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Shadow Theory

Source Lift / The complete argument

Quadratic ultraviolet kernels and comparison · Part 3

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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gives a global floor vI/4. Near zero it adds a nonnegative form. This finite-support correction has a common analytic bound and preserves d∗ 2Qd2 by d3d2 = 0. Increasing the upper cost therefore yields the uniformly positive full-space row with exactly the original curvature contraction on every period, as required in pinning. On a fixed pair of tori let d, d′ be the fine and coarse first derivatives and e the coarse second derivative. For the tower retain fine and coarse full Gram factors W = J∗J, W ′ = J′∗J′, inverses R, R′ and penalty square root s = √ b. The factorized inverse defect is
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gives a global floor vI/4. Near zero it adds a nonnegative form. This finite-support correction has
a common analytic bound and preserves d∗
2Qd2 by d3d2 = 0. Increasing the upper cost therefore
yields the uniformly positive full-space row with exactly the original curvature contraction on every
period, as required in pinning.
On a fixed pair of tori let d, d′ be the fine and coarse first derivatives and e the coarse second
derivative. For the tower retain fine and coarse full Gram factors W = J∗J, W ′ = J′∗J′, inverses
R, R′ and penalty square root s =
√
b. The factorized inverse defect is
W ′ZW ′ = e∗P ∗Pe, Z = R′ −T ∗RT −b−1d′d′∗.
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W ′ZW ′ = e∗P ∗Pe,
Z = R′ −T ∗RT −b−1d′d′∗.
Put Q = J′R′J′∗, X = PeR′J′∗and
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Put Q = J′R′J′∗, X = PeR′J′∗and
M = JRTJ′∗, N = s−1d′∗J′∗, O = stack(I −Q, X).
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M = JRTJ′∗,
N = s−1d′∗J′∗,
O = stack(I −Q, X).

Then Q∗= Q, Q2 = Q, QJ′ = J′, and multiplication gives

M∗M = J′T ∗RTJ′∗, N∗N = J′b−1d′d′∗J′∗, O∗O = I −Q + J′ZJ′∗.
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M∗M = J′T ∗RTJ′∗,
N∗N = J′b−1d′d′∗J′∗,
O∗O = I −Q + J′ZJ′∗.

Substitute Z and Q in their sum to get M∗M + N∗N + O∗O = I on the entire stack source, including harmonic coordinates. The actual transport identities TW ′d′ = WdA0, T ∗d = d′ mean further give MJ′d′ = JdA0, M ∗Jd + N∗(−s mean) = 0,

sN∗= J′d′, NJ′d′ = s−1d′∗W ′d′, OJ′d′ = 0.
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sN∗= J′d′,
NJ′d′ = s−1d′∗W ′d′,
OJ′d′ = 0.
The last uses (I −Q)J′ = 0 and ed′ = 0. These are exact roll identities, not spectral estimates. Weighted convolution at γ = min(σf, σc, ν) > 0 supplies the analytic bounds. The full J stacks √clip. directI and its root row, with bound max(√clip. direct, rootCost). For derivatives the period-independent bound is the finite sum of eγ|z|1 times the absolute stencil coefficients over all orientations and stencil sites. Transposition preserves this bound. Composition of two rows with an intermediate orientation space of size n costs at most n times the product of their entrywise weighted bounds, by the distance triangle inequality and absolute convergence. Apply the convolution rule
Searchable transcription
The last uses (I −Q)J′ = 0 and ed′ = 0. These are exact roll identities, not spectral estimates.
Weighted convolution at γ = min(σf, σc, ν) > 0 supplies the analytic bounds.
The full J
stacks √clip. directI and its root row, with bound max(√clip. direct, rootCost). For derivatives the
period-independent bound is the finite sum of eγ|z|1 times the absolute stencil coefficients over all
orientations and stencil sites. Transposition preserves this bound. Composition of two rows with an
intermediate orientation space of size n costs at most n times the product of their entrywise weighted
bounds, by the distance triangle inequality and absolute convergence. Apply the convolution rule

in order to JR, then its composition with T, then multiplication by J′∗. The mixed row thus retains its L−2 factor, normal and image rows have length-independent bounds, and stacking takes the maximum. Choose one upper price covering these constants and all original donated prices. Keeping A0, T literally unchanged preserves the donated carried-difference bounds L−(1+order) and L−(2+order) at the smaller width and larger price. This is F08; the exact roll equations above give F09 independently of these estimates. Under readout this entire Gram identity is transported on the same section; no component or harmonic coordinate is dropped to obtain positivity.

11.12 Pinning, coefficient symmetries and the inverse-third-power price

Pinning changes only auxiliary tower rows. At a transition assume the beds Hf, Hc have uniform analytic bounds and positive floors, A0, T have their fine-difference bounds, and the inverse defect is d∗ 2Cd2 with C analytic and positive on the d2 image. Extend the defect’s three-form row positively without changing its d2 contraction, take its analytic root P, and form
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Pinning changes only auxiliary tower rows. At a transition assume the beds Hf, Hc have uniform
analytic bounds and positive floors, A0, T have their fine-difference bounds, and the inverse defect is
d∗
2Cd2 with C analytic and positive on the d2 image. Extend the defect’s three-form row positively
without changing its d2 contraction, take its analytic root P, and form
FfBfTF ∗ c , b−1/2d∗ 1F ∗ c , stack(I −FcBcF ∗ c , Pd2BcF ∗ c ).
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FfBfTF ∗
c ,
b−1/2d∗
1F ∗
c ,
stack(I −FcBcF ∗
c , Pd2BcF ∗
c ).

Here Ff, Fc are the full two-form Gram factors of the beds and Bf, Bc their inverse metrics; A0, T are retained literally. These formulas are supplied by common cover rows for all periods. The

auxiliary-row construction supplies a positive width and common costs, including the required fine differences. Their analytic prices are chosen from the fixed clip profiles, estimates and penalty before the particular block length. For each smoothing index first apply positive extension and the analytic root, then choose auxiliary-row prices for each of the two adjacent-stage possibilities. Intersect those widths with the original width and maximize costs. Selecting the corresponding tower at each transition gives the full horizon conditions at these new prices. The Gram proof supplies the roll equations, while unchanged beds and A0, T retain terminal normalization, crossings and floor bounds.

Pilot and wire coefficient bounds use one output cube before L is chosen. The mass filter costs C decayj(l) with positive principal lower bound; rails have common upper and positive principal lower bounds. At nonzero aliases Dl,a and R−1 cost respectively C grade(l)−2 and C grade(l)−4; the l seed representation bounds S. The inverse-alias bounds use the common-collar reciprocal argument and the bulb and bend bounds. The sums H, Ta contain two smoothing factors, contributing decay2j, while each D contributes another inverse grade squared. Each resulting weight is bounded above by decay2 for j ≥4, whose uniform finite-alias sum pays both sums. The quantity U = |m(q0)|2 P a fa + SH is bounded above and has a positive real lower bound from its principal filter and one fa, all summands being nonnegative. The common-collar lemma bounds U−1 and hence the source rests, fled and gains in fixed orientation dimension.
Searchable transcription
Pilot and wire coefficient bounds use one output cube before L is chosen. The mass filter costs
C decayj(l) with positive principal lower bound; rails have common upper and positive principal
lower bounds. At nonzero aliases Dl,a and R−1
cost respectively C grade(l)−2 and C grade(l)−4; the
l
seed representation bounds S. The inverse-alias bounds use the common-collar reciprocal argument
and the bulb and bend bounds.
The sums H, Ta contain two smoothing factors, contributing
decay2j, while each D contributes another inverse grade squared. Each resulting weight is bounded
above by decay2 for j ≥4, whose uniform finite-alias sum pays both sums. The quantity U =
|m(q0)|2 P
a fa + SH is bounded above and has a positive real lower bound from its principal filter
and one fa, all summands being nonnegative. The common-collar lemma bounds U−1 and hence
the source rests, fled and gains in fixed orientation dimension.

For the wire, Λ contains only controlled inverse principal filter and rail factors; B, M, y are bounded products and sums. At nonzero aliases the first stake has one grade-one signal, grademinus-two covariance and smoothing flux; the second has two smoothing factors, grademinus-four inverse, bounded coarse notch and another flux. Each is bounded by C decayj. This proves the asserted per-alias wire bound with constants chosen before L. Periodic regular covariances and patch identities show that a coarse shift 2πx permutes aliases by l 7→l + wrapL(x). For x ̸= 0, if both momenta lie in the enlarged real cube neither is zero, so the original regular-set stationary

matching formulas apply on both sides. For x = 0 the overlap identity is immediate. Reflection follows by even scalar symbols and conjugation of Fourier rows/exterior matrices. At zero use the explicit principal and vanishing nonprincipal branches. Thus pilot(p, l) = pilot(−p, −l) and likewise wire. These are the relay and reflection properties required for the secured symbols and license the common-collar Fourier export of the residue sums.

Finally buckle is the secured product vault(wing, ql) wire(p, l) with toll L−2. Vault and its overlap identities preserve securedness. Thus one actual carried fine difference has exponential coefficient bound CL−3. Its fine curl sums four directions and fixed two-form incidence signs, costing at most 4|Faces2|CL−3; this constant uses only orientation counts and introduces no residue factor. Multiplying by the bounded coarse metric and applying the coarse-column exterior right lift retains the prefactor, so sale costs C′L−3 per residue. Its Gram sum has L4 residues and therefore costs C′′L4(L−3)2 = C′′L−2. This is the precise inverse-square scale used in the positive three-form absorption. The fine difference is taken before the lift and norm estimate; applying an ordinary unweighted bound before the fine difference would lose this power. Readout preserves this order and the actual carried stencil, so no lost block power is replaced by an unproved estimate.
Searchable transcription
Finally buckle is the secured product vault(wing, ql) wire(p, l) with toll L−2.
Vault and its
overlap identities preserve securedness.
Thus one actual carried fine difference has exponential
coefficient bound CL−3.
Its fine curl sums four directions and fixed two-form incidence signs,
costing at most 4|Faces2|CL−3; this constant uses only orientation counts and introduces no residue
factor. Multiplying by the bounded coarse metric and applying the coarse-column exterior right lift
retains the prefactor, so sale costs C′L−3 per residue. Its Gram sum has L4 residues and therefore
costs C′′L4(L−3)2 = C′′L−2. This is the precise inverse-square scale used in the positive three-form
absorption. The fine difference is taken before the lift and norm estimate; applying an ordinary
unweighted bound before the fine difference would lose this power. Readout preserves this order
and the actual carried stencil, so no lost block power is replaced by an unproved estimate.

Recovery of the assembled and pinned tower. Retain the ordered finite schedule, train at every level, metric beds, penalty and response rows as one realized history. Stagewise minima of positive widths and maxima of costs are scalar readers on this record, so their recovery keeps the parameter order and both adjacent stage possibilities. Pinning acts on the auxiliary rows of that same history and leaves its declared beds, A0 and T literally fixed. For the full physical source H of a tower stack and its three output spaces Ha, choose sections j, ja and readouts p, pa with pj = I

and paja = I. Lift its rows Ua = M, N, O by eUa = jaUap. Section adjoints then give
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and paja = I. Lift its rows Ua = M, N, O by eUa = jaUap. Section adjoints then give
X a eU† a eUa = j X a U∗ aUa  p = jp, p X a eU† a eUa  j = IH.
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X
a
eU†
a eUa = j
X
a
U∗
aUa

p = jp,
p
X
a
eU†
a eUa

j = IH.
Thus the entire original stack source, including harmonic coordinates, is recovered. The identity jp acts on the retained section; it need not be the identity on unused redundant directions. Equiv- alently, keeping physical output rows Uap yields Gram form p∗p. Every roll equation recovers by the same intermediate section cancellations and the same real-slice adjoints. The positive exten- sion Q 7→Q + rd∗ 3d3 preserves its curvature contraction under recovery since d3d2 = 0; its full physical three-form lower bound and root are recovered as in the preceding constructions. Fourier normalizations are evaluated before this transport, retaining each coarse/fine volume factor and each installed toll. Consequently the assembled horizon retains its stationary equations, terminal normalization, crossings and donated fine differences, with the original widths and prices in the transported physical norms.
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Thus the entire original stack source, including harmonic coordinates, is recovered. The identity
jp acts on the retained section; it need not be the identity on unused redundant directions. Equiv-
alently, keeping physical output rows Uap yields Gram form p∗p. Every roll equation recovers by
the same intermediate section cancellations and the same real-slice adjoints. The positive exten-
sion Q 7→Q + rd∗
3d3 preserves its curvature contraction under recovery since d3d2 = 0; its full
physical three-form lower bound and root are recovered as in the preceding constructions. Fourier
normalizations are evaluated before this transport, retaining each coarse/fine volume factor and
each installed toll. Consequently the assembled horizon retains its stationary equations, terminal
normalization, crossings and donated fine differences, with the original widths and prices in the
transported physical norms.

11.13 Filling the principal-alias pilot

For a flight put ql = (p + 2π lift(l))/L, q0 = p/L, S = λmass(land)(p), ga = bulb(land, p, a), h = bend(land, p). Retain m(q) = massFilter(q) and fa = bulb(wing, q0, a). For l ̸= 0 set Dl,a = bulb(wing, ql, a)/ blow(mass(wing), p, l) and

Ta = X l̸=0 | pullLinea(ql)|2Dl,a + b−1, ga = | pullLinea(q0)|2fa + STa. (Q3)
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Ta =
X
l̸=0
| pullLinea(ql)|2Dl,a + b−1,
ga = | pullLinea(q0)|2fa + STa.
(Q3)

Absolute squares mean opposite-momentum products offthe real slice. Define

H = X l̸=0 |m(ql)|2 X a Dl,a, U = |m(q0)|2 X a fa + SH.
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H =
X
l̸=0
|m(ql)|2 X
a
Dl,a,
U = |m(q0)|2 X
a
fa + SH.

Away from zero U is S times the original weighted covariance sum. Its first real term has a positive lower bound and the second is nonnegative. The earlier alias estimates bound H, Ta, U on a common collar, and the reciprocal lemma bounds U−1 there. Thus U is the regular denominator replacing the original singular covariance sum. The regular correction coefficients are

G0,a = m(q0)−1(Ta/ga −H/U), " # ( X c Dl,c)/U −Dl,a|RL(ql,a)|2/ga Gl,a = m(−ql) (l ̸= 0).
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G0,a = m(q0)−1(Ta/ga −H/U),
"
#
(
X
c
Dl,c)/U −Dl,a|RL(ql,a)|2/ga
Gl,a = m(−ql)
(l ̸= 0).
Every denominator is regular: ga, m(q0), U and the nonzero-alias blow. Bounds are common for G0 and C decayj(l) grade(l)−2 otherwise. Algebraically Q3 gives El −rl,a = SGl,a, where El is the original normalized covariance weight and rl,a its directional rebate; in particular Q3 yields the dis- played principal formula for G0,a. The isolated longitudinal factor is S−1 times h−1d0(p)∗diag(g−1 a ), so S cancels before Fourier extraction. The filled pilot correction is
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Every denominator is regular: ga, m(q0), U and the nonzero-alias blow. Bounds are common for
G0 and C decayj(l) grade(l)−2 otherwise. Algebraically Q3 gives El −rl,a = SGl,a, where El is the
original normalized covariance weight and rl,a its directional rebate; in particular Q3 yields the dis-
played principal formula for G0,a. The isolated longitudinal factor is S−1 times h−1d0(p)∗diag(g−1
a ),
so S cancels before Fourier extraction. The filled pilot correction is
diag(Gl,a)d(L) 0 (ql)h−1d0(p)∗diag(g−1 a ).
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diag(Gl,a)d(L)
0 (ql)h−1d0(p)∗diag(g−1
a ).
Its leading term is diag(fa) flux1(q0) diag(g−1 a ) at l = 0, and S diag(Dl,a) flux1(ql) diag(g−1 a ) other- wise. Here the flux is the transpose of the directionwise patch multiplier on one-forms, and d(L) 0
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Its leading term is diag(fa) flux1(q0) diag(g−1
a ) at l = 0, and S diag(Dl,a) flux1(ql) diag(g−1
a ) other-
wise. Here the flux is the transpose of the directionwise patch multiplier on one-forms, and d(L)
0

has its prescribed scaled normalization. For the bound, the leading term has decayj(l), while the correction has decayj(l) grade(l)−2 times a signal bounded by a constant times grade(l); all remaining factors are uniformly bounded. Since grade(l) ≥1, the correction has the same decayj bound. This formula supplies the analytic value at zero; no holomorphic extension of the uncancelled isolated longitudinal factor is asserted. This establishes analytic regularity of the filled pilot. Its inputs are all derived within the construction: the covariance recursion in Subsection 11.15, the stationary equations in Subsection 11.10, their finite-torus restoration in Subsection 11.9, and the

reciprocal-shift and reflection identities in Subsection 11.12. Together these arguments establish the analytic predictor and its actual stationary response, including the harmonic fibre. The lifted statement is accordingly unconditional relative to the supplied construction: the realized predictor row is the supplied one, and its readout through pfTZ = Tpc (Theorem A.1) recovers the stationary response on the same domain. The five-row contract of Proposition 11.1 is the form in which these outputs are consumed by the flat convergence argument. The lift retains this cancellation order. In particular, realize the already cancelled predictor Afill as jfAfill l pc. Recovery by pf and jc returns l l , including its value at zero, because pfjf = pcjc = I. No value for the separate factor S−1 at Afill zero is needed in either realization or recovery.
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reciprocal-shift and reflection identities in Subsection 11.12. Together these arguments establish
the analytic predictor and its actual stationary response, including the harmonic fibre. The lifted
statement is accordingly unconditional relative to the supplied construction: the realized predictor
row is the supplied one, and its readout through pfTZ = Tpc (Theorem A.1) recovers the stationary
response on the same domain. The five-row contract of Proposition 11.1 is the form in which these
outputs are consumed by the flat convergence argument. The lift retains this cancellation order. In
particular, realize the already cancelled predictor Afill
as jfAfill
l pc. Recovery by pf and jc returns
l
l , including its value at zero, because pfjf = pcjc = I. No value for the separate factor S−1 at
Afill
zero is needed in either realization or recovery.

11.14 The filled dual predictor on two-forms

Continue the pilot notation, with Hf, Hc the wing/land pulse, d = d1(p), δ = d(L) 1 (q0), Fr = hingesr and Vr = fluxr, transposed at opposite momentum. Here d∗means the Fourier adjoint on real momentum and its analytic continuation using −p. Define
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Continue the pilot notation, with Hf, Hc the wing/land pulse, d = d1(p), δ = d(L)
1 (q0), Fr = hingesr
and Vr = fluxr, transposed at opposite momentum. Here d∗means the Fourier adjoint on real
momentum and its analytic continuation using −p. Define
Λ = m(q0)−1 mask2(RL(q0,a)−1), y = |m(q0)|2hc/hf(q0), xa = Ta/ga,
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Λ = m(q0)−1 mask2(RL(q0,a)−1),
y = |m(q0)|2hc/hf(q0),
xa = Ta/ga,
B = Hf(q0)[diagpairs(xa + xc) −S mask2(x)], M = diag(ga).
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B = Hf(q0)[diagpairs(xa + xc) −S mask2(x)],
M = diag(ga).

The pair diagonal sums the two coordinate entries for each two-form orientation {a, c}. The principal wire is W0 = yV2(q0) + [BΛdM −Hf(q0)Λd diag(Ta)]d∗.

The pairwise identity 1−mask2(1−Sx) = S[diagpairs(xa +xc)−S mask2(x)], gives, by substituting Q3 in each two-form diagonal entry,

SB = Hf(q0) −yV2(q0)Hc(p)F2(q0).
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SB = Hf(q0) −yV2(q0)Hc(p)F2(q0).
Together with F2(q0)Λ = I and F2(q0)δ = dF1(q0) this identifies W0 with the original stationary dual expression offzero. Every displayed regular factor has a common analytic bound, so the filled expression extends through zero. For l ̸= 0 put Rl = blow2 l hf(ql), yl = |m(ql)|2hc/Rl and N = S2I −SHc(p)dMd∗. The actual stake branches are Wl = Hf(ql)d(L) 1 (ql) diag(Dl,a)V1(ql)d∗+ ylV2(ql)N.
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Together with F2(q0)Λ = I and F2(q0)δ = dF1(q0) this identifies W0 with the original stationary
dual expression offzero. Every displayed regular factor has a common analytic bound, so the filled
expression extends through zero.
For l ̸= 0 put Rl = blow2
l hf(ql), yl = |m(ql)|2hc/Rl and N = S2I −SHc(p)dMd∗. The actual
stake branches are
Wl = Hf(ql)d(L)
1 (ql) diag(Dl,a)V1(ql)d∗+ ylV2(ql)N.
The identities portD=scaleD, portR=scaleR and N = S2(1−cloud) identify them with the original dual on the regular set. Only the separated nonprincipal blow is inverted. The nonzero-alias lower bound gives R−1 l its grade(l)−4 bound and Dl its grade(l)−2 bound. These, with signal and flux factors, yield C decayj(l). The previously retained relay/reflection proof exports these bounds to real Fourier rows. Source readout preserves this branchwise matching and its opposite-momentum adjoints; regularity of the filled expression does not assert regularity of each uncancelled term in the original formula. The recovery is applied to the complete regular branch. If Wl maps coarse to fine two-form coordinates, set f Wl = jf,2Wlpc,2. Then pf,2f Wljc,2 = Wl, with the same equality for the
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The identities portD=scaleD, portR=scaleR and N = S2(1−cloud) identify them with the original
dual on the regular set. Only the separated nonprincipal blow is inverted. The nonzero-alias lower
bound gives R−1
l
its grade(l)−4 bound and Dl its grade(l)−2 bound. These, with signal and flux
factors, yield C decayj(l). The previously retained relay/reflection proof exports these bounds to
real Fourier rows. Source readout preserves this branchwise matching and its opposite-momentum
adjoints; regularity of the filled expression does not assert regularity of each uncancelled term in the
original formula. The recovery is applied to the complete regular branch. If Wl maps coarse to fine
two-form coordinates, set f
Wl = jf,2Wlpc,2. Then pf,2f
Wljc,2 = Wl, with the same equality for the

principal value at zero. Every product in the regular matching identities is lifted with the section and readout of its actual intermediate degree; cancellation of pkjk = I recovers the displayed order of multiplication. The analytically continued transpose remains the original opposite-momentum operation before realization, while section adjoints recover the real-slice adjoints. Consequently the lift preserves the denominators that are separated, the branchwise distinction between l = 0 and l ̸= 0, and the dependency on the secured relay/reflection bounds. Fourier export is performed after forming these bounded regular branches.

11.15 Cumulative filters and covariance propagation

A finite history beginning at order j is empty or (L, J; v) with odd L ≥3 and j ≤J ≤j + 1. The tail v begins at order J. Its mass is Mv, the product of its block sides, with empty mass one. For centered box filters write AL(p) = Q3 ν=0 pL(pν); odd-side composition is ALM(p) = AL(Mp)AM(p). Define E∅= 1, E(L,J;v) = AJ−j LMvEv, Sv = Aj MvEv.
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A finite history beginning at order j is empty or (L, J; v) with odd L ≥3 and j ≤J ≤j + 1. The
tail v begins at order J. Its mass is Mv, the product of its block sides, with empty mass one. For
centered box filters write AL(p) = Q3
ν=0 pL(pν); odd-side composition is ALM(p) = AL(Mp)AM(p).
Define
E∅= 1,
E(L,J;v) = AJ−j
LMvEv,
Sv = Aj
MvEv.
Then S(L,J;v)(p) = AL(Mvp)JSv(p). (QF1)
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Then
S(L,J;v)(p) = AL(Mvp)JSv(p).
(QF1)

The exponent J −j is zero or one: only stage changes add extra factors at successive cumulative lengths, while the principal filter retains exactly its original exponent j. Finite sums and induction give entire periodic filters, signed-coordinate symmetry, value one at zero and Sv(p) = Sv(− ̄p). On | Im pν| ≤z/L0, |AM(p)| ≤e4zM/L0; on the fixed real principal domain with M ≤L0, |AM(p/L0)| ≥e−4cboxM/L0. For a history of total mass Mv ≤L0, cumulative masses increase by at least three, giving |Ev(p)| ≤e8zMv/L0, |Ev(p/L0)| ≥e−8cboxMv/L0. (QF2)

Indeed a new factor costs 4zMv/L0 and the prior bound at most 8zMv/(3L0); their sum is at most 8zMv/L0. The lower exponent has the identical calculation; absent factors use mass monotonicity. The principal alias bound |AM(aliasl(p))| ≤B4 mix(l)−1 therefore yields, on taking L0 = Mv in QF2, |Sv(aliasl(p))| ≤B4je8z mix(l)−j, (QF3)

|Sv(p/Mv)| ≥e−4jcbox−8cbox > 0. (QF4)
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|Sv(p/Mv)| ≥e−4jcbox−8cbox > 0.
(QF4)

All constants depend on the initial order, not depth or total dilation, on the common stated neighborhood. In particular all alias-filter functions have a common holomorphic neighborhood and the indicated uniform alias decay. Retain the longitudinal rail and its opposite-momentum product:

Dv,a = SvrMv(pa), PL,J,a = AJ LrL(pa), Y (f)(p) = f(p)f(−p).
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Dv,a = SvrMv(pa),
PL,J,a = AJ
LrL(pa),
Y (f)(p) = f(p)f(−p).

Rail composition gives D(L,J;v),a(p) = PL,J,a(Mvp)Dv,a(p). (QF5)

On real momentum Dv,a(−p) = Dv,a(p), so Y (D) = |D|2 ≥0. The propagated alias operator is

Fv,af(p) = M−2 v X l∈(Z/MvZ)4 Y (Dv,a)(aliasl(p))f(aliasl(p)). (QF6)
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Fv,af(p) = M−2
v
X
l∈(Z/MvZ)4
Y (Dv,a)(aliasl(p))f(aliasl(p)).
(QF6)

Define LL,J,a by the one-patch formula. The radix bijection of aliases for LMv and QF5 prove

F(L,J;v),af = LL,J,a(Fv,af). (QF7)
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F(L,J;v),af = LL,J,a(Fv,af).
(QF7)
The prefactor is exactly M−2 v . The lift retains each history, radix multiplicity and rail in this opera- tion; normalized Haar frames do not modify its dimensional scaling. These finite alias identities pre- cede the remaining covariance recursion and its analytic estimates. The radix proof of QF7 substi- tutes the nested finite sum, factors the outer Y (PL,J,a) by periodicity and uses (LMv)−2 = L−2M−2 v . The empty propagated operator is the identity. Every step adds an independent local covariance k−1. Let Nv,a sum the covariances propagated through all prefixes: the empty history has none, and an extension adds the empty prefix and extends every old prefix by the new patch. Retain them through
Source page 51.
Searchable transcription
The prefactor is exactly M−2
v . The lift retains each history, radix multiplicity and rail in this opera-
tion; normalized Haar frames do not modify its dimensional scaling. These finite alias identities pre-
cede the remaining covariance recursion and its analytic estimates. The radix proof of QF7 substi-
tutes the nested finite sum, factors the outer Y (PL,J,a) by periodicity and uses (LMv)−2 = L−2M−2
v .
The empty propagated operator is the identity. Every step adds an independent local covariance
k−1. Let Nv,a sum the covariances propagated through all prefixes: the empty history has none,
and an extension adds the empty prefix and extends every old prefix by the new patch. Retain
them through
N∅= 0, N(L,J;v),a = 1 + LL,J,aNv,a, Dv,a = Fv,a(λ−1) + k−1Nv,a.
Source page 51.
Searchable transcription
N∅= 0,
N(L,J;v),a = 1 + LL,J,aNv,a,
Dv,a = Fv,a(λ−1) + k−1Nv,a.

Here λ is the initial Wilson quadratic divisor. Linearity and QF7 give the exact feedback

D(L,J;v),a = LL,J,aDv,a + k−1. (QF8)
Source page 51.
Searchable transcription
D(L,J;v),a = LL,J,aDv,a + k−1.
(QF8)

All sums are finite; for k > 0 real nonnegativity holds termwise. Alias permutation proves reciprocal periodicity, evenness and signed-coordinate covariance. This formula accounts for every earlier noise; retaining only the final noise would give a different covariance. Separating the principal alias gives, on the regular denominator set,

Dv,a(p) = Y (Dv,a)(p/Mv) λMv(p) + Hv,a(p) + k−1Nv,a(p), (QF9)
Source page 51.
Searchable transcription
Dv,a(p) = Y (Dv,a)(p/Mv)
λMv(p)
+ Hv,a(p) + k−1Nv,a(p),
(QF9)
Hv,a(p) = X Y (Dv,a)(aliasl(p)) M2v λ(aliasl(p)) . l̸=0
Source page 51.
Searchable transcription
Hv,a(p) =
X
Y (Dv,a)(aliasl(p))
M2v λ(aliasl(p))
.
l̸=0
The principal divisor includes its M2 v scaling. Neither auxiliary divided symbol is declared contin- uous at harmonic momentum; filled regular formulas supply that reconstruction separately. Both remainders are nonnegative on real regular momenta, and QF4 gives the nonvanishing principal numerator. The filters are literal normalized sums rM = M−1 PM−1 u=0 eiuz and
Source page 51.
Searchable transcription
The principal divisor includes its M2
v scaling. Neither auxiliary divided symbol is declared contin-
uous at harmonic momentum; filled regular formulas supply that reconstruction separately. Both
remainders are nonnegative on real regular momenta, and QF4 gives the nonvanishing principal
numerator.
The filters are literal normalized sums rM = M−1 PM−1
u=0 eiuz and
pM = M−1 M−1 X u=0 ei(u−⌊M/2⌋)z.
Source page 51.
Searchable transcription
pM = M−1
M−1
X
u=0
ei(u−⌊M/2⌋)z.

Partition 0, . . . , LM −1 as Mu + v to obtain rL(Mz)rM(z) = rLM(z). For odd sides, M⌊L/2⌋+ ⌊M/2⌋= ⌊LM/2⌋gives the centered composition. The principal lower bound used in (QF2) has coefficient one in front of its exponential, which is essential for arbitrarily long histories. To retain it, choose c± > 0 with c−|z| ≤|eiz −1| ≤c+|z| on a disk excluding other exponential zeros. The quotient extends continuously and is nonzero at zero, so such constants exist. The exact identity MrM(z)(eiz −1) = eiMz −1 gives |pM(z)| = |rM(z)| ≥c0 = min(1, c−/c+) > 0 for real z with M|z| in that disk, including zero. If M ≤L0, L0|z| ≤R0, then |pM(z) −1| ≤k0M/L0, k0 = c+R0. With u = |pM(z)|, x = M/L0, one has u ≥c0, u + k0x ≥1, and, using ey ≥1 + y,

uek0x/c0 ≥u + uk0x/c0 ≥u + k0x ≥1.
Source page 51.
Searchable transcription
uek0x/c0 ≥u + uk0x/c0 ≥u + k0x ≥1.

Hence u ≥e−(k0/c0)x, giving cbox = k0/c0 with no repeated fixed prefactor below one. Bounding each centered exponential by eM| Im z| proves the complex upper estimate. The source lift retains every prefix noise and this principal/remainder decomposition; neither frame integration nor a change of coordinates discards a covariance contribution.

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