large-wave-effect

Rigorous study of the large-wave effect on periodic discrete chains: a localized impulse on a homogeneous one-dimensional lattice amplifies into a peak that grows with the number of nodes — a genuinely discrete resonance with no continuum counterpart.

One paper (paper/), in two parts built on a shared spectral core (the ring Laplacian L_N, Toeplitz/circulant methods):

• Part I — Linear theory (rigorous core). Amplitude A_N = sup_t max_j |u_j(t)| of the velocity Green's function on the ring; cyclotomic independence, the mod-4 ceiling criterion, the discrete Schrödinger contrast, spectral zeta, operator norms, statistics, and the dimensional threshold.
• Part II — Nonlinear extension. The discrete NLS (Peregrine soliton, modulational instability, rogue-wave statistics), discrete breathers and self-trapping, FPUT recurrence, and the route to chaos (Lyapunov exponent).

Every quantitative claim is reproduced by a standalone, version-controlled script.

Main results

Proved results are kept strictly separate from numerical evidence and the (sharply reduced) open problems.

Linear theory (Part I).

• Order law (theorem, all N). A_N ~ (1/π) ln N for every N — the central result, formerly a conjecture. The lower bound comes from a palindromic argument: the first φ(2N)/2 frequencies 2 sin(πr/N) are ℚ-independent, so a (1−o(1)) fraction of the ceiling can always be Kronecker-aligned. Precisely A_N = (1/π) ln N + O(ln ln ln N).
• Exact amplitude / saturation (theorem). A_N = U_N = (1/π) ln N + (1/π)(γ + ln(2/π)) − π/(72 N²) + ⋯ iff N is prime or a power of two; equivalently every integer frequency relation has coefficient-sum ≡ 0 (mod 4) (antipodal variant for even N). Via an explicit root-of-unity obstruction, {N : A_N = U_N} = {primes} ∪ {2^m} (density zero). The orbit-closure subtorus has dimension exactly φ(2N)/2. The theorem is cross-checked by exact Python/GAP scans (verify_classification_proof.py, exact/gap/scan_classification.g).
• Composite deficit. The order theorem gives the general upper envelope 0 ≤ U_N − A_N = O(ln ln ln N). The strict gap A_N < U_N for every composite non-power-of-two follows from the saturation classification, and it is quantitative (Proposition deficit-lower): one relation per residue class gives U_N − A_N ≥ 1/(16p²) for p the least odd prime factor, the vertex-disjoint triplets give U_N − A_N > 0.043 whenever 3 | N (limit ln 3/(4π)), and for N = 2p the deficit is (π³/16) N⁻² (1 + O(1/N)) (verify_defect_lower_bound.py, exact check exact/pari/defect_lower_bound.gp). Sharper small-N gaps are certified by a Lipschitz grid (N = 6, 9, 12, 15) and by exact rational sum-of-squares certificates (exact/pari/defect_sos.gp): U_N − A_N ≥ 0.06717197, 0.14754945, 0.11208183 for N = 9, 15, 21, within 10⁻⁷ of the optimizer. The sharper excess law U_N − A_N = (1/π) ln(m/φ(m)) + O(1) for odd part m remains open beyond prime-power odd part. The per-family limits defect_∞(m) = lim_a (U − A)(m · 2^a) are computed to 10⁻⁸ (0.10368…, 0.03824…) as numerical constants with no closed form.
• Discrete Schrödinger. A localized state is never amplified (unitarity), yet B_N = sup_t ‖e^{−it L_N}‖_{ℓ^∞ → ℓ^∞} = Θ(√N), with liminf B_N/√N ≥ c_0/√2 = 0.861 proved (c_0 = (2/π)^{3/2} · B(1/2, 3/4)). Infinite-time saturation (theorem). B_N = √N iff the orbit closure — a subtorus of dimension exactly φ(N)/2 — contains a flat phase vector (ε-uniform mixing of the quantum walk on C_N). This holds for N = p (the Gauss chirp; Godsil–Mullin–Roy) and for N = p² (an even Frank sequence; Cao–Zhu, arXiv:2607.04207), so limsup B_N/√N = 1; it fails for every even N ≥ 6 (a symmetric periodic Golay-pair obstruction: no ε-uniform mixing on even cycles; for N not an even square this also follows from Monterde's bipartite conditions, arXiv:2603.20977, so the even squares 16, 36, 64, … are the new cases). It fails for N = 15 as well (computer-assisted): an exact Nullstellensatz certificate D y⁴ = Σ N_j m_j f_j (2159 terms) puts y⁴ in the flatness ideal, y a coordinate of the orbit-closure torus (exact/julia/c15_flat_certificate.jl, re-checked in exact/pari/c15_flat_check.gp); the Gröbner basis in Cao–Zhu's Thm 3.2 implies it too, as it lives on the whole closure. It fails for N = 21 (computer-assisted, new): in coordinates adapted to Z_21 = Z_3 × Z_7 the orbit closure is the 6-torus x(1,a) = y_a, x(2,a) = y_{-a}, x(0,a) = 1/(y_a y_{-a}), Π y_a = 1, and an exact invariant certificate 1 = g_0 I_0 + Σ_u u·(g_1 I_1) + Σ_u u·(h_1 II_1) over the autocorrelations (4005 orbit coefficients, 614-digit denominator) shows they have no common zero (numerics/bn_flat_certificate_3q.py, re-checked from the definitions in exact/pari/c3q_flat_check.gp; the same code reproves N = 15). So C_21 admits no uniform mixing, exact or ε-approximate — the case Gray–Patel–Young (arXiv:2607.28345) name as the next open one. It fails for N = 27 (computer-assisted, new), the first prime cube, which Cao–Zhu leave open: in the coordinates of μ_r = 2cos(2πr/27) in the basis 1, μ_1, …, μ_8 the orbit closure is the 9-torus, and a degree-3 Positivstellensatz ρ = m*Qm + Σ g_l R_l + κ with λ_min(Q) ≥ 5.4999 on the 3139 monomials of word length ≤ 3 and ‖ρ‖₁ < 150 shows that the autocorrelations R_1, …, R_13 have no common zero there (the SDP is solved by Douglas–Rachford splitting, numerics/b27_psatz_certificate.py; the certificate exact/pari/b27_psatz_cert.gp.gz is checked exactly by numerics/verify_b27_psatz.py, the identity also from the definitions by exact/pari/b27_psatz_check.gp). The best value found there is 0.99992, so the gap is below 10⁻⁴. That B_N = √N iff N is a prime or a prime square is a conjecture: the other odd composites that are not prime squares (from N = 33 on) are open; numerically B_N/√N < 1 there (verify_bn_saturation.py, exact/pari/bn_saturation.gp; a second implementation of the prime and prime-square cases in verify_bn_prime_flat.py, exact/pari/bn_prime_flat.gp, exact/pari/bn_prime_square.gp). Windowed constant. On the dispersive window t = O(N) the constant splits by parity — numerically even N → c_0/√2, odd N → β_odd = 0.928…, an elliptic-integral constant for which no elementary closed form was found. For odd N and each fixed s = 2t/N ≤ 1 the limiting ℓ¹ profile is the explicit elliptic-integral function F(s) (one-copy term A(s) = ₂F₁ in closed form), and its maximizer max F = F(1) = β_odd is proved by validated interval arithmetic on all of [0, 1] (verify_bn_profile_rigorous.py: a chord minorant β_odd ≥ 0.92735 from AGM-enclosed elliptic integrals, analytic bounds on [0, 0.575], an enclosed tangent majorant on [0.575, 0.90], and F′(s) > 0 on [0.90, 1]). For t > N/2 the whole N → ∞ reduction at each fixed s is rigorous as well — joint equidistribution of the multi-copy Debye phases and its uniform-in-u transfer to the ℓ¹ constant F(s) (panels away from the band edges, O(ε^{3/4}) edge strips; no rate) — leaving two steps: convergence uniform in s (on t ≤ N/2 as on t > N/2), and a single validated residual, the Gaussian-excess inequality E|Z| − E|G| ≤ K⋆ · Σ_2^{−3/2} Σ_ℓ a_ℓ⁴ (K⋆ = (h(1) − √π/2)/2) for the profile's balanced Debye configurations (the live set is an interval of consecutive integers, so the even/odd channel counts differ by ≤ 1; sharp at the two-copy pair — imbalanced configs violate the unrestricted bound) whose integrated majorant clears β_odd — and whose residual is now proved for every configuration with at most three copies by rigorous Arb certificates (the singleton-channel band, dagger_rigorous.py: 111 boxes, worst R ≤ 0.9205 K⋆; the near-pair basin, dagger_nearpair.py: 1231 boxes, convexity around the pair; coverage checked in exact rational arithmetic by verify_dagger_rigorous.py and verify_dagger_nearpair.py), and validated beyond three copies (see open problem (1)). So the windowed constant limsup_N B_N(SN/2)/√N = β_odd (fixed S) holds modulo the rest of this inequality and the s-uniform transfer. On the Dirichlet segment d_j^N ~ (4/π²) ln j for N+1 prime/2^m (the constant C = 4/π² is the published Myshkis–Filimonov 2003 value). Three large-wave laws on one Laplacian.
• Dimension, products & circulants (theorem). The large wave lives exactly at spectral dimension d_s = 1 (O(1) for d_s ≥ 2). A graph-agnostic order criterion plus the embedded ring spectrum give A = Θ(ln N) on every Cartesian product C_N □ H (ladders, prisms, tubes) and, beyond products, on the Möbius ladder C_{2N}({1, N}) via its embedded-ring branch. Multi-jump circulants C_N(S) (|S| ≥ 2) instead saturate (A_N = U_N): for S = {1, 2}, {1, 3}, {1, 2, 3} this is decided exactly for every 5 ≤ N ≤ 126 (exact/pari/kjump_relations.gp). The radical frequencies are not always ℚ-independent — C_N({1, 3}) has two relations at each of N = 28, 56, 84, 112 — but every relation has coefficient sum 0, so the mod-4 test passes. For the nearest case S = {1, 2} saturation is proved for every prime p ≥ 7: up to squares α_r = −(3 + c_r), and ∏ (3 + c_r) = F_p is not a square (Cohn), so by Kummer theory no subset product of the α_r is a square in ℚ(ζ_p) (exact/pari/kjump_nnn_primes.gp checks the identities for p ≤ 211; exact/pari/kjump_kummer.gp certifies the conclusion independently for 7 ≤ p ≤ 97 by quadratic characters). It is an infinite-time ceiling: the finite-time amplitude is Diophantine-limited (recurrence time exponential in N).

Nonlinear extension (Part II), focusing DNLS — now a rigorous chain. The linear large wave persists under weak nonlinearity (Duhamel); the modulational-instability band σ_Q = √(λ_Q (2γ A² − λ_Q)) is read off the L_N spectrum; localized breathers exist (a finite-dimensional variational argument, no MacKay–Aubry continuation); in the anti-continuum regime the site-centred breather is orbitally stable (Vakhitov–Kolokolov / Grillakis– Shatah–Strauss) and the bond-centred one unstable, with both verdicts certified at Ω = 2 in exact rational arithmetic (verify_dnls_breather_cap.py, N = 5 site, N = 6 bond). Full dispersal of a single-site seed is energetically forbidden for γP > 4 (it would force ⟨L_N u, u⟩ → 2P − (γ/2) P² < 0; 4 = 2⟨L_N δ, δ⟩ is the infinite-lattice bandwidth), and for γP > 8 the excitation stays on its site (Shi–Wong prove this above 9). The constant 4 is energy-sharp, not dynamically sharp: dispersal is proved for γP < 0.43, and numerically the seed disperses up to γP ≈ 3.4 and keeps a localized remnant from ≈ 3.5. The remaining strongly nonlinear regime (saturation, rogue-wave focusing), FPUT recurrence and the route to chaos remain computational/exploratory.

Open problems (each reduced to a clean statement, with the proven part marked). (1) The Schrödinger constant. Over infinite time: is B_N < √N for every odd composite N that is not a prime square (known for N = 15, 21, 27), and is liminf B_N/√N = 1? On the dispersive window: the windowed constant limsup_N sup_{t ≤ SN/2} ‖K_N(·,t)‖_1/√N = β_odd for each fixed S. For each fixed s ≤ 1 (odd N) the profile maximizer is proved (max F = β_odd, validated interval arithmetic), and for t > N/2 the entire fixed-s N → ∞ reduction is rigorous (joint equidistribution of the Debye phases and its uniform-in-u transfer to F(s)); a supremum over t further needs convergence uniform in s. What remains beyond that is a single validated inequality: the Gaussian-excess (expected-modulus) bound E|Z| − E|G| ≤ K⋆ · Σ_2^{−3/2} Σ_ℓ a_ℓ⁴ for the profile's balanced Debye configs (live set = consecutive integers ⇒ even/odd counts differ by ≤ 1; sharp at the two-copy pair — imbalanced configs violate it), whose integrated majorant clears β_odd. Whether polynomial windows t ≤ N^C keep β_odd is open. The continuum certificate is operational (verify_dagger_continuum.py). The equal-amplitude singleton zone near a_1 = a_3 ≈ 0.73 that the θ-ball middle could not reach (its rigorous radius blows up past t = 12; the true ratio there is only ≈ 0.88, so it was enclosure-limited, not tight) is now covered by a θ-Taylor + t-collocation enclosure (middle_far_point, validated against the reference in verify_middle_far.py): the previously-failing boxes certify (e.g. (0.70, 0.70) → 0.9915; the hardest boxes sit at the high-λ edge a_3 ≈ 0.61 and close under one refinement, worst leaf 0.9999). That validated 136-box tiling (run_dagger_campaign.py) took the far t-collocation remainder and the stencil's third derivatives from finite differences and maximized the quadratic model on a float grid. It is now superseded by a rigorous certificate (dagger_rigorous.py, run_dagger_campaign.py --rigorous): in u = a_1², v = a_3² the signed excess on [0, 24] is an exact degree-10 Taylor polynomial of a Gauss–Legendre × periodic-midpoint quadrature with analytic (Bernstein-ellipse, strip) error bounds and a Cauchy remainder, its box maximum minus K⋆ D is bounded exactly, the tail uses |J_0(x)| ≤ min(1, √(2/(πx))), and every bound is an Arb ball. All 111 boxes meeting the band prove (worst R ≤ 0.9205 K⋆, 33 s on 8 cores); verify_dagger_rigorous.py re-derives every verdict from the stored balls, recomputes the worst boxes bit for bit, checks the band coverage in exact rational arithmetic and pins hashes and versions in numerics/dagger_rigorous_manifest.json. The near-pair basin λ ≥ 0.82 of the same three-copy family is proved by dagger_nearpair.py: the exact 1-D form E|Z| = 2/π + (1/π) ∫ ĝ(t) (1 − J_0(a_1 t) J_0(a_3 t)) t^{−2} dt (slice b = 1) with the odd-copy kernel ĝ(s) = 2 − 2s I_1(s/2) K_0(s/2), a closed-form leading part, Gauss–Legendre with Bernstein-ellipse bounds, certified convexity of Φ on a rectangle around the pair (308 Hessian boxes) and 923 value boxes elsewhere (1231 boxes, ~75 s on 12 workers); verify_dagger_nearpair.py re-decides every box, recomputes the tightest ones bit for bit, checks the coverage in exact rational arithmetic and pins hashes in numerics/dagger_nearpair_manifest.json. So the inequality is proved (computer-assisted) for every configuration with at most three copies; it stays validated on configurations with more than three copies. The window part of problem (1) is reduced to the s-uniform transfer plus that piece. (2) The excess lemma A_N ≤ L_pre + O(1) (equivalent to a rigorous growing deficit bound) — proved when the odd part of N is a prime power (N = 2p sharp at 1/(2N), N = 4p), and unconditional for any fixed ω(m) via the Kronecker sandwich |A_N − L_pre| ≤ U_N − L_pre = (1/π) ln(m/φ(m)). The open content is an absolute constant uniform as ω(m) → ∞. A proved building block is the per-relation bound (Lemma; verify_excess_decomposition.py): a single support-3 relation ω_c = ω_a + ω_b contributes excess g(b_a, b_b, b_c) ≤ b_c/2, from g(α, β, γ) ≤ (γ²/2)(1/α + 1/β) and 1/b_a + 1/b_b = 1/b_c. The obstruction is the uniform coupling of Θ(N) relations (comb generators + shared phases), the sup-side counterpart of McGehee–Pigno–Smith/Konyagin in the Chowla cosine-extremum circle (where Jin–Milojević–Tomon–Zhang and Bedert recently proved the first polynomial bounds, n^{1/10−o(1)} and n^{1/5−o(1)}, in the opposite direction; Chowla's conjectured √n is still open). (3) A rigorous dispersal proof on γP ∈ [0.43, 4): the constant 4 is energy-sharp, not dynamically sharp — numerically the seed disperses only up to γP ≈ 3.4 and keeps a localized remnant from ≈ 3.5 (Johansson–Hörnquist–Riklund: ≈ 3.45), so a proof can reach at most that crossover, while the rigorous drop is proven only to 0.43. The leading-order drift now vanishes provably (along the linear flow the distance-2 current Im(ā_n a_{n+2}) ≡ 0, so the staggered momentum drifts only at O(γ²) — Maxima + Rocq StaggeredCurrent.v, verify_modified_energy_window.py); what remains is the O(γ²) remainder and the nonperturbative approach to the crossover. (4) Condition (B) of the order criterion for quasi-1D lattices without an embedded ring (the Möbius ladder and multi-jump circulants are now settled; the next-nearest-neighbour degeneracy law holds for all couplings g via Conway–Jones). Priority note (resolved): Filimonov's 1992 C. R. Acad. Sci. note has been obtained and compared on the ring — the rational-independence dichotomy (peaks for N prime/2^m, dependence for even composite, by his cyclotomic argument) and the unbounded ceiling are Filimonov's; the rate (1/π) ln N, the explicit constant, the mod-4 saturation classification, and the all-N rank φ(2N)/2 are new here.

Reproduce

Uses uv. Each numerics/verify_*.py is a PEP 723 standalone script (inline dependencies), so run it with uv run --script.

uv run pytest                                  # unit tests for the core package
uv run --script numerics/verify_theorem1.py    # any verify_*.py re-checks one claim
uv run --script numerics/make_figures.py       # regenerate paper/figures/*.pdf
cd paper && latexmk -pdf large-wave-effect.tex # build the paper (run twice to resolve refs)

Headline checks: verify_order_theorem.py (the order law + palindromic prefix lemma), verify_classification_proof.py (saturation classification), verify_bn_saturation.py (infinite-time Schrödinger saturation), verify_bn_parity.py / verify_beta_odd.py / verify_bn_profile_max.py (the windowed Schrödinger constant, its parity split, and the explicit t ≤ N/2 profile maximizer), verify_defect_certified.py (certified A_N < U_N), verify_excess_smallcases.py (excess lemma for prime-power odd part), verify_product_order.py and verify_kjump_order.py (C_N □ H, Möbius ladder, k-jump saturation), the DNLS chain verify_dnls_persistence.py / verify_dnls_mi.py / verify_dnls_breather_stability.py, and verify_selftrapping_transition.py (the energy-sharp γP = 4 bound and the onset of trapping near 3.5).

For the Schrödinger continuum cover: verify_middle_far.py validates the θ-Taylor enclosure that closes the a_1 = a_3 ≈ 0.73 zone, prove_box_parallel.py certifies a single box on all cores, and the full band is tiled by the adaptive, checkpointed campaign: uv run --script numerics/run_dagger_campaign.py --jobs 14 (writes each box to dagger_campaign.jsonl).

Exact-arithmetic + formal layer (Fedora: dnf install gap pari-gp fricas rocq-prover; Julia from juliaup). The linear-theory arithmetic is reproduced floating-point-free in GAP and PARI/GP, symbolically in FriCAS, the N = 15 exclusion B_15 < √15 is certified in Julia and re-checked in PARI/GP, the N = 21 exclusion B_21 < √21 is certified with python-flint and re-checked in PARI/GP, the N = 27 exclusion B_27 < √27 is a Positivstellensatz checked exactly with python-flint and in PARI/GP, and the mod-4 criterion is machine-checked in Rocq — see exact/ and formal/rocq/. One driver runs the lot and cross-checks against the Python core (missing tools are skipped, not failed):

uv run --script numerics/verify_exact_layer.py   # GAP + PARI + Rocq + Python cross-check
gap -q --nointeract exact/gap/scan_classification.g   # finite exact cross-check to N<=800
rocq compile formal/rocq/Mod4Criterion.v              # machine-checked criterion certificate

Layout

src/large_wave_effect/   shared spectral core (ring, cyclotomic, Schrödinger)
numerics/                verify_*.py (one per claim) + make_figures.py
exact/                   GAP / PARI / Julia / FriCAS exact + symbolic checks (gap/, pari/, julia/, fricas/)
formal/rocq/             Rocq machine-checked mod-4 criterion certificate
tests/                   pytest suite for the core package
paper/
  large-wave-effect.tex  root: preamble, front matter, \part dividers, bibliography
  sections/linear.tex    Part I  — linear theory
  sections/nonlinear.tex Part II — nonlinear extension
  figures/               generated by make_figures.py
  large-wave-effect.pdf  compiled paper

Provenance

This work continues the author's 2015 undergraduate diploma on the large-wave effect; the original C simulation that first observed the peaks is not included here.

License

Apache License 2.0 — code, paper text, and figures. See LICENSE and NOTICE.

Citation

Ilia Gradina, Large-wave amplification on periodic discrete lattices: linear theory and a nonlinear extension (2026). DOI: 10.5281/zenodo.20795861 (concept DOI — always resolves to the latest version). See CITATION.cff.