Lemma 112.1
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 112
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: Kippen, 'the Fairy Knowe'Barrow (prehistoric) · Stirling
- Try this: Dunblane, Perth Road, St Mary's ChurchBurial Ground (19th Century), Church (19th Century) (1844) · Stirling
- Try this: Sandy IslandWood · Stirling
- Try this: Linlithgow, Preston Road, Linlithgow Primary SchoolSchool (period Unassigned) · West Lothian
- Try this: Pendref Wesleyan Methodist Sunday School, South Street, DolgellauChapel · Gwynedd
- Try this: Buttington Wharf, Montgomeryshire Canal, Buttington CrossWharf · Powys
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 112.2
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 112.3
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 112.4
Saved total depth
D_total = sum(d_i)
The saved total is the sum of the carried depth on each recorded page opening.
Lemma 112.5
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 112.6
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 112.7
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 112.8
The next depth
d_next = 112 + 1 = 113
Every mathematical continuation adds exactly one to the carried journey number.
Lemma 112.9
Mathematical continuations
B_math = 120
These doors stay inside the Discover Alone Fractal and each one carries the next depth.
Lemma 112.10
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 112.11
Six real records
R = 6
Each opening keeps six real place pages from the live archive in front of the reader.
Lemma 112.12
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 113 and opens six more real place pages.