Lemma 70.1
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.
Mathematical depth 70
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: William CoxeBirthplace · London
- Try this: William HorsleyBirthplace · London
- Try this: Two Bollards at North East and South East Corners of Junction with Chesterfield HillListed building · City of Westminster
- Try this: Little Rough TorMountain / peak · Cornwall
- Try this: Children's playgroundChildren's playground · East Riding of Yorkshire
- Try this: Priory GardensPark · London Borough of Bromley
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 70.2
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 70.3
Continuation share
120 / 126 = 95.2381%
Most main choices deepen the mathematical web instead of ending the journey after one page.
Lemma 70.4
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 70.5
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.
Lemma 70.6
Possible labelled routes
routes after k mathematical steps = 120^k
The formula counts possible sequences of labelled doors; it does not pretend that extra database records exist.
Lemma 70.7
One row, one public address
public page = page(saved record identifier)
A saved database row and its public page remain the same thing; no second made page is required.
Lemma 70.8
No repeated card
|chosen record identifiers| = 6
A record already selected for the current six is not shown a second time in that same set.
Lemma 70.9
Saved total depth
D_total = sum(d_i)
The saved total is the sum of the carried depth on each recorded page opening.
Lemma 70.10
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 70.11
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 70.12
Average saved depth
D_average = D_total / page openings
This is the average journey number on an opening, not the number of different people.
Continue through 120 mathematical doors
Every door below remains inside Discover Alone, carries depth 71 and opens six more real place pages.