# Pre-fit analysis conventions
Recorded before fitting. India, World Bank EN.ATM.PM25.MC.M3, GBD 2023 vintage. Observed annual national population-weighted exposure: 1990–2023 inclusive, n=34, µg/m³. No imputed years; source estimates are not independent direct monitoring observations.

## Eligible candidate years
A knot is an observed integer year k; require m observations strictly before and m strictly after k. The observation at k belongs to neither side for this count. Continuous hinge model; not a jump.
| Window | n | m=4 | m=5 (primary) | m=6 |
|---|---:|---|---|---|
|1990–2023|34|1994–2019|1995–2018|1996–2017|
|1993–2023|31|1997–2019|1998–2018|1999–2017|
|1990–2020|31|1994–2016|1995–2015|1996–2014|
|1993–2020|28|1997–2016|1998–2015|1999–2014|
All ranges inclusive. Leave-one-year-out uses the remaining observed years, m=5; recalculate eligibility and preserve actual calendar gaps.

## Means and likelihood
Let t=year−first year of each window. Models: line a+bt (no slope change); hinge a+bt+d max(0,year−k); quadratic a+bt+ct² (smooth curvature, no discrete break).
Fit raw exposure and natural-log exposure separately with Gaussian residuals, either independent or stationary AR(1). Covariance is sigma² R, R_ij=rho^abs(year_i−year_j). sigma² is the marginal variance, not innovation variance. Include the initial stationary observation in the likelihood; no dropped first row, conditional likelihood, REML, or differencing.
For AR(1), profile rho on a deterministic grid −0.90 to +0.90 inclusive in 0.02 increments, solve generalized least squares at each grid point, and choose maximum likelihood. This bounded grid is an explicit numerical approximation; report boundary solutions. For actual year gaps Δ use transition rho^Δ and variance factor 1−rho^(2Δ). Independent errors fix rho=0.
q=r'R^{-1}r; sigma²_ML=q/n. −2 log L=n[log(2π)+1+log(q/n)]+log|R|. For log exposure add 2 sum(log(raw exposure)) to −2 log L when reporting likelihood on the original response scale. This Jacobian allows raw/log likelihood comparisons on identical observations.
Parameter counts K include residual variance: line=3, quadratic=4, searched hinge=5 (three mean coefficients, one searched knot, one variance); add one for AR(1). Even though knots and rho use grids, count each as one estimated parameter. AICc=−2logL+2K+2K(K+1)/(n−K−1); BIC=−2logL+K log n. Only compare models using identical observations. Breakpoint-search nonregularity makes these approximate selection scores, not calibrated break tests.

## Search-adjusted and conditional simulations
Set seed 20261008; use 1,999 replicates per simulation. Full-series raw primary comparison: simulate under the fitted no-break line separately for independent and stationary AR(1) errors, using fitted ML marginal variance and Gaussian draws. In each replicate refit the line and all eligible hinge knots including rho search where applicable. Test statistic is max(0, 2[logL_hinge−logL_line]); Monte Carlo p=(1+count(T_sim≥T_obs))/(2000); report Monte Carlo standard error. These are plug-in parametric calibrations with nuisance-parameter estimation, not exact tests or measurement-error propagation.
Separately simulate 1,999 series under each fitted raw hinge error model, rerun knot selection, and retain knot, slopes, and slope change. Report percentile 95% intervals and frequency distributions as conditional on the hinge model, not confidence that a break exists. AR(1) simulations start from the stationary marginal distribution.
Simulation work uses small arrays in the existing local Notebook environment only; no remote computing, installations, or new plugins.

## Sensitivity and interpretation
Cross four specified windows, raw/log scales, independent/AR(1) errors, and m=4,5,6: 48 settings. Include line, hinge, and quadratic in each. Full-series leave-one-year-out repeats raw/log × independent/AR(1) with m=5 and accounts for calendar gaps. Report every setting, minimum-AICc and minimum-BIC model, selected hinge year even when hinge loses, slope-change direction, and rho boundaries. The ±2-year agreement tolerance is descriptive and prespecified, not a hypothesis test.
For raw models report µg/m³/year slopes; log slopes report log units/year and 100[exp(slope)−1]%/year. Check residual autocorrelation, residual magnitude/shape, and influence; avoid causal attribution. A selected hinge can merely approximate smooth curvature. Short series, uncertain source-model errors, unknown measurement covariance, and vintage dependence may justify insufficient evidence even with improved fit.
