Exact limits of order and primitive-kernel reductions

None of these configurations violates the target cardinality bound. They rule out the specific local reductions stated below.

Neither Euclidean replacement of a fixed pair need preserve dimension

Let $P=\{1,4,5,6,7\}$, which has dimension three. Its four-subsets are covered by

and $\{1,4,6\}$ is dissociated. Replacing either $4$ or $1$ by their difference $3$ produces a set containing the dissociated four-set $\{3,5,6,7\}$, whose subset sums are already listed in the convex obstruction note.

This refutes a dimension-preserving replacement guarantee for an arbitrarily prescribed pair. It does not refute the existence of some other usable pair or a more global compression.

No one-unit suffix-gap closure need preserve dimension

Let $P=\{2,3,4,5,9\}$. Every four-subset is dependent, using

while $\{2,3,4\}$ is dissociated. Thus $d(P)=3$.

Consider decreasing an entire suffix by one while retaining positive distinct entries. The only possibilities are

They contain the dissociated sets $\{1,2,4,8\}$ and $\{2,3,4,8\}$, respectively. For the latter the subset sums are

Hence even the existence of a dimension-preserving unit suffix compression is false.

Odd ordered splitting fails in the primitive rank-one regime

Put

Its lower five and upper five points are both dissociated, but $d(O)=5$. For either five-set, its pair sums are distinct and its three smallest points sum to more than its largest; parity rules out other relations.

Here is the exact finite certificate for the upper bound, run during the exploration. Among the 210 six-subsets, just one has every four-subset dissociated:

It has the relation $1+25+31=17+19+21$. The following self-contained checker verifies the finite coverage assertion and both lower bounds.

from itertools import combinations

O = (1, 7, 11, 13, 15, 17, 19, 21, 25, 31)

def dissociated(D):
    S = {0}
    for a in D:
        T = {s + a for s in S}
        if S & T:
            return False
        S |= T
    return True

assert dissociated(O[:5]) and dissociated(O[5:])
exceptions = [
    C for C in combinations(O, 6)
    if all(dissociated(Q) for Q in combinations(C, 4))
]
assert exceptions == [(1, 17, 19, 21, 25, 31)]
assert 1 + 25 + 31 == 17 + 19 + 21

Every four-subset has gcd one: no prime divides more than two entries of $O$. Moreover, its short ternary relations have a primitive, binary-connected rank-one presentation. Nine such relations are

To verify primitivity without a determinant computation, denote the ten generators by $a,b,c,d,e,f,g,h,i,j$ in displayed order. The first seven relations give

The last two then give $c=2b-3a$ and $c=b+4a$, hence $b=7a$, $c=11a$, and all ten generators are the specified integer multiples of $a$. Each elimination has unit coefficient. The quotient lattice is therefore exactly $\mathbb Z$, with no torsion, and reduction modulo two has relation rank nine.

Thus oddness, a primitive short presentation, and primitive nearly maximum subsets do not repair ordered splitting. A successful argument in the remaining rank-one regime must also exploit the cardinality threshold; this ten-point example is far below it.

Full compression to an initial interval also fails

This is stronger than the local suffix-gap obstruction. Let

Then $d(P)=4$, whereas $d(\{1,\ldots,13\})\ge5$. The four-set $\{1,2,4,8\}\subset P$ is dissociated. The absence of a dissociated five-subset is certified by the following exact finite check over all $\binom{13}{5}=1287$ five-subsets. The final assertion verifies a five-subset of the compressed interval.

from itertools import combinations

P = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15)

def dissociated(D):
    sums = {0}
    for a in D:
        translated = {s + a for s in sums}
        if sums & translated:
            return False
        sums |= translated
    return True

assert dissociated((1, 2, 4, 8))
assert all(not dissociated(D) for D in combinations(P, 5))
assert dissociated((6, 9, 11, 12, 13))

The witness was suggested by an online research note on small values of Problem 963; the three assertions above were checked by the author from the definition and are author-recorded. No other claim in that note is needed here.

Thus initial intervals are not universal minimizers of dissociation dimension, even among positive integer sets. This rules out full one-dimensional compression as a general dimension-nonincreasing operation. It is not a counterexample to the requested bound: $13<2^4$.