TROON, Scotland — A fine line separates optimism from delusion, a narrow DMZ where the belief that things will improve collides with immovable facts that simply won’t support buoyancy.

That’s the space where Tiger Woods’ fans have been living for years, and his early but not unexpected departure from the 152nd Open can only render as hollow the arguments of the diehard faithful. It was a performance that leaves a lot of available real estate on the island of believers in Woods’ prospects as an elite force.

By the time he rolled out of this overcast village on Scotland’s western shore, he was at the back end of the leaderboard atop which he once presided. Only five men in the field had a worse two-round total than his 156, a glum number he reached when a Friday 77 was added to his opening 79. He made just three birdies in 36 holes. His Strokes Gained Total statistic shows he lost almost eight strokes to the field, 3.68 with the putter and more than 4 with his approach play. Only around the greens did he creep (barely) into positive numbers.

“Well, it wasn’t very good. Just was fighting it pretty much all day,” Woods said, displaying an admirable gift for understatement.

In the two years since the 150th Open in St. Andrews, Woods has made seven competitive starts. The ledger shows two withdrawals, two missed cuts, a tie for 45th at the Genesis Invitational 17 months ago, and two dead-last finishes, one of which was in his own 18-man Hero World Challenge. His latest effort at Royal Troon continues a well-established chicken-egg conversation.

“I’d like to have played more but I just wanted to make sure I was able to play the major championships this year,” he said. “I’ve got a lot of time off to get better physically better. I’ve gotten better, even though my results haven’t really shown it. But physically I’ve gotten better, which is great. Just need to keep progressing like that and eventually start playing more, start getting into the competitive flow again.”

That narrative has been dispiritingly familiar to golf fans since Woods re-emerged from a 2021 car wreck. At every major he talks about the need for more reps, but the reps never come. He has reasons, of course, none of them unreasonable: young kids, global business, boardroom responsibilities, broken body. Mostly it’s the body. All but the most feverish understand that he’s at the stage of needing to catch two lightning bolts in a thimble in order to win.

After his second round, Woods admitted he doesn’t even plan to compete again for five months, not until December, when he appears at the Hero and the PNC Championship, his annual outing with his son, Charlie. “I’m not going to play again until then and keep working on it. And just come back for our fifth major, the father-son,” he said.

The chicken-egg cycle begins another lap.

Woods has been irrelevant as a competitor for years, certainly since the crash. So in that respect, Colin Montgomerie wasn’t off-base in expressing befuddlement that Woods stays out here, a pale shadow of his once resplendent self. But Montgomerie was myopic in thinking that competitiveness is the measure of Woods’ relevance. What he does inside the ropes is no longer the metric by which his contribution to the product is assessed, and as we know, what matters most these days is “the product.”

In a bitterly divided sport, Woods will play the role of pied piper. He’s on the PGA Tour’s transaction subcommittee that negotiates directly with the Saudis. He’s on the Tour’s Policy Board, the only member with no expiration date on his term. He will help shape the future of the men’s professional game, and be instrumental in selling it to both fans and fellow players. That’s why he was added to both bodies. His public voice matters, perhaps because he hasn’t used it often.

Regardless of the numbers he posts, Woods’ presence is additive to the business, just as Arnold Palmer was even when telecasts stopped showing his scores. The Tour admitted as much last month when it voted him a lifetime exemption into signature events, for which he would not otherwise be eligible. Now, when the whim strikes him, the 874th-ranked player in the world has a guaranteed spot in elite limited fields. It was dressed up in the language of lifetime achievement for those who’ve won more than 80 times on Tour, but if the win total was 79 the line would simply have been moved. And that’s defensible.

Woods is a proud man and his scores must settle somewhere between embarrassing and irksome, but he seems to maintain a belief that there’s another run in him. It’s highly improbable, yet still possible. What isn’t speculative is his value in the here and now. When Woods shows up, he adds eyeballs and bolsters the Tour’s chief constituents — sponsors being asked to pay more, broadcasters airing a diluted product and fans expected to overlook the absence of a handful of engaging stars. Even if his appearances in actual Tour events are scarce.

His value isn’t diminished by the scores at Royal Troon. Not for Tour executives, not for its private equity investors at Strategic Sports Group, and not for the gaggle of pasty-faced kids chasing him around a wet, blustery Scottish links in hopes of a glimpse or an autograph. The kids didn’t seem to care about the number on his scorecard. And the others? They’re focused on the number he adds to a valuation. That’s the long-term outlook that still holds promise.

The USA TODAY app gets you to the heart of the news — fast.Download for award-winning coverage, crosswords, audio storytelling, the eNewspaper and more.