Lemma 1.1
The next depth
d_next = 1 + 1 = 2
Every mathematical continuation adds exactly one to the carried journey number.
Mathematical depth 1
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 / earthen ringfort · Carrowncloghan
- Try this: Fairy fort / earthen ringfort in CartroncarrowntogherFairy fort / earthen ringfort · Cartroncarrowntogher
- Try this: IndicatedFairy fort / earthen ringfort · Galeybeg
- Try this: Fairy fort / earthen ringfort in TullanaglugFairy fort / earthen ringfort · Tullanaglug
- Try this: Holy WellHoly well · Upton (ballaghkeen By.)
- Try this: Boysack QuarryNo Class (event) (period Unassigned) · 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 1.2
Mathematical continuations
B_math = 120
These doors stay inside the Discover Alone Fractal and each one carries the next depth.
Lemma 1.3
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 1.4
Six real records
R = 6
Each opening keeps six real place pages from the live archive in front of the reader.
Lemma 1.5
Two useful kinds of choice
B_total = 120 + 6 = 126
The main web combines mathematical continuation with direct entry into real place pages.
Lemma 1.6
One map point
P = (latitude, longitude)
A trustworthy coordinate pair lets one real record appear at its saved position on the map.
Lemma 1.7
The complete archive
N = 1,118,481
N is read from the same live place database that makes the public pages.
Lemma 1.8
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 1.9
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 1.10
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 1.11
Continuation share
120 / 126 = 95.2381%
Most main choices deepen the mathematical web instead of ending the journey after one page.
Lemma 1.12
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.
Continue through 120 mathematical doors
Every door below remains inside Discover Alone, carries depth 2 and opens six more real place pages.