You're correct, I think the excact rule is at the last day of the training camp, the team should respect the cap.

Thanks!

One additional note: I left the roster at 21 "effectives" to illustrate the cap space ultimately available with the actual squad. Obviously, for the first game, a forward and a defenseman from the Marlies would be brought up.

LTIR does not allow you to exceed cap by the full contract though. You get to exceed by the LTIR Cap Hit - Cap Space at time the contract was put on LTIR. So in your example, they would only get about 1 mill in extra room when they put Horton on LTIR

I think they can put players on LTIR after the first puck drops on the very first game any teams are playing. Teams can also go up to 10% over the cap until a date that I dont remember but it's sometime in September or the same time players can go on LTIR

I believe they can be over the cap by 10% in the off season so I think they can sign Marner and then put Horton on ltir I think this can work not sure though

Well I can't see how that can be true. For instance one year the Leafs had Lupul , Robidas and Horton on LTIR. (13.5m) Let's say the cap was 70m that year. The Leafs weren't limited to have a "active"roster of 56.5 at training cap....and then suddenly after day one they could put 13,5m of player on the roster.
Let's imagine by your scenario that cap strapped Tampa team this summer. Everyone is signed they are under the cap next week. On July 25, Stamkos and Kucherov get season ending injuries while training. Are you saying the Lightining can't sign 18m in replacements for next summer till after the opening game?
Teams are not cap punished for having LTIR. Yes teams have to cap compliant on opening game.....but having LTIR gives a team cap leniency to sign replacements.

Illustration 4 on page 280 in the CBA seems to cover it quite clearly.

Illustration #4: The Upper Limit in a League Year is $70.0 million. A Player who has an SPC with an Averaged Amount of $2.0 million becomes unfit to play on the last day of Training Camp, and on the same day, his Club exercises the Bona-Fide Long-Term Injury/Illness Exception on such Player. On Opening Day, the Club has an Averaged Club Salary of $71.5 million (excluding Earnable Performance Bonuses up to the full amount of the Performance Bonus Cushion). The Club is deemed to have already fully replaced the unfit-to-play Player with any Player or Players on the Opening Day Roster. If these replacements are maintained through the conclusion of the season, the Club's Averaged Club Salary is $71.5 million, as the Club is permitted to exceed the Upper Limit by $1.5 million because of the Bona-Fide Long-Term Injury/Illness Exception.

They'll add him to LTIR on the last day of training camp and be in compliance.

The 10% does not affect LTIR. The 10% allows teams to sign players and make trades during the summer and be over the cap. Then during the summer they have to make deals to be cap compliant.
Also technically many teams are over the cap at training cap. Sending players to the minors by the end of training cap makes them cap compliant.

I get the feeling that the only people who really understand how this LTIR stuff works are the actual professionals and legal eagles.

We just had a nice earthquake near here -- I went on the government's earthquake map (a commonly visited site here in SoCal) and see that there was one today in British Columbia, too.

Now how am I to understand that? Is my bottom line right: that Toronto can sign Marner for anywhere up to $9.75M?

There are some technical issues with LTIR. But then fact is teams are not cap punished for having injured players. And for some reason some folks believe LTIR does not apply during the summer. That's false. Teams get cap freedom when a player is on LTIR....and especially useful if you know the player is going be on LTIR all season.....Horton, Clarkson and Hossa come to mind.
The only really affect having a player on LTIR is in might affect when a team can pay it's performance bonuses....whether that year or the have to deferred to he following. That's the reason Horton was on IR not LTIR ;last year with the Leafs. Leafs wanted/needed to pay their Bonuses last season instead of delaying them to the expensive (Matthews/Marner) this season.

As long as Toronto's total AAV minus Horton's AAV is under $81.5M, they're fine. I calculate their space to sign Marner to be around $9.765M with no other roster changes except for removing one of their extra defensemen.

False. They could sign for 11, 12, 13m . They would have to trade another player to be cap compliant though by first game.

In all likelyhood with a signed Marner the Leafs will be at training camp with 81.5m active payroll plus Horton's 5.3m. (let's assume Dermot and Hyman are healthy). First game...same thing....Game 50 same thing.
Vegas.....81.5m....plus 5.25m Clarkson.. Tampa 81.5m plus 5.8m Callahan. It's really not difficult to understand. It's just that folks make up the 10% rule and the summer rule and relate to the season ending LTIR.

Collin Campbell went on sportsnet a couple of years ago and explained that teams can apply for off-season LTIR. The only benefit to trading someone on LTIR(everyone is trading for Clarkson as they think Vegas needs off-season cap relief) is ro free up a roster spot(as far as i know this is the only benefit)

One thing to note, if Dermott and Hyman are placed on LTIR at the start of the season, it's a different equation than Horton's LTIR. Since Horton remains on LTIR the entire season, his cap hit virtually doesn't affect Toronto. But with Dermott and Hyman, they get placed on what's known as season opening LTIR. On SOLTIR the team gets the cap relief only from the start of the season until the player returns. Once the player returns, only the remaining portion of the season counts towards the cap. But when the cap resets at the end of the season and teams have to be cap compliant, they team gets the full cap hit of the player. If they're over the upper limit they have to pay overages into next season's cap.

Example (don't quote me on exact numbers):
1: Hyman placed on on SOLTIR as he'll be out until Decemberish
2: Leafs receive cap relief of Hyman's contract ($2.25M) until he returns
3: Hyman returns after missing 41 games (just so my math is easier)
4: When Hyman returns, only $1.125M on his contract counts towards the cap while the season is under play as that's roughly what's remaining on his deal
5: When the cap resets at the end of the season, Toronto absorbs Hyman's full $2.25M cap hit on top of whatever their team cap hit is
6: If Hyman's full cap hit pushes Toronto over the upper limit of the cap by, say for example, $1M then Toronto has to pay an cap overage on the cap for the 2020-2021 season, giving them $1M less to spend next season.

For this reason, I wouldn't be shocked if Dubas and company found a way to only have Horton on LTIR and have Dermott and Hyman count towards the cap fully until they return. It's just better for the long term situation of the franchise

That....might be true, I never thought oh how the roster spot works. In Ottawa's case they might want/need Clarkson to hit the floor by using Clarkson's LTIR as an IR, thought he doesn't affect Vegas cap at all.
Last year Hossa and LTIR was traded by the Hawks to Coyotes. The reason I believe the Hawks did this is they were afraid that next season (or the year after) Hossa might be deemed fit. Hawks didn't think they could absorb his cap hit of 5.25m but Arizona could. Hossa was going to stay retired even if was deemed fit since his actually pay 1m. So the trade was made for "what if I am deemed fit to play" purposes not really LTIR reasons.

Season-Opening Injured Reserve (SOIR) only applies to players on two-way contracts that would have a lower salary if they weren't playing in the NHL. In Dermott's situation that would apply here, however Hyman is on a one-way contract and would be subject to his full cap hit during and after his injury.

That's exactly what I got.