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The truth can be hard to find sometimes. Not always is it prominently visible. Often, hard evidence is hard to find, or kept under wraps. And getting confessions is no easy task.

Sometimes you have to dig beyond the basic facts — like the Patriots having at least 11 of 12 official game balls underinflated for last week’s AFC Championship rout of the Colts. Coach Bill Belichick and quarterback Tom Brady have voiced their denials. If the NFL has physical evidence, they haven’t disclosed it, or even suggested its existence. No eyewitness of wrongdoing has come forward.

Without physical evidence, witnesses or a confession, we are forced to turn to the most unreliable of evidentiary standards — the circumstantial variety. If someone with the Patriots purposely was deflating approved game balls, there has to be a reason. Because a less inflated ball is easier to grip, throw and catch, the benefits of such a maneuver presumably would be more apparent in bad-weather games.

Based on that assumption, The Post sought the help of numberFire to track down the stats from such games — which we deemed to be any contest in which the temperature dipped to 40 degrees or below and there was some sort of precipitation. Granted, these parameters exclude last week’s AFC title contest, but that doesn’t preclude similar actions in other games from affecting performances or outcomes.

The results of this statistical analysis reinforce the notion that Brady is a stellar cold-weather QB, in comparison to his NFL quarterback peers, but they don’t offer the kind of statistical outliers that overtly suggest wrongdoing. You be the judge.

Cool customer

Brady wins at a rate far above other contemporary QBs, who have at least 225 pass attempts in games that experience temps 40 degrees or below and have precipitation. Here’s a look:

PLAYER                              W-L               PCT.

Tom Brady                             12-1               .923

Ben Roethlisberger               9-3               .750

Jay Cutler                                5-3               .625

Drew Bledsoe                          4-3               .571

Brett Favre                                8-8               500

Peyton Manning                      3-4               .429

Memorable games

Jan. 19, 2002 — Tuck Rule Game

Patriots 16, Raiders 13 (OT): Tuck Rule overturns a late fumble by Brady, allowing Patriots to kick tying FG then win in OT over the Raiders in the AFC divisional playoffs in Foxborough. Brady goes 32-of-52 for 312 yards with an INT. Rich Gannon throws just 17-31 for 159 yards with a TD.

Jan. 18, 2004 — Brady v. Peyton IV

Patriots 24, Colts 14: After three regular-season victories over Peyton Manning, Brady adds a fourth, and first in the playoffs, going 22-of-37 for 237 yards with a TD and INT at home in the AFC title game to thump Manning (23-47, 237, 1-4) and Co. in a snowy game.

Jan 16, 2005 — Brady vs. Peyton VI

Patriots 20, Colts 3: Brady didn’t need to be great (18-27, 144, 1-0) to knock out Manning (27-42, 238, 0-1) and the visiting Colts in the divisional round on a cool, soggy night.

Oct. 18, 2009 — Whiteout Blowout

Patriots 59, Titans 0: Brady goes 29-of-34 for 380 yards and six TDs, with no INTs, in snow-covered Foxborough. Vince Young and Kerry Collins combine to go 2-of-14 for minus-7 passing yards with an INT.

Dec. 16, 2012 — Sunday Night Stunner

49ers 41, Patriots 34: Despite a furious rally from a 31-3 deficit, led by Brady (36-65, 443, 1-2), Colin Kaepernick (14-25, 221, 4-1) and the visiting Niners did the unthinkable — handing Brady his first loss in cold, wet weather.

Just win, Brady

A look at Brady’s other bad-weather triumphs:

Jan. 6, 2002 — Win, 38-6, at Carolina. Rookie Brady (17-of-29, 198 yards, 1 TD-2 INTS) outduels Chris Weinke (15-36, 144, 0-3) in a cold, rainy season finale.

Dec. 7, 2003 — Win, 12-0, over visiting Dolphins. Brady’s pedestrian 16-of-31, 163-yard game on the snowy night easily dwarfs Jay Fiedler’s 13-31, 111-yard, two-INT outing.

Dec. 14, 2003 — Win, 27-13, over visiting Jaguars. Brady (22-34, 228, 2-0) and Co. dispose easily of the Jags and Byron Leftwich (21-40, 288, 1-2) in heavy snow.

Dec. 4, 2005 — Win, 16-3, over visiting Jets. Brooks Bollinger (15-37, 135, 0-1) is no match for Brady (27-37, 271, 0-0) in snowy conditions.

Dec. 11, 2005 — Win, 35-7, at Buffalo. Brady is stellar in the snow (29-38, 329, 2-2) against the Bills and inept J.P. Losman (10-27, 181, 1-3).

Dec. 16, 2007 — Win, 20-10, over visiting Jets. Brady (14-27, 140, 0-1) and the Pats rely more on defense and the ground game in cool, damp conditions against Chad Pennington (25-38, 184, 0-0) and the Jets.

Dec. 6, 2010 — Win, 45-3, over visiting Jets. The cold and wet conditions didn’t hamper Brady (21-29, 326, 4-0), but the same can’t be said for Mark Sanchez (17-33, 164, 0-3).

Dec. 12, 2010 — Win, 36-7, at Chicago. Jay Cutler has fared well in bad weather in his career, but not in this driving snowstorm (12-26, 152, 0-2). Brady does what Brady normally does (27-40, 369, 2-0).

Over exposure

Brady has won more cold-weather games than any QB since he came into the league in 2001. But he also has played in more. Brady’s 480 foul-weather pass attempts since 2001 are the most outside of Brett Favre. In fact, Ben Roethlisberger (405) is the only other one with more than 250.

PLAYER                                     PA

Brett Favre                                    552

Tom Brady                                    480

Ben Roethlisberger                     405

Jay Cutler                                      247

Peyton Manning                          236

Drew Bledsoe                               229

Best of bad situations

You can measure a player’s productivity by assigning potential points to every play. A stat called the NEP (net expected points) by numberFire illustrates this idea — creating a value of points an average team can be expected to score on a particular drive, based on down, distance, field position, etc., using data from the past dozen years. The NEP changes based on the results each play, and the player responsible for that change is credited, creating an NEP for each player in relation to the expected average.

“By taking the difference between a team’s expected points on two plays,” explained Keith Goldner, numberFire’s chief analyst, “we can see how much a player or play increased or decreased that team’s chances of scoring given the situation.”

Given the small pool with whom to compare, here we have a field of those with 60 or more bad-weather pass attempts. With such a small sample size for some, those near the minimum might not qualify as bad-weather QBs, but merely QB who had a couple of good bad-weather games.

Here, we compared QBs based on pass attempts in normal weather (PA/N) and numberFire’s resulting NEP (NEP/N), the corresponding bad-weather stats (PA/W and NEP/W) to arrive at the net NEP (difference between the two).

PLAYER                            PA/N          NEP/N          PA/W          NEP/W          Net NEP

Jay Cutler                          3,668              .12               247                .24                   +.12

Carson Palmer                  4,862              .13               82                  .18                  +.05

Eli Manning                        5,867              .10              86                  .12                  +.02

Matt Cassel                         2,242              .05              114                 .05                  0.0

Tom Brady                          7,720              .22              480                 .20                -.02

Joe Flacco                           4,019              .12                71                    .06                -.06

Donovan McNabb             5,645              .12               104                   .03               -.09

Drew Bledsoe                      2,576              .06              229                  -.06              -.12

Ben Roethlisberger            5,015              .23               405                   .11               -.12

Philip Rivers                       4,857              .24                101                    .10              -.14

Aaron Rodgers                   3,808              .34                63                     .19               -.15

Drew Brees                          7,850              .23                73                     .07              -.16

Mark Sanchez                     2,273              .02               70                  -.14                 -.16

Matthew Stafford               3,128              .11                   73                  -.11             -.22

Brett Favre                           5,632             .14                   552                -.11             -.25

Peyton Manning                 8,614               .27                  236               -.03            -.30

Kirk Cousins                         323                 .20                 202                -.45            -.65

Anomalies

Andy Dalton                       2,240               .09                     35                    .45           +.36

Alex Smith                          3,232                .09                     20                    .41            +.32

Colin Kaepernick              1,243                  .18                      25                   .47          +.29

Andrew Luck                     2,034                  .11                      46                    .35          +.24

Cool rating

Whatever you think of the validity of quarterback rating stat, glaring differences are just that: glaring. Here is a look at the difference (DIFF.) in QB rating during fair weather (QB/N) and bad weather (QB/W) for those meeting the 60 bad-weather attempts threshold, bad weather defined as 40 degrees or below with some sort of precipitation:

PLAYER                              QB/N                              QB/W                              DIFF.

Jay Cutler                             84.53                              93.16                              +8.63

Carson Palmer                      85.98                              87.25                              1.27

Matt Cassel                             79.53                              78.8                              -0.73

Tom Brady                              95.05                              92.68                              -2.37

Eli Manning                            82.62                              78.59                              -4.03

Joe Flacco                                85.55                              80.37                              -5.18

Ben Roethlisberger                93.10                              86.37                              -6.73

Donovan McNabb                   85.92                              77.72                              -8.20

Drew Bledsoe                            80.01                              70.55                              -9.46

Philip Rivers                               95.11                              80.80                            -14.31

Drew Brees                                  95.7                              79.82                              -15.88

Aaron Rodgers                           105.54                              88.79                            -16.74

Matthew Stafford                        83.97                              66.92                             -17.05

Mark Sanchez                               75.63                              57.62                             -18.01

Brett Favre                                    86.56                              64.96                             -21.6

Peyton Manning                           99.34                              72.14                              -27.2

Kirk Cousins                                  87.69                              37.21                              -50.48

Various anomalies?

— Brian Hoyer’s foul-weather completion percentage, albeit in very limited work in difficult conditions, is 25.52 percent better than is fair weather.

— In a small sampling of bad-weather work, Kurt Warner’s completion percentage dropped 32.4 percent.

— Matthew Stafford, since 2001, has a better bad-weather QB rating (66.92) than Brett Favre (64.96)

— Tony Romo’s QB rating falls 62.47 points (from 97.37 to 34.9) in bad weather, though he has attempted just 16 passes in games with temps 40 or below and precipitation.

What is numberFire?

The stat-heads at numberFire provide statistical analysis for projections and tools for businesses and customers, ranging from “help with March Madness to fantasy sports to implications of breaking news,” Chief Analyst Keith Goldner said.

For the Super Bowl, numberFire is introducing a real-time application that will allow fans to track various analytics during the game, using live action and predictive analysis to determine how likely a team is to score on a given drive, the percentage chance a team has to win at any point in the game, etc.

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