14 The analytic stationary graph, loop moments and the cali- brated coupling history · Part 3

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The frame factors preserve the constant-output statement by their translation convolution identities.
Odd blocking is the hypothesis that makes this centered-gradient readout agree with the installed
fine difference. Exponential column bounds justify passing from the installed periodic identity to
the absolutely summed cover columns.
After substituting these columns in a two-difference pattern, one fixed choice of its two out-
put types has coefficient uf0(z0)f1(z1), where P f0 = P f1 = 0 by (LM3).
Its zeroth total is
u(P f0)(P f1) = 0. Its displacement is (c1 + z1) −(c0 + z0), with fixed carried offsets c0, c1. Split-
ting its first moment gives

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(
)
(
X
f0)
X
z
(c1 + z)if1(z) −
X
z
(c0 + z)if0(z)(
X
f1)
u
= 0.
(LM4)

The absolute first moments exist by the column width bounds. Summing over residues, types and patterns is justified by (LM1) and the charged bound on the full substituted family. Thus inherited loops have the exact two moment identities, with their block price, rather than merely small moments.

Joining two families means enumerating their disjoint union. Absolute sums add, so both moment identities are preserved. Blank initial loops have zero coefficients and satisfy them. The coefficientwise limiting argument (SH1)–(SH3) preserves them as well: first pay the common exponentially weighted tail, then pass to the limit on a finite list. This supplies the full-family justification used to turn retained moments into omitted-tail moments in (TF18). Any fresh loop must satisfy the same identities before it is joined; they cannot be inferred from gauge invariance or the quadratic-potential normalization alone.

Source–readout formulation. The zeroth and first unwrapped displacement moments define a readout of an atom family which forgets everything except five numbers per plaquette and color type. Write for the block transport of the derivation above: is the family obtained by substituting the installed columns into the two-difference patterns of

Proposition 14.10 (The moment readout and the quiet fibre). On the additive monoid of absolutely summable two-curvature atom families with charged exponential bounds (disjoint union as addition), define for each plaquette and color type

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m(A) =
X
a
qa,
X
a
qaδa

∈R × R4.

Then m is additive, the quiet families m are closed under joining, under the block transport of (LM1)–(LM4) (indeed m for every transported two-difference family), and under the coefficientwise limits (SH1)–(SH3). Blank initial loops are quiet. A fresh loop is quiet by (LM0), and quietness cannot be deduced from gauge invariance or the quadratic normalization alone.

Proof. Additivity is additivity of absolute sums over a disjoint union. Transport is (LM3)–(LM4). For limits one first pays the common exponentially weighted tail and then passes to the limit on a finite list. The last sentence records that (LM0) used the order four vanishing produced by subtracting the kinetic readout, which is additional to transversality.

The moment readout is many-to-one. The construction never inverts it; it only proves that every family it produces lies in the fibre over zero, and that this fibre is stable under the three operations applied to atom families (joining, block transport, coefficientwise limits). That stability is the operation-compatibility statement required for the later use in (TF17)–(TF19), where retained moments are converted into omitted-tail moments.

14.7 The one-loop drift used by the finite histories

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The finite coupling drift comes from the committed path coefficient and the bounded scalar reset,
not directly from a formal beta function. At a transition with block length L, put μ = L−1. The
determinant calculation on the retained physical path defines a coefficient αN,i with sign appropriate
to descent from level i + 1 to i. We explain the logarithm and its uniform remainder.
For a fixed color direction, the continuum pole of the relevant second external-momentum
coefficient is
!

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k2
1
|k|6 −4k2
1k2
0
|k|8 −
1
|k|4
pG(k) = 4CA
.
(DR1)

The coefficient includes the vector determinant with weight 12 and the gauge determinant with weight1. These are the same two determinant weights kept in the massive-symbol calculation of the trace. The finite symbol coefficient), on the angular cube [, satisfies

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|fμ(k)| ≤
A
(|k| + μ)4 ,
|fμ(k) −pG(k)| ≤D
|k|3 + Dμ2
|k|6
(|k| ≥μ),
(DR2)

where0 depend on the fixed analytic widths, coefficient and ellipticity bounds, and not on , the torus period or its terminal cutoff. The input paths have their fixed quadratic normalization and the allowed higher soft jets; the latter preserve the continuum pole.

Here is the matrix estimate yielding (DR2). At scale, the zero-field massive matrix, one-field vertex, and two-field vertex have sizes 1. Coercivity bounds the inverse by Differentiate the identity = 1 to obtain and its second derivative, the sum of the two products with two first vertices minus the product with one second vertex. Along a momentum direction scaled by, all jets of each inverse still have size. The lower determinant bubble therefore has size, and its second unscaled external-momentum coefficient has size. The upper insertion obeys the same rough bound. Summing the finitely many matrix

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entries gives the first bound in (DR2).
For |k| ≥μ, take T = |k|. The zero-field symbol differs from its continuum quadratic part
by O(T 3 + μ2), and the first vertex differs from its continuum linear part by O(T 2 + μ2/T). Use
A−1−B−1 = A−1(B−A)B−1 and telescope the factors of the determinant bubbles. Their relative er-
ror has size T +μ2/T 2; their second momentum coefficient consequently has error C(T +μ2/T 2)T −4,
which is the second bound in (DR2). The upper insertion contributes bounds of these same or
smaller orders. The origin-jet cancellations, analytic derivative bounds, inverse formulas and ex-
plicit continuum vertex contraction are developed in the weighted massive-symbol and logarithmic
momentum calculations below. This argument uses them before integration, so its error bound is
absolute.

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To integrate, split the angular cube into |k| ≤μ, μ < |k| ≤1, and its part outside the unit ball.
The first bound in (DR2) gives

Z

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 ≤Aμ−4 π2μ4
2
= π2A
2 .
|k|≤μ
fμ(k) dk

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Outside the unit ball it gives a bound A times the volume of the angular cube. On the annulus the
second bound gives
Z

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μ<|k|≤1
D
|k|3 dk = 2π2D(1 −μ) ≤2π2D,
Z
|k|6 dk = π2D(1 −μ2) ≤π2D.
(DR3)
μ<|k|≤1
Dμ2

All these errors are bounded independently of 0 The pole integral is exact. Rotational symmetry gives, on this annulus,

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Z
k2
1
|k|6 dk = 1
4Iμ,
Z k2
1k2
0
|k|8 dk = 1
24Iμ,
Iμ =
Z
|k|−4 dk = 2π2 log(1/μ).
(DR4)

For the mixed fourth moment, symmetry makes all off-diagonal moments equal. Expanding the fourth powers of and, and rotating these directions to a coordinate direction, shows that the diagonal fourth moment is three times the mixed one. Summing over the four coordinates then gives the factor 124. Substitution of (DR4) into (DR1), with Fourier weight (2, gives

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(2π)−4
Z
μ<|k|≤1
pG(k) dk = −11CA
24π2 log(1/μ) = −s log L.
(DR5)

Equations (DR2)–(DR5) bound the difference between the infinite-volume coefficient and this logarithm by a pre-block constant.

We now connect this integral to the coefficient of the actual stationary path. It is important to distinguish three determinant films:, the film of the actual stationary integration;, its core film before the stationary carrier change; and, the auxiliary rank film. Each film is a matrix of analytic functions of the external momentum, with indices consisting of a direction and a color. Their scalar reading is

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αa(M) = 2 Re leapTip(profitable M)(e0, 1, (a, a)).
(DP1)

The operation leapTip is the coefficient of degree two in that external-momentum direction, with its Taylor factorial; profitable passes from the determinant film to its curvature tensor using its analytic contraction identities. Thus is linear on films satisfying the common bounds, and is the specific normalized kinetic coefficient, not the full film norm.

Here are the quantitative comparisons used in the transition. Fix an analytic width with128, a block patch of stage, its penalty, a positive stationary width, and all the common metric, covariance and path bounds. The required hypotheses are: the input path is attached to its quadratic metric and has its prescribed analytic bounds; that metric has the uniform floor on every torus; the coarse metric has its analytic strip bound; the stationary graph is critical, guessed by the installed tower, and satisfies the gap and heat identities; the same forest and tag are used in its determinant realization; and the relevant torus sides form an auxiliary sequence tending to infinity. These are precisely the linear-horizon and stationary-graph data already constructed. The input loop-tensor norm may have its own bound Λ, whereas the comparison constants below depend only on the fixed analytic and matrix bounds specified before the block length. They do not grow with Λ. Let0 bound the inverse matrices in the determinant calculation, let0 bound the common zero-, one-and two-field film factors, and let0 bound their carrier difference. All three are obtained from the fixed stage, kernel, penalty and common analytic bounds before is chosen. More explicitly the common factor bound is the sum of the absolute bounds for the leg, core, rank and frozen metric factors; the carrier bound is the common bound for the difference between the core and rank factors. Put

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D∗= FK∗F, E∗
= D∗M∗+ FK∗, R∗= E∗+ (E∗F + FM∗D∗)M∗+ FM∗FK∗.
(DP2)

This is the source’s exact reserve polynomial. It arises by subtracting inverse factors with , then telescoping the zero-, one-and two-field determinant products. For

example the first inverse difference costs; multiplying by the common first factor and adding the factor difference costs; the remaining two placements and their cross product give the last line. The trace estimate is a relative rank estimate, so this subtraction introduces no count of all block sites. In detail, if Loc Loc, the individual films are bounded on width

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L4 2r(FM∗+ FM∗FM∗).
(DP3)

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The separate difference estimate is

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Mcore −Mrank
bounded on width h/2 by R∗.
(DP4)

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Here is why (DP4) has no block-volume factor.
The matrix weights are invertible analytic
frames SX(p), SY (q).
A bound for a matrix U means a Euclidean operator-norm bound for
eU = SX(p)−1USY (q). For the difference estimates the stronger bound is used:

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| tr( eUV )| ≤C∥V ∥2→2
for every compatible V.
(DP10)

It is subadditive, is invariant under transposition with dual frames, and multiplies with an ordinary operator bound on either side. At equal source and target frames, taking = 1 bounds the actual trace by. A general matrix with operator norm only has price dim( in (DP10); using that conversion would incur the unwanted

The exact difference between the core and rank quadratic films cancels their common sector and pure Gram terms. Write for the shear row, for its localized emitter correction, and for the fixed block inclusion. Expanding ( gives its reflection part. The complete remaining difference is the sum of the local house film and this reflection part, minus and the finite motive film; is the localized orbit correction. This identity holds separately for the zero-, one-and two-field matrices, with all their Leibniz placements. The common sector cancels before any trace is bounded.

We derive the uniform trace-ideal estimates before using them in the difference. For a finite matrix, set sup

For, the trace is, so its size is at most. Decomposition into columns bounds the size by the sum of their norms. A matrix with a fixed-dimensional source consequently has price. Multiplication on either side by a bounded matrix multiplies this price by its operator norm, by trace cyclicity. Apply these statements after the analytic changes of frame in (DP10).

One further factor has a source dimension growing with, and needs an actual summability argument. Put, let be the lattice Laplacian symbol, and define

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dμ(k) =
μ2
ω(k) + μ2 ,
J = 16
∞
X
n=0
(n + 1)−2 < ∞.

On the strip of width 1128, choose the centered real-momentum representative before enumerating aliases. The massive lower bound gives

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|dμ(kl)| ≤
2
(1 + pdist(l))2 .

At = 0 this is the origin massive bound. At= 0, centered alias separation bounds the symbol scale below by (1 + pdist(; strip ellipticity bounds the denominator below by half the square of that scale. Division gives the inequality. Changing the centered representative permutes the aliases. There are at most (2 aliases at distance, because their lifts inject into [. Therefore

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X
l
|dμ(kl)3| ≤8
X
(2n + 1)4
(n + 1)6 ≤8J.
n≥0

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The alias-diagonal matrix d3
μI has trace-ideal price 8n0J, with fixed n0 = | Loc0 |. The Fourier
frame and inverse frame each cost at most e4/128, leaving the block-independent price

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Q = e4/128(8n0J)e4/128.

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This origin estimate controls the zero-, one- and two-field matrices of the regularizing ninth
power. Write P = μ2(A+μ2I)−1, where A is the sector Gram matrix, and U = P 3. The preceding
estimate applies to U at zero field. If C3 bounds its three ordinary film matrices, the corresponding
three matrices of U3 have trace-ideal price at most 20C2
3Q. In fact every zero-field product has
an unmarked U, every first-field product has two, and every second-field product has at least one.
Put one unmarked factor in the trace ideal and the other two in operator norm. The second-field
Leibniz formula has three placements of a second-field matrix and six ordered placements of two
first-field matrices. Their sum and the lower orders are covered by 20. This never converts an
L4-dimensional bound into a trace bound. The regularizer is μ2P 9, so its price is μ2B, where
B = 20C2
3Q.

We now price each localized composition. We specify how the derivative prices used here are obtained. If an analytic finite-dimensional map is evaluated at, define, for a derivative bound

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m
X
X
π∈Pn
∥D|π|F(z)∥c|π|,
(DP12)
Em(F, z, c) =
n=0

where is the finite set of partitions in the iterated chain rule, with its multiplicities, and contributes the value bound. If an inner map takes the value at the origin and its derivatives through order have the prescribed tested bound, its composite is bounded by (DP12): differentiate successively, grouping the directions according to the block that enters each derivative of the inner map, and sum their norm products. The source’s apollo and encore are precisely these finite sums. For a product use and inner price; inversion of a group matrix is its adjoint and preserves these derivative bounds. The same construction applies to the matrix exponential at zero, the chosen analytic group logarithm at the identity, and the analytic nearest-group projection in its fixed neighborhood.

These rules generate the charter and emission prices from the actual stencil. A link perturbation has its explicit factor. On a length- path, the product rule chooses the differentiated links. If distinct links are differentiated, there are at most choices and their derivative bounds contribute ; all undifferentiated group factors have norm one. The remaining assignments are the finite partitions of the derivative directions. Thus every fixed derivative order has a common bound independent of. Patch averages are probability averages. Moving from the padded patch to its translated subpatch costs at most its fixed stage-dependent cardinality ratio, since both counts are bounded multiples of. The frame estimates already proved give the corresponding finite derivative bounds for the recentered gauge transformations. Form the connector by multiplying the two endpoint frames and the path transport, average it, and apply the nearest-group projection; (DP12) pays all these operations. Multiply by the coarse-link factors and the endpoint frames, and

apply the logarithm. This constructs the finite positive prices denoted springs for the charter and rocket_fare for the emission in FrameEst/Stencils. Every logarithm is evaluated at the identity in this centered calculation. None of these steps takes a supremum over a number of arbitrary independent site values.

Here is the last conversion into row norms, including its block powers. Let2 be the number of inserted field directions and let bound their fixed Fourier-frame readings. Put = muralRoom(, including the fixed stage enlargement of those readings. The scalar directional estimate for the charter is at most

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2n+1| springs(J, n + 1, c′)|;

the corresponding emission estimate, after changing from the patch probability average to the block-site probability average, is at most

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portrait(J)2n+1| rocket_fare(J, n + 1, c′)|.

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The factor 2n+1 pays the real-and-imaginary expansions of the n inserted directions and the one
tested input direction. Conversion of the finitely many color coordinates to their Euclidean norm
multiplies these by √n1 and √n0, respectively. For the emission the probability-average estimate
is a sum of squares divided by L4; removing that denominator and taking a square root contributes
exactly L2. Taking the maximum, or the larger finite sum, over n = 0, 1, 2 gives prices a and h
independent of L, with charter price a and emission price L2h. These are RealGap.CharterR.costs
and Chromatic.costs. The same finite derivative and Fourier-frame bounds give the sector price

; the positive massive inverse and its differentiated inverse identities give the regularizer price . The house kernel has fixed dimension and the already constructed inverse bound, giving its ordinary price Each of thus bounds the zero-, one-and two-field matrices on the same strip. The house kernel has fixed source dimension Loc, so its trace-ideal price is. Film multiplication costs at most four times the product, covering its four second-field Leibniz terms. The house price is consequently

Composition of the regularizer with the site-to-color row cancels = 1. Its trace-ideal price is and its ordinary price 4. The shear row has ordinary price 4, so the corrected row has price. The two reflection products have combined trace-ideal price

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CR = 4rt + 4(4p)r.

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For the orbit term let si bound the fixed stage row in direction i, and put S(z) = P3
i=0 4(4siz)si.
Its unweighted trace price is μ2S(B). The two changes to the dual massive frames cost at most
e4μ−1 each, yielding
CO = e4S(B)e4.

The motive film factors through the fixed-dimensional space, with inclusions of ordinary price one; its trace-ideal price is 4. All widths and derivative orders have been fixed before, and each cancellation of a block power has been displayed. These four estimates supply (DP10). Composing a difference factor with an ordinary bounded factor keeps (DP10). In particular the house film has price; the two reflection products have combined price has price 16, including its two film-composition factors; and the motive film has price 4 Loc. Thus one may use exactly

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K∗= CH + CR + 16CO + 4| Loc0 |.
(DP11)

Each price here is the absolute weighted column sum of its stated local factor with its zero-, one-and two-field placements, not a norm of a full trace. The analytic row and localized column estimates bound those sums by the fixed stage and strip prices. They use the same128, branch bounds and common stationary rows as (DP3), and are uniform in. The inverse identity gives difference price. For a two-field determinant frame, the direct term difference costs . Telescoping its three-factor bubble gives ( Taking its trace through (DP10) gives precisely (DP2) and (DP4). The factor 12 in the coreminus-rank determinant may be discarded when increasing this bound. This proves the relative projection estimate while retaining the factor in the unrelated individual estimates (DP3). The stationary carrier change has zero kinetic reading:

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αa(Mact −Mcore) = 0.
(DP5)

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To see this, write its diagonal film as 1
2(W(p) + W(−p) −V (p)/2). The stationary tax row has
zero jets below order two. Its mixed carrier row has zero jets below order one, by the stationary
graph identity. Hence each summand of W, a product of these two rows, vanishes to order at least
three. Every summand of V contains two tax rows and vanishes to order at least four. These films
are analytic on the common strip, so their external degree-two coefficient is zero. Linearity proves
(DP5).
For an analytic scalar function bounded by A on width r0, two successive Cauchy derivative
estimates give

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∥D2f(0)∥≤
A
r0/2
1
r0/4.

For the normalized reading (DP1), the source keeps the slightly weaker explicit price

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B(r0, A) =
A
r0/4
1
r0/8∥axes(e0)∥2.
(DP6)

The factor two in (DP1) cancels the Taylor factorial. Apply this estimate to (DP4), with and use (DP5). The exact projection bound is

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|αa(Mact) −αa(Mrank)| ≤B(h/2, R∗).
(DP7)

No inverse power of the lattice period and no multiplier occurs in (DP7).

There is one more comparison, which is an equality rather than a bounded error. Let Π be the analytic film of the massive pole test on the fine lattice, with. Its finite-period analytic tensors approximate that test with error cents(, where0, and the rank film approximates the same test pulled back through the stationary uniform map with error cents(, where for fixed. The constants and analytic strip in these approximation statements may depend on. They are used for identification only, not as the uniform bound in (DP7).

Fourier evaluation of the stationary pullback gives the exact alias formula

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H(p) = L4 X
l
S
(p + 2πl)/L

[nl
−(p), nl
+(p)].
(DP8)

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Here S = Πμ, and nl
± are the incoming and outgoing stationary response columns. The analytic
contraction identities of S remove longitudinal columns. For l̸= 0, the remaining response columns
each vanish to order two at p = 0; their product therefore has no degree-two coefficient. For l= 0,
the primal and second-order stationary identities identify its contribution, through degree two, with

). The discrepancy vanishes to order three. Taking two momentum derivatives introduces another, so the coefficient multiplier is = 1. Consequently

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αa(Mrank) = αa(Πμ).
(DP9)

The finite-period approximation just described proves that the analytic pullback in (DP8) is the rank film: let the auxiliary coarse and fine sides tend to infinity at fixed, keeping the same stationary graph and tags. The vanishing approximation errors identify their Fourier tensors and derivatives. The reality, gauge and signed symmetries of the actual pole test give its contraction identities and zero lower jets. No assumption about a momentum continuation between different lattice meshes is used.

Combining (DP7), (DP9), and (DR2)–(DR5) now gives the exact allowance used in the transition: log (DR6)

where0 is the bound for the massive-symbol integral remainder. The stage construction fixes, and hence, before the block length. Afterwards it imposes one lower bound on odd, and post-block bounds on the analytic norm and loop allowance of the output path. In the transition notation these latter bounds have the forms Θ, and ΛΛΛ They may depend on; they do not enter The stationary tensor is produced using arbitrary growing torus sequences satisfying the stated floor and stationary identities; after coefficient identification it is one cover path and works on every realized torus satisfying the construction’s patch and support margins. Thus the auxiliary large-volume limit imposes no extra period-dependent error on (DR6), and the estimate is uniform over the terminal cutoffs whose paths have those same pre-block bounds. The scalar reset has∆1. Since its actual coarse coupling is + ∆, we obtain

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|βN,p,i+1 −βN,p,i −s log Li| = |αN,i + ∆+ s log Li|
≤|αN,i + s log Li| + 1 ≤di+1.
(DR7)

Thus the one-step drift used in the shooting and running arguments includes both the calculated path coefficient and the actual nonlinear reset. The choices of large block lengths and late coupling entry below pay this drift simultaneously with all the local activity and insertion bounds.

Source–readout formulation. The source records here are the three films with their frames, the stationary graph and the tags; the single reader is. Three different relations to the reader occur and must not be merged: (i)ker exactly (DP5): the carrier change is a redundancy of the realization invisible to this reader, although it is visible to the film norm; (ii) is a bounded, generally nonzero discrepancy (DP7): a controlled approximation; (iii) and Π have equal readings (DP9) although the films differ at higher order and at nonzero aliases: equality on the readout, not of sources. Accordingly the status of the slope is the following.

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The number s = 11CA/(24π2) is derived within the supplied construction: it is the exact coefficient
of log L in the annular integral (DR5) of the pole (DR1). The reference clock bβi =  ̄β + sti is a
chosen comparison clock built from that number. The actual couplings are not asserted to equal
the clock: they are produced by the committed path coefficient and the bounded reset, and are tied
to the clock only through the inequality (DR7), which the shooting and running arguments below
then calibrate. The lift transports αa as a scalar reader of the realized film (films are functions
of gauge-invariant stationary data and of tagged frames, read on the identity-frame section), so
N,i = αN,i, dZ = d and (DR7) is recovered with the same constant. No physical interpretation of
αZ
s beyond this calibration role is added.