Let 𝒯 denote the set of all 138 FBS programs in the operative season. For a team i ∈ 𝒯 we write r_i for its net rating (points vs. a mean opponent) and S_i for its published Atlas Score. All standardizations are z-scores taken over the population 𝒯 unless a subscript restricts the reference set; Φ is the standard-normal CDF; 𝟙[·] the indicator; percentile(·) the empirical rank transform onto [0,1]. Small-sample means are shrunk toward their population prior by a phantom-observation count M — a team's estimate is trusted in proportion to the evidence backing it.¹

The published board is a strictly monotone affine transport of the latent rating onto a fixed interval. The anchor 75 and slope 0.8 are structural constants: a league-average program scores 75 in every season by construction, so scores are comparable across years without renormalization.²

The rating itself is a convex, evidence-weighted combination of three independent readings of the same team, each correcting a failure mode of the others:

A rating is only meaningful relative to the schedule it was earned against, but the schedule's strength is itself a function of the ratings — a circularity resolved by treating the rating vector 𝐫 as the fixed point of a contraction 𝒜. Each pass re-expresses every team's performance net of its opponents' current estimates, then rescales the cross-section to the anchored spread σ*. The map is iterated to numerical stationarity; four passes suffice.⁴

where a single application shrinks each team's raw margin signal toward the population and renormalizes to the target dispersion:

Every predictive statement Atlas makes — a single game, a projected record, a national-title share — is a functional of exactly one primitive: the expected point margin, modeled as Gaussian with an empirically fitted dispersion. Rating differentials enter linearly; venue enters through the home-field term of §4.

The choice of σ_μ = 13.74 is not free: it is the value that maximizes out-of-sample pick accuracy on the calibration set, and it silently governs how much a rating gap "should" translate into a probability. A one-score favorite and a three-touchdown favorite differ only in where they land on Φ.

Home field is not a constant. Each venue earns a distinct advantage from a loudness-dominant atmosphere index — a weighted z-blend of measured crowd noise, fill ratio, raw capacity, and a residual "the crowd won games it shouldn't have" term estimated against Elo expectation over 2023–2025.

The atmosphere percentile is then transported onto a home-field band whose mean is pinned at the calibrated 2.5 points, so aggregate prediction accuracy is mathematically unchanged — only its distribution across venues moves. A road team inherits the host's advantage; a louder building is simply harder to win in.⁵

Schedule difficulty is a location-weighted mean of opponent Atlas Scores — a true road game is worth 10% more than the same opponent at home — then min–max renormalized across the cross-section so the hardest slate in the country reads 100 and the softest reads 0.⁶

Reported alongside the index: counts of top-25 and top-50 opponents, road/neutral/FCS splits, the single toughest opponent, and the hardest three-game stretch — the maximum over all contiguous windows of summed opponent score.

A season is a sum of independent, non-identical Bernoulli trials — one per game, each with the win probability of §3. The expected win total, its standard deviation, and the plausible band all follow from the Poisson-binomial in closed form; no simulation is required for the record itself.

Each game is additionally labeled by its probability mass — Favored (≥.85), Likely (≥.65), Lean (≥.50), and their mirror images below the coin flip — so the record carries its uncertainty visibly rather than as a false-precise single number.

Playoff and conference-title probabilities are the empirical frequencies over N = 10⁴ independent season replays. Each replay draws every game against its §3 probability, crowns each conference through a neutral championship between its two best records, and seats a full 12-team field by a record-dominant committee score in which rating breaks ties but never overturns wins.⁷

Players are not ranked on their ceiling — they are ranked on a risk-penalized floor that subtracts one standard deviation of uncertainty from the point estimate. Pedigree without production carries wide variance and therefore pays an explicit tax; the list rewards what a player has demonstrated, not what he might become.

Two corrections keep the grades honest across contexts. Box production earned against a Group-of-Five schedule is discounted a flat −5 points — a G5 sack and a G5 catch are inflated by the same competition gap. Offensive lines, which have no public per-player signal, are graded only at the unit level as a convex blend of measured line stats and the opponent-adjusted SP+ read:

A coaching record is regressed toward .500 by K = 25 phantom games — roughly two seasons of prior — so a small hot streak cannot outrank a long body of work. The regressed rate is then scaled by the level of competition faced, measured as the fraction of games played against Power-4 programs.⁸

The window runs 2015–2025. First-year head coaches with no record are seated correctly but left unranked — Atlas will not fabricate a number for a coach who has not yet coached a game.