Lemma 115.1
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.
Mathematical depth 115
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: Embanked enclosureFairy fort / stone ringfort · Glennanea,turlaghmore
- Try this: Embanked enclosureFairy fort / earthen ringfort · Jamestown
- Try this: Fairy fort / earthen ringfort in Stonepark (ballintober South By.)Fairy fort / earthen ringfort · Stonepark (ballintober South By.)
- Try this: pier/jettypier/jetty · CARROWHUBBUCK SOUTH
- Try this: Hut site in DunsinaneHut site · Dunsinane
- Try this: The Arbroath Cliff TrailVisitor information · Angus
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 115.2
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 115.3
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 115.4
Saved total depth
D_total = sum(d_i)
The saved total is the sum of the carried depth on each recorded page opening.
Lemma 115.5
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 115.6
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 115.7
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 115.8
The next depth
d_next = 115 + 1 = 116
Every mathematical continuation adds exactly one to the carried journey number.
Lemma 115.9
Mathematical continuations
B_math = 120
These doors stay inside the Discover Alone Fractal and each one carries the next depth.
Lemma 115.10
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.
Lemma 115.11
Six real records
R = 6
Each opening keeps six real place pages from the live archive in front of the reader.
Lemma 115.12
Two useful kinds of choice
B_total = 120 + 6 = 126
The main web combines mathematical continuation with direct entry into real place pages.
Continue through 120 mathematical doors
Every door below remains inside Discover Alone, carries depth 116 and opens six more real place pages.