The two formal statements on which this description rests are the following.
Lemma 14.11 (Trace-ideal calculus). Let
= sup
Then
; a matrix with
-dimensional source has
; and
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Proof. tr(uv∗V ) = v∗V u. Decompose U into its n0 columns, U = P j Ueje∗ j, and add. The last claim is cyclicity of the trace, tr(AUBV ) = tr(U BV A).
Theorem 14.12 (The drift allowance). With
the bound for the massive-symbol integral remainder, 
log
and, since the scalar reset has
∆
and the actual coarse coupling is
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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.
The stage construction fixes
and hence
before the block length; afterwards it imposes one lower bound on odd
and post-block bounds Θ
Λ
Λ
Λ
Λ
on the output path, which may depend on
and do not enter
Proof.
). By (DP7) it differs from
) by at most
the latter is
), the integral of
with Fourier weight; by (DR2)–(DR5) that integral differs from
log
by at most
. The stationary tensor is produced with arbitrary growing torus sequences obeying the floor and stationary identities; after coefficient identification it is one cover path and works on every realized torus with the construction’s margins, so the auxiliary limit adds no period-dependent error.
14.8 Massive auxiliary symbols and the uniform alias trace bound
Why the calendar belongs to the witness. The massive regulator and the sequence of purchased scales control which estimates may be combined. Positive regulator mass is fixed before the corresponding volume estimates; the subsequent choices retain this order. The calendar records ratios, stage horizons and delays large enough for all finitely many demands already made. Native source admission supplies no preferred ordering of the realized leaves. The order used here is therefore part of the realized witness, and every later calibration, comparison and limiting argument must use that same calendar.
The massive comparison used in the preceding determinant argument is fixed as follows. With the lattice exterior symbol
1, put
Searchable transcription
s(p) = X a qa(−p)qa(p), dμ(p) = μ2 s(p) + μ2 , cμ(p) = μ2dμ(p)9, μ = L−1. (MA1)
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On real momentum s(p) is the nonnegative squared curl frequency. The scalar massive covariant- site operator at a group field is D∗ UDU + μ2; its normalized inverse is μ2(D∗ UDU + μ2)−1. The auxiliary regulator is μ2 times its ninth power, and the comparison site operator is D∗ UDU plus that regulator. These are ordinary finite positive matrices when μ > 0. Their symbols at the identity are respectively s + μ2, dμ, cμ, s + cμ. The ninth power is part of the construction, not a variable exponent sent to infinity. For link fields, copy the site regulator into each of the four link directions at the link’s initial site. Add it to the physical quadratic Hessian and to DUD∗ U. This defines the positive auxiliary
link matrix. The associated scalar functional is one half of its logarithmic determinant minus the logarithmic determinant of the site comparison operator. The copying operation respects gauges and unsigned permutations of axes. Reversed links move the initial site, so this auxiliary functional is not assigned a signed symmetry by that copying argument. The physical signed laws are established separately.
The symbol at the identity is the physical transverse quadratic symbol plus the longitudinal symbol
plus
) times the identity. On real momentum the gap estimate bounds its quadratic form below by min(
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210 (s(p) + μ2) ∥v∥2 2, (MA2)
where
0 is the physical transverse floor. The scalar companion has the same bound without the transverse factor. Indeed the physical and longitudinal pieces bound their respective curl and gradient components; in the small-frequency region
; in the other region the frequency term already pays a fixed multiple of
. This proves the required positive floor including zero momentum.
To see the uniform trace summation, aliases are
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pl = L−1p + 2πL−1el, l ∈(Z/LZ)4, (MA3)
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using the fixed centered integer lift el. Write r(l) = maxa |ela|. There are at most (2n + 1)4 indices with r(l) = n, by injection into the integer box. Consequently, uniformly in L, X l (1 + r(l))−6 ≤J0, J0 = 16 X n≥0 (n + 1)−2 < ∞. (MA4)
This bound uses a box estimate for each shell; its excess over the actual shell size is harmless. For odd
, on the strip of imaginary width 1
128, first choose the real part of the coarse momentum in its fixed reciprocal fundamental cell. The lattice sine lower estimate on every nonzero alias and the complex perturbation estimate for
give
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|dμ(pl)| ≤ 2 (1 + r(l))2 . (MA5)
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At the zero alias the massive denominator directly gives the bound two. For a nonzero alias, its real frequency is at least (r(l) + 1)μ in the norm used by the sine estimate, and 1 2((r(l) + 1)μ)2 ≤ |s(pl) + μ2|; divide μ2 by this lower bound. A general coarse momentum differs by a reciprocal integer translate; this permutes the alias set. Thus summing (MA5) cubed gives, on that whole strip, X
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l |dμ(pl)3| ≤8J0. (MA6)
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In particular the diagonal alias matrix d3 μ ⊗1Ch has nuclear norm at most 8hJ0. This is the three-massive-arc trace bound used inside the ninth-power regulator. Here are the frames in which it is used. Let TL(p) have entries L−2eipl·exδαβ, with row x, α and column l, β. The finite character sum shows that it is unitary at real p; its explicit inverse is the opposite-phase character matrix. On the fixed complex strip the two matrices have operator norms at most e4/128, by separating their real unitary Fourier factor from the diagonal complex phases. Furnishing an alias symbol means conjugating its diagonal alias blocks by these two matrices. Thus its three-arc nuclear bound is at most
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e4/128 (8hJ0) e4/128. (MA7)
The weighted frames additionally use
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hl(p) = q pebWeight(pl, μ), AL(p) = TL(p) diag(hl(p)−1)TL(0)−1. (MA8)
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Here pebWeight(p, μ) = P a |ei Re pa −1|2 + μ2. Thus the positive weight uses the real part of momentum even when p is complex. Such a weight is a norm-comparison device; it is not asserted holomorphic. For complex differentiation, freeze the weights at the real anchor and use the analytic TL(p) factors. Comparing the original and frozen weights at nearby momenta gives the fixed height allowance. The link-plus-site frame is the direct sum of (MA8) on link components and the unweighted site frame.
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This separation justifies extending the real floor to a complex strip without a hidden dimension factor. Test the conjugated matrix against two fixed vectors and apply the one-variable Cauchy estimate to the resulting scalar analytic function. If its operator norm is bounded by M on a ball of radius R, its derivative is bounded by 2M/R on the half ball. Integrate along the segment from the real anchor to a nearby complex momentum. Choose that distance so that the perturbation norm is less than half the real lower bound in (MA2). The Neumann series then inverts the perturbed matrix with at most twice the anchored inverse bound. Apply this in the frozen frames, then restore the compared positive weights. All constants are fixed before L; none comes from summing matrix entries to estimate an operator norm.
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The derivative films used above are the actual zeroth, first and second coefficient matrices of the local analytic rows. A row’s real directional derivative bound is first converted componentwise, p and then its color-coordinate square sum is taken. For the link row this costs exactly | Loc(G, 1)|; p for the site row it costs | Loc(G, 0)| and the displayed local sampling-density factor. Pulling the site row into the block adds L2, since the normalized site square sum has L4 terms. These are the link and site row allowances used in the weighted operator bounds. The rows are differentiated through third order when constructing a second-order film of their first derivatives; that shift of order is retained.
For the regulator film, form three consecutive cubes of the normalized inverse film in (MA1), giving
, and then multiply by
. At the common center this is exactly
. Its trace-ideal bound uses one unsplit cube as (MA7), while the other factors receive operator bounds. In a first or second product derivative, choose the corresponding three-arc occurrence after applying the exact inverse and product rules; the column placements in (DP10)–(DP12) give the listed alternatives. The resulting finite sum is bounded by the positive sum of their tariffs.
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Finally the house term is the physical Hessian film sandwiched between its two row films; the orbit term is the sum over four initial-site copies of the regulator film. Sandwiching a trace- ideal factor by operator-bounded factors preserves its nuclear bound, with the product of the two operator allowances. Sum the four copied terms and conjugate to the common frame; the fixed Fourier norms cost at most e4 on each side at the wider intermediate allowance. The reflection correction is the sum of its two cross products with the regulated site row, and the fixed extra block contributes 4| Loc(G, 0)|. Addition and the same two-sided multiplication rule give the full trace toll in (DP10)–(DP12). Thus the cancellation there is backed by the massive symbol (MA1), the alias sum (MA6), the real and complex floors, and the actual local row derivative bounds. It does not require treating an arbitrary large block matrix as uniformly trace class.
The complex denominator estimate used in (MA5) can be proved before alias summation. Put 
, and
1. Integrating the derivative of the exponential gives 
, and the same estimate at
. Expanding their scalar product yields
Searchable transcription
|s(p) −s(x)| ≤4t(u + 4t) + 4tu = 8tu + 16t2.
For
128, the right side is at most (
2: use 2
. Since
, the reverse triangle inequality proves
Searchable transcription
|s(p) + μ2| ≥1 2(u2 + μ2) > 0, |dμ(p)| ≤2, |cμ(p)| ≤29μ2. (MA9)
The coarse strip has width 1
128; each alias divides its imaginary part by
, so its fine momentum has exactly the width
128 required here. In particular the inverse denominator is analytic throughout the strip. For completeness the weights in (MA8) obey
Searchable transcription
hl(p) ≥μ, ∥q(pl)∥2 ≤5hl(p)
on the unit coarse strip. At coarse points of distance at most one, the preceding exponential estimate gives
. Using
shows that the two positive weights have ratio bounded by a fixed constant in both directions. Freeze
at a real anchor while retaining 
) and its explicit analytic inverse. These are the analytic frames used in the Cauchy estimate. In ordinary unweighted norm their bounds are
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∥AL(p)∥≤e4σμ−1, ∥AL(p)−1∥≤e4σ sup l hl(p). (MA10)
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For a real anchor, h2
l ≤16 + μ2 ≤17. These estimates display the mass powers; they must not be
silently replaced by two mass-independent unweighted bounds.
Here is an explicit form of the strip-inverse argument. Suppose a matrix K, in the two specified
weighted frames, has norm at most A, its real lower singular bound is a > 0, and the changes to
frozen analytic frames cost at most C on a ball of radius σ. Test the analytic matrix on two unit
vectors and apply the scalar Cauchy estimate. This gives derivative bound k = CAC/(σ/2) on the
half ball. On the strip
η = min{σ/2, (a/2)/(1 + k)}the displacement from its real anchor changes the matrix by at most
The lower singular bound is therefore
2, so its weighted inverse has norm at most 2
. This also proves invertibility and analyticity of the inverse by the determinant formula. The constants are independent of the dimension of the block. In the site application the real lower form bound is proportional to
, and the two inverse weights cancel it; it is that weighted matrix to which the assertion applies.
The remaining mass bookkeeping can now be stated exactly. The normalized inverse film has bounded ordinary operator coefficients because its factor
cancels the two inverse-weight losses. The regulator film has both operator and trace-ideal price
times a fixed price, by its definition 
. Converting it back to the weighted form costs at most
on each side, hence cancels as
this
. This accounts for the mass-independent regulator toll in the link determinant comparison. It does not remove the explicit
in its ordinary frame.
Finally the actual matrices behind the last paragraph of this chapter are as follows. Let
be the fixed inclusion of the link block and let
be the complementary site projection; let
read the initial-site slot of direction
. Write
for the regulator film,
for the differentiated row, and
for the local second variation of the scalar correction. With the rear operation denoting the transpose at reversed momenta, including the reversed external legs, the relevant products are
Searchable transcription
3 X House = RrearHR, Orbit = i=0 Srear i CSi, Motive = P rearP. (MA11)
These are exact products of films, so their first and second coefficients are their ordinary ordered product derivatives. The motive is constant and its finite site rank is the reason its trace allowance
Loc(
. The orbit allowance is the regulator trace allowance multiplied on each side by the Fourier-frame bound. The house allowance follows by first assigning the finite local trace allowance to
, then multiplying by the two row bounds. The two reflection cross products are bounded in the same order. The total toll is the positive sum
Searchable transcription
payLedger + reflectLedger + 4(4 orbitLedger) + popLedger. (MA12)
Each factor in this expression has thus been paid either by a local finite rank, the alias trace sum, or a weighted operator norm; no trace norm of a general extensive Hessian is assumed.
Source–readout formulation. The auxiliary massive objects are realization data that have no target counterpart: the mass
, the ninth-power regulator and the copied link regulator do not appear in any recovered statement. They are spectators in the sense of the method (adding them changes fibres, not readouts), with one caveat stated in the derivation: the auxiliary functional is only invariant under gauges and unsigned permutations, so no signed symmetry of a readout may be inferred through it. What is read out of this subsection is the single inequality (DP7). The frames
are presentation changes (invertible, with the explicit norms (MA7), (MA10)); the nuclear and operator bounds are stated in named frames precisely because they are not invariant under arbitrary changes of frame. Two estimates of the derivation are used again later and are isolated here.
Lemma 14.13 (Complex massive denominator). Let
and
. Then
Searchable transcription
|s(p) + μ2| ≥1 2(u2 + μ2) > 0, |dμ(p)| ≤2, |cμ(p)| ≤29μ2.
Proof. Integrating the derivative of the exponential gives
, and likewise at
Expanding the scalar product,
For
, and 16
1024; the sum is at most (
Since
, the reverse triangle inequality gives the first bound, and the other two follow from (MA1).
Lemma 14.14 (Strip inverse without a dimension factor). Suppose
, in two specified weighted frames, has norm at most
, real lower singular bound
, and the changes to frozen analytic frames cost at most
on a ball of radius
. Put
and
= min
Then on the strip of width
the matrix is invertible, analytic with analytic inverse, and its weighted inverse has norm at most
Proof. Test the analytic matrix against two unit vectors and apply the one-variable Cauchy estimate to the scalar function: a bound
on a ball of radius
gives derivative bound 2
on the half ball, hence
. Integrating along the segment from the real anchor changes the matrix by at most 
2, so the lower singular bound is at least
2. The determinant formula gives analyticity of the inverse. No constant depends on the dimension of the block.
The exactness class is controlled approximation: the massive symbol is compared with the massless one through the bounded discrepancy (MA12), uniformly in the alias and in the period, and the discrepancy enters the drift only through
in Theorem 14.12.
14.9 Constructing the common asymptotic calendar
We give the numerical schedule construction explicitly, since both finite shooting and uniform running use it. For each stage
4, all pre-block parameters have already been chosen: its derivative order, support exponent, analytic width, path bound, column bound and drift allowance 
). Let
Searchable transcription
A(j) = log 1 + paths(j) + width(j)−1 + columns(j) + relaxation(j) + supportExp(j) .
Choose an odd integer block length L
3, at least every previously required lower bound, such that 
log L
min(
log L
(CAL1)
This is possible because the numerators are fixed before this choice and 1
log
0. To make the block lengths nondecreasing, take for each
the maximum of the finitely many lower bounds through
, then an odd integer larger than that maximum. These choices prove the drift and logarithmic-cost inequalities without making the drift allowance depend on the selected terminal cutoff.
After fixing the block lengths choose the positive protection exponents, decreasing with
, small enough for all post-block requests and at most 1
. This can be done by taking successive finite minima of positive admissible upper bounds. Next choose the stage’s coupling-entry and scaleentry thresholds; every analytic estimate proved with these parameters has one eventual threshold, so a finite maximum handles all its requests. Later increases of these thresholds preserve their inequalities.
Choose the initial coupling ̄
1 above the first-stage entry threshold, and sufficiently large that 
exp((log log
̄
(CAL2)
The ratio (log log
log
0 proves this. For fixed
, the required lattice-size bound at logarithmic scale
has the explicit form
Searchable transcription
Vj(u) = Sj 1 + c(u) + c(u + log Lj) qj100(c(u) + 1) + 100 (j + 3)Lj + 1 , (CAL3) c(u) = l log2( ̄β + su) m .
Here
and
are the fixed shell-count and growth requests from the local estimates. The last two terms include the small-field and patch buffers. For each fixed stage,
) and its fixed translate are
); hence
). Choose an integer entrance bound
such that
and
log 3 imply simultaneously
Searchable transcription
̄β + su ≥betaEntry(j), u ≥scaleEntry(j), u > 0, log Lj ≤u/j, Vj(u) ≤eu. (CAL4)
Each condition holds eventually as
log 3
gives one common integer
There are also stagewise comparison-window requests
) satisfying
0, and finite readiness thresholds
. Choose integers
so that for every
and
log 3
log L
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rj ≤u, wj(u) ≤k/2. (CAL5)
Indeed eventually
(2 log L
; multiplication by the upper bound on
proves the second inequality, and the lower bound on
supplies readiness. Integer rounding only strengthens the resulting bound by
We now build the stage clock, rather than presupposing one meeting all these requests. Use an unshifted stage number
0, corresponding to
, and write
for the chosen integer requests after this shift. Define plateau starting indices recursively by
Searchable transcription
a0 = 0, kh = Ch + 2ah + 1, ah+1 = ah + Bh+1 + 2kh + 1. (CAL6)
Set
when
. The sequence
is strictly increasing, so
is defined everywhere, nondecreasing, increases by at most one in a step, and tends to infinity. Also
(1) = 0. Its recursion gives two useful properties:
Searchable transcription
E(i + 1) > 0 =⇒BE(i+1) ≤i, (CAL7)
Searchable transcription
and, for every h, kh ≥Ch and
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E(i) = h whenever kh −⌊kh/2⌋≤i ≤kh + ⌊kh/2⌋. (CAL8)
For (CAL7), the first index of plateau
For (CAL8),
+ 1 places its lower endpoint above
, while
places its upper endpoint strictly before
Thus every stage receives a sufficiently long central comparison window, however large its requests. Define 
L
+ log
(CAL9)
Then
log 3
log L
, by monotonicity of the blocks. The first inequality proves
For every noninitial plateau (CAL7) lets us apply (CAL4) at
. In particular
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0 ≤log Li τi ≤J(i + 1)−1 −→0. (CAL10)
Searchable transcription
This is the gentle-mesh condition. Equations (CAL1) and (CAL9) give (FCA4) and (RUN3). The stage order, power and support exponent stay nondecreasing, the protection exponent decreases, and (CAL2) gives the small-field threshold at every scale. These properties all belong to the same calendar. For volume, let di = Q r<i Lr = eτi. On every noninitial plateau (CAL4) gives VJ(i+1)(τi) ≤di, so any positive physical period pays its lattice-size requirements. Only finitely many steps belong to the initial plateau, since E →∞. Choose an integer M larger than their finitely many size requirements and the required minimum period, and set the physical periods to
Searchable transcription
Pp = 4(M + p + 1), lattice side at (i, p) = Ppdi. (CAL11)
These periods are strictly increasing and divisible by four. They pay the initial exceptions directly and the later requirements by the preceding denominator bound. They also provide arbitrarily large periods divisible by any prescribed positive integer
: take
+ 1), which is at least
, and then
. The selected complete construction may subsequently retain a period subsequence; it keeps all the size inequalities.
Finally use (CAL5) at
, which lies between the two bounds preceding (CAL10). The requested window has width at most
2, and (CAL8) keeps that entire window on stage
Thus the comparison windows, entry delays, local volume margins, gentle mesh and asymptotically small drift are realized together. No selection of a new calendar for a particular history, cutoff, period or observable is required.
Source–readout formulation. The calendar is a source-side object of a different kind from a field history: it consists of numbers and index maps only. Its role in the lift is that of a common-witness datum.
Lemma 14.15 (Plateau properties)
is defined everywhere, nondecreasing, increases by at most one per step, tends to infinity and has
. Moreover
Searchable transcription
E(i + 1) > 0 =⇒BE(i+1) ≤i,
Searchable transcription
and for every h, kh ≥Ch and
Searchable transcription
E(i) = h whenever kh −⌊kh/2⌋≤i ≤kh + ⌊kh/2⌋.
Searchable transcription
Proof. ah+1 −ah ≥2kh + 1 ≥3, so the ah increase strictly and a1 ≥3. If E(i + 1) = h > 0 then i + 1 ≥ah = ah−1 + Bh + 2kh−1 + 1 ≥Bh + 1. For (CAL8), kh −⌊kh/2⌋≥kh/2 ≥ah + 1 2, so the lower endpoint is at least ah (indeed larger), and kh + ⌊kh/2⌋≤3 2kh < 2kh + 1 ≤ah+1.
Theorem 14.16 (One calendar as a common witness). The tuple
Searchable transcription
c = (Lj)j, protection exponents, entry thresholds, ̄β, (Bh, Ch, ah, kh)h, E, J, (Li, τi)i, M, (Pp)p
realizes simultaneously the drift and logarithmic-cost inequalities (CAL1), the small-field threshold (CAL2), the entry and volume conditions (CAL4), the gentle mesh (CAL10), the central comparison windows (CAL5), (CAL8) and the lattice sides (CAL11). It is chosen before, and independently of, every history, cutoff, period and observable. Its readers are
Searchable transcription
Readclock(c)i = bβi = ̄β + sτi, Readstage(c)i = J(i), Readside(c)i,p = Ppdi.
Proof. Each property was derived above from the recursion (CAL6) and the order of choices; none of the choices refers to a history.
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In the language of the method, c is the datum that every later readout shares: lattices, clocks, windows and volumes are all read from the same c, and successful estimates obtained on different calendars are never combined. The calendar is not unique. The admissible calendars form a directed family under the enlargements used below (raising a length floor before the lengths are chosen; raising entry delays with the lengths held fixed), and every inequality listed is monotone under those enlargements. This non-uniqueness is a presentation ambiguity: no recovered statement depends on which admissible calendar is used, but each statement is proved on one. Because the calendar contains no group element, its frame lift is the identity, and it is the labelling datum of the realized family in Section 8: the level index i, the stage J(i), the clock bβi and the side Ppdi of every leaf are read from it. Nothing in the admission of the source support produces these numbers; they are realization data chosen by the supplied construction in the stated order.
14.10 Countable local demands on the same calendar
The calendar construction must also accommodate estimates whose numerical values use the chosen block lengths. We describe this ordering explicitly. First choose the stage profiles, kernel, primary analytic price list, and the secondary price list for the flat covariance and exterior constructions. The three local constructions then supply finite integer lower bounds
) for block lengths at profile
Two further integer lower bounds
) ensure the local
Source: Shadow Theory and the Wilson Yang–Mills Construction: A Source–Readout Reconstruction. Both papers belong to one Zenodo record. No external funding.