Foundations from Minimal Change

Research article · Jinku Guo

Coarse-Graining and Long-Range Response in Generated Systems

arXiv SSRN Zenodo Read manuscript PDF Files and citation

Author: Jinku Guo

Work type: Research article

arXiv: 2608.01989

SSRN: Abstract and download page

Zenodo: 10.5281/zenodo.22093524

Repository snapshot: 23 October 2026

Research overview

Which information can a reduced description discard?

A coarse description replaces a detailed state by fewer variables. Its usefulness depends on the questions those variables can still answer. Matching an equilibrium value does not establish that a reduced model will respond correctly to a new source or preserve a long-range interaction near a critical point. This paper develops coarse-graining around the preservation of future observations and source-dependent response.

The generating framework gives this problem a natural formulation. A state comes with allowed operations, source insertions and observations. Two detailed states belong to the same reduced state precisely when the permitted future procedures cannot distinguish them. This defines an observation-sufficient quotient and its universal property. Information is discarded according to its role in the ensuing construction.

Exact descent and quantitative defects

The first results establish when an evolution and its observations descend to a quotient. In the smooth setting, a bounded surjective linear coarse map between Banach spaces provides the relevant topology. Fibrewise compatibility of the update and observation maps yields a unique continuously differentiable reduced realization.

The response contains four components. The linearized state update determines propagation. A source map specifies how an intervention enters. An observation map specifies what is measured. A direct term records the immediate source response. Exact joint descent preserves this complete response throughout the domain where the associated inverse exists.

For approximate coarse-graining, the paper derives a four-defect identity. It separates the discrepancies in the update, source, observation and direct term while retaining their interaction with the resolvent. The estimates account for displacement of the reference solution and the sensitivity of nonnormal operators. They therefore apply to reductions for which a small entrywise change in the dynamical matrix would otherwise give little information about the measured response.

Contour and Riesz-projection arguments carry these estimates to visible critical poles and residues. In reciprocal Hilbert-space realizations, a positive crossing on a finite-dimensional critical subspace produces a positive finite-rank residue. The source overlap determines which critical directions remain visible. This provides a direct bridge between mathematically controlled reduction and physically observable long-range behavior.

Memory follows from the retained dynamics

Eliminating a hidden variable generally produces both a memory term and a term determined by its initial state. The paper derives these contributions explicitly. They describe how discarded information returns to a retained observation during subsequent evolution.

Reflecting heat-flow and arithmetic-merge examples make this mechanism calculable. Truncating memory changes the pole and residue in different ways. The residue estimate acquires an additional memory-length factor because it involves a differentiated remainder. This distinction is important near criticality, where an apparently accurate stationary reduction can still distort the strength of an observable interaction.

The paper also classifies Gaussian operations in the stated centered, orthogonally invariant, continuous, finite-variance self-similar setting. External averaging, internal relative coordinates and quotient formation have different effects on the retained data. Their separation prevents a smoothing operation from being mistaken for an exact dynamical reduction.

A positive spectral realization connects the reduced response to vacuum spectral measures. With the source pairing and continuation conventions held fixed, the zero-mass atom agrees with the residue selected by the reduced channel. The comparison consequently reaches from the quotient construction to the spectral meaning of a long-range exchange.

The distinctive result

This work turns coarse-graining into a response-preservation problem with explicit mathematical tests. Its main results provide:

The practical consequence is a sharper standard for effective models. A model can be assessed by what its retained variables generate under later operations and by how accurately they reproduce the relevant response. This is especially valuable when the feature of interest is a critical pole, a universal coupling or another observable that depends on more than the present coarse state.

Research subjects and results

Paper keywords

Coarse-graining; Model reduction; Linear response theory; Pole stability; Spectral residues; Memory kernels.

Primary subjects

Coarse-graining; Model reduction; Linear response theory; Pole stability; Spectral residues; Memory kernels; Non-Markovian dynamics; Effective dynamics; Resolvent estimates; Schur complements; Spectral approximation; Error bounds; Long-range interactions.

Research subjects

Reduced-order models; Dynamical systems; Source coupling; Input-output systems; Stationary response; Dynamical closure; Hidden variables; Finite-dimensional critical subspaces; Spectral poles; Finite-rank residues; Reciprocal systems; Vacuum covariance; Positive spectral measures; Response preservation; Memory truncation.

Methods

Projection methods; Operator approximation; Analytic Fredholm theory; Perturbation theory; Resolvent identities; Contour estimates; Block operator matrices; Common-space reconstruction; Differentiable factorization; Defect estimates; Operator norm bounds; Spectral projection; Channel elimination; Matrix-valued response; Finite-memory approximation.

Results and research questions

Response-preserving coarse-graining; Exact error identity for reduced response; Pole and residue stability under model reduction; Geometric convergence with memory length; Positive finite-rank response residues; Source overlap and observable critical modes; Conditions for memoryless reduced dynamics; Identification of response residues with massless spectral atoms.

Chinese subject terms

粗粒化; 模型降阶; 线性响应; 非马尔可夫动力学; 记忆核; 长程响应; 极点稳定性; 留数误差; Schur补; 算子逼近; 约化动力学; 自由度消去.

Public file

This directory provides the public PDF. LaTeX sources and editable research files remain outside this repository.

Citation and archival records

Jinku Guo. Coarse-Graining and Long-Range Response in Generated Systems. 2026. doi:10.48550/arXiv.2608.01989.

The repository PDF is identified by its snapshot tag and checksum. The public article records identify the work or their respective deposited versions.

Research context

Related research and results · Programme overview · Proofs and calculations