Fixed-Scale Persistence Under a One-Sided Residual Budget

**Title:** Fixed-Scale Persistence Under a One-Sided Residual Budget**Author:** Lee Byoungwoo**Version:** v2.0**Current snapshot DOI:** 10.5281/zenodo.18897304**Date:** March 2026 # OverviewThis record releases **v2.0** of a standalone short-window persistence note extracted from the contradiction-window component of the SAPZ Navier-Stokes package. The note isolates a **fixed-scale persistence mechanism** for a scalar defect profile on a finite time window under a one-sided residual budget. Its purpose is to present a reusable local persistence lemma chain in a form that can be read independently of the broader SAPZ threshold/discharge architecture. This record is **not** the primary proof-package record for the full journal-cut SAPZ Navier-Stokes set. Instead, it is a **standalone local persistence note** focusing on one fixed-scale short-window mechanism. --- ## Main content of the noteThe note studies a finite time interval \(I=[t_0-\tau_0,t_0]\), a family of scale-dependent scalar profiles \(\widehat\delta_\varepsilon\), and a nonnegative residual profile \(r_{\varepsilon,+}\). Its main theorem shows that, under:- a near-threshold envelope condition on a positive-measure terminal-time set,- a one-sided residual budget,- and a common-sequence fixed-scale choice, one can extract:- a single contradiction scale \(\varepsilon_\star\),- a terminal time \(t_\star\),- and a short backward parabolic window\[J=[t_\star-\tau,t_\star], \qquad \tau=\theta(\eta)\varepsilon_\star^2,\]such that the fixed-scale profile remains quantitatively close to the threshold throughout \(J\). The analytic chain is organized around:1. a one-sided residual-budget selection lemma,2. a common-sequence scale choice lemma,3. a fixed-scale differential inequality,4. a fixed-scale persistence lemma,5. and a final short-window persistence theorem. --- ## ScopeThis note is a **local persistence statement on a finite time window**. It does **not**:- prove global regularity,- prove a threshold theorem,- prove a continuation criterion,- use numerical or visualization material as proof input. The note should be read as a reusable local lemma chain, not as a standalone global Navier-Stokes resolution claim. --- ## Why this note existsThe larger SAPZ Navier-Stokes package contains a contradiction-window persistence component whose role is to turn an envelope-level near-threshold situation into a genuinely fixed-scale short-window output. This note exists to isolate that component in a form that:- can be read without first reading the full package,- makes the order of choice explicit,- emphasizes common-sequence / diagonal scale selection,- and records a short-window fixed-scale persistence mechanism in standalone form. --- ## Relation to the larger SAPZ packageThis note is extracted from the contradiction-window component of the larger SAPZ Navier-Stokes package. It adds **no new proof input** beyond that package.Its purpose is to isolate one local persistence mechanism in a standalone form. Related records: ### Primary journal-cut proof package- Current snapshot DOI: 10.5281/zenodo.18873659- Concept DOI (all versions): 10.5281/zenodo.15846588 ### Analyst-facing support record- All versions DOI: 10.5281/zenodo.18890268- Support Notes v1.0 snapshot: 10.5281/zenodo.18890269 ### Standalone Route--T bridge note- All versions DOI: 10.5281/zenodo.18892599- Route--T bridge note v2.0 snapshot: 10.5281/zenodo.18892600 The present note may be read independently, but it was originally extracted from that larger modular proof architecture. --- ## Files in this record- `CT3_persistence_note_v2.0.pdf` (Optional, if included)- `CT3_persistence_note_v2.0.tex` --- ## Mathematical positioningThis note is best read as a local statement in the analysis of suitable weak solutions. Its intended mathematical role is:- to isolate a fixed-scale short-window persistence mechanism,- to make the order of scale selection explicit,- and to provide a contradiction-window bridge object that can be discussed independently of the full SAPZ package. It should **not** be read as a one-line claim that the global Navier-Stokes regularity problem is solved inside this note. --- ## Proof disciplineThe note keeps the following distinctions explicit:- the envelope profile \(w(t)=\limsup_{\varepsilon\downarrow0}\widehat\delta_\varepsilon(t)\),- the fixed-scale profile \(\widehat\delta_{\varepsilon_\star}\),- the residual profile \(r_{\varepsilon,+}\),- the short-window budget parameters,- the final contradiction output \((\varepsilon_\star,t_\star,J)\). This is intentional: one of the main reading hazards in the larger package is ambiguity in the order of choice. The present note is written to reduce that hazard by making the scale-last/common-sequence selection explicit. --- ## MSC2020Primary 35Q30; Secondary 35B65, 76D03. --- ## KeywordsNavier-Stokes equations; suitable weak solutions; fixed-scale persistence; one-sided residual budget; common-sequence selection; contradiction window. --- ## Recommended citationLee Byoungwoo, **Fixed-Scale Persistence Under a One-Sided Residual Budget** (Version v2.0), Zenodo, March 2026. Current snapshot DOI: 10.5281/zenodo.18897304 --- ## Record-navigation noteThis record is a **standalone persistence note**. For the full journal-cut proof package, see the separate primary SAPZ Navier-Stokes record. For the analyst-facing support-note collection, see the separate SAPZ Support Notes record. For the fixed-scale transport-extraction component, see the separate Route--T bridge-note record.