Internal proof audit

Date: September 9, 2026. Reviewer: originating agent. This is not independent-agent or human peer review.

1. Exact target and ideals. The primary statement is Problem 19 in the latest arXiv v3. Ideals are ordinary multiplicative s-ideals; the theorem does not restrict to finitely generated or divisorial ideals. The conductor is initially defined in the quotient group, but inclusion of 1 in H places every conductor element in M.
2. Canonical extension and contraction. Divisor reflection proves phi(f)D intersect phi(H)=phi(f). Ideals are invariant under H-units, so image membership is equivalent to membership of every lift. An arbitrary nonsaturated embedding would not justify this step.
3. Prime profiles. Equality of prime colon ideals permits replacing one occurrence of a prime in every context. Finite strings of replacements identify membership for equal aggregate color counts. Dickson's lemma applies in the finite-dimensional count lattice, even if each color contains infinitely many primes. Minimal vectors give a uniform coordinate threshold and an upward-closed acceptance set.
4. Upward extension. If a divisor a of xp does not divide x, remove the distinguished extra p; the resulting divisor b divides x. Congruence gives qb in the recognized set and hence a divisor of xq. This proves that a finite congruence recognizing phi(f) forces finitely many prime profiles for its extension. Repeated prime occurrences are covered by the valuation condition, not treated as distinct variables.
5. All realizations. The classical dense embedding theorem is applicable without seminormality. The complete integral closure is intrinsically H. Uniqueness of divisor theories identifies the free divisor monoid with the prescribed one. Restricting class equivalence to pure monomials gives a finite congruence because only the identity is a pure unit.
6. Unit quantifiers. If a and b are equivalent, epsilonah and epsilonbh have the same M-membership for every ambient unit epsilon and every h in H. Quantifying over all h, including 1, makes this exactly conductor membership; existentially quantifying epsilon makes it membership of the divisor image. This does not assume finite index of unit groups or finiteness of the full class semigroup.
7. Construction and complete closure. For M_V=f union V, the ideal property gives multiplicative closure. A fixed c in f supplies both c and ch, proving quotient-group equality before complete integral closure is used. It also supplies ch^n for every n. Complete integral closedness of H gives the reverse inclusion. Properness excludes all normalization units from the conductor.
8. Factorial embedding. The splitting H=H-units times K follows by splitting quotient groups, since q(K) is a subgroup of a free abelian group. A lift of a positive element differs from an H-representative only by an H-unit, so the section actually takes K into H.
9. Finite class bound. A nonunit ambient element remains nonunit after every multiplication. Membership of that product depends only on the divisor class and truncated counts, independent of its unit coordinate. This gives the displayed bound on nonunit classes, with one extra class represented by 1. M_V meets the factorial unit group exactly in its own unit group.
10. Boundary cases. The empty ideal is excluded by the C-monoid conductor theorem. H is realized by itself. For a nonempty proper ideal, f union {1}=H occurs exactly in the reduced maximal-ideal case and prevents every realization. Nontrivial units remove this obstruction by choosing a proper unit subgroup; its index can be infinite.
11. Examples. The total-degree ideal has infinitely many minimal monomials but one prime profile, with exactly d+1 class residuals for d>=2. In the infinite matching ideal, q_i distinguishes every p_i from all other p_j. The unrestricted conductor construction still works there; no unsupported claim that this unrestricted monoid is Mori is made.

The script proves equality of 30 finite monomial ideals by comparing their exact minimal generators, and checks conductor membership over entire finite quotient monoids. Those finite calculations support the illustrative identities and boundary cases. They do not certify the infinite-rank theorem; its universal quantifiers are discharged in the written argument. No unresolved proof step was found within the stated scope. External novelty and significance assessment remain provisional.