Lemma 113.1
Mathematical continuations
B_math = 120
These doors stay inside the Discover Alone Fractal and each one carries the next depth.
Mathematical depth 113
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: DoocotHeritage attraction · Fife
- Try this: Hatton, FarmhouseFarmhouse (period Unassigned) · Fife
- Try this: Blair PointTrench (20th Century) · Fife
- Try this: Children's playgroundChildren's playground · Fife
- Try this: Càrn an Leth-choinMountain / peak · Highland
- Try this: Arran, Doire Nam PollField Boundary(s) (post Medieval) · North Ayrshire
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 113.2
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 113.3
Six real records
R = 6
Each opening keeps six real place pages from the live archive in front of the reader.
Lemma 113.4
Two useful kinds of choice
B_total = 120 + 6 = 126
The main web combines mathematical continuation with direct entry into real place pages.
Lemma 113.5
One map point
P = (latitude, longitude)
A trustworthy coordinate pair lets one real record appear at its saved position on the map.
Lemma 113.6
The complete archive
N = 1,118,481
N is read from the same live place database that makes the public pages.
Lemma 113.7
Real-page share
6 / 126 = 4.7619%
Six of the main choices open real records, so the archive remains part of every Fractal page.
Lemma 113.8
Distance on the earth
a = sin^2(delta_lat/2) + cos(lat_1) cos(lat_2) sin^2(delta_long/2)
The map uses positions on a globe, so nearby means surface distance rather than a flat drawing.
Lemma 113.9
Wrapped selection
row = ((start + offset - 1) mod N) + 1
When a chosen position passes the end of the database, it continues from the beginning without inventing a record.
Lemma 113.10
Continuation share
120 / 126 = 95.2381%
Most main choices deepen the mathematical web instead of ending the journey after one page.
Lemma 113.11
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.
Lemma 113.12
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.
Continue through 120 mathematical doors
Every door below remains inside Discover Alone, carries depth 114 and opens six more real place pages.