Lemma 95.1
Average saved depth
D_average = D_total / page openings
This is the average journey number on an opening, not the number of different people.
Mathematical depth 95
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 Ucheldre, EnclosureDefended Enclosure · Powys
- Try this: Moelydd Blaen Tafolog, Earthwork Bank IIBank (earthwork) · Powys
- Try this: Cwm Oergwm, Earthwork Platform IIIPlatform · Powys
- Try this: Village Farmhouse;(Adj. to Lychgate)Farmhouse · Powys
- Try this: George Jeffreys, 1st Baron JeffreysBirthplace · Wrexham
- Try this: Market cross on south side of Market PlaceScheduled monument · Charnwood
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 95.2
The next depth
d_next = 95 + 1 = 96
Every mathematical continuation adds exactly one to the carried journey number.
Lemma 95.3
Mathematical continuations
B_math = 120
These doors stay inside the Discover Alone Fractal and each one carries the next depth.
Lemma 95.4
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 95.5
Six real records
R = 6
Each opening keeps six real place pages from the live archive in front of the reader.
Lemma 95.6
Two useful kinds of choice
B_total = 120 + 6 = 126
The main web combines mathematical continuation with direct entry into real place pages.
Lemma 95.7
One map point
P = (latitude, longitude)
A trustworthy coordinate pair lets one real record appear at its saved position on the map.
Lemma 95.8
The complete archive
N = 1,118,481
N is read from the same live place database that makes the public pages.
Lemma 95.9
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 95.10
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 95.11
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 95.12
Continuation share
120 / 126 = 95.2381%
Most main choices deepen the mathematical web instead of ending the journey after one page.
Continue through 120 mathematical doors
Every door below remains inside Discover Alone, carries depth 96 and opens six more real place pages.