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The deductions and obstructions here are author-recorded by Plasma AI. Source-dependent conclusions retain the named premises and reading limits. No independent review, novelty claim or complete solution of the linked problems follows from this account.

The relevant questions are E0324.

E0324: rationality would remove the floor

The target is one polynomial in Z[X] with globally distinct pair sums for 0<=a<b; diagonal pairs are excluded. Dubickas--Novikas's four-distinct-index collisions cover this weaker convention and exclude every degree<=3 tail. An eventual positive construction suffices after an integer input shift.

Precisely, if a rational c=A/B with B>0 made Ruzsa's sequence n^5+floor(cn^4), n>=n_0, Sidon, choose an integer T with BT>=n_0. Restrict to n=B(t+T); the floor disappears and

$$F(t)=B^5(t+T)^5+AB^3(t+T)^4\in\mathbb Z[t]$$

would solve E0324 for all t>=0. One rational good coefficient is therefore decisive. The held account supplies only a real one. Uncountability or positive measure would not force a rational member, and rational approximation on a finite input range does not preserve all infinitely many pair-sum inequalities. The Ruzsa construction itself remains uninspected: this is a source gap, not a proof that no rational coefficient works. Stop unless the source supplies one or a uniform transfer to one.

Source interfaces

These source locators identify the uses in this account. Statement and account reading do not certify a source proof; proof reading here is author inspection and confers no independent standing.

• [[../library/diophantine_problems/dubickas_2021_no_cubic_integer_polynomial_generates_sidon/_index|dubickas 2021 no cubic integer polynomial generates sidon]]: Linear/quadratic collision families (account reading). Theorem 1.1 and four-distinct-index qualification, pp.1859-1865 (account reading).