# Barrier continuity: a two-path teaching model

Scope reviewed with the chapter coverage agent: this study supports protective-film continuity (COAT/SURF), distinct from the previous chapter's intact-film time budget. It does not model electrostatic dust control or road performance.

Imagine non-overlapping patches under the same fixed driving conditions. The intact part occupies fraction1-f and has local flux j. A degraded part occupies fractionf and is assigned local flux r*j. Add their mass flow rates: Q=A*j[(1-f)+r*f]. Compared with an all-intact surface, the normalized flow is1+f(r-1). The degraded region's share of total flow is r*f/[(1-f)+r*f]. These are conservation-based sums of prescribed paths, not a law deriving real-hole leakage from hole area.

All numerical ratios are hypothetical. The study uses r=10,100,1000 and f=0..0.01 in101steps,303rows. For0.1% area with r=1000, total flow is1.999times the intact reference and the small region carries50.025% of it. Equal contributions occur at f=1/(r+1). This demonstrates why measuring continuity can matter as much as measuring the intact material; it does not establish any coating's performance.

Independent validation sums10,000equal cells with the prescribed local fluxes rather than using the closed-form fraction expression. Maximum discrepancy is1.78e-15. Tests cover component addition, f=0, f=1, r=1 and the equal-contribution threshold. The reference intact flux must be positive. There are no experimental error bars.

Assumptions: steady independent parallel patches; identical fixed driving conditions; uniform flux inside each patch. No lateral transport, spreading/access resistance, boundary layers, varying pore geometry, sorption, depletion or interactions. A real defect's leakage may depend on shape, depth, substrate and surrounding transport; area fraction alone is insufficient. No skin, microbial, disinfection, adhesion, abrasion lifetime or shelf-life claim follows.

Primary contextual reference verified in full: Verwolf et al., Journal of Research of NIST120(2015), Potential Stable Low-Permeation Rate Standard Based on Micro-machined Silicon, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=916339 . Their controlled-aperture work includes separate hygrometer and mass-loss measurements and a seal-loss issue, illustrating the importance of identifying transport pathways. No measured values or fitted coefficients from that paper are assigned to this hypothetical model. The model equation follows directly from adding area-times-flux contributions.

Run `python generate.py` with numpy and matplotlib. Outputs:parallel-paths.csv,study.json,original PNG/SVG/PDF publication-style figure. Prepared assets do not establish a completed chapter or a working coating.