Lemma 1.1
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.
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: HMS Pickle GroveVisitor information · Swale
- Try this: Clifford AndrewsBirthplace · Swindon
- Try this: Tesco's Delivery Road BridgeCanal bridge · Tameside
- Try this: Neolithic Long Barrow near AddingtonAncient burial place · Tonbridge and Malling
- Try this: St. John the BaptistGraveyard · Wiltshire
- Try this: Fairy fort / earthen ringfort in CappanagoulFairy fort / earthen ringfort · Cappanagoul
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
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 1.3
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 1.4
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 1.5
Saved total depth
D_total = sum(d_i)
The saved total is the sum of the carried depth on each recorded page opening.
Lemma 1.6
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 1.7
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 1.8
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 1.9
The next depth
d_next = 1 + 1 = 2
Every mathematical continuation adds exactly one to the carried journey number.
Lemma 1.10
Mathematical continuations
B_math = 120
These doors stay inside the Discover Alone Fractal and each one carries the next depth.
Lemma 1.11
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 1.12
Six real records
R = 6
Each opening keeps six real place pages from the live archive in front of the reader.
Continue through 120 mathematical doors
Every door below remains inside Discover Alone, carries depth 2 and opens six more real place pages.