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

Jeremy Rodgers

doi:10.5281/zenodo.24173681

Source Lift / The complete argument

The source support of the realized family, its lift groupoid and its interfaces

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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The source support of the realized family, its lift groupoid and its interfaces

The preceding two sections give the lift of one finite lattice: an admitted antecedent (Proposition 6.1), a redundant realization, its readout and its section (Theorems 7.1 and 7.2). The supplied construction, however, is not one lattice. It is a countable family of finite histories, indexed by a cutoff, a period and a scale level, together with comparisons between them, auxiliary reference volumes, coefficient spaces of field labels, an imported free comparison theory and several successive limits. This section develops the source-side structure that is needed to carry that family, and states what that structure does and does not determine. It has four parts: the admitted support; the lift groupoid of a single leaf; the sectors and interfaces of the supplied construction, with the status of each discrepancy; and the separation of what follows from the source side from what is supplied as realization data. Nothing in this section is used as a hypothesis by the supplied derivations that follow; it is the frame in which their source–readout formulations are stated.

8.1 A fan of marked dependencies

Let $I$ be a nonempty countable set. The fan Φ $I$ has vertices ${$ r $}∪{la : a ∈I}$ , dependency preorder generated by r $≺la$ for every $a$ , and one marked token $ma$ for each $a ∈I$ , with support ${$ r $, la}$ . The lineage, invariant and intrinsic boundary data of $ma$ are those of the walking marked dependency

$X⋆$ , transported along the map $ca : X⋆→$ Φ $I, ca(0) =$ r $, ca(1) = la$ . For a subset $F ⊆I$ write Φ $F$ for the corresponding subfan.

Proposition 8.1 (Admission of the fan). For every countable $I$ the fan Φ $I$ is admitted. More precisely:

1. for finite $F,$ Φ $F$ is obtained from $|F|$ copies of $X⋆$ by successive attachments along the root face ${0}$ , and is admitted;

2. for $F ⊆F ′$ the inclusion Φ $F →$ Φ $F ′$ is a strong map, and Φ $I$ is the directed colimit of the Φ $F$ over finite $F ⊆I$ ; this colimit is stabilizing;

3. distinct leaves are incomparable: for $a ̸= b$ neither $la ⪯lb$ nor $lb ⪯la.$

Proof. The admission construction of [3] is the least isomorphism-saturated class containing the initial witnesses, among them $X⋆$ , and closed under the specified finite attachments along source faces, presentation-preserving retracts and stabilizing directed colimits. The root ${0}$ of $X⋆$ is a source face: it is predecessor-closed, and it is lineage-and boundary-closed because the marked token is not supported in it and the intrinsic boundary is empty. (1) Attaching a further copy of $X⋆$ to Φ $F$ along the identification of its root with r is a pushout along that face. The attaching map is strong and reflects marks, since no marked support lies in the face. The pushout adds one vertex, one strict pair and one marked token, and leaves every existing support, strict pair and lineage datum unchanged; the candidate conditions (finite nonempty supports each containing a strict pair, well-typed inherited data) therefore hold in Φ $F∪{a}$ . Induction on $|F|$ gives admission. (2) An inclusion of subfans preserves the preorder, the marking, the supports with their strict pairs and the inherited data, and is injective; it is therefore strong. Every finite set of vertices and tokens of Φ $I$ lies in some Φ $F$ with $F$ finite, and its preorder, supports and inherited data are the same in every larger subfan; this is the stabilization condition. (3) The preorder is generated by the pairs r $≺la$ , and neither a pushout along the root nor the colimit adds a relation between leaves.

This is the same admission that supports the family of prepared paths in the gravitational reconstruction [4]; only the index set and the realized sector differ. The literal preparation of a leaf is $Ya = Jreal(ΦI, ma) ∼= Y⋆,$

the preparation of Proposition 6.1, by restriction along $ca.$

Proposition 8.2 (What the support determines).

$1. If |I| ≥2$ , the strong automorphisms of Φ $I$ are exactly the maps fixing r and permuting the leaves, with $ma 7→mσ(a)$ ; thus Aut(Φ $I) ∼$ = Sym( $I).$

2. The fold map $φ : ΦI →X⋆, φ($ r) = 0 $, φ(la) = 1, φ(ma) = m⋆$ , is strong, and $φca = id$ for every $a.$

3. Every finite family of marked tokens of Φ $I$ lies in a finite admitted subfan.

Proof. (1) A strong automorphism preserves the preorder, hence fixes the unique vertex that lies strictly below at least two others, and permutes the remaining vertices. It preserves supports, so it sends $ma$ to the token supported on ${$ r $, lσ(a)}$ . Conversely each permutation of $I$ defines such a map, and it preserves the inherited data because these are transported from the same $X⋆$ along the maps $ca. (2) φ$ preserves the preorder and the strict pairs r $≺la 7→0 ≺$ 1, carries each support bijectively onto ${0, 1}$ , so is injective on marked supports, and preserves the inherited data by their definition. (3) is Proposition 8.1(2).

The three statements have direct consequences for the reconstruction. By (1), nothing on the source side distinguishes one leaf from another. Any assignment of cutoffs, periods or scale levels to leaves, any order among them and any relation between two leaves is therefore realization data. By (2), the single-cell presentation of Proposition 6.1, in which one marked occurrence supports the whole realized tuple, is the image of the fan presentation under a strong map; the two presentations carry the same realized record, and every statement of the preceding sections applies leaf by leaf. By (3), every estimate of the supplied construction, which always concerns finitely many histories at a time, is a statement over an admitted finite support, and a limit is a readout of a net of realized data over the colimit; it is not a new source object.

8.2 The lift groupoid of one leaf

Fix one finite lattice (V, E), connected as a graph, with Z = GV × GE, U = GE and π, σ as in
Theorem 7.1. Two groups act on Z: the frame group GV by k · (r, A) = (kr, (kxAek−1
y )e), and
the lifted gauge group GV by (r, A) · h = ((rxh−1
x )x, A). The two actions commute. Let E be the
action groupoid of GV
frame × GV
gauge on Z, and Comp the action groupoid of the gauge group on
U, (h · U)e = hxUeh−1
y . These are the groupoids of realized histories and of allowed comparison
isomorphisms on this branch. — Source page 18.

Theorem 8.3 (The realization is an equivalence of groupoids). The assignment $z 7→π(z), (k, h) 7→$ $h$ is a functor $Π : E →$ Comp, and it is an equivalence: it is surjective on objects, full and faithful. Consequently

$1. for every U ∈U the lift groupoid Lift(U) = E ×h Comp {U} is nonempty and contractible: any two recovery witnesses (z, φ), (z′, φ′) of U are related by exactly one arrow of E compatible with φ, φ′;$

Searchable transcription

1. for every U ∈U the lift groupoid Lift(U) = E ×h
Comp {U} is nonempty and contractible: any
two recovery witnesses (z, φ), (z′, φ′) of U are related by exactly one arrow of E compatible
with φ, φ′;

2. for $z = (r, A)$ the automorphism group of $z$ in $E$ is

AutE(z) = {(k, h) : kxAek−1
y
= Ae, hx = r−1
x kxrx} ∼= StabGV (π(z)), — Source page 18.

and for any base vertex $x0$ evaluation $h 7→hx0$ identifies Stab $GV (U)$ with the centralizer in $G$ of the holonomy group of $U$ at $x0.$

Proof. For $z = (r, A$ ) and ( $k, h$ ) one computes $π(k · z · h)e = (kxrxh−1$ $x$ $)−1kxAek−1$ $y$ $(kyryh−1$ $y$ ) = $hx r−1$ $x Aery h−1$ $= (h·π(z))e$ , so Π is a functor. It is surjective on objects because $πσ$ = id. Fullness: $y$ let $h · π(z) = π(z′$ ). Then $z · h$ and $z′$ have the same readout, so by Theorem 7.1 there is $k$ with $k · (z · h) = z′$ , and ( $k, h$ ) is an arrow $z →z′$ over $h$ . Faithfulness: if ( $k, h$ ) and ( $k′, h$ ) are arrows $z →z′$ , then $k · (z · h) = k′ · (z · h$ ), and freeness of the frame action gives $k = k′$ . An equivalence of groupoids has contractible homotopy fibres, which is (1): an object of Lift( $U$ ) is a pair ( $z, φ$ ) with $φ$ a gauge transformation carrying $π(z) to U$ , and an arrow ( $z, φ) →(z′, φ′$ ) is an arrow ( $k, h) of E$

with $φ′h = φ$ ; here $h = φ′−1φ$ is forced and then $k$ exists and is unique by fullness and faithfulness. (2) An arrow ( $k, h$ ) fixes $z = (r, A$ ) exactly when $kxrxh−1$ $= rx$ and $kxAek−1$ $= Ae$ ; the first equation $x$ $y$ gives $hx = r−1$ $x kxrx$ , and then $hxπ(z)eh−1$ $= r−1$ $x kxAek−1$ $y ry$ $= π(z)e, so h ∈$ Stab( $π(z$ )); conversely $y$ each such $h$ determines $kx = rxhxr−1$ $x$ , which fixes $A$ . Finally, if $h$ stabilizes $U$ then $hy$ $= U−1$ $γ hx0Uγ$ for every path $γ : x0 →y, so h$ is determined by $hx0$ , and applying this to closed paths shows that $hx0$ commutes with every holonomy at $x0$ ; conversely, for $g$ in that centralizer the formula $hy = U−1$ $γ gUγ$ is independent of the path and defines a stabilizing gauge transformation.

The theorem sharpens the statement, made after Proposition 6.1, that the recovery fibres are nonempty: on this realized branch a recovery witness is unique up to unique isomorphism, and

the realization neither adds nor removes symmetry; the reducible configurations, with nontrivial stabilizers, are exactly as reducible upstairs as downstairs. It does not classify other native antecedents of the same target; that question is addressed in the last subsection. The signed lattice permutations may be adjoined to both groupoids, since they act compatibly, $πρZ = ρπ$ (Section 9); the statement and its proof are unchanged.

Corollary 8.4 (The fibre law). Under the law eμ of Theorem 7.2 the frame r and the readout π(r, A)
are independent, r has law ν⊗V and π(r, A) has law μW . For bounded measurable Φ on Z,
Z
Φ deμ =
Z
hZ
GV Φ
�r, (rxUer−1
y )e
dν⊗V (r)
i
dμW (U). — Source page 19.

Proof. This is the change of variables Ae = rxUer−1
y
in the proof of Theorem 7.2, applied to a general
integrand: at fixed r it preserves the product Haar measure and carries e−SW (A) to e−SW (U), so the
joint law of (r, U) is the product ν⊗V ⊗μW . — Source page 19.

A consequence must be stated plainly. For a gauge-invariant function $F$ one has $F ◦π(r, A) =$ $F(A$ ): the lift of $F$ is $F$ itself evaluated on the raw links, and the frame is an independent spectator. On gauge-invariant quantities the frame realization is therefore an identity encoding. It carries real content only where the supplied construction uses objects that are not gauge invariant (charts, axial gauges, predictors, tree slices, potentials), and there the covariance identities of the tangent chapter are what is used. The substantive source–readout structure of the construction does not lie in the frame fibre. It lies in the interfaces listed next, whose fibres are not torsors of a symmetry group and whose descent statements are theorems of the supplied construction.

8.3 Sectors, interfaces and discrepancies of the supplied construction

A realized family over the fan consists of: an injective labelling $λ$ of the leaves by the index set of the construction; for each leaf $a$ a raw datum $za$ in a sector over $Ya$ , with its discrepancy column; and, for each interface $e$ between sectors $s$ and $t$ , maps $Le, Re$ into a common comparison space and the retained discrepancy $δe = de(Lezs, Rezt).$

An interface is exact if $δe$ = 0 identically, controlled if $δe$ is bounded in a declared norm on a declared regime, and selected if it involves a choice (of a subsequence, a complement or a representative) that is retained as a witness. The index set of the supplied construction consists of the triples $(N, p, i$ ) with cutoffindex $N$ , period index $p$ and scale level 0 $≤i ≤N$ , together with labels for the reference tori of the beacons and for the free comparison theory. The numbers attached to a label (lattice side, block length, clock value, stage) are read from the one calendar c of Theorem 14.16.

The sectors are the following.

(S1) Finite Wilson history at (N, p): z = (r, A) ∈ZN,p with the law eμ; readout π. — Source page 19.

(S2) Represented state $at (N, p, i$ ): the tuple $Q$ of Definition 14.19, with readers $β(Q), v(Q$ ), the layout and the evaluated density.

(S3) Stationary corridor of a blocking step: pairs ( $x, v$ ) of retained and vertical coordinates with the residual $S(x, v$ ) (Section 14).

(S4) Reference views: the folded reference tori of the beacons and of the reference chart.

(S5) Label coefficient spaces $Ed$ with their spectator readers, and the physical quotients $Ed/Nd$ (Section 16, Subsection 18.9).

(S6) The free comparison hierarchy: the imported Gaussian theory on the same label algebra.

(S7) The continuum hierarchy: correlations on the physical quotient, its reflection Hilbert space and its characters (Theorems C.1 and D.1).

Table 2 lists the interfaces, the discrepancy that each retains, its status, and the statement in which that status is proved. It is the map of the reconstruction: every source–readout formulation in the following sections refines one row.

Table 2: Interfaces of the realized family and the status of their discrepancies.

$Interface Discrepancy and status Where proved I1 Frame fibre within (S1) None; exact. Fibre a torsor of GV , lifts unique up to unique isomor- phism Theorems 7.1, 7.2, 8.3 I2 Startup: Wilson law at (N, p) to the repre- sented state at level N Zero: an identity of densities, with the representative inside the activ- ity ball Theorem 14.18 I3 Blocking step: level i + 1 to level i Zero: evaluation intertwines the for- ward recipe with block integration; retirement is an exact change of pre- sentation Theorem 14.20; (RET1)– (RET2) I4 Stationary elimination within a step, (S3) Residual S(x, v) retained; vertical distance to the graph bounded by the residual and zero exactly on the graph Theorems 14.4, 14.5 I5 Presentation of a state to its activity data None: the step maps are constant on presentations, hence descend, and are analytic and Lipschitz on the normed data Theorem 14.28 I6 Calibration of a history Bottom coupling equal to ̄β by choice of the bare parameter; the fi- bre is one point within the retained recipe Theorems 14.24, 14.29 I7 Actual coupling to the reference clock Bounded, not zero: |βN,p,i+1 − βN,p,i−s log Li| ≤d; summed to the slope Theorem 14.12; (RUN1)– (RUN5) I8 Finite period to infinite volume, for one film Controlled: at most a strip price times e−σn/4 Proposition 14.6 I9 Reference views to the interior density, (S4) The fourth difference annihilates ev- ery face contribution; boundary in- formation discarded by design Lemma 14.25$

Accessible table transcription

Table transcription, source page 20
Interface	Discrepancy and status	Where proved
	V
I1Frame fibre within (S1)	None; exact. Fibre a torsor of G,	Theorems 7.1, 7.2, 8.3
	lifts unique up to unique isomor-
	phism
I2Startup:Wilson law	Zero: an identity of densities, with	Theorem 14.18
at (N, p) to the repre-	the representative inside the activ-
sented state at level N	ity ball
I3Blocking step: level i +	Zero: evaluation intertwines the for-	Theorem 14.20; (RET1)–
1 to level i	ward recipe with block integration;	(RET2)
	retirement is an exact change of pre-
	sentation
I4Stationary elimination	Residual S(x, v) retained; vertical	Theorems 14.4, 14.5
within a step, (S3)	distance to the graph bounded by
	the residual and zero exactly on the
	graph
I5Presentation of a state	None: the step maps are constant	Theorem 14.28
to its activity data	on presentations,hence descend,
	and are analytic and Lipschitz on
	the normed data
	̄
I6Calibration of a history	Bottom coupling equal toβ by	Theorems 14.24, 14.29
	choice of the bare parameter; the fi-
	bre is one point within the retained
	recipe
I7Actual coupling to the	Bounded,not zero:\|βN,p,i+1 −	Theorem 14.12; (RUN1)–
reference clock	β−s log L\| ≤d; summed to the	(RUN5)
	N,p,ii
	slope
I8Finite period to infinite	Controlled:at most a strip price	Proposition 14.6
	−σn/4
volume, for one film	times e
I9Reference views to the	The fourth difference annihilates ev-	Lemma 14.25
interior density, (S4)	ery face contribution; boundary in-
	formation discarded by design

Interface
Discrepancy and status
Where proved
I10
Atom families to their
moments
Many-to-one; all produced families
lie in the fibre over zero, which is
stable under the operations used
Proposition 14.10
I11
Histories
at
different
cutoffs
Full-sequence
convergence
of
banked limits by contraction; path
books
only
subsequentially,
as
retained witnesses
Subsection 15.11; Proposi-
tion 15.3
I12
Change of background
in the fluctuation chart
Horizontal lift exact on the pro-
tected domain; residual retained off
it
Section 15.5
I13
Jet to lattice cylinder
to slot readout, (S5)
Exact local section; label reading
exact through Taylor order d in the
mesh
(PF7),
(PF10);
Lemma 16.2
I14
Transported to read co-
efficients
Invertible
reader
with
two-sided
bounds at each level; not uniform
in the level
Theorem 16.3
I15
Coefficients
to
the
physical quotient
Kernel Nd; its finite-cutoffshadow,
the calibrated equation image, is
silent in separated tests; operations
descend
Proposition
18.2;
Theo-
rem 18.3
I16
Interacting to free hier-
archy, (S6)
Selected scaling maps; readings con-
verge to all derivative orders along
radii rk; residuals retained
Theorems 23.2, 23.1
I17
Word space to reflec-
tion Hilbert space, (S7)
Null space; insertions descend to
bounded operators
Proposition
19.1;
Theo-
rem C.1
I18
Unselected hierarchy to
its phases
Rooted bounds descend to every
character
Theorem
23.1;
Theo-
rem D.1
I19
Lattice
symmetry
to
Euclidean symmetry
Signed permutations exact on the
source; O(4) derived on the readout
only
Theorem 19.3
I20
Stress and trace
Equation residual retained in the
finite
candidate;
charges
repre-
sentable as fields; full short-distance
limit of the trace
Theorems 23.3, 24.1 — Source page 21.

Two structural conditions of the method are visible in the table. First, all rows are attached to one calendar and, in the limits, to one composite refinement: estimates obtained on different calendars or along incompatible subsequences are never combined. The order of the limits is part of the realized record, with the full limit in the cutofftaken at each fixed period before the limit in the period along the refined sequence. Second, the rows fall into the method’s exactness classes without exception being made for convenience: I1–I3, I5 and I12 are exact; I4 and I6 are on-shell or calibrated restrictions with the off-shell data retained; I15, I17 and I18 are quotient equivalences established by congruence statements; I7, I8, I13, I14 and I16 are controlled approximations with their regimes stated; and I9–I11 record deliberate or unavoidable loss.

8.4 What follows from the source side and what is supplied

The division is the following, and it is used without further comment in the formulations below.

From the source side. The admitted fan provides a support with countably many marked cells (Proposition 8.1); no relation between distinct cells and no preferred labelling (Proposition 8.2); and the reduction of every finite statement to an admitted finite support. On each leaf the realized branch has contractible lift groupoids (Theorem 8.3). These are structural facts. They contain no dynamics.

Supplied as realization data. Everything that makes the family a Yang–Mills construction is chosen in the realized sector: the compact group $G ⊆U(m$ ) with simple nonabelian Lie algebra; the four-dimensional periodic lattices and the Wilson action; the blocking maps and the stationary corridors; the calendar c; the calibration value ̄ $β$ and the blocking penalty one; the label algebra and the free comparison theory; and every estimate. The labelling of leaves by ( $N, p, i$ ), and hence the notions of finer, larger and coarser, are of this kind.

Derived within the realization. Some quantities are neither chosen nor source-side: they are computed from the realization data by the supplied construction. The slope $s = 11CA/(24π2$ ) is the coefficient of the logarithm in the explicit annular integral (DR5); the amplitude $AG$ = 3 dim $R$ g $/π4$

is a Wick contraction (Theorem 24.1); the one-dimensionality of the translation Ward kernel of the free comparison theory (FS10) and the $O$ (4) covariance of the hierarchy (Theorem 19.3) are consequences.

Not obtained. No selection theorem is proved, and none is claimed. The method’s test for a source-derived premise is to look for inequivalent raw realizations over the same admitted support and the same literal preparations. Here they exist in abundance: replacing the group, the dimension of the lattice, or the Wilson action by another gauge-invariant local action gives a different realized sector over the same fan Φ $I$ , with the same lift-groupoid theorem. Admission therefore does not select the gauge group, the dimension, the action or the coupling history. The recovery proved in this paper is conditional in exactly this sense: given the supplied construction as realization data, its objects, operations, limits and conclusions are recovered through the readouts, with the stated exactness classes. This distinguishes an explicit mathematical recovery theorem from selection of the physical realization.

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

Interface	Discrepancy and status	Where proved
I10Atom families to their	Many-to-one; all produced families	Proposition 14.10
moments	lie in the fibre over zero, which is
	stable under the operations used
I11Historiesatdifferent	Full-sequenceconvergenceof	Subsection 15.11; Proposi-
cutoffs	banked limits by contraction; path	tion 15.3
	booksonlysubsequentially,as
	retained witnesses
I12Change of background	Horizontal lift exact on the pro-	Section 15.5
in the fluctuation chart	tected domain; residual retained off
	it
I13Jet to lattice cylinder	Exact local section; label reading	(PF7),(PF10);
to slot readout, (S5)	exact through Taylor order d in the	Lemma 16.2
	mesh
I14Transported to read co-	Invertiblereaderwithtwo-sided	Theorem 16.3
efficients	bounds at each level; not uniform
	in the level
I15Coefficientstothe	Kernel Nd; its finite-cutoffshadow,	Proposition18.2;Theo-
physical quotient	the calibrated equation image, is	rem 18.3
	silent in separated tests; operations
	descend
I16Interacting to free hier-	Selected scaling maps; readings con-	Theorems 23.2, 23.1
archy, (S6)	verge to all derivative orders along
	radii rk; residuals retained
I17Word space to reflec-	Null space; insertions descend to	Proposition19.1;Theo-
tion Hilbert space, (S7)	bounded operators	rem C.1
I18Unselected hierarchy to	Rooted bounds descend to every	Theorem23.1;Theo-
its phases	character	rem D.1
I19Latticesymmetryto	Signed permutations exact on the	Theorem 19.3
Euclidean symmetry	source; O(4) derived on the readout
	only
I20Stress and trace	Equation residual retained in the	Theorems 23.3, 24.1
	finitecandidate;chargesrepre-
	sentable as fields; full short-distance
	limit of the trace