Shadow Theory and the Wilson Yang–Mills Construction: A Source–Readout Reconstruction

Jeremy Rodgers

doi:10.5281/zenodo.24173681

Source Lift / The complete argument

Local nonlinear geometry and Haar realization · Part 6

Shadow Theory and the Wilson Yang–Mills Construction: A Source–Readout Reconstruction

Jeremy Rodgers · Independent Researcher · 2026-11-12
Shared publication DOI: 10.5281/zenodo.24173681

Read this chapter ↓Original PDF ↓Complete Markdown ↓Reading inventory

coefficient, so the covering and anchored-to-hit estimates use the same support currency and parent census.

The covering currency uses periodic maximum-coordinate separation. Its radius- $M$ centers must be connected at range 2 $M$ + 1, so the currency includes connections between disconnected support pieces; it is not an unrestricted ball-counting number. A unit-speed line from $x$ to $y$ has duration separation( $x, y$ ): truncate each centered-lift displacement coordinate to [ $−j, j$ ] at time $j$ , wrap and add $x$ . Each coordinate changes by at most one and reaches its endpoint by the maximum absolute displacement.

For a finite center set $D$ connected at range $b$ , adjoin unvisited centers successively. A first exit along a connecting path supplies an edge from the visited set. Insert the line along this edge and its reverse at the visited endpoint in the current traversal. This adds at most 2 $b$ steps while retaining every earlier visit. Induction gives a unit-speed traversal visiting $D$ of length at most 2 $b|D|$ ; the empty case uses the zero-length trivial traversal. This is the ring-path estimate used for extent and collapse.

A radius- $kB$ halo has a cover whose centers are the image of [ $−k, k]4 ∩Z4$ , scaled by $B$ and translated to the halo center. Division of each centered coordinate by $B$ leaves a remainder of absolute value at most $B$ , proving coverage. Grid steps connect the centers; periodic identification only decreases their number. Thus (2 $k+1)4$ centers suffice. If a center set $D$ gives a radius- $mB$ cover of $X$ , enlarge the balls to radius ( $m + 1)B$ . Original connecting steps obey 2 $mB + 1 ≤2(m + 1)B,$ so these enlarged halos meet along the connections. Tiling each halo and joining at these meetings yields coverNumber $B(X) ≤|D|[2(m + 1) + 1]4.$

The same unit-speed line shows that halos of centers connected at range 2 $M$ + 1 are unit-range connected: the line lies in the two $M$ halos, apart from the possible middle adjacent step. These arguments require no injectivity of periodic halos. In the lift, the walk record retains its ordered visit list and cover coordinates as well as its finite support. Readout therefore recovers both the true length and the connected covering currency, and the displayed induction and counting arguments commute with literal recovery even when wrapping identifies visits.

12.47 Coarse collapse and anchored-to-hit conversion

For block side $L$ , a fine site is within $L/$ 2 of its embedded parent. Centered lifts show that embedded coarse separation is at least $L$ times coarse separation. The triangle inequality yields

L separation(parent(x), parent(y)) ≤separation(x, y) + L. — Source page 92.

For a unit-step path $R0, . . . , RN$ , sample parents every $LB$ steps, $B >$ 0, retaining the final partial segment and allowing repeated samples. The resulting set $D$ is connected at range 2 $B$ , every path parent is within 2 $B$ of a sample, and $LB|D| ≤N +LB$ . Realize a scale- $M$ cover of $X$ by a connected center set $A$ of cardinality coverNumber $M(X$ ). The preceding traversal has length $N ≤|A|(4M +2).$ Any two support sites are therefore separated by at most $|A|(4M + 2) + 2M$ , by joining them to centers and traversing between centers.

Sample this traversal at scale $LB. If M ≤2B$ , a support point’s parent is within 3 $B$ of its center’s parent, hence within 5 $B$ of $D$ . The $sB$ halo of parent( $X$ ) is thus covered by the ( $s + 5)B$ halo of $D$ , retaining connectedness. Tiling costs at most $|D|$ ballCost( $s$ +6). For $B ≥$ 1, the sample count and $M ≤2B$ give

$L$ coverNumber $B$ (halo $sB$ (parent( $X))) ≤[L$ + 10 coverNumber $M(X$ )] ballCost( $s + 6).$

The empty case is immediate. Otherwise at least one center pays the fixed endpoint term. Division
by L recovers the preCost and conversion constants in the coarse hit budget. All block factors thus
arise from parent separation, connected traversal and halo tiling.
Here is the full anchored-to-hit principle used in those budgets. Suppose nonnegative bi satisfy,
at every fine anchor y,
X — Source page 93.

the parent fibers have size at most $P$ , and

x ∈Xi =⇒separation(x, parent(oi)) ≤W(1 + di),
coverNumberB(Xi) ≤v + v′di,
0 ≤gi ≤Cbi(1 + di)k. — Source page 93.

Assume $a ≥0, ε >$ 0 and $av′+2ε ≤A$ . At degree $di = j$ , the new weighted coefficient is bounded by $bieAjC(1+j)keav−2εj$ . Relevant anchors lie above the coarse radius- $W(1+j$ ) ball about $x$ , so there are at most $P$ ballCost( $W)(1 + j)4$ of them. Grouping a finite label set by anchor and applying the old budget bounds the degree- $j$ total by

eavγCP ballCost(W)(1 + j)4+ke−2εj. — Source page 93.

The exponential series gives (1 + $j)4+k ≤ε−(4+k)(4 + k)!eεeεj$ . Summation over $j ≥m$ yields the tail factor $e−εm$ with payment

eavγCP ballCost(W)ε−(4+k)(4 + k)!eε
1 −e−ε
. — Source page 93.

Every finite subset obeys this bound, proving nonnegative countable summability as well. For block parents, decomposition into parent and residue gives $P = L4$ . For old support $Y$ with $M ≤2B$ , put preCost( $l$ ) = ballCost( $l$ + 6). Collapse gives

coverNumberB(halolB(parent(Y ))) ≤preCost(l) + 10 preCost(l)
L
coverNumberM(Y ). — Source page 93.

The extent bound from any chosen anchor is ( $l + 20)B$ (1 + coverNumber $M(Y$ )); these supply $v, v′, W$ . The connected traversal and coarse-collapse arguments above prove these covering and extent bounds; the displayed summation then supplies the hit budget. Inspection adds at most four fine-site steps and hence five parent steps, so increasing $l$ by five includes its enlarged support. The quantifier order is retained: for $A > 2ε$ , choose

L > 10 preCost(l)a
A −2ε — Source page 93.

before choosing $x$ . Then $a$ 10 preCost( $l)/L+2ε ≤A$ . For paths the coefficient is 10 preCost( $l)/(LB);$ with $L$ fixed it tends to zero as $B →∞$ , so any $σ > 2ε$ suffices eventually. The same delay ensures radiusB(betaF) $≤$ 2 radiusB( $x$ ). Fixed powers of these radii and the monitor are subpower in betaF, while $e−c till(x)$ and $e−c monitorz, c, z >$ 0, beat every inverse power. These justify strict-exponent absorption without changing $L$ after the delay. The lifted record retains parent fibers, connected covers, anchors, degrees and the order of parameter choices; literal readout preserves each finite-subset bound and its monotone passage to the countable budget.

Spanning-tree proof of the extent estimate. The centre-traversal construction used in the extent estimate is proved as follows. Realize the scale- $M$ cover of a support $X$ by a centre set $A$ with $|A|$ = cov $M(X$ ) whose union of $M$ -balls is connected by unit steps. Join two centres when their distance is at most 2 $M$ + 1. This centre graph is connected, since a lattice path through the union passes from one ball to another only through overlapping or adjacent balls. A finite connected graph has a spanning tree (add an edge from the reached vertices to a new vertex until all are reached). Traverse the tree out and back: each of its $|A| −$ 1 edges is replaced by a lattice path of length at most 2 $M$ + 1 and is used twice, giving a unit-step path through every centre of length $N ≤2(|A| −1)(2M + 1) ≤|A|(4M + 2).$

The empty cover has the constant path of length zero. Joining two sites of $X$ to covering centres by paths of length at most $M$ gives

diam(X) ≤|A|(4M + 2) + 2M, — Source page 94.

on the periodic lattice as well as on the integer cover. The estimate is a statement about the support column of the realized tuple, which the readout does not alter; the spanning tree is a witness that may be chosen freely (its choice is a presentation ambiguity with no effect on the bound), and no frame data enters.

12.48 Support tariffs and path-tail conversion

For nonnegative envelopes wi, supports Xi and integer extra charges ei, the sales condition includes
summability and, at every coarse site,
X — Source page 94.

i:x∈Xi
wie2εeiea coverNumberB(Xi) ≤γ. — Source page 94.

Disjoint alphabets add their budgets. An additional finite orientation index costs the sum of orientation budgets. Rotation or translation preserves the bound because its site map is bijective and does not increase covering number. For nonempty $Xi$ and $a, ε ≥$ 0, evaluation at one of its sites gives $wi ≤γ.$

Reserve an old anchored support tariffbefore conversion by setting $vi = wie2ε coverNumberM(Yi).$ Exactly, $vie(δ−2ε) coverNumberM(Yi)$ $= wieδ coverNumberM(Yi).$

Thus the original budget is unchanged with the reduced exponent. If Xi lies in the l-halo enlarge-
ment of the parent image of Yi, M ≤2B, and the geometric reserve inequality holds, apply coarse
conversion to vi. This pays the new 2ε tarifffor zi ≤Cwi(1 + coverNumberM(Yi))k. The payment
is the preceding explicit anchored-to-hit payment with child factor L4, extentCost(l, B), degree
k, old budget γ, new support exponent a and geometric preCost(l).
These retain the source’s
(NF9)–(NF10) dependencies.
For rooted bundles start instead from the true-length budget P
i wieδli ≤γ and set vi = wie2εli.
The width left for conversion is δ −2ε. The bundle estimate supplies sales for zroot,i ≤Cwi(1+li)k — Source page 94.

on the allowed enlarged footprint. The root ranges over every fine site, accounted for by originFee and $L4$ . An inspected order- $r$ atom footprint lies in the ( $l$ + 5)-halo of its actual path bundle; applying this transfer to its magnitude and exponential length budget gives the atom payment, retaining its degree- $k$ polynomial mark. For paired tails, the parent-distance inequality puts the $l$ enlargement of pairFoot in the ( $l$ + 2) enlargement of the two-path bundle. This yields the tail payment with $r = 2.$

The omitted-pair small factor is computed before conversion. For actual slot lengths $x, y, a$ nonzero omitted pair has $xd > R$ or $yd > R$ , hence ( $x + y)d ≥R$ . The pair-price bound and half the length reserve give

pairPrice e(σ/2)(x+y) ≤2e−σR/(2d)|ux|eσx|uy|eσy. — Source page 95.

The product of the original budgets, each at most Cpaths, bounds the half-width sum by
2e−σR/(2d)C2
paths. At R = till(x) this has every prescribed negative rate. Under 8ε < σ, the remain-
ing width σ/2 −2ε is positive and supplies the geometric reserve and polynomial mark. Footprint
payments multiply scalar envelopes and old budgets by fixed constants and a four-dimensional
extent factor. If these have rates u, v and extent rate zero, the payment has rate u + v; any strictly
larger exponent absorbs the fixed and subpower factors.
This recovers the inherited six-family
atom and tail rate deductions with the stated geometric conversion dependency. The lifted record
retains extra charges, true lengths, masks, orientations and width reserves.
Readout recovers
each tariffidentity and omitted-pair condition, so support conversion does not discard winding,
multiplicity or polynomial marks.

The preservation of these sales conditions can be expressed without interchanging an uncontrolled series. For a fixed site $x$ and finite label set $F$ , define the nonnegative tariffreader

Tx,F (w, X, e) =
X
wi exp(2εei + a coverNumberB(Xi)) .
i∈F
x∈Xi — Source page 95.

Its lifted version is this reader composed with $π$ , including the recorded site and label addresses. Since readout recovers each support, charge and envelope, every finite sum is identical to its physical reader, and composition with the section recovers $Tx,F$ . Taking the supremum over finite $F$ therefore preserves the full nonnegative sum, its bound and its summability. The reweighting identity above is an equality term by term before this supremum. Disjoint family tags preserve addition of budgets even when two letters have equal numerical coefficients or equal footprints. Thus neither a coincidence of values nor periodic identification merges separately charged labels.

12.49 Inherited envelopes and primitive dictionary

Fix halo $l$ and $ε >$ 0 with 4 $ε$ below the incoming support exponent and 8 $ε$ below both path widths. Choose one sufficiently large $L$ for regular and forced support conversion, retaining the prescribed support price and extra tariff. Use the original envelopes before constructing any new marking reserve. A regular occurrence is bounded by its supplied $bi$ on its healthy support and is zero otherwise; its anchor is unchanged. A forced coefficient uses $di$ without a health restriction. Choose a support inventory point as its anchor: the incoming hit bound implies an anchored bound there. The geometric factors have rate zero for fixed $L$ . The regular rate $−11/$ 10 allows any eventual price betaF $u$ with $u > −11/$ 10; a forced incoming rate $v −$ 1 allows betaF $v$ for any fixed $v.$

An order- $k$ atom multiplied by $|η| ≤D(x$ ) has healthy envelope

zsite,j = D(x) fiery(x)k magnitude(atomj). — Source page 95.

If $D$ and the total atom budget have rates $u, v$ , its preconversion rate is $u −k/2 + v$ . When this is below a target $r$ , choose an intermediate exponent, convert inspected supports and sum finite signed orientations; the strict margin absorbs that count. Retain true path length as extra charge. Omitted pairs pay every prescribed power after any fixed-rate scalar multiplier, using 8 $ε <$ width and the actual cutofftill( $x$ ). Sum plaquette types and orientations first at exponent $r −$ 1, then

absorb their fixed number into exponent $r$ . The detailed atom and tail support-conversion estimates are still the inputs to these two deductions.

The inherited logarithm has six disjoint families: regular occurrences, cubic atoms, quartic atoms, loop atoms, fine tails and coarse tails. Its phase is $θ = −$ orientation $β$ , with $|θ| ≤$ 2 betaF; the coarse tail uses $−θ$ . Cubic and quartic multipliers have rate one and hence rates $−1/$ 2 and $−$ 1 after the field factors. The loop multiplier has rate zero and its own incoming budget; both tails allow arbitrary negative powers. For $r > −1/$ 2, choose $−1/2 < s < r$ . All six prices are eventually at most betaF $s$ , weakening the regular bound to $s$ . Their union costs at most 6 betaF $s ≤$ betaF $r$ eventually. Forced letters keep their independent arbitrary-exponent price. This proves the inherited dictionary bound with the actual fine and coarse beta factors already included. Every footprint is nonempty: regular and forced supports retain their anchor or inventory, and atoms and tails retain their root. These facts support the individual-letter smallness argument used in the optional dictionary assembly.

Before taking connected components the non-warning primitive dictionary has three disjoint kinds: inherited logarithms, local Short corrections and Gaussian chain letters. The last kind has five families: the negative actual log-determinant series, the positive blank-reference log-determinant series, the mean-source series, the first linear mean correction, and the remaining quadratic chain correction. Their exponent coefficients are, in order,

−1
2 ArmA,
+1
2 ArmAblank,
+1
2 ArmS,
qfirst,
chain. — Source page 96.

An inherited logarithm is its actual old exponent on the lifted fine field, multiplied by enforcement of its own full footprint at 2 $a$ ; inherited forced activities use their actual $K$ coefficient with that enforcement. Profiles lie in [0 $,$ 1] and preserve bounds. The disjoint sum is farLog; each optional activity is $efarLog −$ 1. Supports are fixed before evaluation: enlarged old supports, Short sleeves, chain receipts and first-correction towns. Extra charges are respectively the old charges, zero for Short and first corrections, and series indices for other chains. Nonempty supports remain assigned even to zero coefficients. This keeps the support price independent of a field-dependent selection.

Let the chain families have common sales bound $N(x$ ) of rate $−1/$ 2. Apply the estimate independently to blank geometry for its positive reference determinant. Short has envelope $Z(x$ ) and sales bound $X(x$ ) of rate $−1/$ 2. Its caption density is at most $|$ Signed $|$ spots( $L$ ) + 2 itineraryDensity( $L).$ A halo- $l$ sleeve lies within radius ( $l + 1)B$ of its caption, so its price is

ballCost((l + 1)B) captionDensity Z(x)ev ballCost(l+1). — Source page 96.

The census has rate zero. Unite inherited, Short and chain log envelopes as $b0$ , adding sales prices. Choose $−1/2 < s < b <$ 0 and put

T(x) = betaF(x)s + X(x) + N(x). — Source page 96.

It has rate $s$ ; uniformly before choosing geometry or fields, eventually $T ≤$ 1 and 2 $T ≤$ betaF $b.$ Nonempty supports imply $b0(i) ≤T$ . Since $|t| ≤b0(i) ≤1, |et −1| ≤2b0(i$ ). Thus $yi = 2b0(i)$ bounds both the raw logarithm and Mayer activity, with total sales price at most betaF $b$ . Forced envelopes keep their separately prescribed betaF $u$ price. Dummy geometry reads common numerical envelopes only; the bound applies to every admitted geometry, forest and field. This completes the dictionary assembly conditional on its inherited, Short and chain estimates given above. In the source lift, family tags, signs, masks, fixed supports and original envelopes remain in the realized tuple. The inherited log and forced activities have their own preassigned envelopes from the incoming representation. Recovery is componentwise, preserving the disjoint sum, exponential activity and every threshold’s quantifier order.

12.50 Local driving and the Gaussian floor

Fix one real accepted sector and its forest; restrict $u$ to active bonds as $uhot. Define$

Io(u)2 =
X
dist(o,addr(e))≤3B
∥uhot(e)∥2,
Ro(u)2 =
X
dist(o,root(p))≤Rmac+2
∥motion(u)p∥2. — Source page 97.

Each summand norm is at most its corresponding square root, because all summands are nonnegative. Suppose a gauge $z$ makes background links in the 2 $B$ box of $o$ have chord at most $h ≥0.$ The accepted stencil estimate bounds the difference between flat and background-oriented rows on Ad $z uhot by 2CcG(B + 1)hIo(u$ ) for rods rooted within $Rmac$ + 2. The active-bond buffer places the whole selected rod stencil in the clean region. Adjoint action preserves color norm, and gauge covariance identifies the oriented row with its measured motion component. Thus the flat row is bounded by $Eo(u) = Ro(u) + 2CcG(B + 1)hIo(u).$ (NF1)

For the logarithmic color variation $w$ of Ad $z uhot$ , the collected variation vanishes on every forest trunk. The forest link estimate gives, with $o$ = addr( $e),$

∥uhot(e)∥≤K
X
X
α∈PlaqAxes
∥curl w(x, α)∥+ ∥bill w(o, eaxis)∥.
x∈box(Rmac) — Source page 97.

The low face row is √qdir curl w and the pin row is −
√
k bill w, with qdir, k > 0. Their norms are
bounded by Eo(u), and parent distance puts every face root within Rmac + 2. Consequently — Source page 97.

$∥uhot(e)∥≤UEo(u),$ $U = K|$ box( $Rmac)||$ PlaqAxes $|q−1/2$ dir $+ k−1/2.$ (NF2)

This uses no global pure-gauge representation. Clean-background flattening supplies $h ≤Y s(x)$ on actual clean boxes, with $s$ of rate $−1/$ 2. The chord chart writes each transformed link as an exponential of norm at most $s(x$ ) and uses its uniform chord bound; its radius is fixed first. Set $M(x) = 2UCcG(B(x) + 1)Y s(x$ ). Including the zero inactive case gives

∥uhot(e)∥≤URaddr(e)(u) + M(x)Iaddr(e)(u). — Source page 97.

The neighborhood census is swept( $L, d$ ) = ballCost( $d)L4|$ Axis $|$ . Square with ( $a + b)2 ≤2a2$ $+ 2b2$

and sum. When 2M(x)2 swept(L, 3B(x)) ≤1/2, absorption yields
X
e
∥uhot(e)∥2 ≤4U2 swept(L, Rmac + 2)
X
p
∥motion(u)p∥2.
(NF3) — Source page 97.

The census has rate zero and $M$ rate $−1/$ 2, giving this common tail. Constants are chosen after $L$ , before sector and variation. The real plate is the motion Gram matrix plus identity on inactive coordinates, with exact normalization

⟨v, formT form v⟩= scale2 X
p
∥motion(u)p∥2. — Source page 97.

Splitting active and inactive Euclidean squares, a common positive constant $a$ in (NF3) gives

⟨v, plate v⟩≥min(scale2 /a, 1)∥v∥2. — Source page 97.

The floor is therefore derived from the accepted branch’s local forest inequality and flattening. The lift retains the branch, active mask, gauge, rod stencils, motion rows and metric normalization. Readout recovers their exact restrictions; adjoint isometry and literal recovery transport each local bound and the finite census sum. The resulting lower bound applies to the pulled-back form on the section metric, consistently with the operator-versus-form distinction below.

More explicitly, write $H$ for the fixed active-coordinate projection of this accepted sector and $Q$ = plate. For the tangent readout $p$ and section $j$ with $pj = I$ , use $jHp$ for the lifted active-coordinate operation. Then $p(jHp) = Hp,$ $(jHp)j = jH.$

Thus applying the active mask before or after physical readout gives the same $uhot$ ; this statement uses the recorded sector mask. Put $c$ = min(scale $2 /a,$ 1). The recovered floor is exactly

(p∗Qp)(ζ, ζ) = Q(pζ, pζ) ≥c∥pζ∥2,
(p∗Qp)(jv, jv) = Q(v, v) ≥c∥v∥2. — Source page 98.

Here $p∗$ denotes pullback of the quadratic form and the norm on the section is the transported physical norm. The estimate consequently retains the source’s Gaussian constant on recovered physical directions; the pullback vanishes on ker $p$ . The forest-coordinate gap derived next has its own specified inclusion and is recovered on that inclusion.

12.51 Forest differentiation and vertical Hessian gap

For a logarithmic direction $w$ vanishing on forest trunks, put $Ut$ = represent( $tw$ ). This stays on the same forest slice for every real $t$ , and $U0$ = 1. Predictor frame weights are usable near zero: their minimum condition holds in an identity neighborhood and there are only finitely many coarse roots. The nonlinear shell inequality with identity comparison gives

chord( $Ut(e)) ≤K$ port $o(Ut$ ) + chord(prediction( $Ut)(o, eaxis)).$

Bound each port by the full macro-box plaquette-chord sum. Periodic plaquette repetitions only enlarge this nonnegative sum. For an identity-based group curve $Vt$ whose matrix derivative is colorMat( $v$ ), the chord normalization gives

|t|−1 chord(Vt) −→∥v∥. — Source page 98.

Indeed take the norm of the matrix derivative quotient and divide by the fixed color scale. Edge,
plaquette and actual predictor derivatives are the collected edge variation, linear curl and predictor
bill. Divide the shell bound by |t| and pass to zero through nonzero real t, including its finite port
sums. This proves the forest inequality used in (NF2), with the pin term equal to the derivative of
the chosen nonlinear predictor.
For X = box(Rmac), squaring that inequality and applying finite Cauchy–Schwarz gives
X
e
∥w(e)∥2 ≤β
X
b
∥curl w(b)∥2 + γ
X
b
∥bill w(b)∥2, — Source page 98.

where
β = 2K2L4| Axis | |X|2| PlaqAxes |,
γ = 2L4. — Source page 98.

For a fixed offset, the map from coarse root and plaquette type to fine plaquette is injective. Summation over roots therefore costs $|X|$ , even when distinct offsets give the same periodic site. Parent fibers have exactly $L4$ sites; these account for all factors without a period-dependent norm

equivalence. On a vertical forest coordinate v, the stretched collected field is zero on trunks and ve
on complementary bonds. The tested Hessian has lower bound a P
b ∥curl w(b)∥2+k P
b ∥bill w(b)∥2,
where a > 0 is the curvature-metric floor. The pin sign follows from vertical coarse mismatch equal
to minus the bill. Hence — Source page 99.

X
Hess(v, v) ≥min(a, k)
e/∈trunks
∥ve∥2.
(NF4)
1 + β + γ — Source page 99.

Restriction of the Hessian is composition of the joint analytic action with its linear vertical inclusion, so the second derivative is the joint second derivative on two included vertical vectors. These forest verticals remain distinct from the frame-fiber verticals of the source lift. The lifted realization retains the curve, predictor branch, trunk mask, linear inclusion and finite offset census. Readout recovers the curve before differentiation; its finite-dimensional linear differential therefore recovers curl, bill and the restricted Hessian, including the normalization and lower bound above.

12.52 Local stencil transport and constructed logarithmic backgrounds

The local stencil comparison in (NF1) has a separate elementary transport proof. A carrier from $x$ to $y$ has length at most 4 dist( $x, y$ ). On the declared surrounding radius- $d$ box, with link chords at most $h$ and the carrier path contained there, the source’s product-chord estimate gives carrier chord at most $dh$ . Replace exterior links by identity: locality preserves the carrier, so only local hypotheses enter. The adjoint inequality

∥Adg b −b∥≤2cG chord(g)∥b∥ — Source page 99.

bounds straightening of a local variation of size $M by 2cGdhM$ . A stencil of norm at most $C$ transfers this to 2 $CcGdhM$ . Explicitly, cut the variation difference to its lookup neighborhood; its sup norm obeys this local bound and the stencil reads the same difference. A signed coordinate change adds at most 2 $cGhM$ for a reversed link and zero for a forward link. Its transformed lookup lies in the original box enlarged by one. Hence

∥orientedRow(V, u) −orientedRow(1, u)∥≤2CcG(d + 1)hM. — Source page 99.

The box and variation bounds preceding (NF1) give that estimate; adjoint action and signed displacement preserve each individual variation norm. An impulse remains a single impulse after displacement and carrier conjugation, with unchanged color norm. Untransported impulse decay gives

ew dist(e,s focus(p))∥orientedRows,p(V, 1eb)∥≤2Cew(predReach +7)∥b∥. — Source page 99.

Here $w >$ 0 is the common stencil width; the extra one pays a possible signed-link endpoint shift. Thus the local comparison does not introduce a volume factor into the matrix estimate. The same transport calculation preserves the entrywise decay needed for the matrix upper price. For a clean radius-( $d$ +1) box, $d ≤32B(x$ ), axial flattening constructs $z$ with transformed chord on the radius- $d$ box at most

u(x) = 16(2(32B(x) + 1) + 1)5| PlaqAxes | 2 fiery(x). — Source page 99.

The temperature comparison bounds the clean plaquette threshold by 2 fiery( $x$ ). Once $u$ enters a fixed logarithm chart, a whole array $b$ agrees with these links on the box and satisfies $∥b∥≤Clogu(x).$ Radius factors have rate zero and fiery rate $−1/2, so H = Clogu$ has rate $−1/$ 2 and enters the chart eventually. This constructs the small-background function used in (NF3).

Source: Shadow Theory and the Wilson Yang–Mills Construction: A Source–Readout Reconstruction. Both papers belong to one Zenodo record. No external funding.