Local nonlinear geometry and estimates · Part 3

Indeed chord is the Frobenius distance from 1 divided by colorScale, andcolorMat( colorScale. This preserves the physical normalization of the color norm.

Let w be a fine logarithmic field vanishing on the trunks of the given forest. Then exp() lies on that exact tree slice for every t. The actual frame minimum branch is active near = 0 at every coarse site; its frame mass is positive, so the chosen tags are usable there. Apply the nonlinear shell link comparison at the link e to) = exp(), coarse comparison field 1, and its actual predictor ). It gives

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chord(U(t)e) ≤C port(parent(e), U(t))
+ chord
Q(t)(parent(e),axis(e))

.

This upper bound is valid in the internal-block case as well, where adding the last nonnegative term is harmless. The port sum is bounded by the full macro-box sum of plaquette chords, including repetitions on a periodic lattice. Divide by |t| and pass to the limit. Each plaquette derivative is its discrete curl, and the actual predictor derivative was computed above. The finite macro sum commutes with this limit. We obtain

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∥we∥≤C
X
X
p
∥curl(w)(wrap z+embed(parent(e)),p)∥
z∈box(macroR)
+∥probe(w)(parent(e),axis(e))∥.

This proves the local link estimate from the nonlinear shell comparison without a separate linear gauge choice.

The same calculation gives a volume-independent counting bound. Put box(macroR) and. For any plaquette function f, Cauchy–Schwarz on the macro sum costs Nm. For each fixed offset z, the map (coarse site,plaquette z+embed(site),plaquette type) is injective, so summing its squares costs at most the full fine plaquette sum. Summing all N offsets adds the factor N. Each coarse site has q fine children and four outgoing fine-axis choices. Thus squaring the local link inequality with ( gives

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X
e
∥we∥2 ≤β
X
p
∥curl(w)p∥2 + γ
X
b
∥probe(w)b∥2,
β = 2C2 · q · 4 · N2m, γ = 2q.

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This counting uses injectivity in the coarse site for each fixed offset, not injectivity of different
offsets; repeated periodic plaquettes remain allowed.
If the fine quadratic bed has curl floor a > 0 and the predictor penalty is κ > 0, its vertical
Hessian dominates a P ∥curl w∥2 + κ P ∥probe w∥2. The restriction to the vertical tree chart has
the same Hessian as the full germ on directions (0,v), by the linear inclusion chain rule; the collected
vertical field is v offtrunks and zero on trunks. Therefore, with

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η = min(a, κ)/(1 + β + γ) > 0,

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the vertical Hessian is at least η P
free ∥ve∥2, since ηβ ≤a and ηγ ≤κ. This also unfolds the
fixed-block counting-coordinate Hessian floor. The shell comparison supplying C is described next.

Uniform branch cutoffs and continuity of selected minima

The finite-pair uniqueness result gives a positive d depending only on G. Choose the frame cutoff at most d/colorScale. For any patch satisfying the all-minimum criterion, use its center-based axial

frame a and define pair transports travel( . The criterion bounds their chords by the frame cutoff, so their matrix distances from 1 are at most d. For an arbitrary frame b, the transformed array has pair cost exactly the original frame score. A normalized global minimizer of the frame score is a global minimizer of this pair cost: every unnormalized competitor can be multiplied on the left to fix its center, without changing its score. Pair uniqueness then gives uniqueness up to a left multiple, and the center value 1 makes that multiple equal to 1. Existence follows by compactness of the finite product of G with its center fixed. This proves the uniform frame cutoffindependently of patch order and size.

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The predictor cutofffollows from a related weighted-centroid calculation. For real weights wi
summing to one and group points fi, set m = P wi mat(fi). Expansion of the real inner product
gives

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2repDim
X
i
wi price(fiu−1)
=
X
i
wi∥mat(fi)∥2 −∥m∥2 + ∥m −mat(u)∥2.

Thus minimizing the weighted price is equivalent to being nearest to m. For nonnegative weights, the squared norm inequality for a weighted average shows

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∥m −mat(u)∥2 ≤2repDim
X
i
wi price(fiu−1).

If the right side is below the square of the nearest-point tube radius r, m lies within r of the group and has a unique nearest point. Choosing predictor cutoffat most(2 repDim) proves the uniform predictor uniqueness condition. Both cutoffchoices depend only on G.

The selected minimizers are continuous wherever their defining fibers are unique. Here is the compactness argument including arbitrary selections away from that set. Let be the closed minimizer relation, with Y compact. For an open neighborhood V of the unique minimizing value at, project) to X. This projection is closed because Y is compact, and it omits by uniqueness. Its complement is therefore a neighborhood of on which every selected minimizer lies in V. This is continuity at. For frames the relation is closed because it is the intersection of the closed center-normalization condition and the score inequalities against all normalized competitors; scores are continuous finite path expressions. The predictor minimizer relation is closed by the corresponding continuous cost inequalities. Applying the compactness argument proves continuity on the respective minimum branches. Their strict, finite cutofftests make those branches open. These complete the uniqueness and continuity inputs used in the analytic-frame construction.

Minimizer self-consistency and uniqueness

For a finite pair family define term() = price( ), and cost as its average over all three indices. Reversal of the pair exchanges x,y and replaces k by its mirror permutation, preserving the summed term. The diagonal terms vanish because = 1. Consequently, replacing only by a competitor b changes the total cost by twice the change in the off-diagonal x-row. Global minimality implies that this off-diagonal row is minimal. Adding the frozen self term price( ), which is nonnegative and is zero at, proves the stronger comparison with the whole row in which every, including, is held fixed.

The normalized whole-row price is precisely the weighted squared-distance problem to the matrices , up to a positive constant and a term independent of b. Expanding the squared

Frobenius norms, or completing the square about their average, identifies its minimizers with nearest group points to feedback(. Thus is nearest to its feedback. This proves the self-consistency assertion used in both alignment arguments.

A uniform small-transport condition gives uniqueness modulo a common left multiple, independently of the numbers of sites or pair labels. Supposemat(. Comparison with the constant array bounds the average squared pair separation of any minimizing h by 4 nearest to its centroid, averaging this bound at a suitable index givescentroid(. The feedback differs from the centroid by at most d, so every feedback is within 3d of 1.

Choose from the focus two-point estimate with coefficient 1/4, and = min(4). Given two minimizers, multiply each on the left by the inverse of a nearest point to its own centroid. This preserves minimization and makes 1 nearest to both centroids. Let = maxmat(mat( and take an index attaining it. Write = feedback(feedback(. Unitary invariance gives . Subtracting the centroid difference before estimating gives the sharper bound

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∥B −(centroid(h) −centroid(k))∥≤ud,

because each summand factors as (mat(mat())(mat(). Both centroid-minus- 1 matrices are normal to the Lie algebra by the nearest-point normal equation. Their difference has tangent projection zero. Thusprojection(. Self-consistency and the focus estimate, applicable since 3, now imply

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u ≤∥projection(B)∥+ 1
4∥B∥≤ud + u/4 ≤u/2.

Therefore = 0. Undoing the two centroid normalizations shows that the original minimizers differ by one common left group element. A fixed center value removes this last freedom.

Finite-pair alignment and the route census

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The synchronization estimates compare overlapping local predictor charts. They cannot identify
all gauges by declaration: each query has its own route, and every route contributes to the finite
multiplicity count. The common macro gauge is obtained only after these comparison errors have
been estimated. This supplies the coordinate compatibility used by the affine rows and protected
mean in the following steps.
Let Wxyk be a finite family of group-valued pair transports with its reversal symmetry, and let
h minimize the averaged pair price of hx Wxyk h−1
y . Put income(x) = averagey,k chord(Wxyk) and
m = averagex income(x). Comparison with the identity array and chord2 ≤2 chord show that its
minimum price is at most m. Factoring hx h−1
y
through the pair discrepancy gives

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averagex,y ∥mat(hx) −mat(hy)∥2 ≤8 colorScale2 m.

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Indeed the squared triangle inequality bounds each chord(hxh−1
y )2 by 2 times the squared dis-
crepancy plus 4 chord(Wxyk). Averaging and using the minimizing comparison proves the factor
8.
Let centroid(h) = averagex mat(hx) and feedback(h)x = averagey,k mat(hyW −1
xyk). Direct sub-
traction gives

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∥feedback(h)x −centroid(h)∥≤colorScale · income(x).

A minimizer is nearest to its feedback at each coordinate; this coordinatewise variational assertion is the finite-pair self-consistency input. Choose a nearest group point b to the centroid and

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replace h by b−1h. The centroid now has nearest point 1. By choosing one index whose averaged
squared pair distance is at most the global average and applying the squared norm inequality for
averages, its distance from 1 is at most
√
8 colorScale2 m.
Take δ > 0 so that the nearest-point comparison from the focus derivative holds with error
coefficient 1. Put σ = (δ/2)2/(8 colorScale2) and a = δ/(2 colorScale). If m < σ, the centroid is
within δ/2 of 1. At any index with income(x) < a, its feedback is within δ of 1 as well. Apply that
comparison to the nearest points hx and 1 of feedbackx and centroid. The projection has norm at
most one; therefore

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∥mat(hx) −1∥≤2∥feedbackx −centroid ∥,
chord(hx) ≤2 income(x).

At the remaining indices chord() income(). Averaging proves the alignment estimate with max(2, simply choose = 1 and use average chord( Thus the single positive constant

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Calign = max(2/σ, 2, 2/a)

gives average chord( m for every finite pair family, with no smallness hypothesis on its individual transports.

Here is the route census used to bound m. For a radius-r box, replace selected coordinates of an endpoint x by the corresponding coordinates of y; call the resulting site). Swapping those coordinates between x and y is an involution of endpoint pairs. Therefore every site function f satisfies

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X
x,y
f(dp(x, y)) = N
X
x
f(x).

Likewise swapping a site’s j coordinate with an independent height inr,. . . ,r is an involution of (site,height), giving

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X
x,height
f(updatej(x, height)) = (2r + 1)
X
x
f(x).

A straight segment in direction j has chord at most the sum of the internal positive j-link chords on its whole coordinate line. For a positive segment this follows by the product triangle inequality and then enlargement to the full line; for a negative segment reverse the walk and use inversion invariance. Append one new axis j to an ordered route. Its initial part ends at), and its last segment is bounded by that site’s full j-line sum. The two involution identities bound the sum of this added cost over endpoint pairs by + 1) edgeSum(). Induction over the distinct axes proves the route bound length( + 1) edgeSum(); four axes give the constant 4. This establishes the route estimate and the finite-pair alignment used in synchronization. The preceding minimizer self-consistency argument gives the coordinatewise nearest-feedback condition used here.

Predictor query and synchronization estimates

Let be connector values with nonnegative patch weights summing to one. For arbitrary group elements x,y put and

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τ =
X
p
weightp chord(x−1Ypy).

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Cyclic invariance of chord identifies this with P weightp chord(Ypq−1). The weighted matrix
mean m = P weightp mat(Yp) satisfies ∥m−mat(q)∥≤colorScale ·τ. In the best-predictor branch
its value b is a nearest group point to m. Hence

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∥mat(b) −mat(q)∥≤∥mat(b) −m∥+ ∥m −mat(q)∥
≤2 colorScale ·τ.

In the other branch the minimum connector price is at least predictorCut. Since price( chord(chord() for 0chord(2, comparison with q givespredictorCut. The universal chord bound 2 then gives the same common estimate in both branches:

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chord(x−1predict(Y )y) ≤2 max(1, predictorCut−1)τ.

Thus the query estimate uses the actual fallback outcome as well as the nearest-point outcome. We apply the preceding finite-pair alignment estimate to synchronization. The ordered route sum on a radius-r box obeys

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X
p,q
chord(travel(p →qinordero))
≤4N(2r + 1) edgeSum(U), N = numberofpatchpoints.

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It follows by charging each coordinate segment to its internal positive edges; the precise route-
count argument is developed with the finite-pair input below. Averaging this inequality over orders
and endpoints gives N · meanPairChord ≤4(2r + 1) edgeSum(U).
In the minimizing frame branch, gauge by H and set bp = framep H−1
p . This array minimizes the
corresponding finite pair energy: gauge covariance of route products preserves that energy, and any
competing array can be normalized at the center by a common left multiplication without changing
it.
The finite-pair cluster inequality gives c with averagep chord(bpc−1) ≤C meanPairChord.
Therefore

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X
p
chord(c−1framepH−1
p )
≤4C(2r + 1) edgeSum(gauge HU).

The fallback branch is elementary and does not use that alignment inequality. Its positive frame mass makes its axial tag eligible, so its normalized route gauge d satisfies edgeSum(gauge( axialFactor(2 + 1) faceSum(). Put. Edgewise factorization and the chord triangle inequality give

edgeSum(gaugeedgeSum(gauge) + edgeSum(gauge

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For this pure gauge, route transport from p to q is gp g−1
q . Choose q whose sum over p is at
most its average over q, and apply the ordered route sum above. Taking c = gq yields

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X
p
chord(c−1framepH−1
p )
≤4(2r + 1)[edgeSum(gauge HU)
+axialFactor(2r + 1)faceSum(U)].

With = max(4 4), both branches prove the synchronization estimate below. The finite-pair cluster inequality and route census proved in this chapter supply its two inputs.

The patch weight bound used in the macro predictor estimate also has a direct counting proof. Its weight is the distribution of a sum of J independent uniform small-box rolls. Separate the first roll from the other J1 rolls. For every fixed choice of those remaining rolls, at most one first roll can give a specified sum. Since the small box has points (L is odd), that sum has probability at most. Thus weight1, with no assumption that different roll histories give different sums.

Common macro gauge and predictor price

The planting argument uses the axial estimate already proved and a frame synchronization inequality. The latter states that for a usable frame on a patch of radius r and any gauge H there is a constant group element p with

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X
patch
chord(p−1frame(q)H(q)−1)
≤A(2r + 1)[edgeSum(gauge HU)
+axialFactor(2r + 1)faceSum(U)].

The preceding synchronization proof establishes this inequality. Its constant A depends only on the group; the block-length factors are displayed explicitly.

Let j be the smoothing index, set and The macro radius is the patch radius plus 4L; consequently 2 macroR +1 L. Apply the four-axis axial construction on this whole macro box to obtain a single H with edge sum at most D Lport. Every patch translated by, lies in this macro box. Its internal edge and face sums are bounded by the full macro sums by nonnegativity and translation. Apply synchronization separately on each such translated patch using the same H. The displayed bounds give an element with frame error sum at most port. This proves the planting estimate used below and preserves one common macro gauge for all neighboring roots.

To price a predictor connector between adjacent roots y and, factor it as the left frame error, the straight length-L transport in gauge H, and the inverse right frame error. Its chord is bounded by the sum of these three chords. Let the two frame-error sums be at most port and the macro edge sum at most TL port. The sum of the straight-path chord over all patch starting points is at most L times the macro edge sum: each step’s starting-point translation is injective, its links remain in the macro box, and the nonnegative edge census bounds that partial sum. This bound counts all L steps even if they repeat a torus link.

The patch weight bound is weight1. Therefore the weighted connector chord sum satisfies

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L4τ ≤L2(2K + T) port .

The predictor query inequality bounds its chord, in this left and right frame, by 2 max(1 predictorCut. Multiplying by and using 0 proves

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L2 chord(p−1
y Q(y,i)p(y+ei))
≤2 max(1, predictorCut−1)(2K + T) port .

This proves the indicated-predictor estimate from synchronization and the predictor query inequality. It also supplies the coarse curvature comparison. Four such gauged predictors around an elementary coarse plaquette have chord at most port each, where = 2 max(1 predictorCut

). Their endpoint p factors cancel in the plaquette product. Triangle and conjugation invariance yield chord(face port. For any V, compare its four links to Q. The two-factor inequality

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chord(ab(cd)−1) ≤chord(ac−1) + chord(bd−1)

follows by factoring the product into one discrepancy and a conjugate of the other. Iteration, with inversion invariance for the reversed links, gives chord(facechord(face) + fuel( where fuel is the sum of its four wire discrepancies. This is the local curvature comparison used by the clean-patch and shell arguments.

The nonlinear shell link comparison

Fix the forest and its actual slice. To compare a fine link (x,i), close it with the reverse tree hike from its first coarse root to x and the tree hike from to its second coarse root. On the slice both hike transports are 1, so the resulting walk has transport), endpoints the two embedded coarse parents, and length at most 8L+1. Lift the whole walk to the cover starting at 0 over its first root. Every point lies in the macro box.

The planting estimate supplies on this same macro box a gauge H and comparison elements with

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totalEdgeChord(gauge HU) ≤DL · port,
X
patch
chord(p−1
y framey(q)H(q + centerRoot(y))−1)
≤CL2 · port .

The second inequality is read at the distinguished patch point, whose frame is 1, to give chord( port. The indicated-predictor estimate on adjacent roots gives

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chord(p−1
0 Qpei) ≤αL−2 port,
α = 2 max(1, predictorCut−1)(2C + D).

These planting and indicated-predictor estimates are inputs to this step. The triangle inequality along the lifted forest walk bounds its gauged transport by

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u · port, u = (8L + 1)DL.

The period assumption 100 macroR makes the cover wrapping injective on the macro box. If the two coarse parents are equal, the lifted endpoint is therefore 0; conjugation invariance bounds chord()) by u port and its replica comparison link is 1. If they differ, the centered parent-hop identity makes the second parent the positive coarse neighbor, and injectivity identifies the lifted endpoint with), even across a periodic seam. Factoring the gauged predictor through and gives its chord bound

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v · port, v = 2CL2 + α/L2.

Compare the fine link and predictor in that same macro gauge and use the triangle inequality to get chord() port. Comparing Q with the desired coarse link V then adds precisely its wire discrepancy, with coefficient one. This proves the shell comparison with constant u+v uniformly in the period and in the given forest. The room hypotheses supply its size assumptions, so the same constant is available in the local driving inequality.

The local driving inequality

Let lens() be the active part of the fine tangent field. At a coarse site o define current() as the square root of the sum oflens( over the radius-3B fine-link neighborhood, and response( as the square root of the sum of squared selected motion rows over radius macroR+2. A summand’s norm is bounded by the corresponding square root because all squared norms are nonnegative.

Fix a hot fine link e and put = parent(). Suppose a gauge z makes the background links in radius 2B about o have chords at most h. Every row root within macroR+2 of o is selected in the identity orientation: hotness puts e far enough from the seed set, and the room radius leaves this entire neighborhood clear. Its oriented motion norm is bounded by response(). Apply the transported-to-flat row comparison to the gauged background and the gauged lens. The required tangent values lie in the radius-3B neighborhood of o, so their norms are bounded by current( Gauge covariance and norm preservation give, at each such row,

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∥barFieldR(Ad(z) lens(u))∥≤Eo,
Eo = response(o, u) + Kst · 2 colorScale ·(B + 1)h · current(o, u).

The local link estimate for a tree-slice tangent field w with zero trunk values states

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∥we∥≤K
X
X
plaquettesp
∥curl(w)(q,p)∥
qinmacro box
+∥predictorLinearBill(w)(o,axis(e))∥.

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Here K is the common local comparison constant for the actual predictor tags; the gauge-
flattening and local-driving arguments in this chapter supply it.
Apply this estimate to
w = Ad(z) lens(u).
It vanishes on trunks, since these are not hot.
The direct-curvature
rows in the preceding bound imply ∥curl(w)∥
≤
Eo/
√
direct, and the noise row implies
∥predictorLinearBill(w)∥≤Eo/√penalty. All macro-box plaquette roots belong to the radius
macroR+2 neighborhood. Thus

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∥lens(u)e∥≤ZEo,
Z = K| box(macroR)||PlaqAxes|/
√
direct + 1/
p
penalty.

This proves the local driving estimate used in the plate floor. A clean-patch gauge provides h of rate1/2 with its radius factor already included in the flattening bound; hence the coefficient of current tends to zero fast enough for the squared-neighborhood absorption previously described. The tree-slice link estimate in this chapter gives the constant uniformly in the tags.

Transported affine rows

For coarse sites x,y choose the coordinate-order route from 0 to the centered lift of yx and start it at x. Its endpoint is y and its length is at most 4 separation(), since each of its four coordinate lengths is bounded by that separation. Let) be its actual parallel transport in V. For a fine tangent field u, spin each link value at its coarse parent y by Ad()), then apply the cutoff flat linear row rooted at x. This defines the straightened row. The signed oriented row first changes the input coordinates by the actual tangent transformation associated with the replica of V, then applies this straightened row in the signed background.

These constructions are linear in u. The tangent change at a reversed link includes its background adjoint factor and a minus sign; it is not a mere permutation of color coordinates. Both this