Local nonlinear geometry and estimates · Part 5

is at most the sum of coarse-link chord W and predictor chord/2, hence at most. For a fine plaquette based at the chosen parent, replace links outside the radius-B/2 hull by 1. Its four-step product is unchanged, and its chord is. Thus the original fine plaquette toll and the actual wire both meet the small-curvature threshold. The uniform branch-cutoffand predictor estimates in this chapter give this finite-stencil estimate.

Uniform local branch activation

At the identity cover field the minimizing frame is 1 for every frame tag, the connector array is 1, and the minimizing predictor is 1. Both minimum tests are strict there. Continuity of the selected minimizers on their open minimum branches therefore gives a neighborhood of the identity cover field on which the site-frame minimum test holds and every connector predictor lies within any prescribed positive chord tolerance. This neighborhood can be chosen simultaneously for all axes and both endpoint frame tags because these alphabets are finite. On it every predictor fallback tag yields the same best value.

A product-topology neighborhood restricts only finitely many cover links. Choose a finite set I of such links and a common positive chord radius r small enough to meet all their coordinate neighborhoods; the conversion between chord and matrix norm is the fixed factor colorScale. Set d to the largest length of the coordinate-order route from 0 to a root in I. For any torus input, expand it at the chosen embedded coarse root. Every required cover link projects to a fine link whose coarse-parent distance from that root is at most its route length, hence at most d. Thus chord bounds on this parent-distance hull place the expanded field in the chosen cover neighborhood, independently of the period.

The actual connector equals the cover test connector with its actual two endpoint tags. Consequently its predictor chord is. On the minimum branches the normalized site mass and pin mass equal one: their unnormalized masses are respectively the reciprocals of the numbers of frame tags and connector tags. This proves the finite-stencil branch activation used in the local chart-mark argument without requiring any tags or defaults away from the finite hull.

Global mismatch estimates from the flat identities

Let F be a cut transported row and its full flat comparison, with both row and column sums and difference sums. Suppose an input f has linear comparison t with and Chq. The exact decomposition

Searchable transcription

F Tf −F T
0 t = F T(f −t) + (F −F0)Tt

and the column-sum estimates give norm at every output coordinate. For arbitrary bounded input w, applying this argument to and gives

Searchable transcription

∥F TFw −F T
0 F0w∥≤2C2∥w∥∞q0.

Choose C to dominate the fine plotted, coarse plotted, predictor-noise and three kernel constants previously proved. Choose the chart radius as their common minimum, and choose

Searchable transcription

τ = min(τkernel, widthtop, widthbelow/L) > 0,
q0 = widen(R, h, τ).

The plotted-field and predictor Taylor comparisons then all have the common input bounds above; the noise quadratic remainder uses. The F,I,N kernel comparisons likewise have the

common row and column estimates. The full flat cross-roll identity cancels the linear terms in lGap, leaving two transpose/input errors and one coarse plotted error. Hence

Searchable transcription

∥lGap∥≤(4C2 + C)hq0.

The full flat diagonal-roll identity cancels w against the sum of its three flat row-square operators. Applying the second estimate to F,I,N gives

Searchable transcription

∥dGap(w)∥≤6C2∥w∥∞q0.

The common constant 6 bounds both. These estimates use the uncut flat identities, so both the transport error and the cutofferror are included in. They hold for every cutoffR, including zero, and allow winding paths and arbitrarily short periods. The local transfer described next uses exactly these same gaps.

Localizing the mismatch bounds

Assume, in one gauge, both fine and coarse links have chords on the radius-d hull of a tested row, with 3 +4+predictionReach. The bounded logarithm extension constructs globally bounded fields a,b agreeing there with those actual links, and h. Put = max(1 and choosemin(). The global logarithmic mismatch estimate then applies at envelope Xh. Locality identifies its tested gaps with the actual gauged gaps because all their reads are in the hull.

Gauge covariance rotates lGap by the root adjoint. For dGap it rotates both the row and its arbitrary input data w; this preserves the bound. Finally

Searchable transcription

widen(R, Xh, τ) ≤X widen(R, h, τ)

because1 and both summands are nonnegative. Thus increasing the global constant C to gives simultaneously

Searchable transcription

∥lGap∥≤CX2h widen(R, h, τ),
∥dGap(w)∥≤CX2q widen(R, h, τ).

Only the independent supremum bound q on w is used. This matters when w is the actual thinned data: changing the links outside the hull need not preserve that entire data array. The global mismatch estimate and exact local gap identities proved here supply these inputs.

The quartic curvature deficit

The energy estimates separate useful positive income from the remainders that must be paid. “Income”, “price”, “loan” and “tariff” are bookkeeping names for nonnegative numerical bounds in this construction. They do not denote additional physical degrees of freedom. In particular a packet tariffis a summable upper bound on a packet’s contribution after its annular and normalization expenses have been assigned. The inequalities below show exactly which positive curvature terms absorb which remainders. Let denote orthogonal projection onto the real matrix Lie algebra. For every group element U, not only those near 1,

Searchable transcription

Plie(mat(U) −1) = colorMat(curvature(U)).

Indeed subtract the skew part (mat(mat(2 from mat(. The remainder is self-adjoint. Its real Frobenius inner product with every skew-adjoint Lie algebra matrix is zero, by cyclicity of trace. Projecting therefore retains just the Lie projection of the skew part, which is the defining curvature. Pythagoras for tangent and normal projections gives the exact identity

Searchable transcription

colorScale2[chord(U)2 −∥curvature(U)∥2]
= ∥mat(U) −1 −Plie(mat(U) −1)∥2.

The analytic focus fixes each group matrix near 1, and its derivative there at 1 is. Its Taylor remainder consequently gives constants 0 such that

Searchable transcription

∥mat(U) −1 −Plie(mat(U) −1)∥≤C∥mat(U) −1∥2

whenmat(. This is an ambient analytic estimate restricted to group points; it requires no logarithm branch on an entire configuration. Substituting the chord normalization in the squared bound and setting colorScale proves

Searchable transcription

chord(U)2 −K chord(U)4 ≤∥curvature(U)∥2.

Searchable transcription

Choose δ = min(r/ colorScale,
p
1/(4K)). If chord(U) ≤δ, then K chord(U)2 ≤1/4, so the last
bound is at least 3
4 chord(U)2. Both constants depend only on G and are chosen before the block
length. This proves the small-curvature input to the soft and coarse-C estimates.

Soft curvature and noise remainders

) = [chord(curvature(2. It is nonnegative by the curvature norm bound. On a retained fine plaquette whose toll is at most its clipping threshold, expanding the two direct/path moment coordinates gives the exact identity

Searchable transcription

crown = direct · flaw(faceU) +
X
i∈Bool
∥entry(face,i)U∥2/2.

Adding the guide cross term and its square completes each square. Thus workF equals directflaw plus the two half-squares of bill =. The noise identity holds without a retention hypothesis: workI equals penalty) plus the single bill half-square. These are algebraic identities, using the actual = prediction() and the same guide.

The affine comparison already proved bounds entry by) of rate1/2 and billreply by ) of rate1; subtracting the common guide does not change that affine error. The coarse maximum condition bounds guide by Afiery, so bill has norm at most fiery. For vectors v,w with and, factor the difference of squared norms to obtain

Searchable transcription

|∥v∥2/2 −∥w∥2/2| ≤E(2m + E)/2 =: Z.

This has rate The small-curvature estimate supplies 0 and a chord neighborhood with

Searchable transcription

chord(g)2 −J chord(g)4 ≤∥curvature(g)∥2,
3
4 chord(g)2 ≤∥curvature(g)∥2.

Searchable transcription

The local chart-mark estimate places the plaquette holonomy and V Q−1 in that neighborhood.
A scaled curvature bound ∥√k curvature(g)∥≤M therefore gives chord(g) ≤2M/√k and

Searchable transcription

|flaw(g)| ≤(J/2)(2M/
√
k)4.

For the fine and noise terms put respectively

Searchable transcription

X = direct · (J/2)(2M/
√
direct)4,
Y = penalty · (J/2)(2M/
p
penalty)4.

Both have rate2. The common envelope thus has rate3/2 and bounds the difference between each soft energy and its corresponding affine reply half-square sum. The two in the last term pays the two fine coordinates; the noise row needs only one. The small-curvature and local branch-activation estimates above supply this reduction.

Outer N, C and mismatch corrections

On a clean radius-B/4+1 hull, flatten the coarse field and use its bounded logarithmic extension. The guide comparison for the N row has zero linear term and envelope e of rate Gauge covariance preserves its norm. Since workN is its half-square,2 has rate

For the C row, the hypothesis toll(temp makes its clipping multiplier exactly one, also at zero toll. The exact C-row formula and curvature norm bound then show0. The quartic curvature inequality above gives

Searchable transcription

|workC| ≤directbelowJ/2(2 fiery)4,

because temp2 fiery. This envelope has rate For the L and D mismatch rows, use the common local gauge and logarithmic extensions of the actual fine and coarse inputs on their clean grafted hull. The layered extension gives a common envelope W of rate1/2. The interior kernel comparison supplies constants 0 and, writing = widen(

Searchable transcription

∥lGap∥≤CWe,
∥dGap(vision)∥≤Cde, d = A · fiery .

The maximum condition gives Hence Cauchy–Schwarz bounds each L inner product by C W e and each D half-inner-product by C d e/2. There are two Bool coordinates, giving

Searchable transcription

|workL| + |workD| ≤2M + V.

Broadening preserves the rate1/2 of W; both products have rate This proves the asserted outer-envelope assembly from the layered-extension and interior-gap estimates, with actual U,V,Q shared by the two mismatch terms.

Light estimates on the annulus

The light estimate is required in two forms: a lower bound at every row and an absolute bound when that row is selected. The source decomposes it into four already specified energy types: the soft curvature/noise remainder, the N correction, the coarse C correction, and the L/D corrections. Let their envelopes be respectively. The first and fourth have rate3/2; the other two are weakened to that rate. Their sum is nonnegative and has rate

For a selected curvature row, its extracted energy is exactly the sum of its two real-path half-squares. The soft curvature estimate consequently gives the required absolute remainder bound. For a selected noise row its extracted energy is its single real-path half-square, and the exact noise split reduces to the soft noise estimate. At an unselected curvature or noise row the extracted term is zero; the remaining energy is nonnegative, being the sum of a nonnegative flaw cost and half-squares. It therefore satisfies the lower bound without an absolute-value claim. The C row has no extracted parent contribution and is bounded by its C, L and D envelopes; the N row similarly uses its N envelope.

These statements apply to a clean radius-B coarse hull and a grafted tangent bound rate1/2. Under a signed coordinate change first restrict the grafted hull to radius B/2+1, then turn it to obtain radius B/2; the extra site pays the reversed-link shift. The coarse maximum hypothesis is read in the same turned background.

For a packet annulus, take a row in the 200B seed dome but outside the 2B dome, together with the Wide bounds through 300B and tangent control through 304B. Local Interior bounds and shell access give the required grafted condition, provided). The same Wide bounds give the clean hull and make the original debit equal light: retained curvature selectors are one by the actual local toll bounds. Thus on the entire annulus. This uses no derivative of a hard selector and no globally small gauge. The soft-remainder, outer-correction and grafted-signal estimates of this chapter provide these four envelopes and the required shell access.

Physical operator and path-series bounds

Searchable transcription

We record why the allowances and maximum responses in the energy argument are uniform
in the torus and are chosen before L.
Suppose a cover kernel between types α, β satisfies
P
z eη|z|∥Kαβ(z)∥≤C, with η ≥0. Each physical summand is a scalar kernel coefficient times
adjoint transport along its prescribed route, subject to the endpoint congruence and route cutoff.
Adjoint transport is an isometry, including on winding routes; its operator norm is at most one.
Thus the summand is bounded by the absolute cover coefficient with its endpoint indicator.

For a fixed input site and cover displacement, the endpoint congruence selects exactly one output site. For a fixed output site and displacement it selects exactly one input site. Interchanging the finite site sum and absolutely convergent cover sum therefore bounds a physical row by and a column by. This argument takes absolute values before periodization, retaining every displacement when endpoints coincide.

For the fine-to-coarse kernel, the input type includes its fine residue and its two-layer plaquette type. The per-type cover price is; there are residues. The row bound is consequently and the column bound. Noise and tight kernels have row and column bounds at most a common. One may take to be the maximum of one and the cover price multiplied by each of the three finite type cardinalities. This choice is independent of

The support assertion concerns kernel incidence: a nonzero cut term has endpoint distance at most its route length, at most100000, hence at most50. For fine routes use the parent map, which takes adjacent fine sites to equal or adjacent coarse sites. The route from the lifted fine residue to ends at the embedded coarse site parent()+wrap(); this remains true for wrapping routes by additivity of wrapping and the cover route endpoint identity. No larger support claim about all data read by an operation is needed.

For path shortening, retain the entire label index, writing its coefficient and slot as ( Define the nonnegative matrix

Searchable transcription

Mpu =
X
i
|ci| 1{d length(si)≤R, landing(si,p)=u}.

Searchable transcription

The weighted path hypothesis is P
i |ci|eσlength(si) ≤P for each source plaquette type. It gives
row sums at most P.
For a fixed label and source type, the landing equation fixes its source
site uniquely; summing over source types gives column sums at most qP. Absolute convergence
permits all these finite-sum exchanges. Curvature norm is bounded by plaquette chord and adjoint
transport preserves norm, so

Searchable transcription

∥shorten(U, p)∥≤
X
u
Mpu toll(U, u).

If the selector keeps, it bounds the toll at distance by Landing distance is at most slot length, and the stage hypothesis1 makes the weighted path sum applicable. Thus the shortening norm is at most The common stage capacity

Searchable transcription

Astage = (q + 1)P +
√
direct

dominates row sums, column sums, and the direct clamp scale. Both branches of the thinned field therefore have norm at most: its shortening branch uses the kept estimate, and its direct branch uses the multiplier min(1) and curvature norm at most = 0 the curvature

Searchable transcription

vanishes.
The required reach margins hold eventually. If t = temp(x+a) and tek/ρ < 2, then t ≥1/√x + a
implies k/ρ < log x once 2√x + a ≤x. For large log x, this gives k + 1 ≤B/50, since B ≥(log x)2.
The cut path length till(x)+1 also lies below B/50. Parent distances cost at most the fine distance
plus one, by the parent foreshortening inequality. These observations supply both the shortening
stencil reach and the discarded-witness reach used above.

Searchable transcription

Finally a = G.slopeConstant log L ≥0: the minus Killing form is a sum of squares, so the
Casimir and slope constant are nonnegative. At fixed shift a, eventually temp(x) ≤2temp(x + a);
use √x + a ≤2√x and monotonicity of the nonnegative logarithmic factor in the temperature. Put
A = Atop + Abelow, and let C be the common operator allowance. The coarse vision has norm at
most 2At. Applying the fine and noise operator row bounds to it gives

Searchable transcription

∥vision∥≤dt,
∥Mv∥≤dL−2t,
∥Mw∥≤dt,
d = 2A(1 + C) > 0.

Both stage lane counts may use this same. It is fixed before, while the common late cutoff is taken afterwards. These prove the operator and maximum hypotheses used throughout the preceding energy estimates.

Fine and noise income, including discarded rows

Searchable transcription

Here is the summation argument producing the fine and noise inequalities used below. Write a > 0
for the direct quadratic coefficient, b > 0 for its linear coefficient, p > 0 for the noise penalty, and
t > 0 for the fine threshold. Let ri be the plaquette chord, K the listed plaquettes with ri > t,
and Z = P
i∈K ri. The actual plot conditions imply that every listed plaquette outside K has
t/4 ≤ri ≤t. Any plaquette with ri > t whose parent is in the required five-radius dome is listed.
These are local statements about the one configuration under consideration.
The two components of a fine row, with their corresponding coarse responses v0
i , v1
i , are

Searchable transcription

Di = clipt(ri) + 1
2∥v0
i ∥2 −⟨v0
i , shooti⟩,
Ei = 1kept(i)
1
2∥ui∥2 + 1
2∥v1
i ∥2 −⟨v1
i , ui⟩.

Searchable transcription

Here ∥shooti ∥≤√a ri, and clipping is ar2
i /2 for ri ≤t, and btri otherwise. Suppose the previously
constructed response bounds give ∥vν
i ∥≤εt. Completing the scalar norm square gives

Searchable transcription

Di ≥1
2(√a ri −∥v0
i ∥)2
(ri ≤t).

Searchable transcription

Thus, if ε ≤√a/8, each listed small plaquette contributes at least at2/128. If ε√a ≤b/4, each large
listed plaquette contributes at least 3btri/4. Unlisted rows in the summation region have ri ≤t
and hence nonnegative direct part. Consequently

Searchable transcription

X
i
Di ≥3b
4 tZ +
a
128t2|costs \ K|.

It remains essential to bound the discarded second components after summation. Let the fixed relaxation length and PlaqAxes

A row fails the kept condition only if some plaquette satisfies exp(dist(). Such a row belongs to the ball about of radius log(, including all plaquette types. For1, this ball has at most elements: use

Searchable transcription

2⌊ρ log y⌋+ 1 ≤(8ρ + 1)e(log y)/4

Searchable transcription

and raise to the fourth power. The local witness-reach estimate places each witness j in the five-
radius dome, so it belongs to K. The discarded set X therefore satisfies t|X| ≤HρZ. This count
takes place on the original fine lattice and introduces no factor L4.
Let Mij ≥0 be the previously bounded shortening majorant, with both row and column sums
at most C. Its support-reach estimate shows that, for i ∈X, every contributing j /∈K has rj ≤t.
Hence
∥ui∥≤Ct +
X
j∈K
Mijrj,
X
i∈X
∥ui∥≤C(Hρ + 1)Z.

Searchable transcription

Kept rows have Ei = ∥ui−v1
i ∥2/2 ≥0. Discarded rows satisfy Ei ≥−εt∥ui∥. Choosing εC(Hρ+1) ≤
b/4 yields
X
i
(Di + Ei) ≥b
2tZ +
a
128t2|costs \ K|.

Searchable transcription

In particular this argument never presupposes that a discarded row has a good halo.
For noise rows put ni = wirei. Their exact energy is bounded below by 1
2(√p ni −∥vi∥)2. On
listed aisles the plot gives ni ≥wt. The uniform noise-response bound ∥vi∥≤Ct, together with
C ≤√p w/2, therefore gives pn2
i /8 per listed aisle; all other noise rows remain nonnegative.
The order of these choices matters. The response theorem supplies the common constant C
before L is chosen and gives ε = C/L2. First take L large enough for the three displayed restrictions
on ε; next take w ≥2C/√p; finally take the late scale parameter at those fixed values. The fine
coefficient delivered by this construction is exactly a/128 > 0.

Macro multiplicity and the spending census

Searchable transcription

The macro census used here has a period qualification, already included in the step hypotheses.
Put R = j +8. The macro box radius is (j +7)L. If twice this radius is smaller than the fine period,
two displacements in that box with the same periodic image coincide, coordinate by coordinate.
Thus M(i, z) ≤1. A nonzero multiplicity also implies dist(i, parent z) ≤R: multiply the coarse
distance by L, use the distance from the embedded base to z, at most (j + 7)L, and the fine site’s
distance to its embedded parent, at most L/2. The triangle inequality gives the stated integer
bound. Consequently
X

Searchable transcription

i
M(i, z) ≤(2R + 1)4.

The port identity follows by interchanging its finite cover-displacement and plaquette-type sums and grouping identical periodic plaquettes, with their multiplicities. The period hypothesis is used for the column bound, not for that identity.

The store’s base set has at most four times its number of shares: a fine plaquette has at most four corner parents, a noise bond at most two pins, and each flag contributes one site. Hence any collection of coarse faces based in its radius- dome has cardinal at most

Searchable transcription

V (r) shares,
V (r) = 4(2r + 1)4q.

For. This proves the bounds on the outer and affected face sets, using respectively the original store and the spending store. The exact flag condition says each flag is touched by a coarse face with toll above the coarse temperature. Since a plaquette has at most four corners, the number of flags is at most four times the number of those faces.

Searchable transcription

For completeness, every coarse face whose toll exceeds a fixed δ > 0 has an actual spending
witness within distance R, once the scale parameter is late enough. Write T for the large spending
threshold. If no spending base lies in that neighborhood, then every fine input in its macro port has
toll at most T: listed inputs exceeding T would themselves be spending witnesses, while unlisted
inputs have toll at most t/2 ≤T. Likewise every one of its four boundary wires is at most T, using
the unlisted bound 2Wt ≤T. Its toll is then at most


4 + DL−2q| box((j + 7)L)|
T.

At fixed, this is eventually at most, a contradiction. All relevant inputs lie in the plot dome by the distance bounds above, so the local predictor comparison applies.

The limits used here are explicit. With fine inverse coupling whereas ) with smaller eventually than every positive power. For each fixed use and to obtain. Also(log, allowing One common pre- census constant is

Searchable transcription

N = V (R) + V (2) + R + (2R + 1)4.

It simultaneously bounds the macro columns and reach, the number of-large coarse faces by times spending shares, and the outer and affected face counts by times their respective shares. This supplies the census used in the departure and point-charge bounds.

Transfer of affected departure costs

Fix the same plot, and let now denote its affected coarse faces, with. Use constants 0 for the operator allowance, maximum response, and predictor-curvature comparison respectively. These letters are local to this argument. Put

Searchable transcription

high.direct + d,
v =
√
B = L−2
X
swell(t)
toll U +
X
aisles
wire(Q, V ),
u =
p
low.direct + d,
g = √p.

Retain the two nonnegative excess arrays exactly on the fine swell at threshold and on all listed noise aisles. The macro transfer estimate constructs a nonnegative coarse excess such that, in the three-radius dome,

Searchable transcription

toll V ≤rt + H,
X
H ≤b,
r = macroSmall(j, D, L, W),
b = macroLarge(N, D)B.