Further obstructions to cyclic lifting arguments

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The sufficient cyclic assertions in [[research/erdos_963/proof_sketch|the proof sketch]] remain unproved. These exact examples show why two possible repairs of the lifting step do not work. They are not counterexamples to the real-number bound.

A quotient drop of two with all entries odd

Let

$$D=\{9,25,37,61,65,85\}\subset\mathbb Z/128\mathbb Z.$$

Then $d_{128}(D)=6$, but its image modulo $64$ has dimension four.

All six entries are $1\pmod4$. A possible ternary zero relation has support four or six. It must be balanced, an all-plus four-relation, or a five-versus-one six-relation. (Two-term relations are excluded by the entries being distinct and all (1\pmod4).)

The twenty triple sums modulo $128$ are distinct:

$$3,7,19,23,27,31,35,43,47,55, 59,71,83,95,99,107,111,119,123,127.$$

This excludes balanced relations. In particular, any equality of two disjoint pair sums could be extended by a common unused element to an equality of two triple sums.

The total is $282\equiv26\pmod{128}$. The pair sums are

$$18,22,34,46,62,70,74,86,90,94,98,102,110,122,126,$$

none congruent to $26$. Hence no four elements sum to zero. A five-versus-one relation would require $2a\equiv26\pmod{128}$, or $a\equiv13,77$, neither present. This proves dissociation modulo $128$.

Modulo $64$, the four relations

$$\begin{aligned} 25-65+37-61&=-64,\\ 25-65+61-85&=-64,\\ 25-37-9+85&=64,\\ 37+9+61+85&=192 \end{aligned}$$

exclude every five-subset. Their omitted pairs are, respectively, $\{9,85\},\{9,37\},\{61,65\},\{25,65\}$, covering all six possible omitted points. The quadruple $\{9,25,37,65\}$ is dissociated modulo $64$: its pair-sum residues are $10,26,34,38,46,62$, all distinct, and its total is $136\equiv8\ne0$. Since its entries are $1\pmod4$, these checks exclude every possible relation.

Thus the earlier quotient-drop obstruction cannot be repaired just by requiring the original set to be odd.

A prescribed doubled lift need not be consistent

The set

$$T=\{5,15,25,27,41\}\subset\mathbb Z/64\mathbb Z$$

is dissociated. Nevertheless no choices of lifts of $5,25,27,41$ to $\mathbb Z/128\mathbb Z$ give a dissociated set together with both $15$ and $79$.

For the dissociation certificate, change signs individually to obtain $\{5,25,37,41,49\}$. Its entries are $1\pmod4$. Its ten pair sums modulo $64$ are the distinct residues

$$2,10,14,22,26,30,42,46,54,62,$$

and its five four-element sums are $4,24,44,52,56$, all nonzero. These exclude the balanced and all-plus four-relations, the only possible relations.

The two identities

$$2(15)+25+41-5-27=64,\qquad 2(15)+25+41+5+27=128$$

make the lifting contradiction direct. Write the chosen lift of each other element $a$ as $a+64s_a$. Using both $15$ and $79$ with positive coefficients adds $64$ to the displayed left sides. To avoid a zero relation modulo $128$, the first identity requires

$$s_{25}+s_{41}+s_5+s_{27}=1\pmod2,$$

whereas the second requires the same sum to be zero. They are inconsistent.

This refutes consistency for a prescribed doubled element. It does not rule out doubling another element or changing the projected dissociated subset, and it does not give the joint fiber bound needed by the cardinality induction.

A different sufficient cyclic assertion

For a finite cyclic group call $A$ skew if $0\notin A$ and $A\cap(-A)=\varnothing$. The assertion

$$|A|\le2^{d(A)}-1\quad\text{for every skew subset of an odd cyclic group} \tag{1}$$

would imply the stronger positive odd bound. Indeed, if $M=\max O$ for a positive odd integer set $O$, its other elements give distinct nonzero skew residues modulo $M$. Two of them cannot sum to $M$, since that sum is even and $M$ is odd. Any modularly dissociated subset of these residues can be adjoined to $M$ over the reals. Thus its modular dimension is at most $d(O)-1$, and (1) would yield

$$|O|-1\le2^{d(O)-1}-1.$$

Assertion (1) remains unproved here. It cannot be generalized to all finite abelian odd-order groups: the nonzero sign-classes of $\mathbb F_3^r$ have cardinality $(3^r-1)/2$ and dissociation dimension $r$, violating (1) for $r\ge3$. No cyclic counterexample or real-number counterexample was obtained.