The conjecture of OEIS A362534 is false at n = 19

Abstract

At n = 19 the adiabatic polarization-transfer bound g(19) = 215955/131072 has numerator 215955, while the ratio f(19)/g(19) = 92378/43191 has denominator 43191, so the conjecture of OEIS A362534 fails.

Definition 1.1 (The symmetry-constrained bound).

$$\forall n\in \mathbb{N},f\left(n\right)=\mathrm{ite}\left(\mathrm{NatMod}\left(n,2\right)=0,2^{1-n}\cdot n\cdot \mathrm{choose}\left(n-1,\mathrm{NatDiv}\left(n,2\right)\right),2^{1-n}\cdot n\cdot \mathrm{choose}\left(n-1,\mathrm{NatDiv}\left(n-1,2\right)\right)\right)$$

Formalization. D5/S0/Certificates/SabbaPolarizationTransferRefutation.f (✓ std3).

Citation. Mohamed Sabba (2023). OEIS A362534, Numerators of the ratio of the symmetry-constrained bound to the adiabatic bound on polarization transfer in AXn spin-1/2 systems. URL: https://oeis.org/A362534.

Commentary.

For even n the bound is 2^(1-n) n binomial(n-1, n/2), for odd n it is 2^(1-n) n binomial(n-1, (n-1)/2), as rational numbers; NatDiv and NatMod are the natural quotient and remainder.

Definition 1.2 (The adiabatic bound).

$$\forall n\in \mathbb{N},g\left(n\right)=\mathrm{ite}\left(\mathrm{NatMod}\left(n,2\right)=0,2\cdot (1-2^{-n}\cdot \mathrm{choose}\left(n,\mathrm{NatDiv}\left(n,2\right)\right)),2\cdot (1-2^{-n}\cdot \mathrm{choose}\left(n,\mathrm{NatDiv}\left(n-1,2\right)\right))\right)$$

Formalization. D5/S0/Certificates/SabbaPolarizationTransferRefutation.g (✓ std3).

Citation. Mohamed Sabba (2023). OEIS A362534, Numerators of the ratio of the symmetry-constrained bound to the adiabatic bound on polarization transfer in AXn spin-1/2 systems. URL: https://oeis.org/A362534.

Commentary.

For even n the bound is 2 (1 - 2^(-n) binomial(n, n/2)), for odd n it is 2 (1 - 2^(-n) binomial(n, (n-1)/2)).

Definition 1.3 (The conjecture of OEIS A362534).

$$claim\iff (\forall n\in \mathbb{N},(1\le n)\Rightarrow (\mathrm{num}\left(g\left(n\right)\right)=\mathrm{den}\left(\frac{f\left(n\right)}{g\left(n\right)}\right)))$$

Formalization. D5/S0/Certificates/SabbaPolarizationTransferRefutation.claim (✓ std3).

Citation. Mohamed Sabba (2023). OEIS A362534, Numerators of the ratio of the symmetry-constrained bound to the adiabatic bound on polarization transfer in AXn spin-1/2 systems. URL: https://oeis.org/A362534.

Commentary.

For every n at least one, the numerator of g(n) equals the denominator of f(n)/g(n), both in lowest terms (num and den of a rational number).

Theorem 1.4 (A counterexample at n = 19).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S0/Certificates/SabbaPolarizationTransferRefutation.result (✓ std3). ∎

Resolves. Problems/sabba-2023-a362534-polarization-transfer-refutation (refuted) by D5/S0/Certificates/SabbaPolarizationTransferRefutation.result.

Source. Repository-derived.

Acknowledgement. Mohamed Sabba (2023). OEIS A362534, Numerators of the ratio of the symmetry-constrained bound to the adiabatic bound on polarization transfer in AXn spin-1/2 systems. URL: https://oeis.org/A362534.

Commentary.

Since 19 is odd, f(19) = 2^(-18) 19 binomial(18, 9) = 230945/65536 and g(19) = 2 (1 - 2^(-19) binomial(19, 9)) = 215955/131072. Their ratio is 92378/43191 in lowest terms, whose denominator 43191 differs from the numerator 215955 = 5 times 43191 of g(19).

References

• Truth anchor: D5/S0/Certificates/SabbaPolarizationTransferRefutation.claim
• Truth anchor: D5/S0/Certificates/SabbaPolarizationTransferRefutation.f
• Truth anchor: D5/S0/Certificates/SabbaPolarizationTransferRefutation.g
• Truth anchor: D5/S0/Certificates/SabbaPolarizationTransferRefutation.result