Lemma 103.1
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 103
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: PLANTATION VILLAGE: ACTON in ActonPLANTATION VILLAGE: ACTON · Acton
- Try this: RATH in CarrickbroadRATH · Carrickbroad
- Try this: Portal tomb in BallygraneyPortal tomb · Ballygraney
- Try this: Formoyle West CaveCave · Formoyle West
- Try this: Stone CircleStone circle, multiple-stone · Templebryan North
- Try this: Check mapDovecote · Athenry
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 103.2
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 103.3
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 103.4
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 103.5
Saved total depth
D_total = sum(d_i)
The saved total is the sum of the carried depth on each recorded page opening.
Lemma 103.6
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 103.7
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 103.8
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 103.9
The next depth
d_next = 103 + 1 = 104
Every mathematical continuation adds exactly one to the carried journey number.
Lemma 103.10
Mathematical continuations
B_math = 120
These doors stay inside the Discover Alone Fractal and each one carries the next depth.
Lemma 103.11
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 103.12
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 104 and opens six more real place pages.