# Verification and scope report The homogeneous replacement with the constant printed in OWR 24/2023, p1373, formula (9), is false on the first Heisenberg group. One explicit compactly supported real function in D(L) violates it for every 99/100 <= s < 1. A single smooth compactly supported witness also exists on the entire interval. # Detailed proof audit by the originating researcher 1. Quantifiers and target. The source proposes a universal inequality for 0=0, 2k+2-s <=(2-s)(2k+1), hence L_s <=(2-s)^s L^s. Jensen for the normalized L spectral measure then gives the energy inequality E_s <=(2-s)^s E^s M^(1-s). 4. Domain. The witness and its first radial derivative vanish on both bounding surfaces. The norm rho is smooth away from zero, including the vertical axis; the annulus is compact. Piecewise bounded second derivatives and matching first derivatives yield Euclidean W2,2 without surface delta measures. Euclidean mollification in a fixed compact annulus converges in W2,2, hence in the graph norm of L since its coefficients are bounded there. This places f in D(L) and in the source Sobolev domain. All form energies are finite by the multiplier bound. 5. Polar change. Let v=x^2+y^2=r^2 cos(theta) and 4t=r^2 sin(theta). The Jacobian from(r,theta)to(v,t) is r^3/2; dxdy=(1/2)dv dphi, giving dp=r^3/4 dr dtheta dphi. The coordinate degeneracy at x=y=0 has measure zero. The explicit squared horizontal-gradient numerator is rho^4(x^2+y^2), so the angular factors are pi^2/2 for D and pi for E. These factors matter. 6. Energies. With u=log r, g=sin^2(u/3), the integrals are9pi/8, pi/6 and0 for g^2, g'^2 and gg'. Thus D=9pi^3/16, E=31pi^2/24, q=62/(27pi). Support bounds give D_s>=D and M<=exp(6pi)D; these directions are correct. 7. Uniform numerical-free separation. For s>=.99 the quotient is at most [(2-s)q]^s exp(6pi(1-s)). Bound the base by3131/4050<1. Its .99 power is less than39/50 by the explicit Bernoulli certificate; increasing s reduces the power. The exponential is less than175/142. Their product is273/284. Gamma's lower bound, 4^.99>98/25, (1-s/2)^2>=1/4 and concavity of the power1/50 give C(s)>98/101. The gap is exactly259/28684>0. The standard- library checker also verifies the power inequality with integer arithmetic. 8. Smoothness upgrade. Uniformly on[.99,1], L_s<=2(I+L) as forms, so form-norm convergence gives uniform convergence of the quadratic energies by Cauchy--Schwarz. The weighted integrals converge uniformly on the fixed annulus; C(s) is continuous and bounded. The violation has uniform margin at least(259/28684)D. Thus one sufficiently close real smooth approximant works on the entire interval. No explicit mollifier radius is claimed. 9. Scope. Smaller Hardy constants, optimality and all smaller fractional orders remain outside the theorem. The classical horizontal-gradient effect is known. The new explicit fractional counterexample is self-audited; finite symbolic checks are supporting evidence, not formal or independent verification. No absolute historical priority is certified. # Source and literature comparison The primary report OWR 24/2023 was retrieved by verified HTTPS from EMS Press. PDF pages49-51 were rendered and visually read. Printed p1373 formula(9) has 4^s times the square of Gamma((n+1+s)/2)/Gamma((n+1-s)/2), denominator w_s, and real f in H^s. The next sentence proposes replacing w_s with |(z,a)|^(2s). It proposes this specific constant, not just existence of some positive constant. Roncal--Thangavelu, Adv Math302(2016)106-158, arXiv1508.00714, sections2.3/3.1, supplies the exact L_s multiplier and norm. At s=1 the multiplier equals L. This is the sole non-elementary spectral representation used in the proof. The report's extension and nonhomogeneous formulas are not used as a substitute for its homogeneous conjecture. The 2017 H-type extension preprint uses a different weight/constant presentation; we do not identify those constants without normalization, and do not need its theorem for the counterexample. Garofalo--Lanconelli1990 is credited for the classical horizontal-gradient setting, using the bibliographic reference in the primary literature. Its full text was not audited in this turn, and no theorem from it is needed for this proof. Franceschi--Prandi2021 treats the Carnot-Caratheodory distance and classical Hardy constants. The recent d'Arca--Fanelli--Franceschi--Prandi arXiv2603.04086v1 concerns classical unweighted inequalities. We inspected the full primary texts of the latter two. They do not supply the specific uniform fractional counterexample shown here. Classical radial versus horizontal behavior is explicitly known, not a claimed new discovery. Bounded searches on26October2026 combined the exact OWR DOI, Roncal/Thangavelu with conjecture/homogeneous/2025/2026, Heisenberg with fractional Hardy and counterexample, unweighted with sharp constant, and the exact rational bounds. No identical fractional counterexample was found. The source's dated open label is not evidence of current novelty. These searches are not exhaustive priority certification. References are cited in the manuscript; third-party papers are retained for local research and excluded from the upload archive. ## Exact diagnostics ```json { "arithmetic": "integers and fractions.Fraction", "conjectured_constant_strict_lower": "98/101", "energy_over_weighted_norm": "62/(27*pi)", "energy_ratio_strict_upper": "273/284", "finite_numerical_sampling_used": false, "formal_verification": false, "gradient_identity": true, "mean_derivative_squared": "1/18", "mean_g_squared": "3/8", "parameter_interval": "99/100 <= s < 1", "rational_power_checked_exactly": true, "status": "PASS_EXACT_ALGEBRA", "uniform_gap": "259/28684" } ```