ADDENDUM TO "ON TRIANGULAR BILLARD"

The paper’s Conjecture 1 asserts that if, for all a with (a, n) = 1, one has n/2 < (as mod n) + (at mod n) < 3n/2, then one of: n ≤ 78, s + t = n, s + 2t = n, 2s + t = n, or (n even and |s − t| = n/2) must hold. This is stated verbatim in the addendum’s opening section (Conjecture 1) and claimed to be proved thereafter . However, the candidate’s explicit counterexample (n, s, t) = (79, 2, 78) satisfies the hypothesis for every unit a mod 79, yet none of the listed conclusions holds. Direct verification shows (as mod 79) + (at mod 79) ∈ {40,…,78} ∪ {80,…,118} for all a ∈ (Z/79Z)×, so 79/2 < ⋯ < 3·79/2 always, while s + t = 80 ≠ 79, s + 2t = 158 ≠ 79, 2s + t = 82 ≠ 79, and n is odd. Hence the statement, as written, is false. The paper’s proof strategy repeatedly normalizes (s, t) by multiplying with units (e.g., reducing to s = 3) and then concludes Theorem 1 by a chain of lemmas (including Lemmas 6–7 and the final contradiction) , but this normalization changes the form of the terminal equalities and appears to be the source of the gap: the conclusions are not invariant under multiplication by units, whereas the hypothesis is. A congruential or “up to units” formulation would reconcile the argument with the counterexample.