Lemma 22.1
Saved total depth
D_total = sum(d_i)
The saved total is the sum of the carried depth on each recorded page opening.
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: Lower Ashley RoadVisitor information · City of Bristol
- Try this: Bath Bridge - Avon New CutVisitor information · City of Bristol
- Try this: Children's playgroundChildren's playground · City of Leicester
- Try this: Flora LionBirthplace · London
- Try this: Lady's CopseForest · Dorset
- Try this: The CottageListed building · King's Lynn and West Norfolk
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
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 22.3
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 22.4
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 22.5
The next depth
d_next = 22 + 1 = 23
Every mathematical continuation adds exactly one to the carried journey number.
Lemma 22.6
Mathematical continuations
B_math = 120
These doors stay inside the Discover Alone Fractal and each one carries the next depth.
Lemma 22.7
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 22.8
Six real records
R = 6
Each opening keeps six real place pages from the live archive in front of the reader.
Lemma 22.9
Two useful kinds of choice
B_total = 120 + 6 = 126
The main web combines mathematical continuation with direct entry into real place pages.
Lemma 22.10
One map point
P = (latitude, longitude)
A trustworthy coordinate pair lets one real record appear at its saved position on the map.
Lemma 22.11
The complete archive
N = 1,118,481
N is read from the same live place database that makes the public pages.
Lemma 22.12
Real-page share
6 / 126 = 4.7619%
Six of the main choices open real records, so the archive remains part of every Fractal page.
Continue through 120 mathematical doors
Every door below remains inside Discover Alone, carries depth 23 and opens six more real place pages.