Lemma 22.1
Four-country whole
N = Ireland + England + Scotland + Wales
The four country parts add back to the complete live archive rather than creating four separate totals.
Mathematical depth 22
Six real places, with a deeper way through.
The six place choices are real pages from the live archive. The mathematical doors underneath keep the journey inside Discover Alone, add one to its depth and open another changing set of six places.
- Start here: Mapped graveyardGraveyard · City of Bristol
- Try this: Spanish Civil WarMemorial · City of Bristol
- Try this: The Walled GardenGarden · City of Bristol
- Try this: Charles HeaphyBirthplace · London
- Try this: Children's playgroundChildren's playground · Doncaster
- Try this: Children's playgroundChildren's playground · Isle of Wight
The mathematics of finding without an ending
These lemmas describe what the page really does. The formulas change with the carried depth and the live archive total; the facts underneath them do not.
Lemma 22.2
Far-apart starting offsets
offsets = {1, 97, 997, 10,007, 50,021, 100,003}
The six choices are taken from widely separated positions rather than six neighbouring database rows.
Lemma 22.3
Possible labelled routes
routes after k mathematical steps = 120^k
The formula counts possible sequences of labelled doors; it does not pretend that extra database records exist.
Lemma 22.4
One row, one public address
public page = page(saved record identifier)
A saved database row and its public page remain the same thing; no second made page is required.
Lemma 22.5
No repeated card
|chosen record identifiers| = 6
A record already selected for the current six is not shown a second time in that same set.
Lemma 22.6
Saved total depth
D_total = sum(d_i)
The saved total is the sum of the carried depth on each recorded page opening.
Lemma 22.7
Depth survives a handoff
carried d = explicit depth, otherwise depth inside the carried seed
A journey arriving from Dagmor or AI ContactCentre keeps its existing depth instead of restarting at zero.
Lemma 22.8
A bounded spread
0 <= different records after v openings <= min(N, 6v)
Six pages per opening can broaden coverage quickly, while the real database total remains the firm upper limit.
Lemma 22.9
Average saved depth
D_average = D_total / page openings
This is the average journey number on an opening, not the number of different people.
Lemma 22.10
The next depth
d_next = 22 + 1 = 23
Every mathematical continuation adds exactly one to the carried journey number.
Lemma 22.11
Mathematical continuations
B_math = 120
These doors stay inside the Discover Alone Fractal and each one carries the next depth.
Lemma 22.12
Positive-depth share
P_positive = openings with d > 0 / all page openings
This shows what share of the complete page-opening pie certainly followed an earlier page.
Continue through 120 mathematical doors
Every door below remains inside Discover Alone, carries depth 23 and opens six more real place pages.