Affine coherence and a universal coherent-support bound

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These results do not prove the target. They strengthen the available argument for the parameter $c(P)$ from the convex-fiber note.

Affine-coherent extraction

For an integer set $D=\{a_0,\ldots,a_h\}$, call $D$ affine-coherent if $\mathbf1$ is a vertex of

$$\operatorname{conv}\{u\in\mathbb Z_{\ge0}^{h+1}: \textstyle\sum u_i=h+1,\quad\sum a_i u_i=\sum a_i\}.$$

This property is hereditary, by fixing the complementary coordinates at one in any proposed convex decomposition.

Lemma. If the differences $a_i-a_0$, $1\le i\le h$, have distinct $2$-adic valuations, then $D$ is affine-coherent.

In a convex decomposition of $\mathbf1$ into integer fiber points, write $r=u-\mathbf1$ for each participating point. Then

$$\sum_i r_i=0,\qquad \sum_{i=1}^h(a_i-a_0)r_i=0,\qquad r_i\ge-1.$$

Reduction modulo one power of two beyond the least difference valuation makes the corresponding coefficient $r_i$ even. It is therefore nonnegative. Its weighted average over the decomposition is zero, so it vanishes at every participating point. Repeat with the next valuation. All nonpivot coefficients vanish, and the degree equation eliminates the pivot coefficient. The decomposition is trivial.

The binary residue tree used in [[research/erdos_963/proof_sketch|the proof sketch]] has a path with at least $\lceil\log_2 m\rceil$ branching vertices when it has $m$ leaves. Taking its pivot and one point in each sibling branch proves

$$\boxed{c_{\rm aff}(P)\ge\lceil\log_2|P|\rceil+1.} \tag{1}$$

This is a statement about affine coherence, not ordinary dissociation.

Transfer from a congruence class

Lemma. Suppose $D$ is a positive affine-coherent set contained in $a+q\mathbb Z$, where $q\ge2$ and $\gcd(a,q)=1$. If $|D|\le q+1$, then $D$ is coherent.

Write $k=|D|$. Any nonnegative integral representation of $\sum D$ using $D$ has total degree congruent to $k$ modulo $q$. If $k\le q+1$, its only possible degrees below $k$ are zero or one. Neither is possible when $k\ge2$, since $\sum D$ is positive and is greater than every member of $D$. The singleton case is immediate. Thus minimum degree exposes the degree-$k$ face of the full fiber. Affine coherence makes $\mathbf1$ a vertex of that face, hence of the full fiber.

Consequently, for a nonempty positive integer set $B$ in one unit class modulo $q$,

$$\boxed{c(B)\le q\quad\Longrightarrow\quad |B|\le2^{c(B)-1}.} \tag{2}$$

Otherwise (1) supplies an affine-coherent $(c(B)+1)$-subset, and the transfer lemma makes it coherent.

A universal bound

Let $P$ be a finite positive integer set and $C=c(P)\ge2$. Put

$$q=2^{\lceil\log_2 C\rceil}.$$

Order the occupied valuation layers increasingly as $P_1,\ldots,P_r$. The established coherence extension across successive valuation layers gives $r\le C$ and $c(P_i)\le C-i+1$. Scale each layer to odd integers and split it among the $q/2$ odd classes modulo $q$. Each nonempty class has coherent dimension at most $C-i+1\le q$, so (2) gives

$$|P_i|\le(q/2)2^{C-i}.$$

Summation proves

$$\boxed{|P|\le 2^{\lceil\log_2 C\rceil-1}(2^C-1).} \tag{3}$$

For $C=1$, the separate exact bound is $|P|=1$.

For an arbitrary real set $A$ with $d=d(A)\ge2$, first use the established positive integer reduction and then $c(P)\le d$. The right side of (3) is increasing in $C$. Thus

$$\boxed{|A|\le 2^{\lceil\log_2 d\rceil}(2^d-1)+1.} \tag{4}$$

Combined with the previously proved bound, this gives

$$|A|\le \min\{2^{d+\lfloor\log_2(d+1)\rfloor}, 2^{\lceil\log_2 d\rceil}(2^d-1)+1\}. \tag{5}$$

This improves the older bound slightly at some dimensions, but still does not give $|A|<2^{d+1}$.

Exact limitation

The transfer argument stops at size $q+1$: at larger sizes, degree two or larger can occur below the affine fiber. No argument controlling these alternatives across sibling branches of the binary tree has been proved. A long affine-coherent chain alone cannot suffice, as the block construction in the proof sketch already shows for dissociation.