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Shadow Theory

Source Lift / The complete argument

Finite-lattice realization and exact recovery

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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Finite-lattice realization and exact recovery

Let G be the source’s compact gauge group and let (V, E) be one of its finite oriented lattices, with one stored orientation for each unoriented bond. Write U = GE. Introduce the redundant realization

Z = GV × GE, π(r, A)e = r−1 x Aery (e : x →y), σ(U) = (1, U).
Source page 15.
Searchable transcription
Z = GV × GE,
π(r, A)e = r−1
x Aery
(e : x →y),
σ(U) = (1, U).
For reverse orientations set A ̄e = A−1 e and U ̄e = U−1 e . All lattice, cutoffand representation tags are retained with these tuples.
Source page 15.
Searchable transcription
For reverse orientations set A ̄e = A−1
e
and U ̄e = U−1
e . All lattice, cutoffand representation tags
are retained with these tuples.

Theorem 7.1 (Complete frame fibre and section). The map π is onto and πσ = id. Its fibres are exactly the orbits of the free frame action

k · (r, A) = (kr, (kxAek−1 y )e), k ∈GV .
Source page 15.
Searchable transcription
k · (r, A) = (kr, (kxAek−1
y )e),
k ∈GV .

Consequently Z/GV is canonically identified with U, with inverse represented by σ.

Proof. Substitution gives πσ(U) = U and π(k · (r, A)) = π(r, A). If π(r, A) = π(r′, A′), put kx = r′ xr−1 x . The equality of link readouts gives A′ e = r′ xr−1 x Aery(r′ y)−1 = kxAek−1 y , so the two points lie in the same orbit. The equation kr = r forces k = 1, proving freeness. These formulas also prove the inverse identities, not merely a choice of representative for a restricted class of links.
Source page 16.
Searchable transcription
Proof. Substitution gives πσ(U) = U and π(k · (r, A)) = π(r, A).
If π(r, A) = π(r′, A′), put
kx = r′
xr−1
x . The equality of link readouts gives A′
e = r′
xr−1
x Aery(r′
y)−1 = kxAek−1
y , so the two
points lie in the same orbit. The equation kr = r forces k = 1, proving freeness. These formulas
also prove the inverse identities, not merely a choice of representative for a restricted class of
links.

For a path γ : x →y, cancellation of every intermediate frame gives

Uγ = r−1 x Aγry.
Source page 16.
Searchable transcription
Uγ = r−1
x Aγry.
In particular a closed plaquette has conjugate holonomy, and its represented trace and Wilson action agree exactly. The frame action above is distinct from the physical gauge action on readouts. The latter can be lifted by rx 7→rxg−1 x with A fixed, giving Ue 7→gxUeg−1 y .
Source page 16.
Searchable transcription
In particular a closed plaquette has conjugate holonomy, and its represented trace and Wilson
action agree exactly. The frame action above is distinct from the physical gauge action on readouts.
The latter can be lifted by rx 7→rxg−1
x
with A fixed, giving Ue 7→gxUeg−1
y .

Theorem 7.2 (Normalized measure recovery). Let SW denote the supplied finite Wilson action and dν normalized Haar measure. The lattice action and invariant group measure use the classical Wilson and Haar constructions [5, 6]. The probability law

deμ(r, A) = Z−1 W e−SW (A) Y x∈V dν(rx) Y e∈E dν(Ae)
Source page 16.
Searchable transcription
deμ(r, A) = Z−1
W e−SW (A) Y
x∈V
dν(rx)
Y
e∈E
dν(Ae)
has π∗eμ = μW . Every integrable target insertion F therefore satisfies R F ◦π deμ = R F dμW .
Source page 16.
Searchable transcription
has π∗eμ = μW . Every integrable target insertion F therefore satisfies
R F ◦π deμ =
R F dμW .
Proof. At fixed r, change each stored link by Ae = rxUer−1 y . Bi-invariance of normalized Haar measure preserves each link factor. The path identity makes SW (A) = SW (U). Integrating the remaining independent frame factors gives one. This proves both the partition normalization and the integral identity; no unnormalized gauge volume is introduced. The statement extends from nonnegative measurable functions by monotone convergence and to integrable complex insertions by their real and imaginary parts.
Source page 16.
Searchable transcription
Proof. At fixed r, change each stored link by Ae = rxUer−1
y . Bi-invariance of normalized Haar
measure preserves each link factor. The path identity makes SW (A) = SW (U). Integrating the
remaining independent frame factors gives one. This proves both the partition normalization and
the integral identity; no unnormalized gauge volume is introduced. The statement extends from
nonnegative measurable functions by monotone convergence and to integrable complex insertions
by their real and imaginary parts.

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