# Verification and scope report

A normal complex algebraic surface germ has focused scheme-theoretic arc Hilbert-Poincare series (1-q)^(-3) product_{j>=2}(1-q^j)^(-2) if and only if it is an ADE rational double point. More generally, for an isolated normal d-dimensional hypersurface of multiplicity d, the maximal-canonical-multiplicity product is equivalent to canonical singularities.

# Detailed originating-researcher self-audit

Status: PASS for the normal surface criterion and isolated hypersurface converse.
This records a mathematical self-audit, not an independent or formal review.

1. Grading and completion. Centering removes weight-zero coordinate variables.
Every weight is finite dimensional. Formal equations contribute finitely many
terms to each weight. BMS Props3.2-3.3 justify the completed-local presentation.
Nilpotents are retained, as necessary for the arc Hilbert series in the source.

2. Infinite regularity lemma. Put A_i=R/(f1,...,fi), C_i=ann_A_(i-1)(fi).
The exact sequence yields H_i=(1-q^di)H_(i-1)+q^di H_Ci. Iterating is an exact
finite identity. Since di tends to infinity, all terms stabilize coefficientwise.
If a C_i is nonzero, choose the least di+nu(C_i). At that weight only the constant
term one of each subsequent product contributes, so the defect is the sum of
positive dimensions. There is no cancellation at its first weight. This proves
regularity of every prefix from the infinite identity. We do not assert that
all later defect coefficients are positive. No finite truncation is mistaken
for the infinite premise. Generators have positive degree so all quotients are
proper, including in the non-Noetherian infinite polynomial ring.

3. Surface presentation. The q coefficient of the target is3, hence embedding
dimension3. A normal algebraic surface over C has an excellent normal local
ring and normal domain completion. The kernel in C[[x,y,z]] is height-one prime
in a regular local UFD, hence principal. Thus the germ is lci; this property
descends from completion and is open near the point. The free focused algebra
has weight-two dimension9 and the target has8, forcing order exactly2.

4. Jet equations and descent. One coefficient f_j occurs in each weight j>=2.
The target product equals H_S times the factors (1-q^j). The lemma forces
regularity in S. Every finite prefix lies in S_m; the polynomial extension
S_m->S is faithfully flat, so injectivity of the successive multiplication
maps descends. The finite polynomial quotient then has dimension3m-(m-1)=2m+1.
The case m=1 is empty sequence and dimension3, as required.

5. Passage to rational double points. A normal algebraic surface has isolated
singularities locally. Shrink to an integral lci neighborhood with only the
selected singular point. Jets over its smooth locus are irreducible of dimension
2(m+1); every jet component has at least that dimension by the lci equation count.
The exceptional fiber is smaller by one, so no extra component can lie there.
Mustata Prop1.4 applies to every m; Thm4.4 then gives ADE. The converse direction
was already proved in BMS Cor3.8 and is cited, not claimed as new.

6. Higher dimension. Hypersurface, dimension d>=2, order d, normal and isolated
are assumptions. The same lemma gives focused fiber dimension d(m+1)-1 for
m>=d-1. At m<d-1 there are no equations and (d+1)m<d(m+1). The isolated hypothesis
allows the same singular-locus bound for all m. Mustata Thm0.1 gives rational;
Gorenstein hypersurface makes this canonical. BMS Prop3.7 proves forward. No
nonisolated converse is asserted.

7. Nonnormal scope test. For xy=0 the m-jet coefficient locus is the union of
m linear spaces with x1,...,x_(a-1) and y1,...,y_(m-a) zero, 1<=a<=m.
Truncated orders give this union, including zero series. Every component has
codimension m-1, equal to the number of equations, so a homogeneous complete
intersection in the polynomial ring yields the same Hilbert product. This is
BMS Prop3.9 (d=e=2), known, and not counted as our counterexample.

8. A false extrapolation caught. For x^2+y^3+z^6, rad(f2,f3,f4)=(x1,y1,x2),
height3. f5 belongs to this minimal prime and is a zero divisor in the complete
intersection quotient. The lemma excludes equality even though exact finite
coefficients agree through weight9. Weight10 differs by one (1216 vs1215).
This is a diagnostic supporting the proof's need for the full infinite premise.

9. Exact computation. check_graded_pieces.py enumerates every monomial in each
weight <=10 and every monomial multiple of every relation up to that weight.
Sparse elimination uses fractions.Fraction over QQ, no floating point ranks.
A1, E8 and xy match the target through10. Nonreduced x^2 first differs at4;
the cubic cone at2; the simple elliptic example at10, with difference+1.
Assertions pass. These finite statements do not establish the infinite theorem.

10. Source/priority distinction. The normal category is explicit in every
publication description; arbitrary nonnormal/nonreduced germs are not covered.
BMS already settles the unrestricted negative example. No exhaustive novelty,
first historical solution, or original frozen-record closure is counted.

# Literature and source audit

The source report OWR 2023/6, printed372/PDF46, was rendered and read. Its
Question2 follows the ADE formula and does not explicitly specify normality.
Mourtada's rational-double-point paper, Prop4.4/Question4.5, states the earlier
formula and question. Neither a dated dataset label nor an unanswered printed
question certifies that a question is still open today.

BMS (arXiv:1101.4950v2; Ramanujan J30 (2013),9-38) proves the forward
maximal-canonical-multiplicity formula in Prop3.7 and the ADE formula in Cor3.8.
Prop3.9 already gives the same series for a surface normal crossing xy=0.
That example is explicitly attributed; it contributes no new counterexample.
Props3.2-3.3 justify finite-weight stabilization and completed-local invariance;
Prop3.6 is the multiplicity test. The proof below establishes the normal-category
converse rather than repeating the forward theorem.

Mustata (arXiv:math/0008002v4; Invent Math145 (2001),397-424), Prop1.4,
characterizes irreducibility of jets of an integral lci variety by the strict
dimension bound over its singular locus. Thm0.1 gives the rational singularity
criterion; the introduction recalls equivalence with canonical for Gorenstein
lci. Thm4.4 specializes to surfaces and rational double points. All these
hypotheses are verified in the manuscript. Remark4.5 about fifth jets is not
used. The higher-dimensional converse keeps the isolated-singularity assumption.

Bounded searches on 26 October 2026 included: arc Hilbert + characterizes + normal;
arc Hilbert + converse + canonical; arc Hilbert + ADE + 2026;
Hilbert-Poincare + regular sequence + arc + converse; arc Hilbert + normal surface;
arc Hilbert + characterisation; and arc Hilbert + characterization + singularities.
No primary source giving the identical normal converse was identified. This
is not exhaustive priority certification. Irrelevant search matches, including
an OpenAI/math overview without the actual statement, are not evidence.

The graded lemma is proved directly by exact multiplication sequences and a
least-degree annihilator defect. Finite computations through weight ten are
supplementary only. The claimed contribution is the explicit converse and its
isolated extension with a complete argument; no unqualified first-source
resolution is counted.

## Exact diagnostics

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    "E8_x2_y3_z5": {
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    "nonnormal_xy": {
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  "source_sha256": "5e07d0c07f6501af4b1f43bb2189652eb57c68fd1dad1bb94f5a47d99c2503c2",
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}
```