Recentered pivot certificates miss the stronger odd extraction

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_proof: A split perfect binary tree forces the cluster counts in every longest star. The centered cost stays positive, and binary codes realize the construction by odd integers; the 65-element instance has exact defect 892.

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Statement

For a positive odd integer set $O$, a star with $k$ companions is a pivot $a\in O$ and companions $b_j\in O$ at $k$ distinct levels $j=v_2(b_j-a)$. Write $J$ for these levels and $D_j=\{a\}\cup\{b_i:i\in J,\ i\ge j\}$. The star is a recentered graded certificate if, for every $j\in J$ with $2^j\le k+1$, there is a center $c_j>0$ such that

$$\sum_{x\in D_j}|x-c_j|<2^jc_j. \tag{RC}$$

The center may depend on the level. Such a certificate is a dissociated set: its distinct subsets have distinct sums.

For every real $R>1$ there exist an integer $k\ge2$ and a set $O$ of positive odd integers with

$$|O|=2^{k-1}+1,\qquad \max O<R\min O,$$

but no recentered graded certificate with $k$ companions. The integer $k$ may depend on $R$.

An explicit instance is

$$O=\{2049+2t+1792\boldsymbol 1_{8\nmid t}:0\le t\le64\}.$$

It has $65$ elements, ratio below two, and no certificate with seven companions. The minimum of $\sum_{x\in D}|x-c|-2c$ over all eight-element stars $D$ with seven distinct difference levels and all $c>0$ is $892$. Nevertheless

$$\{2049,3843,3845,3847,3851,3859,3875,3907\}$$

is dissociated.

Significance

The stronger odd-set bound $|O|\le2^{d(O)-1}$ would require $k+1$ dissociated elements when $|O|=2^{k-1}+1$. This claim prevents proving that bound by one certificate of the stated kind, even at any fixed ratio above one. It does not assert that the required dissociated subset is absent. In the explicit instance it is present.

The positive formulation of Problem 963 asks for only $k$ dissociated elements at this cardinality. The explicit instance has a seven-element certificate as well as the displayed eight-element dissociated subset. The floor-logarithmic certificate question at bounded ratio therefore remains open. The research sketch records that distinction and the remaining extraction questions.

Argument

The proof, by Plasma AI, is author-recorded at tier 0, with no independent review or formalization. Split one leaf of a perfect binary tree, so only two leaves have enough levels to be pivots. Place their sibling subtrees in two clusters, forcing every large enough star to have too many points on each side for (RC). Binary codes realize this tree by odd integers at arbitrarily close scaled positions. The proof also gives the exact arithmetic for the finite instance and its dissociated subset. No computational premise is needed for the universal construction.

The certificate definition specializes the positive-integer lemma in the certificate note to odd integers. Its sufficiency is proved again in the proof page; no other L-claim is a premise.

Verification and remaining obligations

Read the proof's certificate lemma, two-cluster construction, integer realization and finite instance. The retained exact research evidence also checks the 65-element depth profile, minimum defect $892$, the $256$ distinct subset sums, a seven-element certificate and a passing comparison set. Its executable entry point is wiki/research/sharp_dissociated_subset/evidence/main.py.

No mathematical gap is known in this scoped claim. Independent review has not been commissioned. No conclusion about the truth or falsity of the stronger odd bound or Problem 963 follows from this obstruction.