The Clock: methods for the series

Rendered from methods.template.md. Every number here comes from outputs/final_numbers.csv by key and carries a run ID; the reconcile gate fails if a digit appears that is not on the sheet. The articles are written prose-first and link here instead of carrying their own appendix; the prose gate (gate_prose.py) checks a finished draft's figures against the same sheet.

The four views

Every player gets four impact scores, each in points per hundred possessions. RAPM is a regression that credits each player for how the score moved while he was on the floor, adjusted for everyone else on it. Box is a box-score model. DARKO is a public projection built from box-score trends. Consensus blends RAPM with a public box-score metric. Consensus tracks RAPM at a correlation of 0.977 across players, which means four-way agreement is weaker than it sounds: two of the four are largely the same opinion. So every title probability in the series is the mean of three of the views, consensus, box and DARKO, and a finding still has to hold in all four: the bands, the signs and the count of views clearing a test run over all four, and the band across all four is printed beside a figure where the label asks for it.

The two aging bases

The aged basis is primary (D142): each player is shifted by the expected one-year change for his age, estimated league-wide, and since D142 last season's roster is aged the same way in every counterfactual, so an offseason figure on the aged basis compares like with like. The un-aged basis takes every player at last season's measured level; the articles quote it only where a sentence compares the two, and this page carries both. Tables say "aged" and "un-aged".

The frozen model (D145)

The series quotes the model frozen on 2026-10-11 (D145). What changed from the version running when Part 1 posted, in plain words:

• Three views in the headline. Every player is still scored four ways, and a finding still has to hold in all four; the headline averages consensus, box and DARKO, with DARKO read at the source's full precision and matched by player id.
• The aged basis is primary, and last season's roster is aged in the counterfactuals.
• Last season, like for like. The change from last season is measured against last season's team run through the same minutes engine night by night, on the players who were available each night, instead of against the minutes its coaches played. Part of last season's health luck is then taken back by rule, not chosen from the backtest: the anchor carries 0.71 of last season forward, and team health repeats from one season to the next at a correlation of only 0.11, so the projection takes back that share of last season's health luck, less the part that repeats. The baseline sits 0.63 of the way from last season's expected availability to its realized availability.
• Playoff minutes, in the playoffs only. In the playoffs the model plays the top of each rotation more, on the shape of the last three postseasons' minutes by rank. It is validated against how playoff minutes were handed out, not against playoff results.
• A heavier first season back after an Achilles or an ACL, in the discount section below.
• Regular-season minutes are handed out as before. A version calibrated to game logs reproduced real minutes far better and made last season's projections worse, by more than one paired standard error, so it was dropped for the regular season.

The backtest. The model built as of the 2025-26 preseason, with only what it would have known then, scored against what 2025-26 did: mean absolute error in points of net rating per hundred possessions over the thirty teams, the root mean square error, the rank correlation, and each stage's change against the row above it with its paired standard error (the aged basis on its own changes nothing in the backtest, so the second row's change is the switch of basis). A fix that made the backtest worse by more than one paired standard error stopped the list.

The backtest slightly preferred weight one (3.93) to the rule's 0.63 (3.99), within noise; the weight was set by rule before it was scored, and the rule was kept. The calibrated regular-season minutes were dropped because they made the backtest worse by more than one paired standard error (+0.110 against 0.084), even though they reproduced how real minutes are handed out far better; their playoff shape is kept, because it was checked against playoff minutes, which is what it changes. The heavier Achilles and ACL discount and the playoff shape do not move the backtest: it keeps the first-season-back rule off, because it cannot source injury causes, and it scores regular-season nets only. Scored alone on the frozen construction, the views miss by 3.98 (box), 4.02 (consensus), 4.10 (DARKO) and 4.12 (RAPM).

The playoff shape. Title odds with the shape and with D139's playoff allocation (the regular season's full-strength minutes), on common random numbers, a read of twenty thousand seasons a view; the published figures are the full simulation's. It moves title odds toward Oklahoma City, San Antonio and Denver and away from Detroit, Boston, Houston and Minnesota.

The major lower-body discount, sensitivities. Title odds on the primary basis (a read of twenty thousand seasons a view) with the discount for a ruptured Achilles or a torn ACL at the low sensitivity, the adopted figure and the high sensitivity, magnitudes 1.4, 2.0 and 2.6 points:

The tests against Vegas (D147). One rule for every test of the model against Vegas: after 30 games, whichever expectation the team's net rating is closer to wins, the model's (its season net on the primary basis, the headline mix) or Vegas's (BetMGM's win total of 2026-10-09 converted to net with the backtest's fit, wins = 41.00 + 2.278 x net, fitted on the sixty team-seasons of 2023-24 and 2024-25), and the line printed is the halfway point. Minnesota: the model +0.4, Vegas +3.3, the line +1.8 (Vegas closer above it). Charlotte: +3.1 and -0.7, the line +1.2 (Vegas closer below it). Indiana: -3.9 and +1.1, the line -1.4 (Vegas closer above it). San Antonio: +6.4 and +8.1, the line +7.3 (Vegas closer above it). Charlotte and Indiana are two of the model's largest regular-season disagreements with Vegas among teams Vegas has at thirty wins or more (48.0 wins against 39.5; 32.2 against 43.5); Brooklyn's gap ties Charlotte's, on a line under thirty. The rule replaces the earlier tests, which called it for the market only when a team landed beyond the edge of the model's own range (all four views, both bases) by more than thirty games of noise, so the model had the benefit of the doubt every time; Minnesota's game-twenty band and San Antonio's game-twenty line went with them.

Known limits.

• Williams's minutes, and how the model misses games. Each player's missed games are drawn on his own and spread at random through the season, so the model misses the way real absences come together: rest nights, illness, end-of-season shutdowns. Rebuilt as of last October, the model expected teams to have their whole opening-night top eight available on 5.7% of nights; the real figure was 14.6%. Part 2's depth shares are the eight players and the five players each team used most over the season, by minutes: the share of its games in which all of them played, each player counted only while he was on the roster, for the median team (25.0% for the eight, 57.9% for the five). Available above counts a coach's decision; these count only games played. Picking the players after the season favors the ones who stayed healthy, so the true rate for the eight a team planned around is lower: measured from opening night (the eight highest in each team's preseason order on its opening-night roster, two-way players out, ranked the way the backtest ranks them, and the five who started its first game), the median team's eight all played in 6.1% of its games and its five in 25.6%, though 9 teams started the season with one of those eight already out long-term, which pulls the opening-night figure down for a reason that isn't depth. A player's chance of playing on a night is his cohort's share of team games played, so a coach's decision counts as a missed game, the same as an injury. A version built on clean game-by-game records for eight seasons, with a coach's decision counted as available and games off a roster left out, was tested the same way as every other fix, rebuilt as of last October: it missed each player's games available by 14.4 games on average against the frozen model's 16.4, but leaned the other way about as far (+3.0 games against -2.9), and its team projections were a little worse (4.06 points of net per team against 3.99). It didn't go in. That's why the article judges Williams's minutes against what last season's deep-bench players in his spot actually got, 14.7 a team game, and not against the model's nights: the model expects 19.77, and if it is wrong, it is wrong against Minnesota, by about 1.11 wins. The handout itself is close to how Finch coached: given who was actually out on Minnesota's 59 nights last season with an opening-night top-ten player missing, the model's handout sent 42% of the freed minutes to the rest of the top ten and Finch sent 44%. What runs high is the count of absences (above: the cohort counts a coach's decision as a missed game).
• Beringer's rookie minutes. 37% of his 2025-26 minutes came in garbage time under Cleaning the Glass's definition: the fourth quarter only; a margin of twenty-five or more with twelve to nine minutes left, twenty or more with nine to six left, ten or more with under six left; and two or fewer starters on the floor between both teams (here a boundary minute belongs to the earlier window, the starters are each team's opening five that game, the test runs on every stretch between plays with no carry-over, and overtime is not counted). With the fourth quarter or overtime at a margin of twenty or more it is 34%, and under the project's stricter default (more than fifteen with three minutes or fewer left) 20%. From the play-by-play (beringer_garbage_time.py).
• Indiana is the harshest call on the board: 32.2 wins against a line of 43.5, 0.00% to win the title against the market's 2.00%. Three things stack: the first-season-back discount on Haliburton, the absence cost on a rotation the cohort method expects to miss a lot of games, and aging. Whether the discount and the absence cost both count a returner's lost season is logged for November.
• Charlotte is unexplained: 48.0 wins against 39.5, and no part of the projection that can be named accounts for the gap.
• One season of backtest. Every stage above was scored on 2025-26 alone, thirty teams, and the paired standard errors are wide.
• DARKO is the weakest of the three headline views on the backtest (alone 4.10, against 3.98 for box and 4.02 for consensus); RAPM, outside the headline, is weaker still (4.12).

The first-season-back discount

A player in a 2026-27 rotation who missed 60% or more of his team's 2025-26 games (a share of 0.40 or less; no NBA games after a 2024-25 NBA season counts as zero) and whose lost season is sourced to a major lower-body injury, meaning an injury from the hip down that required surgery or that the team reported as season-ending, with two sources, carries a first-season-back discount of 0.84 points of net per hundred possessions in every view and on both bases, split evenly between offence and defence the way the aging adjustment is. Since D145, a caught player whose lost season was a ruptured Achilles tendon or a torn ACL carries 2.0 points instead, sourced to the published first-season-back drops for those two injuries (The Economist, 2019; Chauhan, 2021; the control-adjusted PER translations; docs/a6_achilles_evidence.md), applied the same way: Jayson Tatum (BOS, Achilles); Kyrie Irving (DAL, ACL); Fred VanVleet (HOU, ACL); Tyrese Haliburton (IND, Achilles); Dejounte Murray (NOP, Achilles); Damian Lillard (POR, Achilles). Its sensitivities, at 1.4 and 2.6, are in the frozen-model section; one season cannot validate the figure. The size is fit, not picked. From 205 player-seasons since 1997-98 in which a rotation player (a season two years before at a share of 0.70 or more of his team's games and 20 or more minutes an appearance, 20 or more games in both seasons compared) lost the season in between, his on-court net in the season back less his on-court net two seasons earlier, against 3,481 age-matched control player-seasons that did not lose the season in between: -0.84 points, standard error 0.46; the raw changes are -1.01 for the returners and -0.12 for the controls. The fit cannot tell an injury from any other lost season and pools every cause, so it is milder than the published first-season-back drops for Achilles ruptures specifically; at stricter lost-season bars it reads -0.60 (standard error 0.67, n 92) and -1.40 (standard error 0.94, n 43), neither one precise, so the rule carries the full-sample figure. Of the 47 rotation players who missed that much of 2025-26 for any reason, the rule catches 18: Jayson Tatum (BOS, 16); Zach Collins (CHI, 10); Dereck Lively II (DAL, 7); Kyrie Irving (DAL, 0); Georges Niang (GSW, 0); Fred VanVleet (HOU, 0); Steven Adams (HOU, 32); Obi Toppin (IND, 24); Tyrese Haliburton (IND, 0); Bradley Beal (LAC, 6); Max Strus (LAC, 12); Scotty Pippen Jr. (MEM, 10); Zach Edey (MEM, 11); Dejounte Murray (NOP, 14); Haywood Highsmith (PHX, 6); Damian Lillard (POR, 0); Domantas Sabonis (SAC, 19); Keegan Murray (SAC, 23); the other 29 lost the season to an upper-body injury, a soft-tissue injury without surgery, a rotation decision or a season abroad. Dejounte Murray is caught by the rule as written although 2026-27 is his second season back from the Achilles; Keegan Murray is caught on the ankle that ended his season and needed surgery, after a thumb surgery cost him the first fifteen games. Before the rule (frozen 2026-10-08T16:19:10Z), the model had Boston at 16.11% on the un-aged basis (then the primary one) and 12.23% corrected for age, by view 22.10%, 22.60%, 12.76% and 13.46%, and Minnesota at 2.79% and 3.73%; the discounted figures are the sheet's current ones: Boston 10.74% and 7.47%, Minnesota 1.38% and 1.79%. A caught player is also ordered on his minutes per appearance from the last season he played 20 or more games (the returners' minutes rule, D137), instead of last season's few games or the fringe prior, so Haliburton, Irving and Lillard enter their rotations at a starter's load with the discount on their impact and, since none is on the injury file, no availability haircut; that is the strong end of a first season back. The rule moves 15 of the 18 caught players (Jayson Tatum 2024-25 35.9; Zach Collins 2024-25 14.8; Dereck Lively II 2024-25 22.7; Kyrie Irving 2024-25 35.6; Georges Niang 2024-25 21.0; Fred VanVleet 2024-25 34.7; Tyrese Haliburton 2024-25 33.0; Bradley Beal 2024-25 31.7; Max Strus 2024-25 24.9; Scotty Pippen Jr. 2024-25 20.8; Zach Edey 2024-25 20.9; Dejounte Murray 2024-25 32.2; Haywood Highsmith 2024-25 24.1; Damian Lillard 2024-25 35.5; Domantas Sabonis 2024-25 34.2); the three whose last twenty-game season is 2025-26 keep their measured minutes. Every rotation row it moved is in d137_returner_minutes_before_after.csv. What the two rules did at first, and why so little (D136 and D137). Until D138 a team's model net was last season's measured net regressed to expectation plus beta times the current roster's rollup less last season's roster's rollup, both at the allocator's minutes, so a player on both rosters cancelled in the difference and neither the discount nor the minutes rule could move a team through a returning player beyond the minutes he took from others; Boston's title odds went from 16.11% to 16.00% across those two decisions. The delta construction since D138. The change is now the minutes-weighted change in every player's contribution: his projected minutes on this season's roster less the minutes he actually played for the team last season, per team game, times his impact, with the discount and, on the aged basis, aging; a returning star registers as his whole load, a departed player as his whole load lost, a stayer as the change in his minutes. A team whose projected minutes equal last season's reproduces its anchor exactly (the gate, checked on every team and view). The transaction-only delta is kept as the recorded sensitivity. Because every team is projected at full health against a season with absences, the league's average delta is +0.08 (spread 1.47) against +0.04 on the transaction basis; the simulation is relative, so the level cancels and the spread is what moves odds. Boston's delta is +0.80 against +1.72 transaction-only, Indiana's +2.52 against -0.28, Houston's -0.08 against +0.21, Minnesota's -2.12 against +0.45. Title odds before and after the fix: Boston 16.00% to 10.74% (-5.26), Indiana 0.00% to 0.01%, Houston 5.38% to 2.44%, Minnesota 2.78% to 1.38% (-1.41). The DARKO view prices 2 rotation players at the mean of the other three views because DARKO carries no current season for them and its prior seasons are not on disk (Georges Niang (GSW); Trey Lyles (MIN)); before D138 the rollup skipped them, which priced them as league average in that view. The ordering benches some players coaches start. The order is half last season's minutes per appearance and half impact, so a starter with a poor rating can fall outside the ten: John Collins (1,837 minutes last season, rated -0.17) and Jerami Grant (1,667 minutes, -0.81) are projected at none, and 13 players with fifteen hundred minutes last season or a star contract sit outside their tens. Charlotte rests on one player. Moussa Diabate at +5.05 in the consensus view over 28.6 minutes a night is 3.01 points of Charlotte's team net, the four views reading +5.05, +5.05, +0.27, +1.51. The honesty recount. Of the 20 largest model-market disagreements after the fix, 8 are returning-star cases, 0 injury-file gaps, 12 a single player the views rate very differently, 0 unexplained; the rank correlation between the model's title odds and the market's is 0.89, from 0.84 before the fix. What carries Boston is the anchor, last season's +8.31 regressed to +6.01, and a summer delta of +0.80 (mean of views; +2.59 in the consensus view), 9th in the league, made of arrivals (Paul George +1.79; Mitchell Robinson +1.15; Mike Conley +0.93) against departures (Jaylen Brown +0.16; Nikola Vučević -0.03; Baylor Scheierman +0.25), each a player's minutes over forty-eight times his consensus net; the four views rate Jaylen Brown at +0.27, -0.67, +0.70 and +2.24. The field is anchored the same way, so a star who sat out last season does not return in the model: Indiana's anchor is -5.45 from a measured -7.85, and its title odds are 0.01% against a market at 2.00%; Dallas 0.02% against 0.41%, Portland 1.90% against 1.01%. That is a limitation of the anchoring, logged as the next version, and partly addressed by the like-for-like baseline in the frozen model. Boston's December test, set before the rules at -1.7, left the watch list with D146 (the frozen-model section has the tests that replaced it).

The two minutes allocators, the four cells and the quotability rule

A roster is ordered by a league-wide score before any minutes are handed out: half the percentile of last season's minutes per appearance and half the percentile of impact for a player who stayed, and one fifth and four fifths for a player who changed teams, the same blend the minutes rule already used for movers. That ordering was adopted on 2026-09-25, before the field was re-run under it (the check that motivated it is commits ce74a2a6 and d3a226f6; the adoption itself is commit 266a2422 at 2026-09-25T17:52:01Z), on the grounds that the mover discount had been applied to minutes but not to ordering. The field under it was first seen at 2026-09-29T09:50:41Z, once the re-run had finished (its last step ended 2026-09-29T09:50:21Z). The flat half-and-half order for everyone is kept as the recorded sensitivity. What that ordering does to Cody Williams, the case Part 2 turns on: ordered flat he is Minnesota's 10th man at 16.1 minutes a night; ordered with the mover discount he is 12th, and 13th on impact alone, outside the ten either way. The base rate for a player in his position: over the last three seasons 34 players changed teams after a season like his, well used on a bottom-ten team at a bottom-quartile rating, and they kept a median 74% of their minutes; the 8 who landed on a top-ten team kept 48%, which for Williams is 11.7 a night, above the 9.2 at which, without the age adjustment, every view says the summer hurt. Priced off the f-curve with the field fixed, Minnesota's title odds are 1.24% with Williams at that role against 1.82% with him outside the rotation, and 0.85% against 1.38% without the age adjustment. Williams's minutes in the frozen model (D139, D146). The availability model is now the model: every player carries a chance of missing games, drawn from the C3 cohort of players like him, and the season is played night by night with whoever is left. Williams, outside the ten at full strength, is expected at 19.77 minutes a team game, nearly all on nights a player above him is out; he plays on 68% of nights. Last season's deep-bench players in his position, ranks eleven to fifteen with a ceiling of twenty-four or more (12 player-seasons, A2), were projected 15.9 and played 14.7. His expected minutes sit 5.02 above that; handed to the rest of the rotation in proportion to their own minutes, the excess costs Minnesota 0.50 points of season net, 1.11 wins, on the headline mix. The fork, without the age adjustment: above 9.2 a night on nights everyone above him is healthy, every view says the summer hurt. On the primary basis no level tested, up to 16 a night, turns every view negative: at that level 2 views say the summer hurt and the headline mix still has it helping by +0.16 points of title odds, from +0.88 with him outside the ten (MIXED at every level; D147). Those healthy nights are 1.8% of the season, so the game-twenty line is his minutes per team game, every minute counted and a game he sits as zero, against the midpoint of the two figures, 17.3. Part 2's bench verdict ("With him on the bench and everybody healthy, the summer helped.") is a different calculation: it prices both rosters at full strength with nobody hurt, and on the primary basis all four views say the summer helped; without the age adjustment, consensus has it slightly worse (-0.12), so all four agree only with the age adjustment; the grid prices the whole season with injuries, so even with no minutes on healthy nights Williams still plays the nights somebody above him sits, and consensus has the summer slightly worse at every level. Two rules hand out minutes. The team-rank allocator, which the headline simulation uses, ranks a roster team-wide and plays ten men. The pooled allocator, which the attribution uses, hands minutes out inside position groups. A verdict ships only if its sign holds under both aging bases and under both allocators, and only if every one of the four views clears its noise floor in all four of those cells. The noise floor is twice the size of change the simulation and its interpolation could produce on their own. The shipping verdicts, in all four cells, mean points of title odds and the views clearing in each:

The parts quote the pooled allocator's figure for Kuminga against the fill, +0.29 points of title odds on the primary basis and +0.19 without the age adjustment, because the slot-aware run is the primary one; it is positive in every cell, but the views clearing the noise floor in the four cells are 2/4, 3/4, 4/4, 4/4, so it leans yes and is not proven: a counterfactual that hands a player's minutes to whoever the model rates highest, regardless of position, is not available to a coach. The team-rank figure for the same comparison, +0.51 and +0.55, is in the table and nowhere in the parts.

The bands and the labels

Every figure on the sheet carries two labels. The first says where it came from: MODELED (a simulation or a fitted model), OBSERVED (counted from box scores, contracts, schedules or documents) or DESCRIPTIVE (a parameter or a diagnostic, not a finding). The second says how it may be quoted: QUOTABLE (the figure stands on its own), QUOTABLE AS BAND (it must be printed with its band or interval, because the views or the seeds disagree by more than the figure's own size would suggest) or DESCRIPTIVE. A modeled title probability is always QUOTABLE AS BAND; the band is the four views, low to high.

The simulation

200,000 seasons per view, on both bases. Attribution prices every combination of Minnesota's offseason moves on a curve built from those simulations, and the curve is checked against the direct simulation every run. Seeding is simulated wins from net rating; the playoff draw is a series model on net rating with a noise term for what net rating cannot see.

The market

Six books' title odds, de-vigged proportionally: the raw prices add to 21.8% over a hundred, and each team's price is scaled back so the league sums to one.

Boston and Philadelphia, from Part 2 (moved here by D161). The bigger disagreements aren't about Minnesota. I spent most of a week on Boston, which used to be the strangest number on the board, and found that its edge was mostly Tatum's return and the ages of the players around him. With the aging built in and the Achilles research behind Tatum's number, the model now sits much closer to the market on the Celtics.

Part of why the model doesn't buy Philadelphia is Jaylen Brown. Boston was +13.4 per hundred possessions with Brown off the floor last season. With him on, +7.1. The model rates him as only a small plus, so his arrival doesn't move the Sixers much, and it has them at 3.20% to win the title against the market's 8.42%. The biggest single piece is Embiid's health.

The September tests (Part 2)

The eight tests. Eight regular-season traits, fixed before any fit (the six style features, the top-three minutes share, and defence share of net rating, the "defense travels" claim as a number), were tested on 195 playoff series from 2014: a trait translates only if, at equal team quality, it improves the prediction of series out of sample and its permutation p clears 0.0063. 0 of the 8 did. "Defense travels" is the near miss, and it points the other way: a team leaning on its defense rather than its offense loses 0.85 points a game in the playoffs for every standard deviation of that lean (standard error 0.34, permutation p 0.011), which is short of the bar we set for calling it real, so we don't.

The matchup test ("who guards Ant"). For Minnesota and each of the eight West teams with the highest modeled title odds, each side's top five by projected minutes is paired with the defender on the other side's projected top eight who guarded him most in 2025-26, in any uniform: 77 pairings with possessions between them. A pairing is judged against the primary-defender norm (a team's heaviest defender holds a high-volume scorer 0.05 to 0.10 points per possession below his own baseline) and counts when it sits two standard errors beyond it, in either direction (the standard error is 0.14 at thirty possessions, measured). Of those, 6 cleared it: 2 on the scorer's side (Aaron Gordon against Jonathan Kuminga and Anthony Edwards against Luka Doncic) against about 2.2 by chance, and 4 on the defender's side (Devin Vassell, Christian Braun and Marcus Smart on Anthony Edwards; Toumani Camara on LaMelo Ball) against about 0.8, 3 of them on Anthony Edwards. Of the six, 2 rest on fewer than thirty possessions (Anthony Edwards against Luka Doncic and LaMelo Ball guarded by Toumani Camara).

What was withheld

The late-clock split. The reconstructed shot clock read within two seconds of zero at recorded violations 79.9% of the time against a bar of 80% fixed before the build, so no late-clock figure appears in the series.

Corrections made during the work

Two bugs in my own data changed figures before publication. The lineup pipeline was crediting some baskets to the wrong team, 3.36% of all points on the 578 team-games I checked, which retracted two sentences (that Reid next to Gobert was clearly better than Randle next to Gobert, and that Kuminga's on-off flipped sign between his two teams) and moved several playoff lineup figures in the postmortem project. The same bug lived one level down, in the possession data that RAPM is fitted on, where it misplaced 2.84% of points across 96 team-games, so RAPM was refit on corrected points. Charlotte, the figure that exposed it, went from 4.66% to 3.28% on the refit; the October 7 re-run on the current rosters (D121) took it to 3.59%, the figure on the frozen model without the age adjustment; on the primary basis the frozen model has Charlotte at 5.75%, against 6.39% before the refit on the same basis. Kuminga's net RAPM went from +1.37 to +1.82 (consensus +1.33 to +1.65) and Ball's from +1.95 to +3.86 (consensus +2.29 to +3.78); Edwards rose as well. The headline went from 1.85% to 1.38%, and from 2.78% to 1.79% on the aged basis, because the whole league was refit with them and it's position in the league that the simulation prices. The attribution model had also been pricing every combination of moves on a roster without Cody Williams; fixing that is what retired three of the verdicts that used to ship.

The champions table

The eleven champions Part 3 draws on, one row each, from c1_champions.csv; prices are the de-vigged preseason title odds from Basketball-Reference's preseason odds pages (courtesy sportsoddshistory.com), net ratings per hundred possessions, the eight by playoff minutes, and the two continuity counts as defined above.

Every champion from 2015-16 to 2025-26, 11 of them, used only after the winner of the last playoff game in the warehouse matched the team Basketball-Reference names as League Champion. The top eight are the eight largest playoff minute totals. How each was acquired is read from his Basketball-Reference transaction log: drafted (draft night, including draft rights traded in that night), traded for, or signed (free agency, waivers, two-way or ten-day), taking the event that opened his current stint with the franchise; a re-signing or a rookie contract inside an unbroken stint does not restart the clock. Age is on the first of February, the Basketball-Reference convention. Continuity is carried two ways. By appearance: how many of the eight appeared for the franchise in the previous regular season, and the share of all playoff minutes that went to such players. By contract: how many of the eight were with the franchise at any point of the previous season whether or not they played, read from the Basketball-Reference transaction log (Jamal Murray in 2022-23 counts by contract and not by appearance), and the share of all playoff minutes that went to such players. The preseason title price for each champion season is Basketball-Reference's preseason odds page for that season (courtesy sportsoddshistory.com), all thirty teams, de-vigged proportionally and ranked with tied prices sharing a rank; each page is cached with its content hash in the manifest, and the three hand-transcribed seasons were checked against the pages team by team. Net rating is the possession-weighted mean of NBA.com per-game team net rating, ranked among the thirty teams; the post-All-Star split starts after the longest gap in the league schedule between the first of February and the middle of March, derived from the schedule itself. The seed is the conference finish on the Basketball-Reference team page, checked against the warehouse standings. Games missed are the team's games after the player's acquisition date minus his appearances. The top-five share is the five largest minute totals over the team's total minutes. Preseason odds columns are open for a pasted source. The paragraphs are written by the script from those fields; no number in them is typed.

The Edwards clock

The departure probabilities are the Part 1 model of "Pricing the LaMelo Trade" (Model B, M2 FINAL): a league-wide discrete-time hazard on 291 star spells, 1,264 player-seasons and 228 departures since the 1990 season, with contract years remaining as a covariate, applied to Edwards's profile under two team paths, a team winning at .600 and one sagging to .450. It scored a C-index of 0.907 on the sealed holdout and a calibration slope of 1.413 against a window of 0.8 to 1.2, a red cell printed as such; nothing was refit for this series. The walk-year multiple, 66 times the odds with two seasons left, is recomputed from the fitted coefficient. The raw rates beside the model are read forward from every star-season at the same contract stage, split by the team's two-year win percentage, with the counts. The contract is from the Basketball-Reference contract book and HoopsHype, the extension date from NBA.com transactions and the Basketball-Reference player page, the rules from the 2023 CBA text (NBPA copy, quoted by page), and the 65-game consequence from two reports.

Ball's games

Appearances are warehouse box scores against Charlotte's schedule, a game counted when he logged minutes. Missed stretches are runs of consecutive team games without him; every stretch carries a cause with two source URLs and a quote, and any stretch without a sourced cause would print as unsourced. The base rate takes every player-season from 2001-02 in which the player was 23 to 27 years old, played at least 24 minutes per appearance the season before, and had at least 3 of the five prior seasons at 60% or less of his team's games, with a season lost entirely counting as zero; the strict variant adds a healthy prior season at 80% or more of the team's games, which is Ball's case, and two looser variants are printed as sensitivity because the strict cohorts are small. The simulation sets the chance Ball is available on a night to games over eighty-two (eighty-two is every night), plays the season night by night with the availability model as the headline does, re-prices Minnesota's regular-season net by the change in its expected minutes, keeps every team's playoffs on its frozen-model playoff net (Ball is there in April), and runs the states on common random numbers. One more state leaves Ball at the availability the model already gives him, 60.7 games, and must reproduce the headline: 1.80% against 1.79% on the primary basis, within the f-curve tolerance, with P(top six) within a point of the seed distribution in every view. The drops the series quotes are from eighty-two games: at fifty, 0.13 points of title odds and 14.5 points of top-six odds on the primary basis, 0.16 and 14.7 without the age adjustment (D146 aligned this simulation to the frozen model; it had been built before the availability model).

The ledger

Every player and pick in and out since the season ended, read from each player's Basketball-Reference transaction log on the player's own leg of each event and from the NBA.com feed, with dollars from the contract book and Spotrac and reported terms with two URLs per move. Grades are listed exactly as written, with what each piece graded, and pieces found but not read are listed with the reason.

Who gets there from here (Part 3)

The price in the opening (D184). Before the 2024-25 season, BetMGM had Indiana at +6600 to win the title. The book's own Pacers preview lists it on October 18, 2024 (BetMGM), and CBS Sports reported it as the Pacers' preseason price on June 12, 2025 (Brad Botkin, "The Pacers are halfway to winning the unlikeliest championship in recorded NBA history"). The opening reads it as one in 67. That is the raw price, with the book's margin still in it, so the fair chance was a little lower.

The exact figures behind the article's words (D185). The article rounds or words these. The exact figures, in the article's order:

• The ladder: Minnesota's net rating last season was +3.12. Of the band's 426 teams, 252 made the playoffs (59%) and 106 won a round (25%). Since 2000-01, 41 of 396 reached a conference final. The model gives Minnesota 27.9% to win a round, 10.0% to reach a conference final and 3.8% to reach the Finals.
• Luck: the bracket's median share is -0.06. The Pacers' chance of a conference final was 10% along their expected path and 66% with their measured luck.
• The named runs, on the corrected split:

• The Lakers series: of Dallas's +9.65 points a game of own-shooting luck, shape is +0.61 and variance +9.04. Of the Lakers' +7.97, shape is +0.05 and variance +7.91. The fourth game holds 71% of Dallas's own-shooting luck in the series.
• Minnesota's 2026 playoffs, each against its regular season: the three-point attempt rate moved -0.075 against Denver and -0.081 against San Antonio, and the free-throw rate -0.018 and -0.025.
• The yardstick for Minnesota: the lines are +0.78 after 20 games and +2.76 in April. On them, the chance of a conference final is 7.8% (80% band 6.8% to 8.7%) and 12.5% (11.4% to 13.5%). The article reads the first as one in 13, computed for it. In April, a team four points per hundred better than its opponents has 21.5%, and one six points better has 44.0%.
• Already below: the lever estimates and their intervals are in the test tables and the notes after them. The median luck shares and the counts are in "The shares, both versions", and the carry is in its own paragraph. The year-to-year shooting correlation is in the persistence table; the shape R-squared is in the corrected split's table.
• The wording keys are on the sheet under "Part 3: the article's wording (D185)", each beside its exact key.

The story draft's figures (D186).

• The two runs in the opening: ranked by luck share among the 116 conference finalists, the 2025 Pacers are 7th and the 2011 Mavericks 8th. Both are clear luck: their 80% intervals sit above 0.5.
• The Mavericks by series: a run's luck share is the bracket plus each series' part, where a series' part is half of (the log chance with its luck less the log chance without), taken with and without shape. So the share splits exactly: the first round +0.11, the Lakers series +0.83, the draw -0.14. The Lakers series alone is more than the whole share.
• The Pacers by piece: each piece's part is its Owen value inside the luck block, the Shapley value among the four pieces averaged over shape absent and present, so the pieces add to each series' part exactly. Opponents' shooting +0.473, of which Cleveland in the second round +0.460; own shooting +0.386; own health +0.062; opponents' health +0.015; the draw -0.122.
• The Lakers series: Dallas won its four games by +14.0 a game on average.
• The shape rule, before and after: the rule set before the run counted 10 shape edges among the conference finalists and 4 in the band. The corrected split counts 1 of 116 and 1 of 45.

The lineup lever, re-estimated (D188, D190).

• The first version (D176): an unforced change of starting five. Every dropped starter was available that night, and the new five was kept five games. Measured over games six to twenty-five against same-quality teams: +0.80 (95% +0.51 to +1.08), over 1,454 changes (1,264 matched), with halves +0.58 and +1.06. It counted under D176's family rule, and the D176 tables below keep it as the record of that test.

• How the examples exposed it: the rule asked only whether the starter going to the bench was available, never whether the player coming in had just come back. Two of the biggest changes by effect were LeBron James's return in January 2015 (8th of 1,264) and Dirk Nowitzki's in January 2013 (13th).

• The clean rule, recorded before the run (D188): the first version's conditions, and for every player moved in and every starter moved out:

  • (a) the promoted player played off the bench in each of the five games before;
  • (b) he had started no more than a fifth of the team's games to date;
  • (c) he was on the roster for those five games, and the dropped starter stays on the roster through the five-game window;
  • (d) the dropped starter starts at most five of the twenty games measured.

• The clean estimate: +0.88 (95% -0.24 to +2.04, p 0.12) over 67 changes (59 matched), with halves +1.37 and -0.44. The interval crosses zero and the halves disagree, so it doesn't hold. Inside the first five games: +2.37 for the first version and +3.84 for the clean one.

• The examples come from a narrower set (D190, D192): the estimate uses all 67 clean changes, the 59 with matched teams. The article's examples were picked only from the 48 with five or more matched teams, and only from those that pass a further check: the starter moved out had started at least half the team's games before, and no regular starter was missing from the game before or the game of the change. Leaving out Dirk Nowitzki's return (below), 16 of the other 47 pass. The check picks the examples; it doesn't change the estimate.

• Counting the first five games: coaches keep lineups that are working, so the first five games of any new five are how the change got picked. They're left out of both estimates, and inside them the jump is always bigger.

• What drove the first version: this split was decided after the run, and is labeled that way. The split recorded before it filed most absences as trades, because older box scores leave some inactive players out.

  • returns from injury or a spell out of the rotation, 70%;
  • returns to a role, 14%;
  • trades or waivers, 10%;
  • the benched starter coming back, 2%;
  • clean, 5%.

  Each type's own effect is in outputs/d188/summary.csv and docs/d188_l1_clean.md.

• The article's two ends: the biggest positive and the most negative of the clean changes that pass that check.

  • Utah, December 16, 1997 (Jerry Sloan; Antoine Carr for Greg Foster): +5.8, 42 matched teams. The Jazz's net rating went from +3.5 to +8.1.
  • Orlando, November 27, 2017 (Frank Vogel; Jonathon Simmons for Terrence Ross): -5.5, 41 matched teams. The Magic's went from -2.8 to -7.9.
  • Both were checked against public box scores (Land of Basketball), with the coaches from Wikipedia's team-season articles.
  • Dirk Nowitzki's January 2013 return passes the clean rule: he had come off the bench for the five games before, after missing the season's first twenty-seven for knee surgery. It is a return from injury in spirit, so it's left out.

• Minnesota's clean changes: 3, all with a negative effect (December 2005, February 2006, January 2010).

• A limit the check exposed: the clean rule compares the five only with the game before. A change made because a regular starter was already out still counts as clean. Oklahoma City, February 2012, is one: Daequan Cook started in place of an absent Thabo Sefolosha, and James Harden went back to his bench role after a one-game fill-in. Cases like it are why most of the 47 fail D190's check. The estimate isn't re-run with it.

The plan and the rules. Part 3 is decisions D176 to D179. Before any data was read, the plan fixed:

• the unit and the strength model with its pass rule;
• the populations, the outcomes and the luck components with their combination rule;
• the "mostly luck" threshold and the persistence test;
• the known cases with their written verdicts;
• the levers, each with its decision date and its holder;
• the test families with their false-discovery rule;
• the yardstick with its pass rule;
• the data checks with their fallbacks;
• the tracker list and the limits.

Every later rule was recorded before the run it governs. Nothing here uses Basketball-Reference.

The unit. Every playoff series from 1997-98 to 2025-26: 435 series and 2,368 games. Rounds are read from dates and checked against the game id and Wikipedia.

The strength model. A playoff game's margin is home court plus a multiple of the two teams' regular-season net rating difference, with noise scaled to that season's spread of regular-season margins.

• The fit: home court is worth 4.0 points; each point of net rating difference is worth 0.97 points of margin; game noise is 0.96 of the season's margin spread.
• Series probabilities come from exact enumeration over each series' format. The first round was best of five through 2002, the Finals used the two-three-two format through 2013, and the 2020 bubble has no home court.
• Validation, leaving one postseason out at a time:
  • log loss 0.536, against 0.581 for home court alone and 0.693 for a coin;
  • calibration slope 0.95 (90% interval 0.78 to 1.18).
• The late-season check: net rating over the last thirty regular-season games did worse (log loss 0.596), so the full season stays.
• One soft spot: among the home-court teams the model rated least likely to win, those it gave 45% won 59%.

Luck. Each run gets three chances of reaching a conference final:

• along its expected path (P0);
• along the path it actually took (P_actual);
• along that path with its measured luck added back as margin shifts, series by series (P1).

The luck share is (log P1 less log P0) over minus log P0, reported unclipped. It is below zero when a run was unlucky and got there anyway.

A run is "mostly luck" at a share of 0.5 or more. The 80% intervals come from a bootstrap that resamples three things: the seasons behind the strength model, the player ratings behind health, and each team's regular-season shooting. When a run's interval spans 0.5, its verdict is "can't tell".

The components, each measured against the team's own norm:

• Health: the top eight's missed playoff games against their regular-season rate. A single missed game between two played ones is rest, not health. Each player is weighted by his share of minutes and his box-score rating above replacement.
• Own shooting: three times the gap between the team's series three-point percentage and its expected percentage, times its attempts a game. The expected percentage is its own regular-season percentage, plus the percentage its opponent allowed, less the league's.
• Opponent shooting: the same, for the other team.
• The bracket: log P_actual less log P0.
• Close games: moved to the lever side after the persistence test below. They're reported both ways.

Shooting luck is measured against each team's own regular season and its opponent's, so it can't tell a hot week from a matchup that suited one side.

Does it persist, from one postseason to the next? 331 pairs; the p-values are one-sided with a Holm correction.

Only close games persisted, and only weakly. The result is fragile:

• the rank correlation is smaller;
• without the five franchises with the most pairs it is smaller still;
• it sits in the seasons after 2012, not before.

Within a postseason (D178). For every team that played two or more series, the shooting luck in one round against the next, over 406 consecutive pairs:

• Own shooting does not carry: r +0.064 (95% -0.034 to +0.160), one-sided permutation p 0.099.
• Opponent shooting carries by the rule: r +0.121 (95% +0.023 to +0.215), p 0.0075. About a third of that carry comes from the team's own regular-season three-point defense, which barely repeats from one season to the next. With that term shrunk, the carry's interval crosses zero.
• Treating the opponent carry as skill (a sensitivity; D176's two-way shares): the median luck share for the conference finalists becomes 0.15, with 3 verdicts changed.

Luck shares with and without close games (D176's two-way shares, before the split below):

Shape and variance (D179, D181). Not all of a team's shooting edge in a series is luck. Some of it comes with playing differently, and that part is called shape.

• The shape regressors, per team-series, each the series value less the same team's regular-season value:

  • the team's three-point attempt rate, free-throw rate, pace, and assisted share of made field goals;
  • the opponent's three-point attempt rate, free-throw rate and assisted share, each against the opponent's own regular season.

  The assisted share comes from the play-by-play, counted only in games whose makes equal the box score's (73,441 of 73,450 team-games).

• The two fits: ordinary least squares with an intercept and robust errors, over all 870 team-series. One regresses own-shooting luck on the team's regressors; the other regresses opponent-shooting luck on the opponent's.

• The shape component is the slopes times the changes. The intercept and the residual are variance.

• The three-way split. Luck is variance plus health plus the bracket; shape is the shape component; the rest is everything the strength model already expected. Luck and shape enter each series' chance as margin shifts, and each one's part is the mean of its two orders, so the three parts add to the whole.

• A shape edge is a shape component of 1.0 point of margin a game or more, averaged over a run's series.

The published version is the corrected split (D181). Made threes are assisted far more often than made twos (0.84 against 0.53). So in the rule set before the run, a series in which a team's threes went in raised its assisted share all by itself, and the assisted-share term mostly measured the makes. The corrected split counts the assisted share with each team's made threes at their expected number, for both teams, and keeps everything else as the rule set it. Inside the bootstrap the corrected share moves with each draw's expected percentage. The article, the luck-and-shape chart and the slide all use the corrected split; the rule's numbers are reported beside it.

The check that found it refit the shape fits two ways: with made threes at their expected number (the published version), and with the assisted share of made twos only.

The shares, both versions.

Why the band's median moved under the rule. On the rule's split, the band's median rose from 0.18 on the two-way share to 0.24, because medians don't add. Taking the rule's shape out raised luck for the 19 runs whose shape share was negative and lowered it for the rest. The middle of the band is crowded, so the run at the middle changed: the 2002 Boston Celtics went from -0.10 to 0.24. On the corrected split, with less shape to take out, the band's median is 0.19.

The opponent-shooting carry as skill (D178's sensitivity, on the corrected split). Count the part of the opponents' shooting that carried from one round to the next as skill, not luck, and the median luck share falls:

• to 0.11 for the conference finalists (24 mostly luck);
• to 0.12 for the band (7 mostly luck).

On the rule's split those were 0.12 and 0.18. The published shares keep the corrected split's definition.

The caveat. Shape is measured in the same series it explains, so it can follow winning as well as cause it. Calling it the coach's and players' lever is a choice. It describes a run; it isn't something to watch for in advance.

The intervals. The 80% intervals for luck and shape come from D176's luck bootstrap, with the shape fits refit inside each draw. Shape's intervals leave out the uncertainty in the fits themselves, so they are narrower than they should be.

The 2011 Mavericks, series by series. In 2010-11 the league shot .358 from three. Dallas made .365 and allowed .343. All figures are points of margin a game.

The known cases, written before the run.

The 2011 Mavericks are the one case that went against what was written. Their second-round sweep of the Lakers is most of why.

The levers: two test families.

• Same quality: every in-season decision is compared with teams of the same quality, measured before it. The match is on net rating over the 20 games before and over the season to date, within 1.0 then 1.5 points, at the same game number within three. The errors come from a cluster bootstrap by team-season.
• The playoff levers: series-level regressions of the margin surprise on the lever, per standard deviation, with heteroskedasticity-robust errors.
• What counts: a test counts if its sign is the same in both windows and it clears Benjamini-Hochberg at 0.10 within its family.
• The predictive family (15 tests, 1 counts) takes only what was measured before the series it predicts.
• The explanatory family (10 tests, 4 count) takes playoff values measured in the same series. It describes; it doesn't predict. P7 is P6a mirrored by construction: every series is in the data from both sides, and a team's opponent three-point rate is the other team's own, so P7's effect is P6a's with the sign flipped. The four that count are three distinct findings.

L1's selection window. The five games that define an L1 event are part of how it was chosen. Its effect is reported after them:

• games five to twenty-four after the change: +0.80 (95% +0.51 to +1.08);
• inside the window: +2.37;
• pure reshuffles, where every new starter played the game before: +1.35 over 478 events.

Rest and the deadline, beside the tests.

• Rest's cost: a team's net rating in games where a top-eight player rested, less its other games, is -1.51.
• The deadline (D172's figures, restated on the round-range band): 36% of the deep runs added a rotation player at the deadline or on the buyout market, against 53% of the rest. In the seasons with an exact deadline date it's 56% against 69%.

The yardstick. A penalized logistic regression (L2, the penalty chosen inside each training fold) of reaching a conference final:

• The unit: every team-season, 1998-99 to 2025-26.
• The fits: one per checkpoint (games 20, 30 and 41, the deadline, game 60 and April), validated by leaving one season out.
• The pass rule: a feature stays only if it improves the log loss of net rating alone by more than the ninety-fifth percentile of the same improvement with the feature shuffled within season. It's Bonferroni-corrected, and its sign must match D172's.
• The three features tried, and their permutation p-values across the checkpoints:
  • a top-ten player by the box-score rating: 0.34 to 0.78;
  • a first-year pairing of the top two creators: 0.92 to 0.98;
  • L1: 0.34 to 0.98.
• The result: none passed at any checkpoint (0), so the yardstick is net rating alone. Its log loss runs from 0.288 at game 20 to 0.246 in April.
• The lines are D172's ranks, read as net ratings: the median net rating of the 14th team after 20 games and of the 9th in April over the same seasons.
• The curve is in outputs/d176/part3/curve.csv.

The band. Teams whose net rating the season before was between -2.5 and +5, and who went out in the first two rounds or missed the playoffs.

• The range is round numbers set without the odds (D176's first call).
• The odds band: it overlaps the band the market's preseason odds would draw by about half, measured in-sample.

Net rating (D189, D192). Part 3's two lines, the yardstick and the champions' bar were drawn on this page's definition, as the champions section states it: the possession-weighted mean of NBA.com's per-game team net rating. Where NBA.com's possessions are missing, the box estimate is used instead: field-goal attempts, plus 0.44 of free-throw attempts, less offensive rebounds, plus turnovers, averaged over both teams. Every card on the Stats page is graded on it.

• The model's own projections use its own average, the plain average of the same per-game numbers. The game-thirty lines in Part 2 and Part 4, Minnesota's +1.8 among them, were set on that scale. Their cards say so, and show the model's number beside this page's only when the two fall on opposite sides of the line.
• The gap: over the 863 team-seasons since 1998-99, the two differ by 0.07 on average and never by more than 0.36. Nothing printed changes: under both, none of the eleven champions was worse than 11th after twenty games, and 10 of 11 finished the season top five.

Sources. Box scores, play-by-play and team ratings come from NBA.com, through our warehouse. Seeds and in-season coaching changes come from Wikipedia's season articles (CC BY-SA 4.0), read through their raw wikitext. The named games, injuries and coaches were checked against the public reports cited in D186, D187, D188 and D190:

• the 2011 Lakers series: ESPN, Wikipedia, the Washington Post, HuffPost and the Spokesman-Review;
• the 2025 injuries: ESPN, NBA.com, CBS Sports and ClutchPoints;
• the coaches: NBA.com, Wikipedia's team-season articles and spurs.com;
• the lineup cases: Land of Basketball box scores;
• the opening's price: BetMGM's Pacers preview and CBS Sports.

Nothing uses Basketball-Reference.

Limits.

• The lineup levers on the play-by-play (P3a, P3b) were dropped as a data failure before any lever result was read. The rebuild for the older play-by-play format is on the November list.
• The band is a proxy: its range is round numbers, not a fitted one.
• Era effects:
  • the 2001-02 defensive rules and the 2004-05 hand-check rules raised pace and scoring, so pace and shot-profile levers are read within era;
  • the play-in changed who reaches round one from 2020-21, and its games aren't in the warehouse;
  • the 2023-24 participation policy changes what rest means;
  • the two lockout seasons and the 2020 bubble are short.
• Conference: bracket luck is measured against each conference's own field.
• Survivorship: deep runs are chosen by their outcome, so nothing is compared within deep runs alone. Every lever is judged on every team that made the decision, against same-quality teams that didn't.
• Health is coarse:
  • only whole missed games count;
  • a player hurt during a game and playing through it isn't caught;
  • a suspension counts as health (the 2011 Lakers' third game is one).
• Small samples: the band has 45 deep runs, and a lever's effect on the band is an extrapolation from all teams.
• Seed luck isn't scored: close regular-season games that moved a team's seed are folded into home court.