A tower of congruences for Domb numbers: proof of a congruence in Z.-W. Sun's Conjecture 77(iii)

Let D_n = Σ_k C(n,k)² C(2k,k) C(2n−2k,n−k) be the Domb numbers (OEIS A002895: 1, 4, 28, 256, 2716, …). This note proves that for every odd prime p and every integer n ≥ 1, (1/n³) · ( Σₖ₌₀ᵖⁿ⁻¹ (2k+1) D_k/(−8)^k − (p/3)·p · Σₖ₌₀ⁿ⁻¹ (2k+1) D_k/(−8)^k ) ≡ 0 (mod p³), where (p/3) is the Legendre symbol. This is the third congruence of Conjecture 77(iii) in Z.-W. Sun's list "Open conjectures on congruences" (Nanjing Univ. J. Math. Biquarterly 36, 2019). Equivalently, T(mp^r) ≡ (p/3)·p·T(mp^(r−1)) (mod p^(3r)) for all m, r ≥ 1, where T(N) = Σk<N (2k+1) D_k/(−8)^k; for p = 3 it says ν_3(T(N)) ≥ 3ν_3(N). The exponent 3 cannot be improved in general: ν_p(T(p) − (p/3)p) = 3 exactly for p = 3, 7, 11, 13, 17, 19, 23. What is new, and how it is proved. The theorem follows by combining three facts: (a) an identity of Chan, Tanigawa, Yang and Zudilin (Adv. Math. 228, 2011, (2.27)): D_n/8^n is the Legendre transform of the sequence f_k(−1/8)^k, where f_k = Σ_j C(k,j)³ are the Franel numbers; (b) a transfer principle (R. Blanco Gómez, doi:10.5281/zenodo.23692226, Theorem 1.3): a tower of congruences modulo p^(2r) for the alternating partial sums of a sequence implies a tower modulo p^(3r), with constant multiplied by p, for the (2k+1)-weighted alternating partial sums of its Legendre transform — there it was observed that this reduces the Domb congruence to Sun's congruence (4.6) for the Franel numbers (Corollary 1.5); (c) that congruence, Σk<pn f_k/8^k ≡ (p/3) Σk<n f_k/8^k (mod p^(2+2ν_p(n))), which is now a theorem for every odd prime p (R. Blanco Gómez, doi:10.5281/zenodo.23753301, Theorem 1.6). The note records the resulting unconditional statement and gives a self-contained account of the deduction, with complete proofs of the transfer principle and of its two lemmas; only (a) and (c) are quoted. We have not found a proof of this congruence in the literature. Related results on Domb numbers (Liu; Mao and Liu; Calderón) concern other sums. Verification. The script verify_domb.py (Python 3, standard library only, about 10 seconds; 2282 checks, no failures) checks in exact rational arithmetic the identity of Chan–Tanigawa–Yang–Zudilin for n < 200, the two lemmas of the transfer principle, the Franel tower for p ≤ 13 and mp^r < 700, and the Domb congruence for every odd prime p ≤ 23 and every n with pn < 700 (694 congruences), together with the sharpness of the exponent. The proofs do not depend on these computations. Authorship. The author of this work is Roberto Blanco Gómez. The result was obtained with SyntheticMind, the mathematical reasoning engine that the author created, designed and programmed himself; its version 49, which incorporates the results of the Franel note, derived the theorem automatically from the two earlier notes. The AI system Claude (Anthropic) collaborated as an assistant, under the author's direction, in the checking of the proofs, the literature search, the verification script and the writing. The author has checked the arguments and takes full responsibility for the content. Files: Blanco_Gomez_2026_Domb_Sun77_EN.pdf: the note (English). Blanco_Gomez_2026_Domb_Sun77_ES.pdf: the Spanish version, with the same content. verify_domb.py: the verification script (Python 3, standard library only). verify_domb_log.txt: its output. sources_tex.zip: the LaTeX sources.