Local nonlinear geometry and Haar realization · Part 7

For completeness the analytic product chart) = log(exp exp) satisfies 0) = 0 and by matrix multiplication and the local inverse relation. A bounded second derivative on a small closed ball yields

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∥B(a, b) −(a + b)∥≤C∥(a, b)∥2,
∥B(a, b)∥≤C∥(a, b)∥.

The exponential recovers the product there. Coordinatewise application gives uniform finite-lattice sup-norm constants and a genuine left chart with this quadratic error. The fixed chart has

log 1 = 0 exp(log log chord(

The last bound follows from chart Lipschitz continuity andmat( chord(). Choose the radius within both the Lipschitz and right-inverse neighborhoods. Set = log on requested edges and = 0 elsewhere. Then and its exponential agrees on the requested set; no global logarithm or continuous cutoffis needed. Finally, for unitary

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UXU∗−X = (U −1)XU∗+ X(U∗−1).

Frobenius unitary invariance and submultiplicativity bound this by 2; division by the color scale proves the physical adjoint bound. Induction using product triangle inequality and inversion invariance proves the path-chord bound, including repeated edges. The realized record retains local lookup sets, carrier words, signed endpoint shifts and chart domains. Recovery therefore preserves each local comparison and impulse bound without extending local hypotheses to the whole lattice.

The local logarithm construction has an explicit recovery domain. For its recorded edge set and fixed chart radius, let

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(
log Ue,
e ∈D,
0,
e /∈D.
UD = {U : chord(Ue) < ρ (e ∈D)},
(LDU)e =

Choose within both chart neighborhoods just specified. Its source domain is) and its source reader is. For the coordinate restriction and coordinatewise exponential

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RDELDπ = RDπ,
LDπσ = LD.

These identities prove exact patch recovery and section recovery. The bound

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∥LDπ(z)∥∞≤C max
e∈D chord(π(z)e)

preserves the same constant. The construction is applied to the gauged physical links appearing in the local comparison. Their gauge and the set remain recorded data. Consequently the recovered logarithm has precisely the stated local chart domain and the zero extension used by the source construction.

12.53 Curvature remainder and common non-packet envelopes

On a fixed sup-norm ball the local vertex estimate is

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∥curvature(face(exp b)) −curl b∥≤C∥b∥2
∞,

with a lattice-independent constant. Indeed the analytic four-coordinate map

curvature(exp exp exp() exp(

vanishes at zero and has derivative. Bounded-second-derivative Taylor expansion gives the quadratic remainder and a uniform linear-part bound. Substitute the actual face coordinates, each bounded by, including when periodicity identifies coordinates.

The regular-log and forced-letter assembly uses one envelope independent of sector, forest and Gaussian variable. Fix below one quarter of the support exponent and one eighth of each stencil width, footprint weight0 and retained order Choose the block side for inherited footprints, then fix the positive Gaussian floor, matrix upper prices, chart constants and remaining real-sector parameters. Assume covariance-window error of rate1. For 0 and any forced exponent, common nonnegative weights satisfy

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| bagi(U)| ≤wi,
| farLogp(z)| ≤yp,
|efarLogp(z) −1| ≤yp.
(NF5)

Their weighted footprint sums have budgets and , respectively. Specified footprint exponentials, overlap prices, radius charges and retained derivatives are paid before reducing to these powers.

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To prove the bookkeeping, combine inherited physical, finite local and covariance-window letters
into b(0)
p . Its budget is T(x) = βF (x)s + X(x) + N(x), with −1/2 < s < b and X, N of rate −1/2.
Thus T has rate s and eventually T ≤1, 2T ≤βb
F . Evaluate uniform estimates on an empty-seed
dummy sector at the same radius and cutoffand retain their background-independent weights. Its
hypotheses hold and the estimates quantify over all real backgrounds; the weights are chosen before
the integrated Gaussian value. Nonnegative weighted sales bounds imply b(0)
≤1 individually. The
p
inequality |eq −1| ≤2a for |q| ≤a ≤1 permits yp = 2b(0)
p ; the arranged factor-two reserve pays
its budget: multiplying the sales budget by two costs at most βb
F .
Forced letters retain their
independent price. Real localization factors in [0, 1] can only decrease these bounds, proving (NF5)
uniformly for logarithms and Mayer factors.

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The actual finite local types are row-energy remainders indexed by signed orientation and row,
diagonal logarithmic corrections indexed by fine bond, and Haar-chart corrections indexed by active
fine bond. Alongside inherited logarithms, the five chain types are real-plate principal log chains
with coefficient −1/2, blank-plate chains with +1/2, covariance correction chains with +1/2, affine
first corrections and residual quadratic chains. Chain length is the extra charge; the affine bond
term has charge zero.
Supports are full receipts enlarged by lB, including every lookup.
The
lifted record retains all these tags, supports and parameter choices; readout recovers the uniform
weights before field evaluation. The local four-coordinate Taylor map is recovered before restricting
periodic coordinates, so its bound and (NF5) transport without treating coincident coordinates as
independent physical variables.

12.54 Explicit Short letters and their local census

The row Short exponent at center vanishes within 200 of seeds and otherwise is

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−βθ [go(u) lighto(u) + extractedo(μ)],
0 ≤go(u) ≤1,
0 ≤θ ≤1.
(NF6)

The diagonal term vanishes if either bond endpoint meets the 200 seed dome and otherwise equals

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1
2
 joint(plateblank) −joint(plate)


e.

The Haar term is the localized chart logarithm on active bonds outside the 304 seed dome, zero elsewhere. These are exponent signs; the activity is1. On an intact real sector, light equals the actual debit. Outside the dome the local slice is interior and the predictor alive. Clean plaquette bounds supply the kept-path test locally; outside, its exponential threshold is at least two, so the universal chord bound supplies it. Thus the exact row-energy split applies with the configuration’s existing choices.

A nonzero localization factor supplies the grafted-input variation bound, yielding light ) of rate2. Protected means bound each of at most two parent path values by) of rate 1. The extracted term consists of at most two nonnegative half-squares, hence is at most Therefore

bounds the row contribution and has rate2. Local flattening and principal-inverse comparison for real and blank plates with the same windows give a diagonal bound of rate2. The Haar bound is fiery() of rate2. Their nonnegative sum gives a common short-letter bound. The caption census and radius-( support containment give sales budget

ballCost(( Signed spots() + 2 density(

It retains rate2 and is independent of total volume. Before union, inherited regular activities use healthy-sector sup envelopes and vanish offthe healthy set. A forced activity is anchored at a point of its nonempty inventory. Incoming support budgets then imply anchored budgets. Inspected-footprint conversion pays overlap and charge weights once 4 is below the incoming exponent and exceeds the fixed threshold. This yields regular price for10 and forced price for every prescribed. A degree- translated atom with coefficient rate and aggregate magnitude rate has envelope equal to coefficient

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bound times fieryktimes fixed magnitude; its sales exponent can be any value strictly greater than
a−k/2+b. Finite signed orientations cost a fixed factor. Cubic and quartic coefficient rate one, and
loop quadratic coefficient rate zero, place all three below every exponent greater than −1/2, leaving
room for conversion and polynomial charges. True tails, including lower-scale tails, have arbitrary
power saving at positive retained cutofforder, with finite orientation and plaquette sums. Choose
an intermediate s < b for the six inherited families; the factor six is absorbed by βb
F /βs
F →∞.
This supplies the inherited part of (NF5). In the lift, seed domes, masks, chart domains, window
geometries, exact exponent signs and inherited family tags are retained. Literal readout recovers
the actual debit identity and each nonnegative census term before taking the uniform sales bound.
The inherited letters satisfy their own support estimates before they are joined to these local types.

12.55 Ordered-chain census and exact chart localization

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A route consists of a starting fine bond and an ordered list of successors.
Allowed hops have
endpoints within coarse distance d, so centers are within d + 2.
The successor count is Pd =
ballCost(d + 2) density(L), where density(L) = L4| Axis |. Induction gives P n
d routes of n hops,
retaining repetitions. Attach the mB halo of both endpoints at every node. If d+2 ≤mB, successive
halos meet and their union has B-cover number at most (n + 1) steady(m), with steady(m) =
ballCost(m+1). Each receipt point lies within (n+1)(mB +1) of the start. For a letter with n+u
hops, charge n and magnitude Cqn on allowed routes, the number of starting bonds for a receipt
containing a fixed x is at most ballCost((n+u+1)(mB+1)) density(L). Using n+u+1 ≤(u+1)2n

gives the four-dimensional factor 16. Including and the footprint price, the order- sum is at

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most HC(qR)n, where

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H = ballCost((u + 1)(mB + 1)) density(L)P u
d ev(u+1) steady(m),
R = 16Pdev steady(m)+2ε.

For2 the sales bound is 2, uniformly in site and period. Every endpoint and full lookup halo is counted before shortening. Finite route sums followed by the nonnegative order sum justify the bound, as does bounding every finite subset. This independently recovers the ordered-chain support estimate used in the matrix assembly. Record the exact localization identities

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γa(u, e) = χinner(at, ∥ue∥),
Γa(u, Y ) =
Y
e∈Y
γa(u, e).

Both are in [0 1]. For implies= 0 implies and hence Thus nonzero Γ bounds every selected variation and gives Γ = 1 on the same set. For Haar density, the localized logarithm is log= 0 and 2 lies in the chart, exponentiating this logarithm gives exactly. For its all-argument bound, if = 0 the localized logarithm is zero; otherwise. Positivity, analyticity and(0) = 1 give log once 4 is in the fixed neighborhood. Multiplication by the cutoffpreserves the bound, proving the uniform Haar contribution of (NF6) on one common tail. This size bound holds at all arguments, without assuming the argument is in the chart. The lift retains ordered route coordinates, complete receipts, selected edge sets and chart thresholds. Literal recovery preserves these exact cutoff implications and the density identity, as well as every term of the nonnegative census sum.

For a precise source domain, interpret the localized logarithm as the piecewise reader

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(
γ2a(u, e) log J(ue),
γ2a(u, e) ̸= 0,
0,
γ2a(u, e) = 0.
la(u, e) =

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On the common tail 4at is within the positive chart, so the first branch is defined wherever selected.
This defines the zero extension without requiring evaluation of the chart logarithm outside its
domain. With u = pζ, put lZ
a (ζ, e) = la(pζ, e) and define γZ
a , ΓZ
a by the same composition. Then

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lZ
a (jv, e) = la(v, e),
γZ
a (ζ, e) ̸= 0 =⇒elZ
a (ζ,e) = J((pζ)e).

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Finite products on the recorded edge set recover Γa componentwise. The same implication and
bound |lZ
a | ≤4Cat therefore hold on the whole realized tangent domain. These identities preserve
the actual localized density factor used in the correction exponent.

12.56 Exact warning factors and quiet-box witnesses

The five detector addresses are a fine plaquette, two distinct coarse-bond types, a coarse site and a fine site. Their raw factors are respectively fine cutoff, predictor-noise cutoff, pin-frame mass, site-frame mass and contracted flag charge. A fixed district contains each focus and lies within one coarse step of it; warning support is its halo. A sixth type is the fine-bond logarithmic-cutoff complement with its two-endpoint support. A detector whose district meets the 300 seed dome is compulsory and its optional letter is zero. Otherwise the letter is exactly

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Γ2a(u, plaza(haloB(district)))(rawStorm(u) −1).
(NF7)

Fine-bond warnings are1 on active bonds outside the 304 dome, zero otherwise. Put FrameTag Points. Raw detectors lie between zero and this constant: cutoffs are in [0 1], frame masses are bounded by their finite summand counts and contracted charges are zero or one. Hence all warnings have absolute value at most. For, quietness of the radius- box about means

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|ze,α| ≤ξt
p
β
if dist(o, addr(e)) ≤D.
(NF8)

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If ∥μe∥≤Mt, the Euclidean norm bounded by the coordinate absolute sum gives ∥ue −μe∥≤nGξt
and ∥ue∥≤(M + nGξ)t, without coordinate independence.
For background guide bound A > 0, choose first τ = √qdir/32, then L so that 2(A/L2 +
τ)/√qdir < 1/4, then W so that 2(A + τ)/
√
k < W. For uniform row price Y , a concrete choice is

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!
(nG + 1)−1,
τ
3nG scale−1 |Y | + 1
ξ = min
> 0.

This ensures1 and noise row response at most3. The other two row-bill contributions, nonlinear-to-linear error and protected row at, have rate1 and are eventually at most each since has rate2. Thus actual bills are at most. Adding the fine guide bounds weighted curvature by ( ; curvature-to-chord conversion and the fine quota give chord below4. The predictor guide and penalty similarly give mismatch chord below. Small-field tag identities make both frame masses exactly one. These statements hold outside the 30 dome on quiet radius- boxes with the actual predictor choices. If the 2 box about a detector focus is quiet, a compulsory letter is already zero. Otherwise the focus is outside the 300 dome. Every point within is outside the 30 dome and has a quiet box. Thus fine and noise cutoffs are on unit plateaus and frame masses equal one. A nontrivial contracted flag would give a touched plaquette of chord larger than whose parent is in that same neighborhood, contradicting its chord below4. Its charge is consequently one too, proving (NF7) vanishes. The fine-bond complement vanishes when mean and noise put its variation on the unit plateau. Therefore any nonzero optional warning has a coordinate violating (NF8) in its fixed enlarged support. This holds for each processed warning regardless of other cutoffvalues; no product of independent warning probabilities is used. The source lift retains detector type, district, compulsory mask, actual predictor tags and the coordinate witness. Literal recovery preserves each raw factor and the quiet-box implication, supplying the same witness to the joint Gaussian catastrophe estimate.

The witness transport can be stated as a set inclusion. For a processed warning, let be its recorded finite set of eligible bond–color coordinates in the enlarged support, and write for its actual optional coefficient, including its compulsory mask. With all the preceding mean and threshold hypotheses retained, define

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Ee,α = {z : |ze,α| > ξt
p
β}.

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The vanishing argument gives
{z : Wi(μ + β−1/2z) ̸= 0} ⊆
[
(e,α)∈Ai
Ee,α.

Realize the full noise input and let read its physical variation. Taking preimages under preserves this inclusion and the finite union exactly; composing with recovers it because. Thus the witness address is recovered with its actual coordinate threshold. This is an inclusion for each warning coefficient, available before any joint Gaussian integration or selection of several warnings.

12.57 Explicit inherited support-hit payment

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For completeness retain the source’s explicit normalization of the preceding anchor conversion. Let
p have fibers of cardinality at most P, with anchors oi, nonnegative weights bi and integer sizes di
satisfying P
oi=y bieAdi ≤γ at every fine site. Assume

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x ∈Xi ⇒dist(x, p(oi)) ≤W(1 + di),
covB(Xi) ≤v + v′di,

and 0 . Then, with fixed,

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X
giea covB(Xi) ≤eavγCP(2W + 1)4ε−(4+k)(4 + k)!eε
1 −e−ε
e−εm.
(NF9)
i:x∈Xi
di≥m

At degree, there are at most possible anchors. Sum their anchored budgets and use

One term of the exponential series at) yields (1 +. The degree- bound is therefore the numerator in (NF9) times. The resulting geometric tail proves (NF9). Every finite address subset obeys it, so infinitely many variants are allowed by nonnegativity. For block ratio, parent and four residue coordinates uniquely specify a fine site, giving For halo- enlargement of the parent image of, put = cov. Then

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v = (2(l + 6) + 1)4,
covB(Xi) ≤v + 10v
L di,
W = (l + 20)B.

Each point is within) of the parent of any chosen anchor in. Absorb the extra tariff into the old weight first, replacing exponent by. The exact reserve condition is consequently

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a(10v/L) + 2ε ≤δ −2ε.

This retains the child factor and polynomial-degree payment explicitly. For the direct geometric proof, with the block ratio and the periodic maximum-coordinate distance used throughout, centered residues place a site within2 of its embedded parent, while embedded coarse distances scale by. Thus dist(dist( (NF10)

Centered representatives ensure this also holds on winding paths. The lift retains residues, anchors, degree and old exponent as separate data; recovery therefore preserves the two successive 2 reserves and the explicit payment rather than replacing them by an unspecified constant.

12.58 Realizing the geometric input to (NF9)

Choose a realizing-cover of with = cov ). Its centers are connected at range +1. Starting at one center, repeatedly attach an unvisited center along an edge from the visited connected set and insert an out-and-back detour of length at most 2(2 + 1). This constructs a unit-step walk visiting every center with + 2). Treat the empty support separately. Sample at times, including zero. Parent samples form with consecutive samples and any intervening walk-point parent versus its preceding sample are within . A point of is within of a cover center, so its parent is within 3 of that center’s parent for 1. Thus halo ) lies within ( of with the required connected centers.

To replace this larger-radius cover by a-cover, tile each radius- ball with the coordinate grid spaced by in the four coordinate directions, with + 1. Integer division puts every point within of a grid point. Grids and their center connections give a connected union of size at most; passage to torus images only reduces cardinality. For

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covB(halolB p(Y )) ≤|D|(2(l + 6) + 1)4.

Since 4, multiplication by and the sample count yield

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L covB(halolB p(Y )) ≤(L + 10d)(2(l + 6) + 1)4.

The traversal also gives dist( for . Combine this with (NF10) and the halo radius to obtain the anchor-distance bound with, completing the geometric input to (NF9). The lifted record carries the realizing center set, traversal and sampling times, not merely the resulting number. Readout recovers their torus images and incidences, so connectedness and both cardinality bounds survive periodic identifications and repeated samples.

For clarity, the map in (NF9)–(NF10) is the lattice parent map; write it as in the recovery identities here. It is distinct from the tangent readout used for variations. The finite geometric record contains fine addresses, their residues and parents, the centers, the traversal and the sample times. Realization retains those discrete data, so its address readout satisfies

(halo )) = halo

The identity-frame section recovers the original record. In particular

and are identical on recovery. First reserving the charge and then applying (NF9) consequently transports precisely the condition

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δ ≥4ε + 10av/L,

which is the displayed two-reserve condition rewritten without changing its order of use. This also preserves the full factorial and geometric-tail payment in (NF9), including its degree parameter