Finally, we're at the point where things can get assembled without anyone screaming Spoiler! Oh, how I've waited.
Here's a summary of the travels of the Napoleon cards, with page numbers:
1. Mick has them, alluding to Ada's aid in acquisition. (28)
2. Sybil sends them to Paris and picks them up there. (57)
3. Sybil gives the box to the Fils de Vaucanson, specifically Theo Gautier. (392)
4. Theo runs them through the Napoleon and affects its powers of higher reasoning. (386)
5. The box is stolen by Flora Bartelle and brought back to Britain. (386)
6. Mallory is given them by Ada for safe-keeping; he hides them in the Brontosaurus. (94; 216)
7. The box, revealed by Flora and the Marquess, is given to Oliphant at the crime scene, with a note from Ada to Flora and Collins. (377 and 375)
8. Oliphant gives the cards to Keats, to find out what they are. (415)
Keats, of course, doesn't tell us, but Ada does, sort of, a bit. In the very last section of the novel, the Queen of Engines gives us a long-awaited acknowledgement of the Modus and its operation, which I now think of as something like the computer running through infinite tic-tac-toe sequences at the end of Wargames (anyone else?):
"And yet the execution of the so-called Modus Program demonstrated that any formal system must be both incomplete and unable to establish its own consistency. There is no finite mathematical way to express the property of 'truth.' The transfinite nature of the Byron Conjectures were the ruination of the Grand Napoleon; the Modus Program initiated a series of nested loops, which, though difficult to establish, were yet more difficult to extinguish. The program ran, yet rendered its Engine useless! It was indeed a painful lesson in the halting abilities of even our finest ordinateurs.
"Yet I do believe, and must asset most strongly, that the Modus technique of self-referentiality will someday form the bedrock of a genuinely transcendental meta-system of calculatory mathematics. The Modus has proven my Conjectures, but their practical exfoliation awaits an Engine of vast capacity, one capable of iterations of untold sophistication and complexity."
This is a partial list of the varying theories and explanations of the Napoleon cards:
"... No one can get it to run." (22)
"A certain nested series of mathematical hypotheses." (27)
"Amuse [Babbage]...." and "gambling system...." (30)
"It is gambling-trouble. Lady Ada has a Modus.... It is a legend in sporting circles, Dr Mallory. A Modus is a gambling-system, a secret trick of mathematical Enginery, to defeat the odds-makers. Every thieving clacker wants a Modus, sir. It is their philosopher's stone, a way to conjure gold from empty air! .... I'm no mathematician, but I know there's never been any betting-system that worked worth a damn. In any case, she's blunded into something nasty again." (188-189)
The supposed mathematical dysfunction created in the Napoleon is probably easily recognized by people better at pure mathematics than I -- but I know enough about programming to know that anything resembling the organic world would be massively complex and self-referencing, constantly checking and adjusting untold variables. Suffice it to assume that the program is correct in a way simpler minds cannot comprehend, that the so-called "error" in the Napoleon is actually just un-computer-like behaviour.
Does said effect take 140 years to germinate, culminating in the self-creation of an omniscient artificial intelligence? Well, maybe. The authors certainly seem to hope so. Is the program, in fact, a gambling aid, capable of analyzing probability via game and set theories? Probably not, when you think about it. What kind of conversation must Ada have had with Mick or his clacker associates when the cards were first created? How could knowledge of the program have gotten to such unsavoury characters as the tout and the tart? What are we missing?