Internal proof and reproduction audit
September 9, 2026. Review by the originating agent. No human or separate agent review is claimed.
Logical checks
- Decomposition: distinct triples give incomparable primes, unique isolated irreducible primary components, and no embedded primes. Powers of a pure-power component are primary. Thus symbolic powers are exactly the intersections of component powers.
- Localization: invert variables outside a support union, then descend from the Laurent extension by faithful flatness. Exactly the components inside that union survive. No exponent changes.
- Zero specialization: membership of a monomial in remaining variables is unchanged under setting other variables to zero. This proves commutation with monomial intersections; ideal powers commute under a surjection. The statement explicitly requires all image components to be proper nonzero and to have incomparable distinct radicals after removing identical duplicates. Both applications verify this requirement.
- Dimension one: all-power Simis equalities give no embedded primes in one-dimensional power quotients, hence depth one at the homogeneous maximal ideal. Generic complete intersection and grade at least two satisfy Branco Correia–Zarzuela Theorem 6.4 in rank one. The local minimal-generator count equals the monomial count. A shared variable between two generators would lower the height. The resulting disjoint supports in h+1 variables give at most two minimal primes and uniform component exponents. This uses all powers, not merely the square.
- Support reduction: a discrepancy between two triples cannot occur on a two-variable overlap. A single-variable overlap localizes to five variables. Any third component containing that variable would bridge two overlaps and contradict the discrepancy. Inverting the disputed variable leaves at most two extras by dimension-one rigidity. Two omitted extra vertices either lie in the same pair or different pairs, giving exactly the four cases in the paper.
- No extras: the stated monomial belongs to both component squares. Contraction to the two-variable subring commutes with monomial powers; all three possible generator products fail to divide the witness.
- One extra: the indicated zero specialization gives three distinct height-two primes on three variables; this contradicts the dimension-one bound on the number of primes.
- Cross-pair extras: their overlap forces equality of both exponents in the duplicated image ideal. Equality of radicals alone would not justify deleting a duplicate; the proof checks equality of ideals.
- Same-pair extras: three components share two variables, whose exponents agree. Their eight-parameter normal form is exactly the one in Proposition 6. The two vertex formulas are feasible and are uniquely determined by five active linear equations. Both r>1 and r<=1, including equality, place the first three coordinates strictly below the first component's thresholds.
- Polyhedral obstruction: denominator clearing uses the exact floor membership test to obtain a monomial in the Nth symbolic power. Simis makes it an ordinary-power monomial, proving NP=IP. A vertex of NP is a generator exponent, whereas the constructed vertex is outside even the first component's exponent orthants. No assertion about ordinary powers is inferred from integral-closure equality.
- Conclusion: common component exponents commute with the monomial intersection under weighting. MPV's prior invariance theorem gives the radical's Simis property and the converse.
No unresolved proof step was identified in this internal audit. That statement is not an independent correctness certificate.
Exact supplementary calculations
Run python3 check.py using only Python's standard library. The output
is recorded in check-results.json.
| Check | Count |
|---|---|
| All nonempty families of triples on five labeled vertices | 1,023 |
| Families satisfying the four-variable at-most-two condition | 252 |
| Potentially discrepant marked component pairs | 165 |
| No-extra / one-extra / same-pair / cross-pair patterns | 15 / 60 / 30 / 60 |
| Rational vertex parameter choices | 2,943 |
| Vertex cases r>1 / r=1 / r<1 | 2,912 / 4 / 27 |
| Explicit two-component symbolic-square witnesses | 792 |
The support checker independently enumerates the finite set system and checks the image supports and duplicate intersections. The vertex checker uses exact fractions for feasibility and Gaussian elimination to verify the active equations determine the displayed vertex. Extra larger denominators ensure that r<1 is tested. None of these bounded samples proves the arbitrary-exponent statement; the hand argument does so.
Artifact checks
The LaTeX compiles to seven pages. Every final page was rendered and
visually inspected. The full Markdown conversion verifies all source
math expressions, theorem and equation references, and a semantic
roundtrip. The math renderer reports zero errors. Hashes and detailed
results are in artifact-qa.json, markdown-conversion.json,
markdown-math-check.json, and manifest.json. The GitHub live Markdown
renderer was not separately inspected.