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Non-attained Pell phase supremum and unit asymptotic tail decay for two-level trees

Abstract

Across all two-level rooted trees T(k, l) with k >= 2 and l >= 1 and all real times t, the scaled root-to-leaf Laplacian transfer probability n^(3/2) P_{k, l}(t) has global supremum C_* = 2 + 6 sqrt(5) / 5, realized already by the time supremum on the single five-vertex tree T(2, 1) along an explicit Pell phase-alignment sequence t_j -> +infinity yet strictly unattained at every finite time because of an integer parity obstruction. Restricting to trees of order n >= N drives both the scaled spectral-bound tail supremum and the scaled transfer-probability tail supremum to 1 as N -> +infinity, yielding uniform (1 + epsilon) n^(-3/2) decay while ruling out O(n^(-beta)) decay for every exponent beta > 3/2 via an infinite even-parameter family attaining peak root-to-leaf transfer at time pi.

Definition 1.1 (Global set of scaled root-to-leaf transfer probabilities).

Formalization. D5/S3/Quantum/Dynamics/LaplacianTwoLevelTreeSharpTransferDecay.scaledTransferSet (✓ std3).

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

The set scaledTransferSet collects all real values n^(3/2) P_{k, l}(t) as (k, l) ranges over integers k >= 2 and l >= 1, (i, j) ranges over leaves of T(k, l), and t ranges over all real times, where n = treeOrder k l = 1 + k(l + 1).

Definition 1.2 (Scaled root-to-leaf transfer set on T(2, 1)).

Formalization. D5/S3/Quantum/Dynamics/LaplacianTwoLevelTreeSharpTransferDecay.scaledTransferSet21 (✓ std3).

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

For a fixed leaf (i, j) of T(2, 1), whose order is n = 5, scaledTransferSet21 i j collects all values 5^(3/2) P_{2, 1}(t) as t ranges over the real line.

Definition 1.3 (Pell-type integer recurrence approximating sqrt(5)).

Formalization. D5/S3/Quantum/Dynamics/LaplacianTwoLevelTreeSharpTransferDecay.pellPair (✓ std3).

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

Starting from (p_0, m_0) = (2, 1), where 2 - sqrt(5) < 0 has unit norm in Z[sqrt(5)], multiplying repeatedly by the totally positive fundamental square unit 9 + 4 sqrt(5) = (2 + sqrt(5))^2 defines (p_{j + 1}, m_{j + 1}) = (9 p_j + 20 m_j, 4 p_j + 9 m_j).

Definition 1.4 (Numerator sequence of the Pell phase approximation).

Formalization. D5/S3/Quantum/Dynamics/LaplacianTwoLevelTreeSharpTransferDecay.pellP (✓ std3).

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

The first coordinate p_j = (pellPair j).1 is even for every j and satisfies p_j - sqrt(5) m_j = (2 - sqrt(5))(9 - 4 sqrt(5))^j -> 0 as j -> +infinity.

Definition 1.5 (Denominator sequence of the Pell phase approximation).

Formalization. D5/S3/Quantum/Dynamics/LaplacianTwoLevelTreeSharpTransferDecay.pellM (✓ std3).

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

The second coordinate m_j = (pellPair j).2 is odd, strictly increasing, and bounded below by 9^j, ensuring that 3 m_j - p_j is an odd integer for every j.

Definition 1.6 (Upper positive root-supported Laplacian eigenvalue on T(2, 1)).

Formalization. D5/S3/Quantum/Dynamics/LaplacianTwoLevelTreeSharpTransferDecay.bStar (✓ std3).

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

On T(2, 1) the trace parameter is S = 5 and the discriminant is d = sqrt(5), so the upper positive root-supported eigenvalue is b_* = (5 + sqrt(5)) / 2, while the lower eigenvalue a_* = (5 - sqrt(5)) / 2 satisfies a_* / b_* = (3 - sqrt(5)) / 2.

Definition 1.7 (Pell phase-alignment times on T(2, 1)).

Formalization. D5/S3/Quantum/Dynamics/LaplacianTwoLevelTreeSharpTransferDecay.approxTime (✓ std3).

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

Setting t_j = 2 pi m_j / b_* aligns the upper frequency exactly at exp(i b_* t_j) = exp(2 pi i m_j) = 1 while driving the lower frequency a_* t_j = pi(3 m_j - p_j) + pi(p_j - sqrt(5) m_j) modulo 2 pi toward pi.

Definition 1.8 (Tail set of scaled squared spectral bounds above threshold N).

Formalization. D5/S3/Quantum/Dynamics/LaplacianTwoLevelTreeSharpTransferDecay.tailBoundingSet (✓ std3).

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

For a real order threshold N, tailBoundingSet N collects all scaled squared bounding entries n^(3/2) B_{k, l}^2 over parameters k >= 2, l >= 1 with n = treeOrder k l >= N and leaves (i, j).

Definition 1.9 (Tail set of scaled root-to-leaf transfer probabilities above threshold N).

Formalization. D5/S3/Quantum/Dynamics/LaplacianTwoLevelTreeSharpTransferDecay.tailTransferSet (✓ std3).

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

For a real order threshold N, tailTransferSet N collects all scaled transfer probabilities n^(3/2) P_{k, l}(t) over parameters k >= 2, l >= 1 with n = treeOrder k l >= N, leaves (i, j), and real times t.

Definition 1.10 (Tail supremum of scaled squared spectral bounds).

Formalization. D5/S3/Quantum/Dynamics/LaplacianTwoLevelTreeSharpTransferDecay.tailBoundingSup (✓ std3).

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

The function tailBoundingSup N = sSup(tailBoundingSet N) records the supremum M_B(N) of n^(3/2) B_{k, l}^2 over all two-level trees of order at least N.

Definition 1.11 (Tail supremum of scaled root-to-leaf transfer probabilities).

Formalization. D5/S3/Quantum/Dynamics/LaplacianTwoLevelTreeSharpTransferDecay.tailTransferSup (✓ std3).

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

The function tailTransferSup N = sSup(tailTransferSet N) records the supremum M_P(N) of n^(3/2) P_{k, l}(t) over all two-level trees of order at least N and all real times t.

Definition 1.12 (Discriminant lower-bound threshold as a function of tree order).

Formalization. D5/S3/Quantum/Dynamics/LaplacianTwoLevelTreeSharpTransferDecay.tailThresholdD (✓ std3).

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

For N >= 4, every two-level tree with k >= 2, l >= 1, and 1 + k(l + 1) >= N satisfies k + l + 1 >= 2 sqrt(N - 1) and (k - l)^2 + 4 l >= 2(k + l + 1) - 3, so its discriminant specD k l is bounded below by D_N = max(sqrt(5), sqrt(4 sqrt(N - 1) - 3)).

Theorem 1.13 (Global supremum C_*, strict finite-time non-attainment, and Pell convergence).

Proof. Machine-checked in Lean as D5/S3/Quantum/Dynamics/LaplacianTwoLevelTreeSharpTransferDecay.sharp_transfer_supremum_and_pell_approx (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

The approximation times t_m = 2 pi m_m / b_* are strictly increasing and diverge to +infinity. For every k >= 2, l >= 1, leaf (i, j), and real time t, one has the strict inequality n^(3/2) P_{k, l}(t) < C_: when (k, l) != (2, 1), the uniqueness clause of sharp_global_spectral_transfer_bound gives n^(3/2) B_{k, l}^2 < C_; when (k, l) = (2, 1), expanding B_{2, 1}^2 - P_{2, 1}(t) as the sum of three nonnegative cosine-gap terms 2 c A_0(1 + cos(a_* t)) + 2 c B_0(1 - cos(b_* t)) + 2 A_0 B_0(1 + cos((a_* - b_) t)) shows that equality would force a_ t = (2 u + 1) pi and b_* t = 2 v pi for integers u, v, leading to the impossible integer parity equation (4 v - 6 u - 3)^2 = 5(2 u + 1)^2 whose difference is 4(2(2 v^2 - 6 u v - 3 v + 2 u^2 + 2 u) + 1) != 0. Conversely, at t_j on T(2, 1), the odd parity of 3 m_j - p_j yields 5^(3/2) P_{2, 1}(t_j) = (C_* / 2)(1 + cos(pi delta_j)) with delta_j = (2 - sqrt(5))(9 - 4 sqrt(5))^j -> 0, which converges to C_* and establishes nonemptiness, boundedness above, and least-upper-bound equality sSup(scaledTransferSet) = sSup(scaledTransferSet21 i_0 j_0) = C_*.

Theorem 1.14 (Unit asymptotic tail constant, uniform (1 + epsilon) n^(-3/2) bound, and sharpness).

Proof. Machine-checked in Lean as D5/S3/Quantum/Dynamics/LaplacianTwoLevelTreeSharpTransferDecay.sharp_asymptotic_tail_decay (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Commentary.

For every positive even integer s = 2(m + 1), choosing l_s = s(s + 1) and k_s = l_s + 1 produces odd lower eigenvalue a_s = s^2 + 1, even upper eigenvalue b_s = (s + 1)^2 + 1, and discriminant d_s = 2 s + 1; at t = pi the phases exp(i pi a_s) = -1 and exp(i pi b_s) = 1 align constructively so that P_{k_s, l_s}(pi) = B_{k_s, l_s}^2 and n_s^(3/2) P_{k_s, l_s}(pi) = H(2 s + 1) > 1. Because n_s >= m can be chosen above any threshold M, both tail sets are nonempty and bounded above by C_*, and 1 <= M_P(N) <= M_B(N) holds for all N >= 4. Conversely, specD k l >= D_N and antitonicity of G and H give M_B(N) <= max(G(D_N), H(D_N)); for d >= 128 one has G(d) <= 1 < H(d) <= 1 + 175 / d, while D_N -> +infinity as N -> +infinity, forcing both M_B(N) -> 1 and M_P(N) -> 1. This yields for every epsilon > 0 a threshold N_0 above which P_{k, l}(t) <= B_{k, l}^2 <= (1 + epsilon) n^(-3/2), whereas for any exponent beta > 3/2 the even-s family satisfies n_s^beta P_{k_s, l_s}(pi) = n_s^(beta - 3/2) H(2 s + 1) > n_s^(beta - 3/2) -> +infinity, refuting every global O(n^(-beta)) upper bound.

References