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 4

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

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thick family A, retain the support exclusion indicator χA(p), so the restricted entries are χA(p) ̄ap.
Readout preserves this indicator and the omitted connected labels, giving the same restoration
factor
! — Source page 130.

−
X
q
missingA(q, ·) ◦π
C ◦π exp
. — Source page 130.

Absolute majorants justify the displayed sum. The thick/thin decomposition and its inverse union preserve every nonzero family’s labels and grounds, so its density identity is recovered term by term. Finally each literal score coefficient is composed with readout together with its viewed full home and mask. Section recovery therefore returns the same three path bills and channel decomposition with their original seat coefficients.

13.22 Detector-to-warning implication with separate error bills

A radius- $B$ quiet watch at $o$ bounds fluctuation coordinates by $|zi| ≤ξt/g$ . For field $μ + gz$ , color dimension $d$ and $∥μe∥≤Mt$ , coordinate absolute sums give

∥(μ + gz)e −μe∥≤dξt,
∥(μ + gz)e∥≤(M + dξ)t. — Source page 130.

A 2B watch supplies B watches at nearby sites. Choose the stationary guide bound A first, then
τ = √cdirect/32, then L with A/L2 ≤τ, then W > 2(A + τ)/√cpenalty. These ensure — Source page 130.

2
√cdirect
(A/L2 + τ) < 1
4,
2
√cpenalty
(A + τ) < W. — Source page 130.

Keep three discrepancy contributions separate: fluctuation response, protected mean and nonlinear path-approximation remainder. Choose $ξ >$ 0 with $dξ ≤$ 1 and response at most $τt/$ 3, by the source’s linear coefficient bound. Mean and approximation errors have reference-coupling order $−1;$ each is eventually at most $τt/$ 3 since $t$ is at least inverse square-root coupling. Total discrepancy is at most $τt$ . Outside mandatory seeds, the same-link clean-patch and grafted-field estimates put site and pin tags on their plateaus and identify rows with actual fine and mismatch curvatures. Add guide bills $At/L2$ and $At$ . Curvature comparison gives fine chord below $t/$ 4 and mismatch below $Wt$ by the displayed quotas. Fine and noise cutoffs and both tags equal one. A flag would require a nearby chord greater than $t$ , contradicting $t/$ 4; the forced-charge detector equals one too. Thus each nonmandatory detector, a bounded auxiliary cutofftimes a raw factor minus one, vanishes under the 2 $B$ watch. Mandatory detectors are zero by definition; active-link complements vanish on their plateau using the mean bound and $dξ ≤$ 1. This is the required warning implication, dependent on protected means, physical path accuracy, clean-patch tags and curvature comparison. Those local geometric estimates remain dependencies of this paragraph; the Gaussian probability argument does not supply them. The lift retains those separate local error records and plateau predicates; readout preserves the implication with the same witness neighborhoods and ordered parameter choices.

13.23 Joint Gaussian warning estimate

Let centered finite-dimensional Gaussian $z$ have covariance $A ≥$ 0 with $vT Av ≤M∥v∥2.$ For coordinate set $D$ and signs $εi$ , choose $vi = εi$ on $D$ , zero elsewhere. Then

Eesv·z = es2vT Av/2 ≤eMs2|D|/2. — Source page 130.

Markov’s inequality for $s ≥$ 0 on each signed orthant, followed by the 2 $|D|$ sign union, gives

P(|zi| ≥t ∀i ∈D) ≤2|D|e(−st+Ms2/2)|D|. — Source page 131.

This includes degenerate covariance and empty $D$ . Suppose at most $m$ warning addresses and $p$ coordinate addresses lie over one site. Each warning has a coordinate witness within $R$ of magnitude at least $t$ . For finite $W$ there are at most ( $VRp)|W|$ assignments, where $VR = (2R + 1)4$ . Their distinct-coordinate image $D$ satisfies $|D| ≤|W| ≤VRm|D|$ , since one coordinate witnesses at most $VRm$ addresses. Choose $H ≥$ 0 with $M/2 + HVRm ≤t$ and use $s$ = 1. The exponential is at most $e−H|W|$ and the sign factor at most 2 $|W|$ . Thus

P(every warning in W has a witness) ≤(2VRpe−H)|W|. — Source page 131.

Shared witnesses and overlapping neighborhoods are counted explicitly. For the actual family $R = 2B$ , its census is bounded by

m = L4|PlaquetteAxes| + (|Axis| + |Axis| + 1 + L4) + itineraryDensity(L). — Source page 131.

Coordinate count p is the same block’s tubular census, and M is its actual covariance bound. Its
address census is the sum of the fine plaquette types, two coarse bond types, coarse and fine site
types, and fine-link itinerary types. The warning threshold is ξtfield/g, g = β−1/2, with tfield/g ≥
P/2 for fine monitor P.
Put ε = ξ/4 and H = ε
√
P.
Eventually V2Bm ≤
√
P and M/2 ≤
εP, because the monitor dominates the fixed covariance cost and the polynomial width cost, so
M/2 + HV2Bm ≤2εP ≤ξtfield/g. The warning allowance is therefore — Source page 131.

E = 2V2Bpe−ε
√
P ,
P(all witnesses) ≤E|W|. — Source page 131.

$Monitor growth makes E smaller than every prescribed reference-coupling power after fixed block and stage choices. This is the rapidly decaying warning allowance used by the activity prices. A nonzero detector violates its quiet watch and supplies precisely such a coordinate within 2B. Thus simultaneous nonzero detector products lie in this joint event. The preceding plateau calculation proves the implication: under the quiet watch the curvature is at most one quarter of its threshold, the prediction mismatch is within its plateau, no nearby flag can occur, and every nonmandatory detector is a cutofftimes a vanishing plateau complement, while a mandatory detector is identically zero. Taking the contrapositive gives precisely the witness used in the probability bound. The lift retains covariance, address censuses and witness assignment maps; readout preserves their multiplicity bounds and the Gaussian moment-generating identity. The same joint estimate is recovered without selecting a new Gaussian for individual molecules.$

Searchable transcription

Monitor growth makes E smaller than every prescribed reference-coupling power after fixed block
and stage choices. This is the rapidly decaying warning allowance used by the activity prices. A
nonzero detector violates its quiet watch and supplies precisely such a coordinate within 2B. Thus
simultaneous nonzero detector products lie in this joint event. The preceding plateau calculation
proves the implication: under the quiet watch the curvature is at most one quarter of its threshold,
the prediction mismatch is within its plateau, no nearby flag can occur, and every nonmandatory
detector is a cutofftimes a vanishing plateau complement, while a mandatory detector is identically
zero. Taking the contrapositive gives precisely the witness used in the probability bound. The
lift retains covariance, address censuses and witness assignment maps; readout preserves their
multiplicity bounds and the Gaussian moment-generating identity.
The same joint estimate is
recovered without selecting a new Gaussian for individual molecules.

The Gaussian event reader uses the physical noise projection RX for the fixed local data X,
with (RX)∗bγX = γX. For a finite warning family W, let EW be the event that every warning has
a coordinate witness in its recorded radius-2B neighborhood. The local plateau implication, with
its three separate error bills, gives




Y

⊆EW ,
bγX(R−1
X EW ) = γX(EW ) ≤E|W|.
j∈W
dj ̸= 0
 — Source page 131.

Here the realized detectors are the original coefficient readers composed with $RX$ , with their other local inputs retained. A witness assignment keeps its warning-to-coordinate map and distinct-coordinate image; readout preserves both cardinality inequalities used above. For the exponential moment calculation the test vector $v$ remains a physical coordinate vector, so its covariance is the same $A$ and its upper bound the same $M$ . The section recovers the original event and moment identity. Thus the allowance transports through exact Gaussian pushforward with shared witnesses counted, on the original fixed choice of local data.

13.24 The primitive integrand and its local extension

Here are the finite objects to which the integration estimates apply. Fix a coarse-to-fine step, its fine
tree F, a seed inventory, a coarse configuration V , local discrete choices k, and the off-tree group
variables Y . The active off-tree links are determined by the tree and seed inventory. For a color
array u, the fine configuration has link exp(ue)(replica V )e on an active link, the value supplied by
Y on an inactive off-tree link, and the fixed tree value on a tree link. The logarithmic signal equals
u on active links, zero on tree links, and the chosen cutofflogarithm of Ue(replica V )−1
on inactive
e
off-tree links. Thus the group configuration and the logarithmic signal are different objects. Their
identification by exponentiation is asserted only on the interior domain where the cutofflogarithm
agrees with the genuine chart inverse.

A letter is of one of four disjoint types: a block packet $A$ , an old forced index $q$ , a regular logarithmic index $λ$ , or a warning index $w$ . A molecule $p$ is a finite set of letters; repeated copies of a letter are not allowed. Its forced inventory is the set of its forced indices. Its seed inventory is the union of the seeds of its packet letters. The ground $H(p$ ) is the union of the enlarged read footprints of all its letters. In particular it contains every seed and every forced read. Admission requires all of the following: the overlap graph of letter footprints is connected, the packet letters form the permitted layout of their combined seed inventory, the forced inventory satisfies its sealing and incidence conditions with those seeds, and

flags(V ) ∩H(p) = flags(seed inventory of p).
(PRI1) — Source page 132.

We spell out the discrete tests in this assertion. A seed is a member of the disjoint union of fine plaquettes, coarse bonds and coarse sites. These types remain distinct even when their spatial positions coincide. A store $A$ is the triple of finite sets ( $Aplaq, Abond, Aflag$ ); its number of seeds is the sum of their cardinalities. The read set of a plaquette seed is its coarse projected start set, that of a coarse-bond seed its two pin endpoints, and that of a site seed its singleton. The place of $A$ is the union of these read sets; its dome $A[r]$ is their coarse radius- $r$ halo. Give the seed set the graph in which two seeds are adjacent when their radius- $lB$ halos intersect. A connected packet means a nonempty connected set in this graph.

A permitted layout $L$ for $A$ at parameters $l, B$ is a finite set of stores with the following three properties: each $A′ ∈L$ is a connected packet $,$

(A′)[lB] ∩(A′′)[lB] = ∅
(A′ ̸= A′′),
seeds(A) =
[
A′∈L
seeds(A′).
(PRI6) — Source page 132.

It exists by taking the connected components of the seed graph and converting each component back to its store. Conversely any permitted layout is this one: an edge cannot connect distinct members because their halos are disjoint, and each member is connected. Thus the packet layout is determined by the combined seed inventory. It does not introduce a multiplicity of choices or a choice of a graph witness.

For a finite set $I$ of old forced indices write $Pi$ for its actual fine support and $Ji ⊂Pi$ for its declared inventory. Sealing means $Pi ∩Pj = ∅$ for distinct $i, j ∈I$ . Incidence means

∀i ∈I, ∀y ∈Ji,
∃p ∈Aplaq with y ∈touched(p).
(PRI7) — Source page 132.

Here touched( $p$ ) is the fine set inspected for that plaquette seed in the original sector partition. This condition uses the inventory, whereas sealing uses the full supports. Their roles cannot be interchanged.

The selector used in the packet integrand is defined at each fine site $y$ by

(
0,
y ∈flagsθ(U) \ S
i∈I Pi,
1,
otherwise.
OI,Z(U) =
Y
y∈Z
cI(U; y).
(PRI8)
cI(U; y) = — Source page 133.

For the packet, $Z$ consists of fine sites whose parents are inside its 300 $B$ dome in the prescribed sense. For the full gas take all fine sites. If a product of old forced factors is nonzero, their individual permission identities give $Ji = Pi ∩$ flags $θ(U$ ). On a sealed family, the full selector is one exactly when the union of those inventories is all flags. Otherwise it is zero. This proves the exact rewriting of the old gas occurrence as the sealed indicator times (PRI8) times its old forced product. If the product is zero, both expressions are zero without using a permission identity for a vanishing factor.

The footprint of a packet letter is its enlarged dome; that of every other letter is its prescribed inspected, projected halo. Define two letters to be adjacent when these coarse footprints intersect. The molecule ground is their union, as in (PRI1). Its admission is now completely specified by nonempty connectedness in this graph, the layout (PRI6) of its packet letters, sealing and incidence (PRI7) of its forced indices, and (PRI1). The empty seed inventory is allowed when there are no packet letters; its permitted layout is empty.

Compatible molecules have disjoint grounds. For such a finite family, admission of its union is equivalent to admission separately in every member, together with the global flag coverage. To prove the nontrivial localization of incidence, suppose a forced inventory site in one molecule is touched by a seed attributed to another. The parent of that site lies in the forced footprint of the first molecule and in the seed footprint of the second. Disjointness of grounds forces the two molecules to coincide. The same parent argument localizes the sealing test. Layout restrictions localize because the enlarged packet domes are contained in their grounds; joining the local layouts gives a global layout. Equation (PRI1) localizes by intersecting the global flag set with disjoint grounds. Finally every finite set of letters has a unique partition into connected components of its footprint graph. These statements prove that expansion into finite letter sets and regrouping into compatible connected molecules are inverse operations, with multiplicity one. Absolute summability proved below permits this finite combinatorial identity to pass to the countable alphabets. All these admission tests remain discrete masks and are not differentiated.

For the common lifted fluctuation $u = μ + β−1/2z$ , the four vertex values are, respectively,

vA = 1{packet connected}1{packet eligible}SA(u),
vq = Kq(U(u)) E2a,q(u),
vλ = eLλ(u,z) −1,
vw = Dw(u).
(PRI2) — Source page 133.

Here $E2a,q$ is the product of fluctuation cutoffs on the entire forced read footprint. Packet eligibility requires its place to lie in the chosen seed place and its declared flags to agree with the actual coarse flags throughout its 300 $B$ neighborhood. To define the profile factors, let $χ$ be the fixed smooth cutoffand put $hs(r) = χ((r/s)2$ ). For a tested fine plaquette use $ht/4$ (toll( $U, p)) if p /∈Aplaq,$ and its complement if $p ∈Aplaq$ . For a tested coarse bond use $hWt$ (wire(prediction( $U, k), V ; b)) if$ $b /∈Abond$ , and its complement otherwise. Multiply these factors once over every plaquette whose projected start set meets $A[300B]$ and every bond whose pin set meets that dome. These are the profiles in [0 $,$ 1] in the following formula. Let $nF , nP$ be the cardinalities of the finite frame-tag and predictor-point sets. At a site $x,$ the ticket factor is $nF$ times the actual frame-choice probability at $x$ . At a bond it is $nP$ times the actual predictor-choice probability for its connector. Multiply the site factors over $A[300B]$ and the bond factors over all pin sets meeting it. They are nonnegative and bounded respectively by

$nF , nP$ , rather than by one. With the uniform average over the complete choice space,

Y
x
siteMass(U, k, x)
Y
b
pinMass(U, k, b).
(PRI9)
likelihood(U, k) =
1
|K| — Source page 134.

Indeed $|K| = n|Λc|$ $F$ $n| Bondc |$ $P$ , which cancels the inserted cardinality factors. This identity also includes zero-likelihood choices; their ticket product is zero. The frame probability is uniform over the admissible minimizing tags when that set is nonempty, with the prescribed uniform fallback; the predictor probability is its prescribed nonnegative normalized weight with its corresponding fallback. Thus the ticket retains the actual choice denominators, including the fallback branch.

The packet value is explicitly

SA(u) = e
−β(Qnear(A,u)+Qfar(A))+P
visit200B A ce+P
active304B A je(u)−NA log mβ
× OA(U) TA(U, k) PA(U, k, V ) Ea,A(u),
(PRI3) — Source page 134.

$√ 2π β−1/2)dim g, and NA counts the off-tree frozen links in the 10B visit where mβ = chart scale ( set. The terms ce are the link condensation terms; je are the local logarithmic Haar and cutoff corrections. The far term is the signed-orientation average of the extracted nonnegative row costs on the 200B posted set. The remaining four factors are the forced-order selector, the finite layout ticket, the product of sector profile factors, and the active cutoffproduct on the 304B neighborhood. The selector, profiles and cutoffs are in [0, 1]; the ticket has its finite combinatorial toll bound. The normalization deficit in (PRI3) is per frozen link, and is retained before that toll is estimated.$

Searchable transcription

√
2π β−1/2)dim g, and NA counts the off-tree frozen links in the 10B visit
where mβ = chart scale (
set. The terms ce are the link condensation terms; je are the local logarithmic Haar and cutoff
corrections. The far term is the signed-orientation average of the extracted nonnegative row costs
on the 200B posted set. The remaining four factors are the forced-order selector, the finite layout
ticket, the product of sector profile factors, and the active cutoffproduct on the 304B neighborhood.
The selector, profiles and cutoffs are in [0, 1]; the ticket has its finite combinatorial toll bound. The
normalization deficit in (PRI3) is per frozen link, and is retained before that toll is estimated.

$The regular logarithms Lλ comprise the old regular activity and cubic, quartic, loop and quadratic-tail energy pieces, the remote row and local chart terms, and the finite ledger terms from Gaussian completion and covariance reweighting. Their signs are the signs in the exponent of the original density: in particular an old regular occurrence contributes its negative value. A warn- ing vertex is zero when mandatory; otherwise it is its fluctuation cutofftimes the corresponding sector-profile complement. The Gaussian warnings are also retained in this warning alphabet. All factors in (PRI2) use this single configuration, signal and Gaussian variable. Put bp(Y, z) = Q α∈p vα(Y, z). The primitive value before the finite choice average is the literal integral$

Searchable transcription

The regular logarithms Lλ comprise the old regular activity and cubic, quartic, loop and
quadratic-tail energy pieces, the remote row and local chart terms, and the finite ledger terms
from Gaussian completion and covariance reweighting. Their signs are the signs in the exponent of
the original density: in particular an old regular occurrence contributes its negative value. A warn-
ing vertex is zero when mandatory; otherwise it is its fluctuation cutofftimes the corresponding
sector-profile complement. The Gaussian warnings are also retained in this warning alphabet. All
factors in (PRI2) use this single configuration, signal and Gaussian variable.
Put bp(Y, z) = Q
α∈p vα(Y, z). The primitive value before the finite choice average is the literal
integral

ap =
Z
bp(Y, z) dνF (Y ) dγC(z),
(PRI4) — Source page 134.

with the product tree-gauge probability measure and the one covariance $C$ prepared for this sector. There is no normalization by the mass of an admissible event. Local discrete choices are subsequently averaged with weight 1 $/|K|$ ; their probabilities do not replace any of the profile or ticket factors.

For extension to coarse configurations outside the healthy domain, use the same finite formulas with the extended local row expressions and retain the admission masks at their actual enlarged output footprints. For a chosen ground $H$ , retain only the finite Gaussian coordinates $I(H$ ) read there. The submatrix $CI$ is positive definite at an actual admissible sector and has density

(2π)−|I|/2 exp

−1
2 log det CI −1
2zT C−1
I z

dz.
(PRI5) — Source page 134.

Integrate the extended product against this density and the tree-gauge measure, then take the uniform finite choice average. Set it to zero if the structural connectedness, layout or incidence tests fail. The marginal extension has its own fixed support and is not presumed positive or energetic at other coarse configurations.

On a healthy ground this agrees with (PRI4). The integrand is constant on fibers of the coordinate projection to $I(H$ ); its Gaussian pushforward has covariance $CI$ . The remaining local data agree on the ground. Integrating over these fibers and using (PRI5) proves the identity. The same finite-choice fiber argument proves independence of choices outside $H$ . For disjoint grounds the finite choice space is a product; writing its cardinality as the product of the cardinalities of the factors proves that the average of their product is the product of their averages. The tree variables have the analogous product-measure identity. No independence of overlapping Gaussian warning events is used.

The later continuity passage uses these same primitive integrals. Along an ultrafilter all finite seed stores and discrete tags can be fixed on one member of the ultrafilter. Keep the tree-gauge input $Y$ fixed, and compare the two histories on this fixed finite data. Their covariance densities and vertex factors have the asserted pointwise limits. The primitive bounds below give a common integrable majorant, so dominated convergence applies to each molecule. Since this holds along every ultrafilter tending to infinity, the difference of the primitive values tends to zero in the ordinary sequential sense. This explains why the fixed finite data in the continuity argument can be selected after fixing $Y$ , without claiming that a continuously varying configuration has a globally constant sector.

Source–readout formulation. The source object of this subsection is the indexed family ${($ P $, p)}$ of realized primitive data with their molecules; the readout is $p 7→ap$ together with the retained type, ground, layout and admission flags of $p$ . Relative to the frame realization of Theorem 7.1 the fine configuration $U(u$ ) is carried on the identity-frame section, $z(u) = σ(U(u)),$ and the tree values and $Y$ are part of the integration variable, not frame data: the tree gauge is a slice of the target, while the frame fibre is a redundancy of the realization. Every vertex function is composed with $π, so vZ$ $α(σ(U(u)), . . . ) = vα$ and $aZ$ $p= ap$ by Theorem B.1. The operations

whose compatibility with the readout is needed are: the passage from letters to stores and layouts (Lemma 13.1), the rewriting of the old forced gas occurrence (Lemma 13.2), the regrouping into molecules (Proposition 13.3), the choice average (Lemma 13.4) and the marginal extension (Proposition 13.5). Each is stated here in the form in which it is used, with the proof given in the derivation above.

Lemma 13.1 (Uniqueness of the layout). The permitted layout exists and is unique: it is the set of connected components of the seed graph, each converted back to its store.

Proof. This is the connected-component construction and converse immediately following (PRI6). That argument proves all three layout tests before the molecule integrand is introduced, so the same unique layout is retained here.

Lemma 13.2 (Exact rewriting of the old gas occurrence). For a finite family $I$ of old forced addresses,

1{supports disjoint, S
i Ji = Fθ(U)} Q
i∈I Ki(U) = 1{I sealed} OI,all(U) Q
i∈I Ki(U). — Source page 135.

$Proof. If the product vanishes both sides vanish, and no permission identity is used for a vanishing factor. Otherwise each factor is nonzero and (AB9) gives Ji = Pi∩Fθ(U). On a sealed family the full selector equals one exactly when no flag lies outside S i Pi, that is when Fθ(U) = S i(Pi ∩Fθ(U)) = S i Ji. Sealing is the disjointness of supports.$

Searchable transcription

Proof. If the product vanishes both sides vanish, and no permission identity is used for a vanishing
factor. Otherwise each factor is nonzero and (AB9) gives Ji = Pi∩Fθ(U). On a sealed family the full
selector equals one exactly when no flag lies outside S
i Pi, that is when Fθ(U) = S
i(Pi ∩Fθ(U)) =
S
i Ji. Sealing is the disjointness of supports.

Proposition 13.3 (Multiplicity-one regrouping). Compatible molecules have disjoint grounds. For a finite compatible family, admission of the union is equivalent to admission of each member together

with global flag coverage. Every finite letter set has a unique partition into connected components of its footprint graph. Hence expansion into finite letter sets and regrouping into compatible connected molecules are mutually inverse, with multiplicity one.

Proof. Only the localization of the tests needs an argument. If a forced inventory site of one molecule were touched by a seed attributed to another, the parent of that site would lie in the forced footprint of the first and the seed footprint of the second, contradicting disjoint grounds; the same parent argument localizes sealing. Layout restrictions localize because enlarged packet domes lie inside their grounds, and the union of local layouts is a global layout, unique by Lemma 13.1. Equation (PRI1) localizes by intersecting the global flag set with disjoint grounds. Uniqueness of the component partition is a property of finite graphs.

Lemma 13.4 (Ticket identity). With the uniform average over the complete choice space $K,$

likelihood(U, k) =
1
|K|
Y
x
siteMass(U, k, x)
Y
b
pinMass(U, k, b). — Source page 136.

Proof. The cardinality cancellation, zero-likelihood choices and both fallback branches are proved directly after (PRI9). Applying that identity to the same complete choice space gives the displayed formula; there is no additional normalization by the eligible event.

Proposition 13.5 (Marginal recovery and factorization). On a healthy ground the marginal extension equals (PRI4); it is independent of the discrete choices outside $H$ ; and for molecules with disjoint grounds the choice average and the tree-gauge integral of a product are the products of the averages and integrals.

Proof. On a healthy ground the integrand is constant on the fibres of the coordinate projection $z 7→zI(H)$ . The pushforward of the centred Gaussian with covariance $C$ under a coordinate projection is the centred Gaussian with covariance the principal submatrix $CI$ , whose density is (PRI5); the remaining local data agree on the ground. Integrating over the fibres proves the first claim. Summing over choices outside $H$ of a function that does not read them multiplies by their number, which the normalization 1 $/|K|$ removes; this is the second claim. For disjoint grounds $K$ is a product and its cardinality the product of the factor cardinalities, so the average of a product of functions of different factors is the product of averages; the tree-gauge measure is a product measure and Fubini gives the same for it. No independence of overlapping Gaussian warning events is used or asserted.

Three losses of the readout p 7→ap are explicit.
(i) The number ap forgets which letters
produced it; the letter set is kept beside it, and by Proposition 13.3 the regrouping is a bijection
with multiplicity one, so no term of the old expansion is counted twice or dropped.
(ii) The
marginal extension integrates out every Gaussian coordinate outside I(H); by Proposition 13.5
this loses nothing for functions of the ground, and nothing else is ever evaluated on it. (iii) Outside
the interior chart domain the logarithmic reader is the cutofflogarithm and does not determine the
group reader; the discrepancy ∆= exp(Readlog)(replica V ) Read−1
grp is the recorded group-valued
residual there. Indeed, writing B = replica V , U = Readgrp and l= Readlog, the protected interior
identity is exp l= UB−1, so the ordered product gives — Source page 136.

∆= exp(l)BU−1 = UB−1BU−1 = 1.
(2) — Source page 136.

The factors are not commuted. Offthat domain ∆is retained as its actual value; neutrality is not asserted. The exactness class is therefore exact recovery of each primitive value on the healthy ground, with an explicitly retained residual outside the interior chart.