Lemma 84.1
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.
Mathematical depth 84
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: South HillStructure(s) (period Unassigned) · Shetland Islands
- Try this: HestinsetterHead Dyke (post Medieval), Township (period Unassigned) · Shetland Islands
- Try this: Fair Isle, Homis DalePlanticrub (period Unassigned) · Shetland Islands
- Try this: FardenBuilding (period Unassigned) · South Ayrshire
- Try this: Eisteddfa Gurig Track IITrackway · Ceredigion
- Try this: Dingestow Court Garden, DingestowCountry House Garden · Monmouthshire
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 84.2
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 84.3
The next depth
d_next = 84 + 1 = 85
Every mathematical continuation adds exactly one to the carried journey number.
Lemma 84.4
Mathematical continuations
B_math = 120
These doors stay inside the Discover Alone Fractal and each one carries the next depth.
Lemma 84.5
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 84.6
Six real records
R = 6
Each opening keeps six real place pages from the live archive in front of the reader.
Lemma 84.7
Two useful kinds of choice
B_total = 120 + 6 = 126
The main web combines mathematical continuation with direct entry into real place pages.
Lemma 84.8
One map point
P = (latitude, longitude)
A trustworthy coordinate pair lets one real record appear at its saved position on the map.
Lemma 84.9
The complete archive
N = 1,118,481
N is read from the same live place database that makes the public pages.
Lemma 84.10
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 84.11
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 84.12
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.
Continue through 120 mathematical doors
Every door below remains inside Discover Alone, carries depth 85 and opens six more real place pages.