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Statement and conventions
For a finite set A of real numbers, a subset B is dissociated when its subset sums are pairwise distinct, including the empty subset with sum zero. Let d(A) be the maximum size of such a subset. Write [13] = {1,2,...,13}, and set
Then
In the notation of Problem 963, this gives f(13) <= 4. It does not establish f(13) = 4 or a universal lower bound of four. Since floor(log_2 13) = 3, the example does not refute the catalog's proposed lower bound. The mathematical point is that initial intervals need not minimize the largest dissociated-subset size among sets of a fixed cardinality.
Source
BAKKAOUI reported the example in post 8701, 14:52 on 3 September 2026, and corrected its scope in post 8709, 17:31 the same day. The source excerpt retains both posts from the pinned saved thread and the author's disclosure of AI assistance. The correction explicitly separates f(13) <= 4 from the reported window search. The live source and external links were not checked.
Proof
Every subset of a dissociated set is dissociated: equal sums of two distinct subsets of the smaller set would also be equal sums of distinct subsets of the larger set. Thus excluding all dissociated five-element subsets of A* excludes every larger dissociated subset too.
The set W4 = {1,2,4,8} is contained in A*. Its 16 subset sums are distinct by uniqueness of binary expansion, so d(A*) >= 4. The fixed-instance evidence checks every five-element subset of A*. There are
For each candidate, the checker generates all 32 subset sums in binary-mask order and finds two distinct masks with equal sums. It independently recomputes both sums from those masks before accepting the collision. Complete coverage and a valid collision for every candidate establish d(A*) <= 4.
For the interval, use W5 = {6,9,11,12,13}, which is contained in [13]. The checker validates its cardinality and membership and checks that all 32 subset sums are distinct. This witness is supplied by the present reconstruction; the saved posts did not specify it. Hence d([13]) >= 5.
For the reverse inequality, take any six distinct integers from [13] and let their total be T. Then
If the six integers were dissociated, their 64 subset sums would be distinct integers in [0,T]. This forces T = 63 and occupation of every integer from 0 to 63. Equality in the total-sum bound forces the six integers to be exactly {8,9,10,11,12,13}, but that set has no subset sum equal to 1. This contradiction excludes dissociated six-element subsets. Heredity excludes larger ones, so d([13]) <= 5. This upper-bound proof is supplied in the reconstruction and uses no enumeration over six-element subsets.
Finally, A* is a 13-element set of positive integers, hence is one of the real sets quantified over in the definition of f(13). If that universal guarantee exceeded four, A* would contain a dissociated subset of at least five, which has just been excluded. Thus f(13) <= 4 < d([13]). No reduction from arbitrary real sets to integer sets is needed for this upper bound.
Current verification and limits
The frozen statement, proof, owner checker and exact input received the independent mathematical verdict refutation-failed. A fresh-context independent reviewer and distinct graders checked its essential deductions, finite facts and report contract. The review record identifies their exact subjects, corrected report, completed-record grading and scope limits. The mathematical sections above retain the reviewed text; the input retains its reviewed bytes, and the owner checker's docstring was edited after the review of the checker as it stood on 2026-09-10 (an expected-runtime sentence replaced the external supervisor limit; its checks are unchanged).
The owner checker, preserved independent program and shared-harness adaptation have passed full normal and optimized author runs. The execution account records their finite coverage and failure controls. Historical independent runs still apply to the preserved program. These later author runs do not independently certify the adaptation or extend the frozen finite report's scope. A computational success alone does not verify the heredity, interval upper bound or implication for the universal guarantee.
The required computational inputs are only n=13, A*, W4 and W5; [13] is derived from n. No larger-window data, collision table, private review JSON or source program is an input. The source's searches through n=16 and its exclusion of 13-element sets with d <= 3 inside [34] remain unverified reports here. They do not settle f(13), prove smallest positive-integer failure at 13, or support an exceptionality claim. No native claim, status change, formal verification or current literature-search conclusion is asserted.
Bears on. Problem 963, by ruling out the initial interval as a universal minimizer; it supplies no catalog solution.