A certified continuation machine for Mahler measure identities at CM points, with twelve new proofs of conjectures of Samart
Abstract
We axiomatize the differential-comparison continuation method of [Zf] into a general machine for proving Mahler measure identities at CM points: a modular parametrization with rigorous tail bounds, a single-valued holomorphic Mahler differential on the complement of the critical image, a propagation lemma replacing all monodromy and universal-cover arguments, interval-arithmetic path certification, and an exact three-track CM evaluation. Applied to Samart's family n4(s)=4m(x4+y4+z4+1+s1/4xyz), the machine proves twelve further entries of his 2015 table [Sa15, Table 6], including the first identities in this family proved at non-degenerate CM points of class number two (attached to ℚ(√(−6)) and ℚ(√(−15))) and of class number four (attached to ℚ(√(−21)), with four newforms and the genus group (ℤ/2)2). New ingredients include exact twist levels via grossencharacter conductors (Hecke's theorem), with the interval-locked Fricke ratio as an exact root-number sign lock; a twist-level trap (g16⊗χ8 has level $64$, not $32$); a genus-character decomposition of the lattice sums in class number two; and the discovery that Samart's applicability boundary {Im τ=1/√2} is not the topological boundary of the continuation region. Every identity is certified by an exact algebraic track, rigorous final-identity interval locks (half-widths 4.6×10−53 to 5.1×10−52), and independent 50--60-digit cross-checks; all scripts are public. The evaluation track also yields six new identities at Heegner points (D=11,19,27,43,67,163), stated as conjectures with 60-digit evidence; we close with an umbrella conjecture organizing the proved and conjectured identities, and a precise analysis of the five remaining open entries.
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BibTeX
@misc{zheng2026certified,
title = {{A certified continuation machine for Mahler measure identities at CM points, with twelve new proofs of conjectures of Samart}},
author = {Huimin Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26120},
eprint = {2609.26120},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26120},
}
RIS
TY - UNPB
TI - A certified continuation machine for Mahler measure identities at CM points, with twelve new proofs of conjectures of Samart
AU - Huimin Zheng
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26120
UR - https://arxiv.org/abs/2609.26120
AB - We axiomatize the differential-comparison continuation method of [Zf] into a general machine for proving Mahler measure identities at CM points: a modular parametrization with rigorous tail bounds, a single-valued holomorphic Mahler differential on the complement of the critical image, a propagation lemma replacing all monodromy and universal-cover arguments, interval-arithmetic path certification, and an exact three-track CM evaluation. Applied to Samart's family n_4(s)=4m(x^4+y^4+z^4+1+s^1/4xyz), the machine proves twelve further entries of his 2015 table [Sa15, Table 6], including the first identities in this family proved at non-degenerate CM points of class number two (attached to ℚ(√(−6)) and ℚ(√(−15))) and of class number four (attached to ℚ(√(−21)), with four newforms and the genus group (ℤ/2)^2). New ingredients include exact twist levels via grossencharacter conductors (Hecke's theorem), with the interval-locked Fricke ratio as an exact root-number sign lock; a twist-level trap (g_16⊗χ_8 has level $64$, not $32$); a genus-character decomposition of the lattice sums in class number two; and the discovery that Samart's applicability boundary {Im τ=1/√2} is not the topological boundary of the continuation region. Every identity is certified by an exact algebraic track, rigorous final-identity interval locks (half-widths 4.6×10^−53 to 5.1×10^−52), and independent 50--60-digit cross-checks; all scripts are public. The evaluation track also yields six new identities at Heegner points (D=11,19,27,43,67,163), stated as conjectures with 60-digit evidence; we close with an umbrella conjecture organizing the proved and conjectured identities, and a precise analysis of the five remaining open entries.
ER -