Wang Conjecture 3.26: finite algebraic constant does not bound the relative constants

Guoqing Wang, Semigroup rings and algebraically independent sequences with respect to idempotents in commutative semigroups, arXiv:2509.18724v1, Conjecture 3.26, states:

Let S be a commutative periodic semigroup, and let R be a commutative unitary ring. If d(S,R) < infinity then sup {D_bar(a)(S/H): a in S and <a> is not a group} < infinity.

Conjecture 5.3 is a separate finite-semigroup assertion and is not a second settlement of this question.

| Source clause | Formal counterpart | | --- | --- | | every commutative semigroup, including infinite and nonunital ones | S : Type, [AddCommSemigroup S] | | periodic | Periodic S: each native AddSubsemigroup.closure {a} is finite | | positive cyclic generation | cyclic a, without adjoining a zeroth multiple | | actual cyclic idempotent | IsCyclicIdempotent and cyclicIdempotent; pointwise existence uses finite cyclic closure | | every commutative unitary ring | R : Type, [CommRing R]; no field hypothesis | | ordinary semigroup algebra | upstream AddMonoidAlgebra R S, retaining all labels | | all independently nonzero positional coefficients | GoodSequence: every c : N -> R, with c i != 0 for each occupied position | | finite algebraic constant | algebraicD S R < top, supremum of finite good lengths | | actual principal ideals | principalIdeal a = {a} union {a+c : c in S} | | Green quotient | native quotient of greenCon S, whose relation is equality of those ideals | | cyclic nongroup, with its own possible identity | negation of CyclicIsGroup a, quantifying identity and inverses inside cyclic a | | relative constant at the canonical image | relativeD ((greenCon S).toQuotient a) | | relative irreducibility | no shorter sublist, including empty, has the same sum | | one uniform bound over all filtered elements | nongroupSup S < top | | whole implication and exact negation | closed claim : Prop; designated result : Not claim |

All supremums are in ENat. Infinity means unbounded finite lengths; no infinite product or finite truncation substitutes for the source assertion. Nonempty products require no algebra unit. Length zero contributes only zero to the length supremum. The empty sequence sums only to an existing two-sided identity. In a monoid this is exactly the usual list sum, so the empty proper subsequence is included in the counterexample's irreducibility. Lists enumerate the source's unordered sequences and their submultisets; commutativity makes enumeration irrelevant.

The algebraic constant measures cancellation of products, while relative irreducibility can also detect independent semilattice coordinates. The counterexample separates these two effects inside one semigroup.

The proposed implication requires one bound uniform over all nongroup cyclic elements. Pointwise finite cyclic closures give no uniform bound on the number of independent union coordinates.

The carrier is one fixed infinite commutative monoid

$$S=\operatorname{Finset}(\mathbb N)\times{0,1,2},\qquad (A,k)+(B,l)=(A\cup B,\min(2,k+l)).$$

The injection i maps to ({i},0) proves infinitude. If E(A,k)=(A,0) for k=0 and E(A,k)=(A,2) otherwise, every positive cyclic subsemigroup is exactly {x,E(x)}. Thus S is pointwise periodic. The proof never instantiates an interface requiring globally finite S. The level operation is min-capped addition, not the modular addition of Fin 3.

For R=Q choose every positional coefficient to be one. An idempotent term has zero factor. Otherwise two level-one terms have all four convolution labels equal and their coefficients cancel. Every product of length at least two therefore vanishes for an allowed assignment. The sufficient bound is d(S,Q)<=1. An exact value is not required or asserted by this unit.

The principal ideal of (A,k) consists exactly of (B,l) with A subset B and k<=l. Equality of principal ideals forces both coordinates equal, so the actual quotient map is injective on the entire infinite carrier.

For each n, put a_n=({0,...,n-1},1). The sequence T_n has one marked term (empty,1) and all singleton terms ({i},0) for i<n. Its length is n+1 and its sum is a_n. The weight

$$w(A,k)=|A|+\mathbf 1_{k\ne0}$$

is subadditive. Each term of T_n has weight one, and w(a_n)=n+1. A sublist with sum a_n must therefore have at least n+1 terms. This proves full relative irreducibility and passes through the injective actual Green quotient. Consequently D_bar(a_n)(S/H)>=n+1 for every n within the same S. The positive cyclic subsemigroup at a_n has no identity even of its own: neither its level-one nor its level-two element fixes a_n. Every a_n belongs to the source's nongroup filter. The filtered supremum is therefore top.

A globally finite carrier, modular level addition, contracted convolution, a shared coefficient for equal labels, or a family of different finite semigroups would fail to match the source. Failure to preserve proper subsequences or the actual principal-ideal quotient would also invalidate the refutation.

The formal owner is WangConjecture326Refutation, with closed claim : Prop and designated result : Not claim. Pointwise periodicity, the convolution upper bound, principal-ideal coordinates, quotient injectivity and the unbounded irreducible family form its proof path. The primary versioned HTML supplies the whole source statement. The related source note records the versioned author-source hashes and the separate 5.3 statement.

The mechanism separates pairwise cancellation in the ordinary algebra from the arbitrarily many independent union coordinates needed by relative sums. No exact formula for individual relative constants, minimality assertion, or classification of repairs to the conjecture is supplied.

Proved by the displayed construction: finite algebraic constant alone does not control the number of independent union coordinates in relative sums. The finite-semigroup restriction makes the filtered set of elements finite, but a separate bound on each relative constant would still be needed; this unit supplies no theorem classifying that restriction. The source's Remark 3.27 invokes Conjecture 3.26 to infer an equivalence; this derivation cannot use the refuted implication. The converse direction and other hypotheses of Theorem 3.25 remain separate source statements.

Unresolved here: which additional uniform structural hypotheses restore the implication, exact individual relative constants, and the independent status of Conjecture 5.3.

Ordinary convolution and congruence quotients are reused from pinned Mathlib. The positive cyclic closure is the native subsemigroup closure, allowing pointwise finite cyclic sets inside infinite S. The proof uses neither globally finite cyclic-idempotent interfaces nor field-only definitions in its universal claim. The shared paper note retains the separate Conjecture 5.3 statement and six-label mathematical argument.

The bounded literature context is the source and preregistration. Novelty, priority and exhaustive later-literature absence are not established. The formal theorem concerns Conjecture 3.26 only. The shared note's separate six-label argument about Conjecture 5.3 is not an additional formal conclusion of this theorem.