restores the finite-torus normalizations; and Section 5.20 gives reciprocal-shift and reflection compatibility. Together these prove the analytic predictor and its actual stationary response, including the harmonic fibre.
The regular dual predictor on two-forms
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Continue the filled-potential-response notation. Let Hf(q) and Hc(p) be the fine and coarse pulse metrics, d = d1(p) and δ = dL 1 (q0). The adjoint d∗on real momentum is continued analytically using −p. Write Fr(q) = hingesr(q) for the patch multiplier on r-forms, and Vr(q) = fluxr(q) for its transposed symbol at −q. Define
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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).
Here diag
sums the two coordinate entries in each two-form orientation. The principal wire is the regular expression
Searchable transcription
W0 = yV2(q0) + BΛdM −Hf(q0)Λd diag(Ta) d∗.
For a vector
), let mask
) have diagonal entries
at orientation
, and let diag
) have entries
. The entrywise identity
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I −mask2(1 −Sx) = S diagpairs(xa + xc) −S mask2(x)
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and covariance identity (3) prove
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SB = Hf(q0) −yV2(q0)Hc(p)F2(q0),
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by substitution in each two-form diagonal entry. Together with F2(q0)Λ = I and F2(q0)δ = dF1(q0), this identifies W0 with the original stationary dual expression for p ̸= 0. Every factor of the displayed regular expression is uniformly analytic, so it extends the response through zero. For l ̸= 0, set Rl = blow2 l hf(ql) and yl = |m(ql)|2hc/Rl, and put
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N = S2I −SHc(p)dMd∗.
The remaining wire branches are
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Wl = Hf(ql)dL 1 (ql) diag(Dl,a)V1(ql)d∗ +ylV2(ql)N.
These are the nonprincipal dual branches, called stake in the source. The identities portD = scaleD, portR = scaleR and
cloud) on the regular set identify them with the original dual response. Their nonzero-alias divisor is separated, so no principal denominator is inverted. The lower bound gives
(grade(
) and
(grade(
) with the common analytic bounds. Signal and smoothing-flux factors then give
decay
), as for the bounded principal branch. Section 5.20 gives the reciprocal-shift and reflection identities needed for Fourier export. The algebra here identifies the regular symbols themselves.
Cumulative filters and propagation of the quadratic covariance
The cumulative-filter calculation closes the quadratic argument. Each one-step identity is composed along the same scale history, and the weighted alias sums remain absolutely convergent. The resulting covariance is therefore attached to the constructed history, rather than to an independently chosen continuum kernel. These identities are later used as the fixed linear comparison in the nonlinear predictor and in the free hierarchy.
A finite linear history begins at smoothing order
. It is either empty, or consists of a patch with odd block side
3, new order
with
+ 1, followed by a history beginning at
Let
be the product of its block sides, with
= 1. For the centered one-dimensional box filter 
, write
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3 Y AL(p) = ν=0 pL(pν).
Its exact composition identity for odd sides is
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ALM(p) = AL(Mp)AM(p).
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Define an extra filter recursively by E∅= 1 and
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E(L,J;v)(p) = ALMv(p)J−jEv(p), Sv(p) = AMv(p)jEv(p).
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The composition identity gives
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S(L,J;v)(p) = AL(Mvp)JSv(p). (QF1)
The exponent
is zero or one. Thus extra smoothing factors enter at the successive cumulative lengths, while the principal filter has exactly the original exponent
The filters are entire, periodic under reciprocal integer translations, invariant under signed coordinate permutations, equal to one at zero, and satisfy
These statements follow first for the finite box sums, then for
by finite products and induction.
There is a bound independent of both the number of steps and the total dilation. On the strip 
Im
, a box filter satisfies
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|AM(p)| ≤e4zM/L0.
When
, at real principal momenta
in the fixed domain one also has
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|AM(p/L0)| ≥e−4cboxM/L0.
For a history of total mass
, induction using each block side at least three gives
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|Ev(p)| ≤e8zMv/L0, |Ev(p/L0)| ≥e−8cboxMv/L0. (QF2)
To check the induction, a new extra factor costs 4
in the upper exponent, and the preceding bound costs at most 8
Their sum is at most 8
If no extra factor occurs, monotonicity of the mass suffices. The lower bound has exactly the same calculation with negative exponents and
in place of
At an alias indexed by
, the principal box estimate is
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|AM(aliasl(p))| ≤B4 mix(l)−1.
Combining this with (QF2), with
, proves
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|Sv(aliasl(p))| ≤B4je8z mix(l)−j. (QF3)
At the real principal alias the lower bound is
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|Sv(p/Mv)| ≥e−4jcbox−8cbox > 0. (QF4)
These bounds are uniform over all finite histories, with constants depending on the initial order only. In particular all alias-filter functions have a common holomorphic neighborhood and the indicated uniform alias decay. Include now the longitudinal edge box filter
), and set
Searchable transcription
Dv,a(p) = Sv(p)rMv(pa), PL,J,a(p) = AL(p)JrL(pa), Y (f)(p) = f(p)f(−p).
The exact rail composition and (QF1) imply
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D(L,J;v),a(p) = PL,J,a(Mvp)Dv,a(p). (QF5)
On real momenta
For any scalar symbol
, define its propagated alias sum by
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Fv,af(p) = M−2 v X l∈(Z/MvZ)4 Y (Dv,a)(aliasl(p))f(aliasl(p)). (QF6)
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The normalization is M−2 v . Define LL,J,a by the same formula for one patch. The radix bijection between aliases of LMv and pairs of aliases of L and Mv, together with (QF5), gives the exact identity F(L,J;v),af = LL,J,a(Fv,af). (QF7)
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Indeed substitute the nested alias formula in the sum, factor out the outer Y (PL,J,a) by periodicity, and use (LMv)−2 = L−2M−2 v . The empty history has F∅,af = f. Every renormalization step adds an independent local covariance k−1. Let Nv be the sum of the covariances of these noises propagated through their respective prefixes. More explicitly the empty history has no prefixes; extending a history introduces the empty prefix and extends each old prefix by the new patch. Thus
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N∅= 0, N(L,J;v),a = 1 + LL,J,aNv,a.
If
) is the initial Wilson quadratic divisor, the cumulative diagonal covariance is
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Dv,a = Fv,a(λ−1) + k−1Nv,a.
Linearity and (QF7) prove the feedback equation
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D(L,J;v),a = LL,J,aDv,a + k−1. (QF8)
This formula accounts for every earlier noise; retaining only the final noise would give a different covariance. All sums are finite. Real nonnegativity follows term by term when
0, and reciprocal periodicity, evenness and covariance under signed coordinate permutations follow by the corresponding permutation of aliases.
Separating the principal alias in (QF6) yields the pole decomposition
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Dv,a(p) = Y (Dv,a)(p/Mv) λMv(p) + Hv,a(p) + k−1Nv,a(p), (QF9)
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where λMv includes the M2 v scaling of the principal divisor and
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Hv,a(p) = X Y (Dv,a)(aliasl(p)) M2v λ(aliasl(p)) . l̸=0
This is an algebraic identity on the regular set of the denominators. The auxiliary divided symbols are not asserted to be continuous extensions at harmonic momenta. Their regular parts and their harmonic reconstruction are handled by the pole-removal construction already given above. Both 
and the propagated noise are nonnegative on real regular momenta, and (QF4) supplies the uniform nonvanishing principal numerator required there.
The one-dimensional filters in this construction are literal normalized finite sums:
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M−1 X u=0 eiuz, pM(z) = 1 M−1 X rM(z) = 1 u=0 ei(u−⌊M/2⌋)z. M M
Partitioning the indices 0
proves
). For odd sides the centers satisfy
, proving the corresponding identity for
The principal lower bound used in (QF2) has coefficient one in front of its exponential, which is essential for arbitrarily long histories. To derive it, choose a disk on which
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c−|z| ≤|eiz −1| ≤c+|z|.
Such positive constants exist because the ratio extends continuously with nonzero value at zero, and the disk excludes the other zeros. The geometric-sum identity
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MrM(z)(eiz −1) = eiMz −1
gives
for real
with
in that disk, where
= min(1
and
, the centered finite sum also gives
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|pM(z) −1| ≤c+R0 M/L0.
Put
, and
. Then
and
1. Since
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uek0x/c0 ≥u + uk0x/c0 ≥u + k0x ≥1.
Thus
. This proves the stated principal estimate with
, without multiplying a fixed prefactor less than one at every step. The complex-strip upper bound follows simply by bounding every exponential in the centered average by
Source: A Wilson Yang–Mills Construction: A Natural-Language Mathematical Exposition. Both papers belong to one Zenodo record. No external funding.