# Verification and precise scope

# Canonical source and pairing audit

OWR-3389-011 is record30001159 in the frozen UnsolvedMath v1.6.0
commit c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e.
The original source is Nadirashvili's Conjecture3 on printed
page408 of Low Eigenvalues of Laplace and Schrodinger Operators,
Oberwolfach Reports06/2009, DOI10.4171/OWR/2009/06. The preceding
page407 identifies the proposer. The displayed page408 was
rendered and visually read. It has C_{lambda1(g)}, not C*lambda1(g).
The norm is printed H^-1(S^2,g), not H^-1(S^2,g0).
The frozen background discusses the neighboring Question2
about smoothness of plane-domain local minimizers; it does not
review this conformal-factor conjecture.

Three mathematically different formulations must be kept separate:
1. Scalar s in the intrinsic dual H^-1(g_s), with pairing
   integral f*s*dA_gs = integral f*s^2*dA0. The manuscript gives
   a complete counterexample even with lambda1 exactly6*pi and
   optimization over every conformal or anticonformal parametrization.
2. Area measure s*dA0 in the fixed round dual H^-1(g0). After
   Hersch balancing it has a bound at fixed positive lambda1.
   KNPS2025 provides stronger quantitative stability. The paper
   gives an elementary comparison estimate, without novelty credit.
3. Area measure dA_gs in H^-1(g_s). Its norm is exactly1 for unit
   area and the inhomogeneous H1 norm. This is not formulation1.

The result closes formulation1 as explicitly defined. It does not
establish the source author's intended pairing or count an
unqualified original-source/frozen-record closure. Record one
complete scoped counterexample, zero original closures.

# Mathematical self-audit

The counterexample fixes BOTH area and the exact first eigenvalue.
This is not just a sequence with eigenvalues bounded away from0.
Round g0 has area4*pi and first eigenvalue2; q0=1/(4*pi)
therefore has metric eigenvalue8*pi. All q_t have area1 and
q_t>=1/(8*pi). The odd test z has q1-variance11/21 and energy
8*pi/3, so lambda1(q1)<=56*pi/11<6*pi. This is only a Rayleigh
upper bound, not an eigenfunction claim.

For probability densities, reciprocal lambda1 is the supremum
of weighted variance over round-mean-zero, unit-energy tests.
Round Poincare and H1 embeddings into L3 and L6 uniformly control
the difference of variances by the L^(3/2) difference of densities.
This proves the needed continuity without an assumed smooth
eigenvalue branch, simplicity, or numerical eigenvalue computation.
The polynomial caps b_n=(2*n^5+1)z^(2*n^5)/(4*pi) have mass1.
Mixing with weight n^-2 is O(n^-1/3) in L^(3/2), uniformly in t,
while the squared L2 norm is at least n/(4*pi). Endpoint strict
inequalities persist for all sufficiently large n. Continuity
then supplies an exact t_n in(0,1); no explicit threshold or
closed formula for t_n is claimed. Each density is smooth and
strictly positive; the caps need not individually be positive.

Every round conformal presentation is related by a Mobius or
anti-Mobius diffeomorphism because an isometry between conformal
metrics is conformal relative to g0. The pushforward density
satisfies integral(s^Phi)^2=integral s^2/J_Phi. The reciprocal
Jacobian is(a+b.x)^2 with a^2-|b|^2=1. Antipodal symmetry of s
annuls the linear moment of s^2, leaving only nonnegative terms.
This proves the lower bound for ALL conformal parametrizations,
including arbitrarily large boosts. It is not merely a bounded
coordinate-family assertion. The unit test has intrinsic H1
norm1 because area is1, and pairs with s as integral s^2*dA0.

The positive round-background comparison uses three coordinate
Poincare inequalities and weighted balance. It does not change
the pairing in the negative theorem and is not counted as new.
The area measure itself in the varying H1 dual has norm1.

Exact Fraction diagnostics check102 concentration parameters,
510 mixtures,7650 boosted moments and30 stereographic identities.
They support formulas, not an all-n computational proof. The
written continuity and group arguments establish the universal
claims. This is the originating system's detailed self-audit,
not independent review or formal machine verification. All five
PDF pages were visually inspected, with no remaining layout issue.

# Bounded literature comparison

Read the original OWR source pp407-408 and visually inspect the
displayed norm and the subscript on C. The 2009 source does not
state a universal bound proportional to lambda1. The neighboring
plane-domain problem in the frozen background is not this question.

Karpukhin--Nahon--Polterovich--Stern, Stability of isoperimetric
inequalities for Laplace eigenvalues on surfaces, Journal of
Differential Geometry129(2)(2025),415-490, DOI10.4310/jdg/1738163208,
proves quantitative fixed-background W^-1,2 stability after
conformal normalization. Read the author manuscript Theorem1.2,
Lemma2.1 and its quadratic-form argument, Section2.2 and Remark2.3.
The latter directly bounds lambda1 times the measure in a round
Sobolev dual. These results concern a measure relative to a round
background; they do not identify scalar s*dA_g with s*dA0.
Only these relevant parts were audited, not the entire article.
Publication metadata is verified at UCL Discovery. Direct UCL
PDF download returned403; the author-hosted PDF downloaded with
verified TLS and is retained with its hash.

Searches on26October2026 included Nadirashvili conformal factors
dualH1; weak coercivity sphere conjecture; conformal factor
negativeSobolev firsteigenvalue; intrinsic negativeSobolev norm
fixedeigenvalue; spectrally fixed conformal factors unbounded;
and conformal factor L2 firsteigenvalue bounded. No earlier
identical fixed-eigenvalue, all-parametrization counterexample
was located in this bounded search. This is not an exhaustive
priority certification or proof that the result is absent from
the literature. Standard variational, Sobolev and Mobius facts
are not claimed as new. The known positive estimate is credited.