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

Jeremy Rodgers

doi:10.5281/zenodo.24173681

Source Lift / The complete argument

Marked integration and coupling reset · Part 5

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

For a marked team, the recorded distinct-key removal defines the same remaining body and paid in-
gredient product before and after readout. The identical pushforward calculation therefore recovers
its WM
Q
i∈p\keys(M) ti bound. The section recovers the original integrands. Finally the unchanged — Source page 138.

nonnegative majorant WMe
P
i ti bounds the molecule series on both sides, so its summable integral
norms justify recovery through termwise integration and the permitted address regroupings. — Source page 138.

13.26 Regular-gas factorization and exact forced restoration

Split primitive labels by empty versus nonempty flag inventory. For fixed compatible forced set $A,$ its enlarged-support union $YA$ excludes intersecting regular labels. If $Z0 = eL0$ is the full regular gas and $MA$ sums regular occurrence clusters meeting that exclusion region, then

Z0,A = eL0−MA = Z0e−MA. — Source page 138.

Cutting labels decreases absolute budgets; subtraction of absolutely convergent log sums leaves exactly missing clusters, with original occurrence indices and factorials. Multiply by the forced product and sum sets of the required inventory. Expand $e−MA$ by finite positive multiplicities of missing addresses, giving factors ( $−h)r/r$ !. Each missing cluster meets the forced region. Absolute restoration bounds, or the fixed-volume exponential of the absolute missing series, justify regrouping. Therefore grouping these terms does not change their sum. Decompose atom supports into overlap components. Each component with a missing cluster contains a forced atom; forced-only components are allowed. Components have disjoint supports and partition inventories. Conversely, union recovers the forced set and every cluster multiplicity uniquely. Factorials split over components since no atom address belongs to two components. This weight-preserving bijection factors the original gas as $Z0$ times the connected restored forced gas, retaining actual exclusions and exact inventories. This proves the regular-log factorization and forced regrouping identity once the primitive Mark expansion has been supplied. It uses exact inventories and actual overlap exclusions; the more permissive repetition count used for the absolute bound does not alter this equality.

For switched leaves the same argument holds for every 0 $≤x ≤$ 1 with fixed supports, masks and inventory tests and uniform paid bounds. Two fusions put components in the final dictionary, recording switch-independent attached logs. Unmarked scalar normalization then gives exactly

eEA(U)eL0(x,R)−L0(0,∅) GasA(K(x, R); U) = P(Buy(x, R))(U). — Source page 138.

The real gas/log theorem proves positivity of $Z0$ on every branch, justifying the regular-gas ratios. The primitive Mark expansion, dictionary fusion and unmarked scalar normalization proved in this chapter establish the scalar equality, with the stated exclusion restoration and component regrouping. The unmarked normalization identity says that the final coarse exponent, its fusion logarithms, and $L0(0, ∅$ ) reproduce the original scalar prefactor. The lift retains original occurrence addresses, positive multiplicities and inventory partitions. Readout preserves the component bijection and factorial products; retained absolute majorants permit regrouping, recovering the switched equality with the same scalar normalization.

For explicit recovery, record a restoration term as a forced label set A and a finite-support
positive multiplicity function r on the missing-cluster addresses. Its coefficient has the form
Y
!
Y
(−hq)r(q)

i∈A
Ki
r(q)!
.
q:r(q)>0 — Source page 138.

Realization retains $q$ as its full original occurrence address, including its own occurrence factorial. It also retains the additional multiplicity $r(q$ ), so these two factorials remain separate. Connected-component recovery partitions the atom addresses, with every repeated copy of a fixed address

belonging to its unique component. Thus the product of multiplicity factorials splits exactly over the recovered components. Composition with readout preserves the displayed coefficient and the exclusion predicate, and section recovery reconstructs ( $A, r$ ) by union. The absolute restoration majorant justifies summing this equality. Likewise the positive regular-gas reader gives

Z0(x, R)
Z0(0, ∅) = exp{L0(x, R) −L0(0, ∅)}, — Source page 139.

and its pullback preserves the denominator and the switch-independent fusion logs. This recovers the stated switched Mark identity with the original scalar normalization.

13.27 Paid cluster terms and positive multiplicities

A switched primitive activity is a finite ticket sum with dependence $x|D|1D⊆R$ . Each ticket’s scalar bound is the product of marked leaf prices and unselected primitive prices; its switching model is concentrated at $D.$ Summation gives the augmented-tariffproduct. Mixed-activity support and polynomial-moment weights enlarge this paid bound, with zero assigned on failed eligibility, uniformly in configurations. A regular occurrence cluster has paid bound

n
Y
|φ(z)|
j=1
uij.
n! — Source page 139.

For primitive budget $P0$ and tree ratio $ρ <$ 1, rooted estimates give sitewise support-weighted sum $P0/(1−ρ$ ). The omitted-cluster restoration and primitive-integral arguments in this chapter control their support-weighted sum. Subtracting its unmarked value doubles the bill; these differences are the switched regular log increments. Forced restoration multiplies forced primitives with negative log clusters with factors $hr/r$ ! for multiplicity $r ≥1.$ Paid products and powers retain these factorials. The paid product and power bounds therefore give exactly the corresponding product of envelope powers divided by factorials. Connectedness, presence of a forced atom and exact inventory restrictions are switch independent, so their indicators preserve paid bounds.

Restoration indices are finite distinct atom sets $D$ with positive multiplicities $ri$ , different from logarithm occurrence tuples:

Wh(D, r) =
Y
hri
i
ri! ,
N(D, r) =
X
i∈D
ri.
i∈D — Source page 139.

Absent atoms are omitted from $D$ . For 0 $≤hi ≤σ < 1,$

X
hri
i
ri! ≤
hi
1 −σ,
ri≥1 — Source page 139.

using ri! ≥1 and hri
i ≤hiσri−1. Products reduce the bound to distinct sets with inflated weights
hi/(1−σ). Sitewise smallness implies individual hi ≤σ by testing one site in its nonempty support.
The distinct connected-set tree estimate then bounds the sum of repeated connected terms whose
union contains a fixed site by η/(1−q′), provided the inflated vertex activities satisfy the readiness
condition with budget η ≤σ and tree ratio q′ < 1. Connected-cover subadditivity pays actual
union weight; each positive multiplicity increases available exponent sum. All successive sums are
nonnegative. For omitted restoration, forced atoms have budget c at exponent A + δ′, and regular-
log occurrence atoms have budget γ/(1 −q) there. Forgetting exclusions and permitting repeated
forced atoms enlarges the absolute majorant. For size tilt t ≥1 and forced tilt μ ≥0, multiply — Source page 139.

all weights by t and forced ones additionally by eμ. After multiplicity allowance their combined
budget is at most
teμc
1 −σ +
tγ
(1 −σ)(1 −q). — Source page 140.

The lift retains the distinction between occurrence tuples and distinct addresses with positive multiplicities, together with their switching models and factorials. Readout preserves each product and switch-independent restriction; nonnegative majorants justify the same summations and tilts.

13.28 Forced restoration size tail

Require the tilted combined budget above to be at most $η ≤σ$ , with 0 $≤σ < 1, δ′ >$ 0 and $4(2B + 1)4η/δ′ ≤q′ <$ 1. The repeated connected estimate applies to augmented activities. A term of total size $N ≥m$ gains $tN ≥tm$ ; a forced atom guarantees at least $eμ$ even when its multiplicity exceeds one. Dividing gives

X
eAcB(∪i∈DXi)Wh(D, r) ≤t−me−μ
η
1 −q′ . — Source page 140.

$(D,r$ ) connected $, N(D,r)≥m$ at least one forced atom $, x∈∪i∈DXi$

Applications take t = 2, σ = 1/2, q′ = 1/2; at m = 0 the price is 2e−μη. The compulsory atom is
charged at least once, independently of multiplicity. The distinct connected-set readiness estimate
is the remaining rooted-tree input. The lift retains size and forced-atom counts separately; recovery
preserves both tilt inequalities and hence this support-weighted tail on the same restored address
family.
The tilt reader has an exact multiplicity identity.
Put Nf(D, r) = P
i∈D forced ri and ehi =
teμ1i forcedhi. Then
Weh(D, r) = tN(D,r)eμNf(D,r)Wh(D, r). — Source page 140.

Realization retains $D$ , every $ri$ and each forced/regular tag, so readout preserves both counts and this equality, including all factorials. A permitted term in the size tail has $N ≥m$ and $Nf ≥1,$ which yields the same $t−me−μ$ gain after undoing the tilt. The union-support reader depends on $D$ and is unchanged by multiplicity; positive $ri$ preserve the vertex support-weight majorant. Taking finite nonnegative subsums and then their supremum therefore transports the readiness bound and the entire restoration tail. Section recovery returns the original signed term, its paid envelope and its switch-independent eligibility predicate.

13.29 Real switching and nilpotent substitution

For finite switches $I$ , a modeled real family has

f(x, R) =
X
A⊆R
x|A|gx(A),
0 ≤x ≤1,
gx(A) = x−|A| X
R⊆A
(−1)|A|−|R|f(x, R)
(x > 0). — Source page 140.

The inverse follows either by finite Möbius inversion or subset induction. Assume gx(A) →g0(A)
as x ↓0; uniqueness for x > 0 makes Jf(A) = g0(A) model independent.
A paid model has
|gx(A)| ≤b(A) and P
A b(A) ≤c uniformly in configurations, so |f(x, R)| ≤c. For square-zero
marker elements zi, define
Sz(f) =
X
A⊆I
Jf(A)
Y
i∈A
zi. — Source page 140.

Addition is immediate. Product arrays obey (g ∗x h)(A) = P
B∪C=A x|B∩C|g(B)h(C). At zero
only disjoint pairs remain; overlapping substituted products vanish by z2
i = 0. Hence substitu-
tion is multiplicative, maps constants to scalars and multiplies paid bounds. For countable sums
with summable paid bounds, coefficientwise dominated convergence and the finite subset sum give
Sz(P
n fn) = P
n Sz(fn). Apply this to exponential terms bounded by cn/n! to obtain — Source page 141.

For zero scalar output this is the finite nilpotent exponential, requiring no density division. If
∥zi∥w ≤1, bound each output-mode coefficient by the sum over switch subsets D of b(D) times the
absolute corresponding coefficient of Q
i∈D zi. Submultiplicativity bounds total weighted price by
P
D b(D) ≤c. For zero-scalar zi, the scalar output is f(0, ∅), and singleton coefficients arise only
from size-one switch subsets. Switch independence at i makes all Möbius coefficients containing
i zero.
Exterior silence therefore restricts nonzero coefficients to leaves whose required parent
supports lie in the output support, placing their scoped marker pins there. Finite substitution

preserves gauge invariance, locality and measurability. Vanishing offa healthy region or inventory event passes to coefficients and limits, proving correction permissions. These statements supply the support and permission properties of the marked corrections. The lift retains real switch values, paid arrays and finite inversion as recoverable data. Readout commutes with the finite Möbius transform and, by the retained majorants, its coefficient limits and summable series; thus it recovers this algebra map and its local price bounds exactly.

To express switching recovery directly, let $fZ(x, R) = f(x, R) ◦π$ . For $x >$ 0 its finite inversion gives $gZ$ $x$ $(A) = gx(A) ◦π,$ $JfZ(A) = Jf(A) ◦π.$

The second equality follows by taking the coefficient limit at each realized input; the paid majorants are unchanged. If the marker readers are also realized coefficientwise, the algebra-map property gives $SL z(fZ) =$ L $(Sz(f)).$

Section composition recovers every coefficient, including the scalar and singleton readings. For a
countable family with paid bills cn, the coefficient tail before taking x ↓0 is bounded by P
n>N cn,
independently of x and configuration.
This same tail bound holds after readout and tends to
zero, justifying the recovered sum and exponential laws.
Finally the switch-independence and
permission identities are finite equalities of these arrays, so their vanishing coefficients and scoped
pins are retained through the limit and section. — Source page 141.

13.30 Reading the marked Mark-step output

The retained reserve is built for the actual finite leaf inventory. Each leaf has its fee and normalized square-zero $zi$ , with parent support containing all scoped pins and $∥zi∥w ≤$ 1. The marked activity expansion produces a family ( $K(t, A$ )) indexed by subsets of the finite leaf inventory. Mixed jets evaluated on products of $zi$ retain only distinct-leaf derivatives. Triangle inequality and submultiplicativity turn mixed-jet bounds into local polynomial bounds. Group forced corrections by coarse support-and-inventory barcode; their scalar mode stays the background profile. Only single-leaf jets contribute to singleton marker $v$ , and

where $Dj$ is that same leaf’s transported hinge. Multiplication by its normalized monomial cancels the fee, including zero charges. The sound-door estimate bounds each hinge’s weighted forced

mass by a fixed multiple of leaf load. Absolute summability therefore permits finite-leaf summation, barcode grouping and coefficient extraction, recovering exactly the sum of quoted singleton transported profiles. This proves the singleton-profile identity used in the restoration step.

The coarse evaluation identity is Mark push of priced tickets equal to exponential output times the coarse field polynomial, with the fixed scalar displacement factored out. The exponential has zero scalar mode. Scoped mixed-jet bounds give the reserve’s served and field envelopes; hinge additivity preserves summably grouped leaves. Explicitly at $U$ , write $a = eEA(U)$ , regular increments $L(x, R$ ) and forced gas $G(x, R$ ). The established real identity is

aeL(x,R)G(x, R) = P(Buy(x, R))(U),
0 ≤x ≤1. — Source page 142.

Apply $Sz$ , commuting through the paid summable regular and gas series and using product and exponential laws. Since the regular increment vanishes at the unmarked switch, its substituted scalar is zero. The left side becomes

X
!
a Exp
S
HS(U)
Field(K(U), FA(U)). — Source page 142.

On the right each finite ticket has switching jet concentrated at its leaf set D, so substitution
gives its scalar coefficient times Q
i∈D zi. These coefficients are Mark-integrable; finite linearity
commutes substitution and push. This is exactly the priced polynomial inventory push, proving the
marked identity with all infinite sums controlled by paid bounds. The switched marker expansion,
primitive integral and connected-cluster arguments in this chapter construct the marked activity
jets and prove their exact Mark identity. The lift retains the finite inventory, barcode fibers, hinges
and actual Mark kernel. Recovery commutes with the finite coefficient substitution and probability
integral, while retained paid majorants justify its series passages and singleton extraction.
For the Mark operator, denote fine and coarse readouts by πf and πc. The realized kernel c
Kz
retains the actual kernel Kπc(z) through — Source page 142.

For an integrable ticket coefficient $bD$ , this gives

bP(bD ◦πf)(z) = P(bD)(πc(z)). — Source page 142.

Apply finite linearity to all tickets and then the switching recovery identity already established. One obtains coefficientwise

bP Lf(Sz(Buy)) = Lc
�P(Sz(Buy))
. — Source page 142.

Here $Sz$ uses the fixed normalized marker monomials; the argument $z$ of the realized kernel is a separate coarse source point. Section recovery returns the original marked identity. The fee– monomial product and its zero-charge branch are recovered term by term, and the sound-door majorant permits the same barcode grouping of singleton hinges. Thus this kernel identity preserves both exact evaluation and the served and field bounds on the actual finite inventory.

13.31 Shipping price and the numerical retained step

For fine leaf cover cost $v = cB1(S$ ), the birth-support bound is

(q + 1)cB∗(birth(S)) ≤(q + 1)Cfoot + 5
4εv. — Source page 142.

The tariffspends another 2 $εv$ . With $s1 ≥$ 100 the selected parameters obey 13 $ε/4 ≤s1 −4, so$ the fine weight pays both, leaving $e(q+1)Cfoot$ . Charge leaves to scoped pins and use parent-pin occupancy $M$ . For inventory input budget $v′$ , shipping requires

M siteFactor(B∗)e(q+1)Cfootv′ ≤ribbon . — Source page 143.

Grouping equal labels sums nonnegative prices and preserves this bound. Adding the unmarked reserve makes the marked tariffadmissible, with birth geometry and scalar activity estimates as the retained inputs.

For fine and coarse exponents $s1, s0 ≥$ 100, widths $B1, B0$ and occupancy $M$ , choose 0 $< d ≤1$ and shrink marker weights $Iv$ to $dIv$ . Higher corrections have degree at least two, so their weights decrease by at least a factor $d$ , reducing unit price to at most $d$ . At exponent $s1 −$ 4 the finite inventory costs $T + eT T$ , where

T = M siteFactor(B1)d + 4d. — Source page 143.

Buy a reserve covering this input with output field budget u. Response prices satisfy 2Ivav ≤1,
Ivbv ≤1, hence after shrinking 2dIvav ≤d, dIvbv ≤d, as required. Set Q = u + M siteFactor(B0)d
and require
(2B0 + 1)4Q ≤1
2,
eQ −1 ≤1. — Source page 143.

These pay closure exponent loss and final unit correction price. Matching coarse weights $Jv$ costs a further half exponent unit with multiplier at most one. The reserve supplies exponential exponent $s0 + 1/$ 2 and field exponent $s0$ + 1; both half-unit expenditures leave at least $s0$ . Singleton feed and restoration identify the coarse primary and forced fields, proving the retained unit-price served conclusion with the constructed inventory and reserve. The lift retains shrunk weights, occupancy counts and each exponent expenditure. Recovery preserves these numerical inequalities and the singleton identity, so the retained step uses the same purchased reserve and output budget.

Weight recovery is explicit on every allowed marker mode $A:$

tdI(A) = d|A|tI(A),
|A| ≥2 =⇒d|A| ≤d2 ≤d. — Source page 143.

The source lift keeps the mode, pins and chosen $d$ in its realized data, so coefficientwise readout preserves this rescaling and the reduced correction price. Likewise it retains the fine support $S,$ its birth support and both cover readers. Adding the two shipping expenditures uses exactly $5ε/4 + 2ε = 13ε/$ 4 per fine cover unit; the recovered inequality therefore leaves the same fixed footprint multiplier. Pin occupancy then uses the same parent addresses and bound $M$ , preserving the ribbon demand after any equal-label grouping. Finally $T$ and $Q$ are scalar readers of these retained choices. Their readiness tests and the two coarse half-unit expenditures are recovered before applying the marked kernel. Section recovery consequently returns the same retained step, with its original unit output prices and prescribed singleton functions.

13.32 Separation pays for complex inflation

Let $B ≥$ 8, and let marker set $A$ have at least two pins separated by at least $B2$ in torus maximum distance. Any support $S$ containing them obeys $B|A| ≤64cB(S$ ). Indeed a minimizing cover with $D = cB(S$ ) centers has a unit-step center traversal of length $N ≤D(4B$ + 2). Choose for each pin a center visit within $B$ . By the distance triangle inequality and the unit-step walk bound, distinct chosen times are separated by at least $g = B2 −2B >$ 0; their integer quotients by $g$ are distinct, $so (|A| −1)g ≤N$ . Therefore

|A|B2 ≤4(|A| −1)g ≤4N ≤4D(4B + 2) ≤64DB. — Source page 143.

Divide by $B$ . The traversal need only visit covering centers. A nonzero higher correction has a nonzero mode with at least two scoped pins in its support, hence $cB(S) ≥B/$ 64. Put $SB = eB/64.$ For complex $|z| ≤SB$ , multiply each nonzero correction envelope by $|z|$ , setting identically zero corrections’ envelopes to zero. Then

|z|aie(e−2)cB(Si) ≤aie(e−1)cB(Si) ≤aieecB(Si). — Source page 144.

Thus complex inflation preserves its price budget at a support-exponent loss of two. Scalar multiplication preserves locality, gauge invariance, measurability, scope and permissions, and introduces no scalar or singleton mode. This proves precisely the inflation estimate needed for holomorphy and retained transport in the omission comparison. The primary and forced singleton insertions retain their values. The large analytic disk in the omission argument is a geometric consequence of separated markers. The lift retains separated pins, minimizing covers and center visits. Readout preserves the distance and traversal estimates, so the same complex disk and exponent payment apply to the recovered correction family and its omission comparison.

For the inflation map on a higher correction family $Y = (Yi$ ), write $IλY = (λYi$ ) with $|λ| ≤SB.$ The coefficientwise lift satisfies

L (IλY ) = Iλ(L Y ),
(L (IλY )) ◦σ = IλY. — Source page 144.

Pin addresses, their separation and support-cover witnesses are unchanged by either operation, so the displayed envelope inequality holds with the same $SB$ uniformly on the whole disk. Higher-mode projection commutes with this scalar multiplication. In particular scalar and singleton coefficients remain zero for every $λ$ , and the prescribed primary and forced singleton insertions are recovered unchanged. Each individual correction depends linearly, hence holomorphically, on $λ$ ; the uniform paid bound is preserved by readout for its subsequent summed omission argument. This identifies the inflation domain and its price payment as part of the recovered family, including identically zero corrections with zero envelope.

13.33 Positive denominator from a Haar shelter

Let normalized Haar measure be $μG$ , ambient real matrix dimension $dM$ and chord normalization chord( $g) = |g −I|/γ, γ >$ 0. For 0 $< r ≤1,$

μG{g : chord(g) ≤r} ≥(r/4)dM . — Source page 144.

Indeed mutually $rγ$ -separated group matrices have disjoint ambient balls of radius $rγ/$ 2 inside radius 2 $γ.$ Volume comparison gives $q(r/4)dM ≤1.$ A maximal finite collection exists by this bound; its closed $rγ$ balls cover the group. Their group intersections are left translates of the chord shell, by unitary norm invariance. Haar invariance and union bound give 1 $≤qμG$ (shell( $r$ )). For the four-dimensional torus shelter with every positive link of chord at most $s/4, 0 < s ≤$ 1, product Haar measure is at least $(s/16)dM|E|,$ $|E| = 4|$ Λ $|.$

Product chord subadditivity and inversion invariance make every plaquette chord at most $s$ . For $s ≤θ$ there are no flags and the activity gas is one. Signed translations preserve the shelter. For path bound $C$ , loop cap Λ $0$ , regular allowance $δ$ , forced allowance $κ$ , coupling $β ≥$ 0 and direct clip coefficient $Cdir$ , put

l= β(Cθ3 + Cθ4) + Λ0θ2,
e0 = δ + l,
b(s) = |P|(Cdirs2/2 + C2s2/2). — Source page 144.

Healthy degree- $k$ atoms cost coefficient magnitude times $θk$ ; transport preserves curvature norm. Absolute atom sums bound cubic, quartic and loop remainders per site by $Cθ3, Cθ4,$ Λ $0θ2$ . The quadratic base is nonnegative, giving channel action at least $−|$ Λ $|l$ . On the shelter, path curvature legs are at most $s$ , the linear path field at most $Cs$ , and the base per site at most $b(s$ ). Channel action is at most $|$ Λ $|(l+ βb(s$ )). Charging regular occurrences to their roots bounds the regular exponent by $|$ Λ $|δ$ . For vacuum scalar $v0,$

ER(U) ≤v0 + |Λ|e0,
ER(U) ≥v0 −|Λ|(e0 + βb(s))
on the shelter. — Source page 145.

The density is nonnegative from bare Wilson positivity and positive Mark kernels by downward induction. Integrating over the shelter, where gas is one, gives

ZR ≥exp{v0 −|Λ|(e0 + βb(s)) −4|Λ|dM log(16/s)} > 0. — Source page 145.

The lift retains normalized product Haar, shelter, thresholds and positive kernels. Its exact mea-
sure pushforward recovers shelter mass and exponent bounds, and therefore this strictly positive
denominator with the same vacuum scalar and constants.
Write Hs for this physical shelter and let μZ be the realized prior with π∗μZ = μ. Its exact
lifted shelter is bHs = π−1(Hs), so — Source page 145.

μZ( bHs) = μ(Hs) ≥(s/16)4dM|Λ|. — Source page 145.

This is a condition on the recovered physical links, retaining the original plaquette threshold. Put
RZ = R ◦π. The same prior pushforward and nonnegativity give
Z
RZ dμZ = ZR ≥
Z
b
Hs
RZ dμZ ≥ev0−|Λ|(e0+βb(s))(s/16)4dM|Λ|. — Source page 145.

On that preimage the recovered gas is exactly one. The section recovers both the shelter and its exponent reader. Thus the denominator estimate is an identity of the normalized integration measures followed by the original shelter lower bound, with unchanged link census, vacuum scalar and ambient-dimension cost.

13.34 Normalized insertion bill and stagewise treasury

Combining the denominator with the insertion norm majorant gives

tw(A)|
R [X(U)]A dμ(U)|
ZR
≤exp {|Λ|[2e0 + κ + βb(s) + 4dM log(16/s)] + |V |(x + y + 2f)} . — Source page 145.

The vacuum scalar v0 cancels. The numerator retains its full law and integration domain. For
selected history take s = bgk = bβ−1/2
k
and β ≤2 bβk. Monotonicity of the nonnegative loop and base
prices gives the explicit volume coefficient — Source page 145.

τk = 2{δk + lk(2 bβk)} + κk + 2 bβkbk(bgk) + 4dM log(16/bgk). — Source page 145.

This is the extensive root estimate for omission. Its extensivity is explicit; no volume factor is silently absorbed in an error tending to zero. For $P(s$ ) = exp ${$ (log log $s)2}$ and Θ( $s) = s−1/2P(s),$ each fixed $b$ and $p >$ 0 satisfy eventually $P(s)b ≤sp$ , since $b$ (log log $s)2/$ log $s →$ 0. Hence eventually

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