Pith Number

pith:CULIC2ES

pith:2026:CULIC2ESDP2MTT4NSWALAJMOIM

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Color--Phase Separation for Mixed Random Operators in Two-Speed Stochastic Klein--Gordon Systems

Guangqian Zhao

In two-speed stochastic Klein-Gordon systems, color labels organize contractions while phase labels separate Duhamel channels via speed-induced frequency gaps.

arxiv:2604.21884 v4 · 2026-04-23 · math.PR

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Claims

C1strongest claim

The mixed paracontrolled random operators exhibit a color-phase separation mechanism: color labels determine Wick contractions and covariance blocks, while phase labels record the Duhamel-source phase difference between the outer propagator and the stochastic high-frequency leg produced by the low-high bound |ω_i(ℓ+q)−ω_j(ℓ)| ≳ N^α for i≠j.

C2weakest assumption

The model is restricted to diagonal independent colors, fixed distinct speeds, and pure cross interaction u1u2; the phase difference bound |ω_i(ℓ+q)−ω_j(ℓ)| ≳ N^α for i≠j is assumed to hold and is used to separate same-color and cross-color terms (abstract, paragraph on color-phase separation).

C3one line summary

Proves local paracontrolled solutions and Galerkin convergence for a two-speed stochastic Klein-Gordon system with cross interaction via color-phase separation of mixed operators, for 12/13 < α ≤ 1.

References

17 extracted · 17 resolved · 0 Pith anchors

[1] Da Prato and A 2003

[2] Hairer,A theory of regularity structures, Invent 2014

[3] M. Gubinelli, P. Imkeller, and N. Perkowski,Paracontrolled distributions and singular PDEs, Forum Math. Pi3(2015), e6 2015

[4] M. Gubinelli, H. Koch, and T. Oh,Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, J. Eur. Math. Soc.26(2024), 817–874 2024

[5] Y. Deng, A. R. Nahmod, and H. Yue,Random tensors, propagation of randomness, and nonlinear dispersive equations, Invent. Math.228(2022), 539–686 2022

Receipt and verification

First computed	2026-05-20T00:00:39.330454Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

15168168921bf4c9cf8d9580b0258e43210df2bac15cc1267dfdbf050ff9386b

Aliases

arxiv: 2604.21884 · arxiv_version: 2604.21884v4 · doi: 10.48550/arxiv.2604.21884 · pith_short_12: CULIC2ESDP2M · pith_short_16: CULIC2ESDP2MTT4N · pith_short_8: CULIC2ES

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/CULIC2ESDP2MTT4NSWALAJMOIM \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 15168168921bf4c9cf8d9580b0258e43210df2bac15cc1267dfdbf050ff9386b

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b469acd12529c981f470c40e4cabc58416748966e8dc3d80bf18dad195fdb039",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/publicdomain/zero/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-04-23T17:28:45Z",
    "title_canon_sha256": "68ea0d175d966ea48a9740242efd002ff6add42c36111b627fe3cdf591c38d5a"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.21884",
    "kind": "arxiv",
    "version": 4
  }
}