Mathematical results

Can explicit finite-group families refute specified machine-generated inequalities?

Unitriangular and dihedral family arguments address C20–C22 and C27–C28 in the specified Graffiti³ source version. Polynomial dependencies and version conventions remain part of the claim. Exact finite checks corroborate instances; they do not prove an infinite family.

Verification and scope

Verification

The public manuscript contains the family arguments and hypotheses. Exact arithmetic and bounded checks corroborate particular instances; family proofs carry the general claims.

Corrections

An incorrect first-failure forecast was revised. A replay harness capable of reporting false success was corrected before the retained release.

Boundary

Version-scoped refutations do not resolve every Graffiti³ conjecture. Novelty and external specialist acceptance remain unestablished.

How complex can a planar projection of represented-semiorder ideals become?

For the stated inclusive positive-threshold representation, the note proves h ≤ 1 + n + inc(P), with width form h ≤ 1 + rn − r(r−1)/2 for r = min(n,w). The manuscript defines the companion vertex bound and geometric conventions. These class-specific bounds do not solve unrestricted parametric closure.

Verification and scope

Verification

The written proof carries the all-size statement. The frozen finite verifier checks the declared corpus; repeated representations are not independent experiments.

Corrections

The bounded theorem was separated from the unrestricted target. A source-mutating replay wrapper was replaced by a zero-mutation replay before the retained publication package.

Boundary

The unrestricted parametric-closure problem remains open. Comparison with existing bounds must preserve their different parameters and assumptions.

What structure would a smallest counterexample to the conjecture require?

Choose a counterexample with minimum order and then minimum size. Let A contain degree-three vertices, B the vertices of degree at least four, b = |B| and s = Σ_v∈B(d(v)−4). The resulting bound is |A| ≥ 2b + s + 4. Its minimality assumptions are essential.

Verification and scope

Verification

The conditional all-size proof rests on minimality, auxiliary-graph counting and parity. Separate finite evidence is not substituted for this argument.

Corrections

The publication scope separates the structural theorem from a different bounded computational packet with unresolved raw-evidence gaps.

Boundary

The Erdős–Gyárfás conjecture remains open.

Can one pointwise totient-iteration result be checked independently?

Iterate φ(n)+1 until the first prime, using the released stopping convention. The exact result F(400000287233629)=104 implies sup_n F(n) ≥ 104. This is a pointwise lower bound; the global Erdős #409 problem remains open.

Verification and scope

Verification

The frozen trajectory, arithmetic certificate and evaluator establish 104 transitions to the first prime under the specified rule.

Corrections

Earlier F=68/F=71 artifacts remain historical. The current claim is bound to the F=104 release.

Boundary

The certificate establishes a pointwise stopping time and a lower bound of 104 on the supremum; the global problem remains open.