CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A fixed unbalanced one-variable substitution need not preserve lexicographic leading terms

Statement refuted

A fixed substitution $x_{1} = t^{w_{1}}, \dots, x_{n} = t^{w_{n}}$ with positive weights always turns the lexicographically leading monomial of a polynomial into its highest power of $t$ .

Facts & Assumptions

Given: Lexicographic order on exponent pairs and the substitution $x = t$ , $y = t^{100}$ .

[L1]

In lexicographic order, exponent tuples are compared at the first coordinate where they differ (Lexicographic order on exponent tuples and the multidegree of a nonzero polynomial).

Counterexample

technique · direct

1.1

For $f (x, y) = x^{2} + x y^{100}$ , the tuple $(2, 0)$ is lexicographically larger than $(1, 100)$ , so $x^{2}$ is the leading monomial.

givenL1

2.1

Under $x = t$ and $y = t^{100}$ , the polynomial becomes $t^{2} + t^{10001}$ , whose highest power comes from $x y^{100}$ , not from $x^{2}$ .

step 1.1algebra

3.1

Thus the fixed weighted substitution reverses these two terms and does not detect the lexicographic leader. Degree-dependent separated weights can encode any one finite support, but this fixed choice cannot replace lexicographic descent.

step 2.1∎

Depends on

Lexicographic order on exponent tuples and the multidegree of a nonzero polynomial

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 4 results over 4 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

K. Conrad, Symmetric Polynomials, Section 2 (standard reference, not scraped)