,Q} = e^{t\lambda/Q}-1. $$ The Taylor remainder is: $$ \log(1+z) - T_m(z) = (-1)^m \frac{z^{m+1}}{m+1} + O(z^{m+2}). $$ Therefore: $$ t\lambda - Q T_m(z_{t,Q}) = \frac{(-1)^m}{m+1} (t\lambda)^{m+1} Q^{-m} + O(Q^{-m-1}). $$ Define: $$ \varepsilon_t = t\lambda - L_{t,m,Q}. $$ then: $$ \boxed{ \varepsilon_t = c_m (t\lambda)^{m+1} Q^{-m} + O(Q^{-m-1}), } $$ where: $$ c_m = \frac{(-1)^m}{m+1}. $$ --- # 2. Expand the quadratic defect Let: $$ a=\lambda, $$ $$ b=s\lambda, $$ $$ c=s^2\lambda. $$ Thus: $$ b^2=ac. $$ Also: $$ L_A=a-\varepsilon_1, $$ $$ L_B=b-\varepsilon_s, $$ $$ L_C=c-\varepsilon_{s^2}. $$ Therefore: $$ \Delta_{m,Q}^{\log} = (b-\varepsilon_s)^2 - (a-\varepsilon_1) (c-\varepsilon_{s^2}). $$ Expanding the exact quadratic terms: $$ \Delta = -2b\varepsilon_s + a\varepsilon_{s^2} + c\varepsilon_1 + \varepsilon_s^2 - \varepsilon_1\varepsilon_{s^2}. $$ --- # 3. Leading coefficient Substituting the first-order remainders gives: $$ -2b\varepsilon_s + a\varepsilon_{s^2} + c\varepsilon_1 $$ the leading coefficient: $$ c_m \lambda^{m+2} \left( -2s^{m+2} + s^{2m+2} + s^2 \right). $$ The factor in parentheses is: $$ s^{2m+2} - 2s^{m+2} + s^2 = s^2(s^m-1)^2. $$ Thus: $$ \boxed{ \Delta_{m,Q}^{\log} = c_m \lambda^{m+2} s^2(s^m-1)^2 Q^{-m} + O(Q^{-m-1}). } $$ --- # 4. No fixed-order cancellation Since: $$ s $$ is transcendental: $$ s^m-1\neq0. $$ Thus, for every fixed: $$ m, $$ there exists: $$ Q_0(m) $$ such that: $$ Q\ge Q_0(m) $$ such that: $$ \boxed{ \Delta_{m,Q}^{\log}\neq0. } $$ Ordinary Taylor truncation therefore cannot accidentally become an exact quadratic relation in a sufficiently flat regime. --- # 5. Algebraic field Let: $$ K = \mathbb Q(A,B,C), $$ $$ d_0=[K:\mathbb Q]. $$ All: $$ \widehat L_{A,m,Q}, \widehat L_{B,m,Q}, \widehat L_{C,m,Q} $$ all lie in: $$ K_Q = K(A_Q,B_Q,C_Q). $$ Therefore: $$ \boxed{ \widehat\Delta_{m,Q}^{\log} \in K_Q. } $$ The trivial radical upper bound is: $$ \boxed{ [K_Q:\mathbb Q] \le d_0Q^3. } $$ Round 29 further proved: $$ [K_Q:K] = \Theta(Q^3). $$ --- # 6. Height of the unscaled approximants Absolute logarithmic Weil height satisfies: $$ h(A_Q)=\frac{h(A)}Q. $$ and: $$ h(A_Q-1) \le h(A_Q)+\log2. $$ Thus: $$ h(A_Q-1) \le \log2+O(Q^{-1}). $$ analogously for: $$ B_Q-1, \quad C_Q-1. $$ --- # 7. Height of the truncation polynomial Clear the denominators of: $$ T_m(z) $$ the truncation polynomial. One can use: $$ \operatorname{lcm}(1,\ldots,m) \le m! $$ to obtain the polynomial coefficient height: $$ \boxed{ h(T_m) = O(m\log(m+1)). } $$ The standard polynomial-evaluation height inequality therefore gives: $$ \boxed{ h( \widehat L_{A,m,Q} ) \le C_A m\log(m+1) + C_A' } $$ uniformly in: $$ Q. $$ Likewise for: $$ B, C. $$ --- # 8. Defect height From: $$ \widehat\Delta = \widehat L_B^2 - \widehat L_A\widehat L_C $$ and the standard height inequalities: $$ h(xy)\le h(x)+h(y), $$ $$ h(x-y)\le h(x)+h(y)+\log2, $$ we obtain: $$ \boxed{ h( \widehat\Delta_{m,Q}^{\log} ) \le C_1 m\log(m+1) + C_2. } $$ where the constants depend only on: $$ A,B,C. $$ In particular, for fixed: $$ m, $$ the height is uniformly bounded in: $$ Q $$ that parameter. --- # 9. Scalar Liouville lower bound If: $$ 0\neq\alpha\in\overline{\mathbb Q}, $$ of degree: $$ D, $$ the product formula / standard Liouville inequality gives: $$ \boxed{ \log|\alpha| \ge -D\,h(\alpha) } $$ up to harmless normalization constants depending on the chosen absolute-value convention. Apply this to: $$ \alpha = \widehat\Delta_{m,Q}^{\log}, $$ and: $$ D \le d_0Q^3, $$ we obtain: $$ \boxed{ \log| \widehat\Delta_{m,Q}^{\log} | \ge - C_3 Q^3 \left( m\log(m+1)+1 \right) } $$ when the defect is nonzero. --- # 10. Analytic upper scale The exact asymptotic for fixed $m$ gives: $$ \log| \widehat\Delta_{m,Q}^{\log} | = -(m+2)\log Q + O_m(1). $$ More generally, if: $$ m=m(Q) $$ and: $$ Q $$ is sufficiently large that: $$ |e^{t\lambda/Q}-1| \le \frac12 $$ for: $$ t=1,s,s^2, $$ the Taylor tail bound gives: $$ \boxed{ | \widehat\Delta_{m,Q}^{\log} | \le C_4 \left( \frac{C_5}{Q} \right)^{m+2} } $$ up to a harmless factor polynomial in: $$ m. $$ Thus: $$ \boxed{ -\log| \widehat\Delta_{m,Q}^{\log} | \gtrsim m\log Q - O(m+\log m). } $$ --- # 11. Taylor–Liouville mismatch The scalar Liouville exponent cost is: $$ \boxed{ Q^3m\log(m+1) } $$ versus the Taylor analytic gain: $$ \boxed{ m\log Q. } $$ Their ratio is roughly: $$ \boxed{ \frac{ m\log Q }{ Q^3m\log(m+1) } \le \frac{ \log Q }{ Q^3 } } $$ up to slowly varying factors. Thus: $$ \boxed{ \frac{\log Q}{Q^3}\to0. } $$ --- # 12. Taylor–Liouville Barrier ## Theorem/Diagnosis R33.2 Using only: 1. The radical-field degree upper bound: $$ D_Q=O(Q^3); $$ 2. The standard height bound: $$ h(\widehat\Delta) = O(m\log(m+1)); $$ 3. Scalar Liouville / field-norm lower bounds; ordinary truncated-log quadratic defects cannot be forced to contradict their analytic Taylor upper bounds by any choice of truncation order: $$ m=m(Q) $$ through this scalar comparison alone. This is not a general impossibility theorem. It says only: $$ \boxed{ \text{the available scalar lower-bound scale is structurally too weak for this construction}. } $$ --- # 13. Why increasing $m$ does not repair the scale Increasing: $$ m $$ does make: $$ |\widehat\Delta| $$ smaller by approximately: $$ Q^{-1} $$ per truncation order. However, each added order also increases algebraic-height complexity. The scalar product-formula cost is multiplied again by Kummer degree: $$ \asymp Q^3. $$ Thus: $$ m $$ appears in both upper and lower exponents and cannot remove: $$ \boxed{ Q^3/\log Q } $$ this scale gap. --- # 14. Fixed-$m$ bounded-height interpretation Fix: $$ m. $$ then: $$ h( \widehat\Delta_{m,Q}^{\log} ) = O_m(1), $$ And: $$ \deg \widehat\Delta \le O(Q^3). $$ At the same time: $$ |\widehat\Delta| \asymp Q^{-(m+2)}. $$ If: $$ D\asymp Q^3 $$ represents the ambient arithmetic-degree scale, then: $$ |\widehat\Delta| \asymp D^{-(m+2)/3}. $$ This is only smallness polynomial in degree. It remains far from the exponential-in-degree small-value regime often encountered in general algebraic-independence criteria. --- # 15. Why a scalar defect loses too much information $\widehat\Delta$ is a single scalar polynomial value of: $$ (\widehat L_A,\widehat L_B,\widehat L_C) $$ three approximants. Once compressed into one algebraic number: $$ \widehat\Delta, $$ we retain only: - Its degree; - Its height; - Its absolute value at one place. Much of Type P's original structure: - Three logarithmic coordinates; - One quadratic relation; - Trdeg-$2$ geometry; - Kummer rank-$3$ structure; has been discarded. The failure of the scalar norm method is therefore not accidental. --- # 16. Multivariate Small-Value Principle This round proposes a methodological principle: ## MSVP At the trdeg-$2$ BADT layer, if an auxiliary construction ultimately retains only one algebraic scalar: $$ \alpha_Q $$ and uses a generic lower bound in terms of: $$ \deg(\alpha_Q), \quad h(\alpha_Q), \quad |\alpha_Q| $$ those quantities, it usually pays the full ambient Kummer-degree cost. To cross the: $$ Q^3 $$ barrier, one must retain: $$ \boxed{ \text{multivariate vanishing / multiplicity / determinant structure} } $$ until the zero-estimate or algebraic-independence-criterion stage. --- # 17. Relation to Philippon's methodology Philippon's general framework from 1997 explicitly combines: - Auxiliary analytic functions; - Effective zero estimates; - Approximation properties; to derive transcendence and algebraic-independence results. His method does not first compress all information into an arbitrary algebraic scalar and then apply a generic Liouville inequality. This agrees with the diagnosis of Round 33: $$ \boxed{ \text{BADT next step should be an auxiliary-function / zero-estimate problem, not a better scalar Taylor remainder}. } $$ --- # 18. Relation to Roy–Waldschmidt The approximation approach of Roy–Waldschmidt 1997 is: - Construct / characterize algebraic approximations; - Retain simultaneous geometry; - Use approximation criteria to deduce algebraic independence. Their logarithm results address small transcendence degree and prove that trdeg-$1$ logarithm fields cannot accommodate certain rational quadratic relations. Round 33 now reaches the next natural dimension: $$ \boxed{ \operatorname{trdeg}=2 } $$ where a scalar Taylor defect alone fails to capture enough geometry. --- # 19. A sharper efficiency invariant Define scalar truncation efficiency: $$ \mathcal E_{m,Q} = \frac{ -\log| \widehat\Delta_{m,Q}^{\log} | }{ \deg(K_Q/\mathbb Q) \left( h( \widehat\Delta_{m,Q}^{\log} )+1 \right) }. $$ using the available scales: $$ -\log|\widehat\Delta| \sim m\log Q, $$ $$ \deg K_Q \asymp Q^3, $$ $$ h(\widehat\Delta) = O(m\log(m+1)). $$ Thus, heuristically / at the upper-scale level: $$ \boxed{ \mathcal E_{m,Q} \lesssim \frac{ \log Q }{ Q^3\log(m+1) }. } $$ so: $$ \boxed{ \mathcal E_{m,Q}\to0. } $$ This precisely expresses that simple truncation moves increasingly far from the scalar Liouville threshold. --- # 20. Why Padé of one variable is not obviously enough An ordinary one-variable Padé improvement can raise approximation order from: $$ m $$ to a constant multiple of: $$ m. $$ However, its coefficient complexity also usually grows with: $$ m $$ that order. Improving only a fixed constant factor therefore does not remove the: $$ Q^3/\log Q $$ gap. To genuinely match Kummer rank $3$ in dimension, one needs an auxiliary space whose number of adjustable coefficients / vanishing conditions grows in a genuinely multivariate fashion. --- # 21. Cubic-Dimension Auxiliary Requirement Round 29 proved that the first three BADT constants: $$ A,B,C $$ generate multiplicative Kummer rank: $$ 3. $$ Their radical extension has degree: $$ \Theta(Q^3). $$ This suggests a dimension-matching target: $$ \boxed{ \text{auxiliary parameter space should itself have cubic-scale combinatorial capacity}. } $$ For example: - Three-variable polynomial families; - Hermite–Padé systems with multiple indices; - Interpolation determinants with three independent exponent indices; - Group-variety zero estimates retaining all three logarithmic coordinates. This round calls it: ## Cubic-Dimension Auxiliary Principle (CDAP) > Rank-$3$ Kummer arithmetic cost should be confronted by an auxiliary construction whose independent coefficient / vanishing-condition budget scales cubically, rather than linearly with a single truncation parameter. This is a methodological target, not a theorem. --- # 22. A natural exponential auxiliary family One candidate family is: $$ F(z) = \sum_{ 0\le i,j,k **Round 33 inserts the original Type-P quadratic-logarithm relation directly into algebraic truncated logarithms and obtains an exact, nonvanishing defect whose leading term is proportional to $s^2(s^m-1)^2Q^{-m}$. This shows that ordinary Taylor truncation has no hidden cancellation. More importantly, scalar arithmetic comparison has a fundamental scale mismatch: analytic gain per order is only $\log Q$, while the nonzero-algebraic-number lower bound pays the full $Q^3$ Kummer-degree cost; increasing truncation order also increases height, so it cannot remove the $Q^3/\log Q$ gap.** Thus, ACTM's approximation route should now formally shift from: $$ \boxed{ \text{one scalar very small} } $$ to: $$ \boxed{ \text{many algebraic conditions retained simultaneously} } $$ that is, the auxiliary-function / interpolation-determinant / zero-estimate layer where classical deep transcendence machinery is genuinely effective. --- # References 1. Damien Roy and Michel Waldschmidt, **Approximation diophantienne et indépendance algébrique de logarithmes**, *Annales Scientifiques de l'École Normale Supérieure* 30 (1997), 753–796. DOI: 10.1016/S0012-9593(97)89938-7. https://numdam.org/articles/10.1016/s0012-9593%2897%2989938-7/ 2. Damien Roy and Michel Waldschmidt, **Simultaneous approximation and algebraic independence**, *The Ramanujan Journal* 1 (1997), 379–430. Develops algebraic-independence criteria from simultaneous Diophantine approximation. 3. Patrice Philippon, **Une approche méthodique pour la transcendance et l'indépendance algébrique de valeurs de fonctions analytiques**, *Journal of Number Theory* 64 (1997), 291–338. DOI: 10.1006/jnth.1997.2130. General auxiliary-function, zero-estimate and approximation framework for transcendence / algebraic independence. 4. 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. Supplies the rank-$3$ Kummer degree scale used in Rounds 29–33. 5. Lars Kühne, **Logarithms of algebraic numbers**, *Journal de Théorie des Nombres de Bordeaux* 27 (2015), 499–535. DOI: 10.5802/jtnb.912. A modern Padé/Siegel-lemma approach to transcendence of logarithms of algebraic numbers, useful methodological comparison for future auxiliary constructions. --- **End of ACTM Round 33**