The lowest-mode Alcaraz–Ramos prediction fails

Abstract

The homogeneous N=3 lowest quasienergy cannot have the predicted conformal limit.

Definition 1.1 (The polynomial recurrence).

$$((\mathrm{poly}\left(0\right)=1)\wedge ((\mathrm{poly}\left(1\right)=1)\wedge (\mathrm{poly}\left(2\right)=1)))\wedge (\forall n\in \mathbb{N},\mathrm{poly}\left(n+3\right)=\mathrm{poly}\left(n+2\right)-\mathrm{Polynomial}.X\cdot \mathrm{poly}\left(n\right))$$

Formalization. D5/S3/Quantum/SpinChains/AlcarazRamosConformalTowerRefutation.poly (✓ std3).

Citation. Francisco C. Alcaraz, Lucas M. Ramos (2024). Conformally invariant free parafermionic quantum chains with multispin interactions. DOI: 10.1103/PhysRevE.109.044138. URL: https://arxiv.org/abs/2402.09936v3.

Commentary.

Page 4, Eqs. (26)–(27): “P_M^{(N)}(z) = P_{M−1}^{(N)}(z) − zλ_M^N P_{M−N}^{(N)}(z),” followed by “with the initial condition” and “P_M^{(N)}(z) = 1, for M ≤ 0,”. Here N=3, λ_j=1, and poly(n)=P_{n−2}; the three initial indices represent nonpositive M. Polynomial.X is the polynomial indeterminate.

Definition 1.2 (Nonnegative lattice sizes).

$$\forall M\in \mathbb{N},P\left(M\right)=\mathrm{poly}\left(M+2\right)$$

Formalization. D5/S3/Quantum/SpinChains/AlcarazRamosConformalTowerRefutation.P (✓ std3).

Citation. Francisco C. Alcaraz, Lucas M. Ramos (2024). Conformally invariant free parafermionic quantum chains with multispin interactions. DOI: 10.1103/PhysRevE.109.044138. URL: https://arxiv.org/abs/2402.09936v3.

Commentary.

Page 4, Eq. (26): “P_M^{(N)}(z) = P_{M−1}^{(N)}(z) − zλ_M^N P_{M−N}^{(N)}(z),”. P(M) is P_M^{(3)} at homogeneous couplings λ_j=1, expressed through poly with its index shifted by two. Thus P(0)=1.

Definition 1.3 (Ordered positive-root quasienergies).

$$\forall M\in \mathbb{N},\forall i\in \mathbb{N},\mathrm{quasienergy}\left(M,i\right)=(\mathrm{List}.\mathrm{getD}\left(\mathrm{Multiset}.\mathrm{sort}\left(((a:\mathbb{R})\mapsto ((b:\mathbb{R})\mapsto a\ge b)),\mathrm{Multiset}.\mathrm{filter}\left(((z:\mathbb{R})\mapsto 0<z),(P\left(M\right)).\mathrm{roots}\right)\right),(i-1),0\right){)}^{-\frac{1}{3}}$$

Formalization. D5/S3/Quantum/SpinChains/AlcarazRamosConformalTowerRefutation.quasienergy (✓ std3).

Citation. Francisco C. Alcaraz, Lucas M. Ramos (2024). Conformally invariant free parafermionic quantum chains with multispin interactions. DOI: 10.1103/PhysRevE.109.044138. URL: https://arxiv.org/abs/2402.09936v3.

Commentary.

Page 4, immediately before Eq. (26): “Apart from the zero modes they are given by Λ_k = 1/z_k^{1/N}, where z_k are the roots (P_M^{(N)}(z_k)=0) of the characteristic polynomial”. Eq. (38), pages 5–6, orders the quasienergies by i=1,2,… . Positive real roots are sorted in decreasing order with multiplicity. The index i−1 uses natural subtraction, truncated at zero. List.getD supplies zero when the index is absent. The outer power is Real.rpow, so the exponent is the real number −1/3.

Definition 1.4 (The fixed-index leading prediction).

$$\mathrm{claim}=(\forall i\in \mathbb{N},(1\le i)\Rightarrow (\mathrm{Tendsto}\left(((m:\mathbb{N})\mapsto (3\cdot m:\mathbb{R})\cdot \mathrm{quasienergy}\left((3\cdot m),i\right)),\mathrm{atTop},\mathrm{nhds}\left(\mathrm{Real}.\mathrm{pi}\cdot (2\cdot \mathrm{Real}.\mathrm{sqrt}\left(3\right))\cdot (\frac{7}{6}+(i:\mathbb{R})-1)\right)\right)))$$

Formalization. D5/S3/Quantum/SpinChains/AlcarazRamosConformalTowerRefutation.claim (✓ std3).

Citation. Francisco C. Alcaraz, Lucas M. Ramos (2024). Conformally invariant free parafermionic quantum chains with multispin interactions. DOI: 10.1103/PhysRevE.109.044138. URL: https://arxiv.org/abs/2402.09936v3.

Commentary.

Page 5, Eq. (38), continuing on page 6: “1/z_i^{1/N} = ε_i = π A_i^{(N)}/M = π A^{(N)}/M (x_s^{(N)} + i − 1),” and “for i = 1, 2, …”. Page 6: “Our numerical solutions for polynomial roots with M up to 10⁹ corroborate the conformal invariance prediction (38) for the roots. We conjecture the following exact values for N = 2, 3, 4 and 6”. Eqs. (39) and (41) include “A^{(3)} = 2√3” and “x_s^{(3)} = 7/6 (mod(M,3)=0)”. The leading behaviour is expressed as Tendsto along M=3m for every fixed natural i≥1. The coefficients involving i are real; the lattice-size multiplication 3*m is natural before its cast to ℝ. The TeX labels (4.7), (4.8), (4.10) correspond to printed Eqs. (38), (39), (41).

Theorem 1.5 (Refutation at the lowest mode).

$$\neg (\forall i\in \mathbb{N},(1\le i)\Rightarrow (\mathrm{Tendsto}\left(((m:\mathbb{N})\mapsto (3\cdot m:\mathbb{R})\cdot \mathrm{quasienergy}\left((3\cdot m),i\right)),\mathrm{atTop},\mathrm{nhds}\left(\mathrm{Real}.\mathrm{pi}\cdot (2\cdot \mathrm{Real}.\mathrm{sqrt}\left(3\right))\cdot (\frac{7}{6}+(i:\mathbb{R})-1)\right)\right)))$$

Proof. Machine-checked in Lean as D5/S3/Quantum/SpinChains/AlcarazRamosConformalTowerRefutation.result (✓ std3). ∎

Resolves. Problems/alcaraz-ramos-2024-conformal-tower-lowest-mode (refuted) by D5/S3/Quantum/SpinChains/AlcarazRamosConformalTowerRefutation.result.

Source. Repository-derived.

Acknowledgement. Francisco C. Alcaraz, Lucas M. Ramos (2024). Conformally invariant free parafermionic quantum chains with multispin interactions. DOI: 10.1103/PhysRevE.109.044138. URL: https://arxiv.org/abs/2402.09936v3.

Commentary.

It suffices to use i=1. The recurrence gives the coefficient formula (−1)^j choose(n−2j,j). Reversing the finite sum at M=3m gives a scaled polynomial Q_m. Binomial asymptotics and dominated convergence imply Q_m(v_m) → F(u) whenever v_m → u and eventually |v_m|≤5, where F(u)=Σ_{k≥0}(−1)^k u^{3k}/(3k+2)!. The alternating-series lower bound through k=5 proves F(u)>0 throughout [4.2321,4.2323]. Bounds on π and √3 place 7π/(3√3) in that interval. The predicted quasienergy limit would produce positive roots with Q_m(v_m)=0 eventually, forcing F(7π/(3√3))=0, a contradiction. The argument does not establish existence or the exact value of the actual lowest-mode limit, equal spacing of its first zeros, other N or residue classes, or asymptotic large-index spacing.

References

• Truth anchor: D5/S3/Quantum/SpinChains/AlcarazRamosConformalTowerRefutation.P
• Truth anchor: D5/S3/Quantum/SpinChains/AlcarazRamosConformalTowerRefutation.claim
• Truth anchor: D5/S3/Quantum/SpinChains/AlcarazRamosConformalTowerRefutation.poly
• Truth anchor: D5/S3/Quantum/SpinChains/AlcarazRamosConformalTowerRefutation.quasienergy
• Truth anchor: D5/S3/Quantum/SpinChains/AlcarazRamosConformalTowerRefutation.result