Searchable transcription
weights bi, and integer sizes di, with X
Searchable transcription
i:oi=y bieAdi ≤γ for every fine site y.
Suppose their coarse supports
obey, for
Searchable transcription
dist(x, p(oi)) ≤W(1 + di), covB(Xi) ≤v + v′di.
Let
, where
0, and
, then
Searchable transcription
X giea covB(Xi) ≤eavγCP(2W + 1)4ε−(4+k)(4 + k)! eε 1 −e−ε e−εm. (NF9) x∈Xi di≥m
The sum on the left is over labels
, with
fixed. To prove it, first restrict to
. All anchors lie over the coarse ball of radius
) about
, so there are at most
possible fine anchors. Sum the given anchored budget over these anchors and use
Searchable transcription
giea covB(Xi) ≤bieAdiC(1 + di)keav−2εdi.
Searchable transcription
The inequality (1 + j)4+k ≤ε−(4+k)(4 + k)! eεeεj
follows by retaining one term in the exponential series at
). The degree-
bound is therefore the numerator in (NF9) times
Sum its geometric tail. This proof also covers infinitely many address variants, since every finite partial sum has the same bound and all summands are nonnegative.
For a block ratio
: a fine site is uniquely specified by its coarse parent and its four residue coordinates. For the halo enlargement of the parent image of an old support, the geometric cover estimate used earlier reads
Searchable transcription
covB(Xi) ≤v + 10v L di, v = (2(l + 6) + 1)4,
where
= cov
, and the enlargement has halo radius
. Each point of it is within 
) of the parent of any chosen anchor in
. Thus (NF9) applies with
and
. In the inherited sales bound the extra tariffis
; absorb it first into the old weight, replacing the anchored exponent
by
. The reserve condition is then exactly
Searchable transcription
a(10v/L) + 2ε ≤δ −2ε.
This gives the stated payment, with its
child factor and polynomial-degree cost, rather than an unspecified enlargement constant.
The geometric cover estimate just used has the following direct proof. Write
for the block ratio and use the periodic maximum-coordinate distance throughout. A site and the embedded parent of that site are at distance at most
2. Embedded coarse distances are multiplied by
The triangle inequality therefore gives
Searchable transcription
L dist(p(x), p(y)) ≤dist(x, y) + L. (NF10)
These statements use centered residue representatives and remain valid on winding paths.
Choose a realizing
-cover
of
, with
= cov
). Its centers are connected at distance 2
+ 1. A unit-step walk visits them all in at most
Searchable transcription
N ≤d(4M + 2)
steps. One way to construct it is to start at one center and repeatedly attach an unvisited center adjacent to the visited connected set, inserting an out-and-back detour of length at most 2(2
For the empty cover use the empty-support case separately.
Sample this walk at times
, including time zero. Its parent samples form a coarse set
with 
Successive samples, and the parent of any intervening walk point and its preceding sample, are within 2
by (NF10). A fine point of
is within
of a visited cover center, so its parent is within 
of the parent of that center, since
and
1. It follows that the halo of radius
around
) lies within (
of
, with the required connected covering centers.
To replace this larger-radius cover by a
-cover, tile each such ball with points spaced
apart in the four coordinate directions. For a ball of radius
, the integer coordinate box with
suffices: coordinatewise integer division puts every point within
of one of its grid points. The grids are connected, and the connections between their centers join their union. Their combined cardinality is at most
; taking images on a torus only decreases cardinality. With 
+ 5, this gives cov
(halo
Finally, 4
. Multiply the last inequality by
and use the sampled-walk bound to obtain 
cov
(halo
as claimed. The same walk bound gives dist(
for
. Combining it with (NF10) and the halo radius proves the anchor-distance estimate with
. This completes the geometric input to (NF9).
Source: A Wilson Yang–Mills Construction: A Natural-Language Mathematical Exposition. Both papers belong to one Zenodo record. No external funding.