with
0 the fixed penalty, and likewise its bill norm is at most
). All the face roots lie within
, by the parent-distance bound. Thus
box(
PlaqAxes
dir 
(NF2)
No global pure-gauge representation of the background has been assumed.
The clean-background flattening estimate supplies the gauge in (NF1) with
), uniformly over the actual clean boxes, where
) has rate
in the source’s rate calculus. The local chord chart supplies this bound by writing each transformed link as the exponential of a Lie-algebra element of norm at most
), then using its uniform chord bound. The chart radius is fixed before taking
large. Put
Searchable transcription
M(x) = 2UCcG(B(x) + 1)Y s(x).
Equation (NF2), and the trivial zero bound on inactive bonds, give
Searchable transcription
∥uhot(e)∥≤URaddr(e)(u) + M(x)Iaddr(e)(u).
The neighborhood census bounds the sums of the squared local quantities by their global sums times swept(
) = ballCost(
Axis
Searchable transcription
Squaring the preceding inequality with (a + b)2 ≤ 2a2 + 2b2, summing, and requiring 2M(x)2 swept(L, 3B(x)) ≤1/2, yields X e ∥uhot(e)∥2 ≤4U2 swept(L, Rmac + 2) X p ∥motion(u)p∥2. (NF3)
Searchable transcription
This smallness holds on a common tail: the radius census has rate zero, while M has rate −1/2, so the absorbed quantity has negative rate. All constants are chosen after the block side L, before the sector and the variation. Finally the real plate is the motion Gram matrix plus the identity on inactive coordinates. The exact normalization is ⟨v, formT form v⟩= scale2 X p ∥motion(u)p∥2.
The Euclidean square norm splits into its active and inactive squares. If
0 is a common constant in (NF3), the plate therefore obeys
Searchable transcription
⟨v, plate v⟩≥min(scale2 /a, 1)∥v∥2.
This derives the uniform Gaussian floor from one accepted branch’s local forest inequality and clean-background flattening, rather than assuming a floor for the stacked action.
The forest inequality used in (NF2) is obtained directly from the nonlinear slice-shell inequality. For a logarithmic direction
that vanishes on the forest trunks, consider
= represent(
). Then 
remains on the same forest slice for every real
, and
= 1. All predictor frame weights are usable near
= 0: the minimum condition holds at the identity and on a neighborhood, and there are finitely many coarse roots. The shell inequality, with comparison background the identity, gives
chord(
port
) + chord(prediction(
Each local port is bounded by the full sum of plaquette chords over the macro box. This may overcount periodic plaquettes, which only enlarges a nonnegative sum.
If a group-valued curve
starts at the identity and has matrix derivative colorMat(
), the normalization of chord and the color norm imply
Searchable transcription
|t|−1 chord(Vt) −→∥v∥.
Indeed the derivative quotient of its matrix is colorMat(
); take its norm and divide by the fixed color scale. Apply this fact to the edge, to each plaquette holonomy, and to the actual predictor. Their derivatives are respectively the collected edge variation, its linear curl, and the predictor bill. Divide the shell inequality by
, let
0 through nonzero real values, and pass through the finite port sums. The resulting inequality is exactly the forest inequality used in (NF2). In particular its pin term is the derivative of the nonlinear predictor chosen by the construction.
Searchable transcription
This argument also gives the fixed-block vertical Hessian gap. If X = box(Rmac), squaring the forest inequality and applying Cauchy–Schwarz to the two finite sums yields X X X ∥w(e)∥2 ≤β ∥curl w(b)∥2 + γ ∥bill w(b)∥2, e b b
Searchable transcription
with β = 2K2L4| Axis | |X|2| PlaqAxes |, γ = 2L4.
For fixed offset the map from coarse root and plaquette type to the fine plaquette is injective, so summation over coarse roots costs
, even when different offsets represent the same periodic site. Parent fibers have exactly
elements. These two observations account for all factors and avoid any period-dependent norm equivalence.
On a vertical forest coordinate
, the stretched field has collected value zero on trunks and
on the complementary bonds. The Hessian of the tested action is bounded below by
Searchable transcription
a X b ∥curl w(b)∥2 + k X b ∥bill w(b)∥2,
where
0 is the curvature-metric floor. The pin term has this sign because the vertical coarse mismatch is minus the bill. Therefore
Searchable transcription
X Hess(v, v) ≥min(a, k) e/∈trunks ∥ve∥2. (NF4) 1 + β + γ
The Hessian restriction follows by composing the joint analytic action with the linear vertical inclusion; its second derivative is the joint second derivative evaluated on two vertical vectors.
The local stencil comparison in (NF1) has a separate elementary transport proof. A carrier from
to
has path length at most 4 dist(
). When all links on a surrounding radius-
box have chord at most
, and that path lies in the box, the carrier’s chord is at most
, by the product chord inequality. To apply it using only local hypotheses, replace all links outside the box by the identity; locality leaves the carrier unchanged. The matrix adjoint estimate
Searchable transcription
∥Adg b −b∥≤2cG chord(g)∥b∥
then bounds straightening of a local variation of size
. A bounded stencil of norm at most
, depending only on its declared lookup neighborhood, transfers this to a row error at most 
. One can see the locality step explicitly by cutting the difference of the two variations to that neighborhood; its sup norm is at most the local bound, and the stencil sees the same difference.
Under a signed coordinate change, a reversed link contributes a further adjoint correction of at most 2
; forward links contribute none. The transformed lookup is contained in the original box enlarged by one. The total comparison is thus
Searchable transcription
∥orientedRow(V, u) −orientedRow(1, u)∥≤2CcG(d + 1)hM.
Taking the box and variation bounds specified before (NF1) proves the asserted estimate. Adjoint action and the signed displacement preserve each individual variation norm.
The same transport calculation preserves the entrywise decay needed for the matrix upper price. A single-bond impulse remains a single impulse after the signed displacement and carrier conjugation, and its color norm remains unchanged. The untransported stencil’s impulse decay therefore implies
Searchable transcription
ew dist(e,s focus(p))∥orientedRows,p(V, 1eb)∥≤2Cew(predReach +7)∥b∥,
where
0 is the common stencil width. The extra one in this exponent pays the possible signedlink endpoint shift. Thus the local comparison does not introduce a volume factor into the matrix estimate. Finally the small logarithmic background in (NF1) is constructed, not assumed. For a clean radius-(
+ 1) box,
), axial gauge flattening gives a gauge
with chord at most
Searchable transcription
u(x) = 16(2(32B(x) + 1) + 1)5| PlaqAxes | 2 fiery(x)
on the radius-
box. The accepted temperature comparison bounds the clean plaquette threshold by 2 fiery(
). A fixed local logarithm chart extends these transformed links to a whole logarithmic array
, agreeing on this box, with
), once
) lies in its fixed chart radius. The radius factors have rate zero and the fiery threshold has rate
2; hence
) has rate
2 and eventually lies in that chart. This supplies the function
) used in (NF3).
The logarithmic product estimates used here follow from the analytic local map
log(exp
exp
). It satisfies
0) = 0 and
, by differentiation of matrix multiplication and the local inverse relation between logarithm and exponential. A bounded second derivative on a sufficiently small closed ball therefore gives
Searchable transcription
∥B(a, b) −(a + b)∥≤C∥(a, b)∥2, ∥B(a, b)∥≤C∥(a, b)∥.
The local exponential recovers the product on that ball. Applying this at each link gives the same constants for every finite lattice in the sup norm, since the operation is coordinatewise. This justifies use of a genuine left group chart in the comparison, including its quadratic error.
The extension of local logarithms used above has a particularly simple form. The analytic matrix logarithm chart satisfies log 1 = 0 and, on a fixed neighborhood,
Searchable transcription
exp(log U) = U, ∥log U∥≤C chord(U).
The second inequality follows from the local Lipschitz bound of the analytic chart and
mat(
chord(
). Choose the chord radius small enough to lie both in the Lipschitz neighborhood and in the neighborhood of the right-inverse identity. On the requested local set of edges define 
= log
, and outside it define
This is a whole finite array with
; its exponential agrees with
on the requested set. No continuous cutoffor global logarithm of
is involved.
The adjoint estimate in (NF1) is likewise a direct matrix inequality. For a unitary
and a color matrix
Unitary invariance of the Frobenius norm and submultiplicativity bound its norm by 2
Dividing by the color scale proves the stated physical-color inequality. The path chord inequality follows by induction on path length, using the product triangle inequality and invariance of chord under inversion; it also applies to paths with repeated edges. A related estimate needed in the local vertex remainders is
Searchable transcription
∥curvature(face(exp b)) −curl b∥≤C∥b∥2 ∞
on a fixed sup-norm ball, with the same constant for every lattice. To prove it consider the analytic finite-dimensional map
curvature(exp
exp
exp(
) exp(
It vanishes at zero and its derivative there is
. Taylor’s theorem with bounded second derivative gives the quadratic error and a uniform bound on this linear part. Substituting the four actual face coordinates gives the result; their maximum norm is at most
, even when periodicity identifies two or more of these coordinates.
Common envelopes for the non-packet activity letters
The regular-log and forced-letter estimates must use one envelope independent of the background sector, its forest, and the Gaussian variable. The assembly is as follows. Fix
0 below one quarter of the support exponent and below one eighth of each of the two stencil widths, a nonnegative footprint weight
, and a retained order
4. Choose the block side large enough for the inherited footprint estimates. After this choice fix the positive Gaussian floor and matrix upperprice parameters, the local chart constants, and the remaining real sector parameters. Suppose the covariance-window error has rate
For every exponent
0 and any prescribed forced-letter exponent
, there are common nonnegative weights
for forced letters and
for the other logarithmic letters such that
Searchable transcription
| bagi(U)| ≤wi, | farLogp(z)| ≤yp, |efarLogp(z) −1| ≤yp. (NF5)
Searchable transcription
Their weighted footprint sums have respectively budgets βu F and βb F . These are the sales inequalities used in the connected expansion: the weights include the specified footprint exponential, overlap price, radius charge and retained derivatives. All the ancillary costs enter before the budget is reduced to the displayed powers. To see the exponent bookkeeping, the local estimates give a base envelope b(0) p , obtained by combining the inherited physical letters, the finite local logarithmic letters and the covariance- window letters. Their combined sales budget has the form
Searchable transcription
T(x) = βF (x)s + X(x) + N(x), −1/2 < s < b,
Searchable transcription
where X, N have rate −1/2. Thus T has rate s; eventually T ≤1 and 2T ≤βb F . The envelope is chosen by evaluating the uniform estimates on an empty-seed dummy sector at the same radius and cutoff, then retaining their background-independent weights. This is legitimate because the sector hypotheses hold there and the estimates themselves quantify over all real backgrounds. It makes no choice based on the subsequently integrated Gaussian value.
The base envelope bounds each logarithm and is individually at most one, since a nonnegative weighted sales sum bounds each of its summands. For
1, the elementary bound
supplies the Mayer envelope. Consequently set
; multiplying its sales budget by two costs at most
, as just arranged. The forced-letter envelope keeps its separately prescribed budget. The real localization factors all lie in [0
1], so multiplying by them can only decrease these pointwise bounds. This establishes (NF5) with one set of weights usable in every sector and for both the raw logarithms and their Mayer factors.
Searchable transcription
The letters combined in (NF5) are the following actual terms. Besides the inherited activity logarithms, there are three finite local types: row-energy remainders indexed by a signed orientation and a row, a diagonal logarithmic correction indexed by a fine bond, and a Haar-chart correction indexed by an active fine bond. There are also five chain types: the principal log chains for the real plate with coefficient −1/2, the corresponding blank-plate chains with coefficient +1/2, the covariance correction chains with coefficient +1/2, the affine first-order correction, and the residual quadratic chain. Their chain lengths are their additional charges; the affine bond term has charge zero. Their supports are the corresponding full receipts enlarged by the halo parameter lB, including every lookup used by the term.
For a row with coarse center
, the actual short exponent is zero within distance 200
of the seeds, and otherwise is
![−βθ [go(u) lighto(u) + extractedo(μ)] ,
0 ≤go(u) ≤1,
0 ≤θ ≤1.
(NF6)](/publications/source-lift/yang-mills-companion/math/p108-002597.svg)
Searchable transcription
−βθ [go(u) lighto(u) + extractedo(μ)] , 0 ≤go(u) ≤1, 0 ≤θ ≤1. (NF6)
Searchable transcription
The diagonal term is zero when either endpoint of its bond meets the radius-200B seed dome, and otherwise equals 1 2 joint(plateblank) −joint(plate) e.
The Haar term is the localized chart logarithm on active bonds outside the radius-304
seed dome and is zero otherwise. These signs are exponent signs: the activity is
On an intact real sector the row quantity light equals the actual debit. Indeed outside the dome the local slice is interior and the predictor is alive. The kept-path test is satisfied on the local region by the clean plaquette bound; outside it the exponential path threshold is at least two and the universal chord bound supplies the test. The exact row-energy splitting can therefore be applied to the actual configuration, with its existing choices.
Here is the size estimate for (NF6). A nonzero localization factor supplies the local variation bound required by the grafted-input energy estimate. That estimate bounds
) light
by a common
) of rate
2. The protected-mean estimate bounds each of the at most two parent path values of
by
), of rate
1. The extracted term is a sum of at most two nonnegative half-squares, so it is at most
. Since
), its row contribution is bounded by
Searchable transcription
W(x) = 2βF (x) J(x) + A(x)2,
of rate
2. The diagonal-log difference has a common bound
) of rate
2, obtained by local gauge flattening and the principal-inverse comparison for the real and blank plates with the same window geometry. The localized Haar logarithm has bound
fiery(
), also of rate
Adding these three nonnegative bounds gives a common bound for every short letter. To turn this pointwise bound into its sales bound, the center census is
Searchable transcription
| Signed | spots(L) + 2 density(L).
Every support is contained in the radius-(
neighborhood of its center. Thus a common short-letter envelope
) has weighted budget at most
Searchable transcription
ballCost((l + 1)B) | Signed | spots(L) + 2 density(L) Eshort(x)ev ballCost(l+1).
Its rate remains
2. This is a local census bound, independent of the total number of sites. The inherited letters satisfy their own support estimates before they are joined to these local types. For the incoming regular activities, use their common healthy-sector sup envelopes; the masked activity is zero offthe healthy set. For forced activities choose an anchor in each nonempty inventory, which lies in its support. The incoming support budget then becomes an anchored budget at that anchor. The footprint conversion expands each old support to its inspected coarse footprint, paying the retained overlap and charge weights. Its exponential reserve is available once 4
is less
than the incoming support exponent and
exceeds the fixed conversion threshold. These choices give the regular budget
for every
10 and the forced budget
for every prescribed
A degree-
translated local atom, with coefficient bound of rate
and aggregate magnitude budget of rate
, has pointwise envelope equal to that coefficient bound times fiery(
times its fixed atom magnitude. Its sales budget therefore has any exponent strictly greater than
Searchable transcription
a −k/2 + b.
Searchable transcription
Finite signed orientations only multiply the budget by their fixed cardinality. Applying this rule to cubic and quartic terms with coefficient of rate one, and to the loop quadratic term with coefficient of rate zero, places all three below every exponent strictly greater than −1/2. The strict inequalities leave room for footprint conversions and their polynomial charges. True path tails, including the lower-scale tail, have an arbitrary power saving after their positive retained cutofforder is imposed; their additional orientation and plaquette-type sums are finite. The six inherited classes can consequently be combined with an intermediate exponent s < b, and the fixed factor six is absorbed by βb F /βs F →∞. This supplies precisely the inherited part of the common envelope construction (NF5).
The ordered-chain support estimate can be stated and proved independently of the matrix formulas. A route of length
is an ordered list of
fine bonds after a starting bond. A hop is allowed if an endpoint of one bond is within coarse distance
of an endpoint of the next. Hence their coarse centers are within
+ 2. The number of allowed next bonds is at most
Searchable transcription
Pd = ballCost(d + 2) density(L), density(L) = L4| Axis |.
Searchable transcription
Induction on length bounds the number of allowed routes from any starting bond by P n d , including routes with repeated bonds. Attach to every route node the radius-mB halo of both bond endpoints. The full receipt is the union of these halos. If d + 2 ≤mB, successive halos meet. Each halo has B-cover number at most steady(m) = ballCost(m + 1). The union therefore has cover number at most
Searchable transcription
(n + 1) steady(m).
Every point in it is within (
+ 1) of the starting center, by the triangle inequality along the route.
Suppose a chain letter has
hops, with
fixed obligatory hops, charge
, and magnitude at most
on allowed routes and zero on the others. Fix a coarse site
and sum over all such letters whose receipts contain
, including their charge factor
and footprint factor
. Its possible starting bonds number at most
Searchable transcription
ballCost((n + u + 1)(mB + 1)) density(L).
The elementary inequality
gives, in four dimensions,
Searchable transcription
ballCost((n + u + 1)(mB + 1)) ≤16n ballCost((u + 1)(mB + 1)).
Combining the starting-point count, route count, magnitude, charge and cover costs bounds the length-
contribution by 
Searchable transcription
where
Searchable transcription
H = ballCost((u + 1)(mB + 1)) density(L)P u d ev(u+1) steady(m), R = 16Pdev steady(m)+2ε.
Searchable transcription
If qR ≤1/2, summing the geometric series gives the sales bound 2HC, uniformly in x and in the lattice period. This counts every endpoint and its complete lookup halo before any geometric shortening. Summing over the finite route coordinates at fixed n, then over n, is justified by nonnegativity; finite partial sums obey the same bound. The localization and chart factors used in these estimates have exact identities worth recording. Let γa(u, e) = χinner(at, ∥ue∥), Γa(u, Y ) = Y e∈Y γa(u, e).
Both lie in [0![, 1]. If at >](/publications/source-lift/yang-mills-companion/math/p110-019565.svg)
0, then
implies
= 1, and
= 0 implies
, hence 
= 1. Thus a nonzero Γ
bounds every variation on its selected edge set and gives Γ
= 1 there. For the Haar chart density
, the localized logarithm is
log
= 0 and 2
lies within the chart radius, its exponential equals
exactly.
Its all-argument size bound follows without assuming the argument is in the chart. If
the localized logarithm is zero. Otherwise
. Since
(0) = 1, positivity and analyticity on a fixed chart give
log
, once 4
lies in that fixed neighborhood. Multiplication by
1 preserves the bound. This proves the uniform Haar contribution used in (NF6), with the shrinking threshold chosen on one common tail.
Exact warning factors and their vanishing region
There are five detector addresses: a fine plaquette, two distinct types of coarse bond, a coarse site, and a fine site. Their raw factors are respectively the fine cutoff, the predictor-noise cutoff, the pin-frame mass, the site-frame mass, and the contracted flag charge. Each address has its fixed district containing its focus, and every district point is within one coarse step of that focus. The warning support is its radius-
halo. There is one further warning type, the complement of a fine-bond logarithmic cutoff, with the two-endpoint support used above.
A detector is compulsory when its district meets the radius-300
seed dome. Its optional warning letter is then zero. Otherwise it is
Searchable transcription
Γ2a(u, plaza(haloB(district)))(rawStorm(u) −1). (NF7)
The fine-bond warning is
1 on active bonds outside the radius-304
dome, and zero on all other bonds. Set 
FrameTag
Points
Every raw detector lies between zero and
: the two cutoffs lie in [0
1], each frame mass is bounded by its finite number of summands, and the contracted charge is zero or one. Thus 
rawStorm
. Since all localization factors lie in [0
1], every warning has absolute value at most
Write the Gaussian-coordinate input as
. A quiet box of radius
about
means
Searchable transcription
|ze,α| ≤ξt p β whenever dist(o, addr(e)) ≤D. (NF8)
Searchable transcription
If ∥μe∥≤Mt, this implies
Searchable transcription
∥ue −μe∥≤nGξt, ∥ue∥≤(M + nGξ)t,
Searchable transcription
where nG is the number of color coordinates. The estimate follows by bounding the color Euclidean norm by the sum of the absolute coordinate values. It uses no independence between coordinates. Here are the choices that make every ordinary detector equal to one on a quiet box. Fix the background guide bound A > 0. First choose τ = √qdir/32 > 0, then take L large enough that
Searchable transcription
2 √qdir (A/L2 + τ) < 1 4.
Searchable transcription
After L, take W large enough that 2 √ k (A + τ) < W.
Choose
0 so small that
1 and the row-stencil response to the noise in (NF8) is at most 
3. A concrete admissible choice is
Searchable transcription
! (nG + 1)−1, τ 3nG scale−1 |Y | + 1 ξ = min ,
where
is the uniform row-price bound.
The remaining row bill has three contributions: the nonlinear-to-linear row error, the row change from
to
, and the protected row at
. The first and third have rate
1 and hence are eventually at most
3 each, since the threshold
has rate
2. The middle contribution is at most
by the choice of
. Thus every relevant actual row bill has norm at most
For a fine plaquette add its guide norm at most
; its weighted curvature is at most 
. The local curvature-to-chord inequality and the displayed fine quota give its chord strictly below
4. For a predictor bond the same argument, with guide norm at most
and penalty
, gives mismatch chord strictly below
. The local small-field tag identities give site-frame and pin-frame masses exactly one. These estimates apply outside the radius-30
dome on a quiet radius-
box and use the actual predictor choices.
To deduce vanishing of (NF7), suppose the radius-2
box about its focus is quiet. If the detector is compulsory, the letter is already zero. Otherwise its focus is outside the radius-300
dome. Every point within distance
is then outside the radius-30
dome, and its radius-
box is quiet, by the triangle inequality. The preceding conclusions apply there. The fine and noise cutoffs are on their unit plateaus, and both frame masses are one. Finally a nontrivial contracted flag would supply a touched plaquette with chord greater than
. Its parent lies within the same radius-
neighborhood, where its chord is less than
4, a contradiction. Hence the flag charge is also one. Each of the five raw detectors is therefore one, so its warning letter vanishes.
The fine-bond complement similarly vanishes whenever its protected mean and noise place
on the unit plateau of
. Consequently a nonzero optional warning has a coordinate violating a bound of the form (NF8) within its fixed enlarged support. This is the witness statement used by the joint Gaussian catastrophe estimate. It is valid for an individual processed warning regardless of the values of the other cutoffs, so no product of independent warning probabilities is assumed.
From anchor bounds to support-hit bounds
Here is the summation underlying the inherited support budget. Let a map
from fine to coarse sites have at most
fine preimages per coarse site. Labels
have fine anchors
, nonnegative
Source: A Wilson Yang–Mills Construction: A Natural-Language Mathematical Exposition. Both papers belong to one Zenodo record. No external funding.