# Verification and scope

The centered, unshifted Prékopa-Leindler stability conjecture printed in OWR2009/53, p2863, Conjecture1 is false. In every dimension and at lambda=1/2, explicit strictly positive real-analytic isotropic probability densities have L1 distance at least11/8 and an admissible majorant of integral1+2k/(1+k^2), k>=3. A related majorant gives vanishing deficit for every fixed lambda in(0,1), uniformly on compact subintervals.

# Detailed self-audit of the written argument

1. Quantifiers: a fixed lambda=1/2 suffices to refute any gamma_n>0.
The literal source has no log-concavity assumption. The conclusion is
about the unshifted truncated L1 norm, not an optimized translation.

2. Mixture bound: for0<alpha<1, power subadditivity is in the upper
direction (sum A)^alpha<=sum A^alpha. Multiplying the two bounds and
applying Gaussian log-concavity termwise gives a global pointwise
majorant, not just sampled inequalities. The Gaussian normalization
factor has total exponent1 and so is unchanged.

3. Centers and weights: for p=k^2/(1+k^2),q=1/(1+k^2), centers are
+4,-4k^2 for f and -4,+4k^2 for g. Their mean is zero. Midpoints are
0,0,+2(1+k^2),-2(1+k^2), with coefficients p,q,sqrt(pq),sqrt(pq).
Thus the central coefficient is exactly1; no cross term is omitted.

4. Integrals: every translated phi_n has mass1. The majorant mass is
1+2k/(1+k^2), and the covariance of each input is
diag(1+16k^2,1,...,1). The first variance includes within-Gaussian
variance1; the mean is exactly zero, not asymptotically zero.

5. L1 lower bound: on x1>0 the main-component difference is>=7/8
by Markov applied to the standard Gaussian square. The tail difference
is>=-1. Since k>=3 implies p>=9/10,q<=1/10, the half-space difference
is>=11/16. Equal total mass gives L1>=2 times this difference. Thus
the truncated distance is exactly1 for every k, with a rigorous gap.

6. Normalization: densities are transformed as det(A) f(Ax), not
f(Ax) alone. Covariance becomes A^{-1}Sigma A^{-T}=I. L1 and masses
are invariant. PL is preserved because det(A)^lambda times
det(A)^(1-lambda)=det(A), and A is linear.

7. General lambda: integrate the four-term mixture, obtaining
p^lambda q^(1-lambda)+q^lambda p^(1-lambda) as the excess over1.
For lambda in[tau,1-tau], it is<=2q^tau. Each fixed interior
parameter therefore fails independently of endpoint behavior.

8. Translation check: g(x-8e) aligns the main component with f(x).
The two residual densities have massq, yielding distance<=2q. The
isotropic shift is8A^{-1}e. This is compatible with modern translated
stability. No assertion about Gaussian-mixture log-concavity or a
uniform bound on higher moments is used.

9. Regularity and priority: finite positive mixtures are strictly
positive real analytic and have finite moments of all orders for
each k. No uniform higher-moment claim is made. The construction is
elementary; no historical-priority or independent-review claim is made.
The exact rational script supplements, and does not replace, the
continuum inequalities and all-k written proof.

# Primary-source comparison and bounded novelty search

The exact conjecture was read on rendered PDF13 of OWR2009/53. It compares
the unshifted densities, assuming zero means and no log-concavity. The
following theorem explicitly requires log-concavity. The present result
is a counterexample to that printed formulation, without reconstructing
what extra assumptions the proposer might have intended.

Ball--Boroczky, Mathematika56(2010)339-356, arXiv0909.3738, Theorem1.2,
allows a translation parameter and assumes a log-concave majorant.
Remark1.3 permits replacing that assumption by log-concavity of both
inputs. Neither is contradicted. The downloaded arXiv PDF has a
2009v1 margin stamp and an October2018 typesetting date; the publication
is cited as2010, and no inference is made from that typesetting date.

Figalli--van Hintum--Tiba, arXiv2501.04656v2, revised6October2026,
Corollary1.7 proves sharp square-root deficit stability modulo a
translation, with constants depending on n and lambda. The explicit
translation definition in footnote4 was visually checked. This result
does not replace translation minimization by barycenter equality.
Our Gaussian construction has an explicit aligning translation with
L1 error at most2/(1+k^2), consistent with their conclusion.

On26October2026 bounded web searches combined Prekopa/Prékopa/Leindler
with centered/centred/zero mean/barycenter/isotropic/counterexample and
stability, plus the exact OWR page/conjecture and the named authors.
The original source and two primary papers were retrieved and compared.
No identical explicit isotropic Gaussian family was located. This does
not establish historical priority. Escaping small mass and its effect
on barycenters is elementary, not claimed as a new general mechanism.
The initial possible literature-closure route was rejected because its
translation quantifier does not match the source's unshifted distance.
Third-party PDFs are kept locally and excluded from source uploads.

## Exact diagnostics

```json
{
  "arithmetic": "exact rational",
  "families": [
    {
      "L1_lower_bound": "11/8",
      "deficit": "3/5",
      "k": 3,
      "p": "9/10",
      "q": "1/10",
      "translated_L1_upper_bound": "1/5",
      "variance": 145
    },
    {
      "L1_lower_bound": "173/101",
      "deficit": "20/101",
      "k": 10,
      "p": "100/101",
      "q": "1/101",
      "translated_L1_upper_bound": "2/101",
      "variance": 1601
    },
    {
      "L1_lower_bound": "68599/39208",
      "deficit": "99/4901",
      "k": 99,
      "p": "9801/9802",
      "q": "1/9802",
      "translated_L1_upper_bound": "1/4901",
      "variance": 156817
    },
    {
      "L1_lower_bound": "1749998/1000001",
      "deficit": "2000/1000001",
      "k": 1000,
      "p": "1000000/1000001",
      "q": "1/1000001",
      "translated_L1_upper_bound": "2/1000001",
      "variance": 16000001
    },
    {
      "L1_lower_bound": "1749999999998/1000000000001",
      "deficit": "2000000/1000000000001",
      "k": 1000000,
      "p": "1000000000000/1000000000001",
      "q": "1/1000000000001",
      "translated_L1_upper_bound": "2/1000000000001",
      "variance": 16000000000001
    },
    {
      "L1_lower_bound": "1749999999999999998/1000000000000000001",
      "deficit": "2000000000/1000000000000000001",
      "k": 1000000000,
      "p": "1000000000000000000/1000000000000000001",
      "q": "1/1000000000000000001",
      "translated_L1_upper_bound": "2/1000000000000000001",
      "variance": 16000000000000000001
    }
  ],
  "family_checks": 100,
  "formal_verification": false,
  "independent_review": false,
  "proof_obligations_not_replaced_by_tests": [
    "power subadditivity for all inputs",
    "Gaussian density normalization and moments",
    "Markov inequality and total variation",
    "common-linear-change covariance and PL invariance",
    "all-parameter limit and quantifier contradiction"
  ],
  "quadratic_identity_checks": 1805,
  "status": "PASS"
}
```