Local nonlinear geometry and estimates · Part 6

This uses the actual usable-plaquette inequality tollfuel( port(). The macro transfer estimate has the following explicit form. LetPlaqAxes, and let) be the integer macro-path multiplicity, counting cover displacements separately even when their periodic images coincide. Its row sum is at most box((

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while the macro-store census gives P
i M(i, z) ≤N on the physical torus. The port identity is
port(i, U) = P
z M(i, z) toll(U, z). For the retained arrays E, F, set

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H(p) =
X
e∈∂p
F(e) + DL−2 X
z
M(p.1, z)E(z).

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The boundary sum has four terms, retaining repetitions. For each fixed plaquette type and each
one of its four boundary positions, the map from its base site to that bond is injective, including
on the torus. Hence its total is at most 4q P F. Summing the macro term and using its column
census gives
X
p
H(p) ≤4q
X
F + DqNL−2 X
E ≤(4q + DqN)B.

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Thus macroLarge(N, D) = 4q + DqN.
In the five-radius dome, the plot conditions give toll U ≤t + E and wire(Q, V ) ≤2Wt + F.
For unlisted inputs these follow from the plot’s upper thresholds; for listed inputs they follow by
retaining the full input size in the excess. Every face in the three-radius dome is usable: its required
coordinate box has radius one and is inside the plot’s listing dome. Each of its four boundary bonds
is in the five-radius dome. The four wire terms therefore contribute at most 8Wt + P
∂p F. The
macro stencil also stays inside the five-radius dome, by its reach bound and B ≥1 + N; its row
sum bounds the small part of the port by Sj,Lt. The predictor-curvature comparison now proves
the asserted split with
macroSmall(j, D, L, W) = 8W + DL−2Sj,L.

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The row count follows simply by summing the defining indicator over all cover displacements and
plaquette types. It allows coincident periodic images, so no injectivity of the cover box is assumed.
The operator row and column bounds transfer the local input split to the 2m tested Boolean
slots as follows:
X
∥Φ∗Xf∥≤2m aL2ut + 2auB,
X
∥(Φ′)∗Z∥≤2m ag(2Wt) + agB,
X
∥Xn∥≤2mvrt + 2vb.

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For the first inequality, separate the uniform small input ut from the plotted nonnegative excess.
The wide column bound costs aL2 on the small input, while the row bound a/L2 multiplies the
total excess, at most 2u P E. Thus the large-input term retains its L−2 saving. The noise argument
uses narrow row and column bounds a, and the coarse argument uses the plotted excess bound
directly. In each case the stencil radius takes the tested two-radius dome into the input three- or
five-radius dome, so only the given local split is used.

The departure is the slotwise inner product of the vision, of norm at most, with Φ . Cauchy–Schwarz gives its summed lower bound by minus times the sum of the three right sides above. The remaining departure requires no large-input saving. Indeed its residual vector has norm at most

since the wide factor cancels the response bound, while the narrow and tight terms cost and. Pairing with the vision and summing the two half-weighted slots gives

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workD(p) ≥−d0t2,
d0 = d(d + ad + ad + a2d).

All third-range coarse faces charged by the payments relation are contained in. Their toll sum is at most. As the coarse temperature is at most 2, their linear charge can consequently be included in the same bound. Writing= lowlinear, the result is

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ltemp
X
tides
toll V −talll(s) ≤Cpre tB + Cseat t2m,

where the exact available choices are

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Cpre = d(2au + ag + 2v macroLarge(N, D)) + 2lmacroLarge(N, D),
Cseat = 2d(aL2u + 2agW + vr) + d0 + 2lr.

The first constant is fixed before; the second may depend on both. The store census gives shares(spend). Thus the seat term is bounded by ( shares(spend), with a positive constant. This is precisely the affected-departure charge needed in the point-charge assembly. In particular the fine excess here is cut at, while only the census uses the larger spending threshold.

The three coarse clipping ranges

The curvature estimate above yields the coarse row bounds without differentiating its selector branches. Let = toll( = temp( = lowdirect 0, and = lowlinear. Put if = 0, and otherwise = min(1). The shortening components cancel exactly: on a kept row their contribution is its shortening half-square, and on a discarded row both tested shortening terms vanish. Expanding the remaining direct component gives

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workC(i) = a(α −α2/2)∥curvature(V, i)∥2 −clips(r).

Choose 0, depending on the group alone, so that for

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r2 −Kr4 ≤∥curvature ∥2,
3
4r2 ≤∥curvature ∥2.

If, then = 1 and workC( If, then 0 2. Therefore

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workC(i) ≥as
2r∥curvature ∥2 −bsr.

In the middle range, the curvature lower bound and the clip hypothesis8 give workC(4. In the outer range, discard the nonnegative curvature term to obtain workC(

Finally eventually. The supplied temperature estimate temp(1 implies this whenever. Thus the same late-scale choice gives all three bounds, with small-range debit coefficient2 and a group-dependent fixed middle-range cutoff

Threshold ratios and absorption of stitch errors

All limits in this paragraph are taken after fixing. Set, where is the fixed coupling shift,(log

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t = temp(y),
T = y−49/100.

The temperature is exp((log log) for positive in these late-scale calculations. For every fixed real shift, temp(temp(1. Indeed its logarithm is

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(log log(x + k))2 −(log log x)2 −1
2(log(x + k) −log x).

) the derivative of (log has absolute value at most two. The mean value estimate bounds the first difference by twice log(log, which tends to zero. Exponentiation proves the ratio limit.

For every 0 and integer ), where the latter notation means 0. To see this directly, eventually2(log, and every fixed power of log is smaller than by a factor tending to zero. The fixed shift does not affect that comparison. The monitor bound gives, for every temp(

eventually. It follows thattemp(0 and

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Bnt/T →0,
BnT →0,
BnT 2/t →0.

For the first limit choose200, leaving the decaying power; for the last use the lower bound on, leaving. In particular the square of, rather than itself, is the quantity small relative to

For each eventually. In fact + 100000, while log2 log eventually. Since (log, the quadratic growth of this lower bound for dominates 2 log Thus

Define the stitch size for fixed 0, and its widening error. The preceding limits, with + 1 in place of, give

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Bnh →0,
Bnh2/t →0,
Bne →0,
Bnhe/t →0.

For the last two statements use, expand, and bound each nonnegative summand by one of the already established limits; in the mixed tail term also use0. These are the precise limits that pay the unaffected-row stitch losses after multiplying by the dome census. They involve no torus period or number of store shares.

Finite-store budget inequalities

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Write a = lineartop, p = penalty, t = fiery and u = largeT ≥t. Let Ft be the listed fine plaquettes
with toll > t, Fu those with toll > u, and R = P
Ft toll. Then t|Ft| ≤R and u|Fu| ≤R. The
supplied fine tally bound is (a/2)tR+d t2|costs minus Ft|. If m ≤a/8, m ≤d and cF ≤a/4, use
2m + cF ≤a/2 and split costs into Ft and its complement to obtain

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tallyF ≥mt2|costs| + mtu|Fu| + cF tR.

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This separately pays all original fine shares, the larger spend subset, and the fine mismatch fee.
Every listed noise bond satisfies wire ≥Wt ≥t.
With RI = P
l isted wire2, this implies
t2|aisles| ≤RI. For the spend subset wire > u, also tu| spenda isles| ≤RI. If C ≤pW/16, then
C t · wire ≤(p/16) wire2 bondwise, since Wt ≤wire. Using m ≤p/32, the supplied noise bound
tallyI ≥(p/8)RI therefore gives

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tallyI ≥mt2|aisles| + mtu| spendaisles |
+Ct
X
listed
wire .

The listed-wire lower bound comes directly from Plot read at that bond’s endpoint, which belongs to the store dome.

For the coarse row split outs into three ranges: toll = temp, toll, and toll where. The coarse row estimate gives respectively a lower bound, a positive bound at least 4m whendirect64 and, and a lower boundlinear stoll. Sum these disjoint ranges, enlarging the small-range count to all outs because its error is nonnegative. This proves

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tallC + linearbelows
X
toll>δ
toll
+Bquartics4|outs|
≥4mt2 · number(s < toll ≤δ).

Finally Payments bounds flags by four times the number of s-large tides and bounds the-large tide count by K shares(spend). Since the latter tide set is a subset of the former, subtract it and obtain

toll shares(spend)]

These are exactly the four finite-store inequalities used in the assembly. They are algebraic consequences of the fine/noise/coarse producers and Payments, with no compatibility assumption on the periods. In particular the fine and noise spend gains are paid at tu, while the original seed shares are still paid at

Assembly of the point-charge gain

Use the same store, actual input, predictor and background in all estimates below. Put = fiery( = temp(), and). The fine energy estimate bounds its tally below by

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lineartop/2t
X
largefinetolls
toll
+dt2 · number(otherlistedfinetolls),

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with d > 0 after choosing L,W. The noise estimate gives tallyI ≥(penalty/8) P
l isted wire2.
The coarse estimate, store census and hit-row mismatch estimate supply the remaining terms
described below.
Choose a positive m no larger than lineartop /8, d, penalty/32 and directb elow/64. The hit
mismatch estimate has fine-loss coefficient cF = C/L2 and noise-loss coefficient C:

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tallLD(hit) + cF t
X
large
finetolls + Ct
X
listed
wire
+Zt2B4 · shares(spend)
≥linearbelow · s
X
δ−largecoarsetolls
toll .

Choose L first so thatlinear4. Choose W afterwards with1 and 16. Since every listed noise wire is at least Wt, its quadratic positive energy absorbs the linear C t wire loss. The fine positive energy similarly absorbs the loss. The coarse inequality pays the large coarse-toll term, leaving only the small quartic coarse error. The flag census is

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number(flags) ≤4[number(tidessminustidesδ)
+K · shares(spend)].

The output-row census is at most K shares. Choose x large so that, and

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(4mK + ZB4)t2 ≤mtu,
Cquartics4KB4 ≤(m/4)t2.

The first inequality lets the large toll/noise spend pay both its own flags and the mismatch remainder; the second pays the small coarse error by one quarter of the desired gain. The unaffected mismatch estimate costs at most another ( shares. The N tally is nonnegative.

To obtain these last two scale inequalities, use0 with0, and Dividing the first desired inequality by tu reduces it to a quantity tending to zero. For the second use and the vanishing. These choices are uniform in the live period.

Split the L/D tally into hit and unaffected rows. Add the fine, noise and coarse budget inequalities, the flag census, and the two quarter-error bounds. Their positive remainders pay m per listed fine/noise/flag share before the two quarter losses. The result is

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assessed ≥(m/2)t2 shares .

The exact fine/noise/coarse budget rearrangements supplying these positive remainders are developed next. This assembly fixes the quantifiers: the fine coefficient is paid by L, the noise coefficient by W, then 0 is fixed, and finally the common delay pays Z and the quartic and unaffected errors. No later change of L or W is used.

Unaffected mismatch costs in the point-charge proof

The point-charge producer combines the E01–E09 energy estimates with one additional unaffected-row estimate. The latter is derived from the mismatch bounds proved above. Fix an unaffected coarse plaquette row, and let = fiery. The coarse maximum gives t. The stitching estimate supplies a gauge in which the actual fine and coarse links are bounded on its radius- 2B/5 hull by), with no change of forest or tags. Choose the delay so that this hull contains 3 till()+4+predictionReach and h lies in the interior chart radius. Put widen(till(

The local gap bounds and Cauchy–Schwarz give

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workL + workD ≥−loss,
loss = 2(At)(Che) + (At)(CAte).

The factor two pays the two L coordinates; the two D half-inner-products together give the last term. This estimate applies to each row outside the hit subset of the store’s output rows. The store census bounds their number by N shares. The stitching-scale limits give

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B4he/t →0, B4e →0.

Indeed after dividing N loss by its two terms are exactly 2NAC and Hence, for every prescribed 0, the total unaffected mismatch cost is at least shares eventually. Sum the row bounds with the unchanged actual U,V and predictor. This proves the additional unaffected-row estimate rather than assuming it in the point-charge chain. The stitching bound, its two displayed scale limits, and the census and maximum estimates of this chapter supply its inputs.

The final point-charge statement has order of choices: choose a block lower bound n, then choose W sufficiently large, then a positive; finally choose x sufficiently large, uniformly over all periods with coarse side at least 100(1+B), all funded step data, forests and actual Plot inputs. It asserts shares pointwise. Passing from the supplied grant to this statement just intersects its delay with positivity of x and t and specializes its universal query. The fine, noise, coarse-clipping and flag budgets in this chapter prove this combination of energy estimates.

Exact assignment of the energy to packet cores

A row location has four types: a fine plaquette, a coarse noise bond, a coarse plaquette carrying C together with L and D, or an N row. Their principal values are respectively workF, workI, workC+workL+workD and workN, all at the same actual input, coarse background and predictor. Their coarse anchor is the parent of the fine plaquette root or the coarse root of the other types.

A fine plaquette has two parent moment rows; a noise bond has one; the remaining row locations have none. Each moment row occurs as a parent exactly once. Select a signed row when its signed coarse anchor lies outside the 6B ambient seed dome. Its surcharge is the corresponding real affine path half-square if selected and zero otherwise. Extracted is the sum of its parent surcharges. Put and liftSq equal to the orientation average of all extracted rows.

The actual kernel localization identity, including the invariant noise sum, now gives

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wholeGap +
X
noise
monetary
= averages
X
all
rowlocationsdebits + liftSq .

This is exact recombination: finite subtraction cancels the extracted rows, while C,L,D remain bundled at their common coarse plaquette. Noise covariance under a signed change is accompanied by the actual transformed tags, and its finite bond sum is invariant under that bond permutation.

For a store Z contained in the ambient seed place, every posted 2B core row lies inside the ambient 6B dome, so it is unselected. Its extracted value is zero, and its debit equals its principal energy. Summing the four principal types over this dome is exactly): expand the finite sums and group C,L,D at their anchor. This proves the core identity used in the packet gain without discarding any extracted energy.

Finally, a Wide condition through implies Plot on its 5B restriction and is preserved by a signed rotation of fields, tags, forest and store together. Apply the point-charge inequality in each rotated tree slice to get shares. Rotation preserves shares. Averaging gives the mean assessed bound used above. This completes the assignment and averaging step, using the preceding point-charge estimate.

Paying a packet's annulus and obtaining its tariff

For a store Z contained in the ambient seed place, the core identity identifies its assessed energy with the sum of debits on the posted 2B rows. Its assigned energy sums the same debits through 200B. Subtract the two finite sums. Every annulus row has debit at least), by the annulus estimate already proved. The number of posted rows is at most

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shares(Z) ballCost(200B + 1)spots(L)

in each orientation, where) counts the row types per coarse site. Consequently assessed) shares(, with ballCost(200). The radius factor has rate zero, so E retains rate The point-charge estimate on the actual wide sector gives mean assessed energy at least

shares(). Average the preceding inequality over the same signed orientations. This proves

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quoteNear(Z, u) ≥[μt2 −E(x)] shares(Z).

Its hypotheses include Wide through 300B, tangent control through 304B, and the same shell-access bound. No connectedness of Z was required. The exact core assignment and point-charge inequality proved in this chapter supply this step.

The ancillary packet expenses can be bounded uniformly before using this gain. The all-field joint diagnostic bounds the condensation by a constant. The Haar logarithm is eventually bounded by one. Each swept neighborhood and the ticket fee is at most a fixed constant times bill(, and the negative logarithm of the frozen measure is at most a fixed constant times bill(). Thus for one fixed

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steamLog +tollFee · shares
≤−βquoteNear + Kup bill(x)5 shares .

Now consider an admitted nonzero packet. Its seed-consistency indicator supplies containment in the ambient seed place and the correct actual flags. Nonzero ticket and sector profiles imply the Wide hypotheses on its 300B dome. Its nonzero enforcement profile supplies hot tangent control 2a t through 304B. The width condition supplies the matching frozen tangent bound. Hence the lower bound for quoteNear applies with that same field, forest and tags.

Since E has rate3/2 and2 betaF, eventuallybill. Combining the last two bounds and exponentiating yields

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|Pack(Z, u)| ≤exp [1 + Kupbill(x)5 −βμt2] shares(Z).

This is the packet income with cap 1, per-share expense 1+ and per-share gain zero packet satisfies it immediately. The estimate is pointwise before the frozen-Haar and Gaussian integrations and applies to the entire packet factor once, including its own tag and guard costs.

Compulsory packet factors and normalization expenses

For a seed store A, let visit) be the fine bonds with at least one coarse endpoint in its radius-d dome, and) the nontrunk bonds in visit. Let

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mg(β) = chartScale(
√
2π β−1/2)dim(Color).

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The packet exponent is defined exactly by

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steamLog = −β[quoteNear(A, u) + farBills(A)]
+
X
e∈visit200B(A)
condensation(e)
+
X
e∈plaza(dome304B(A))
gentry(e, u)
−|pens(A)| log mg(β).

Here farBills is the signed-orientation average of the extracted mean half-squares at the posted 200B rows; it is nonnegative. The entire packet factor is exp(steamLog) times the contraction-order profile, the actual tag tickets on the 300B dome, the sector profiles there, and tangent enforcement on the 304B plaza. It is nonnegative. A nonzero factor forces every individual enforcement cone in that plaza to be nonzero.

The order, sector and enforcement profiles lie in [0,1]. The tickets have upper bound exp(tollFee( shares()). Hence

exp[steamLog + shares(

This bound keeps the ticket cost of the whole packet; it is not repeated separately in neighboring log components. Suppose, andlog, with these bounds nonnegative. Endpoint census gives

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| visitd(A)| ≤shares(A) ballCost(d + 2)L4|Axis|.

The gentry plaza is contained in visit), and pens in visit). Dropping the nonpositive term farBills yields

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steamLog ≤−βquoteNear
+ shares(A)[Q0swept(200B + 2)
+C0swept(304B + 2) + K0swept(10B + 2)],

where) = ballCost(. This is the exact ancillary expense to be compared with the packet’s core gain. The measure expense has a uniform logarithmic bound over 02 betaF. Expand

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−log mg = −log chartScale
−dim(Color) log
√
2π + (dim(Color)/2) log β.

Its upper bound depends only on log(2 betaF), so eventually it is at most .bill() for one positive constant. No lower beta-band bound is needed for this upper estimate. The point-charge core bound and the packet annulus estimate show that the gain in quoteNear dominates these counted expenses.