Activity domains, ordered derivatives and absolute budgets

Activity domains, ordered derivatives and absolute budgets

The supplied construction fixes, before any estimate is used, what an activity estimate means (tags (AB1)–(AB12)). Every later regular, anchored, early or forced-hit budget refers to these objects. This section gives those definitions in full and proves that each is carried, without loss, by the frame realization of Theorem 7.1.

9.1 Local activity domains, derivatives and absolute budgets

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We fix the meaning of all activity estimates before using them. At a fixed finite torus Λn = (Z/nZ)4,
let Cn = GΛn×{0,1,2,3} be the configuration space and An = gΛn×{0,1,2,3} its real space of left-link
directions. A function is local on a site set X if its value depends only on links with both endpoints
in X. Gauge invariance refers to Ux,i 7→kxUx,ik−1
x+ei.
For a plaquette p, write V (p) for its four visited vertices and Up for its holonomy. At threshold
0 < θ ≤1 define

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Fθ(U) = {x : ∃p, x ∈V (p), chord(Up) > θ},
Hθ(X) = {U : X ∩Fθ(U) = ∅}.
(AB1)

Thus equality with the threshold is healthy. Each flag event is open, because it is a finite union of strict inequalities for continuous plaquette functions. Each healthy domain is closed and Borel. The mask is the literal function; its boundary is not differentiated in any regular budget. A coefficient which also reads the final flags in is required to agree whenever both the links internal to and) agree. Its actual link read set is bounded by

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Xins = X ∪
[
p:V (p)∩X̸=∅
V (p).
(AB2)

Agreement on internal links of this enlarged set fixes every plaquette incident to, hence all flags inspected there. This is why flag-dependent payloads are assigned enlarged physical footprints even though compatibility in a gas is tested on the originally assigned supports. For, set

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(inc(U, tA))e = exp(tAe)Ue,
∇AF(U) = d
dtF(inc(U, tA))

t=0
.

For an ordered list), define recursively

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∇∅F = F,
∇A::lF(U) = d
dt(∇lF)(inc(U, tA))

t=0
.
(AB3)

The order matters; no commutation of left derivatives is assumed. These derivatives are required to exist where tested. A regular representative on is a globally Borel, gauge-invariant local function such that(inc( near zero for every), and its recursively specified left derivatives have the derivative identities in (AB3) through order. In particular the smooth germ is required also at a healthy configuration lying on the threshold boundary. Off-domain values of the representative remain available for locality and measurability, but do not enter its regular derivative price.

A budget point consists of integers1, support exponent 0, and threshold 1. We denote it by p). For anchor, its allowed directions satisfy

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|Ae| ≤θ(1 + dist(e−, o))p
for every link e.
(AB4)

Hereis the initial vertex of the stored positive link, and the distance is the periodic maximum-coordinate distance. A nonnegative number is a derivative envelope for

whenever and everysatisfies (AB4) (AB5)

Norms replace absolute values for vector-valued coefficients. The support price is

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massp(X, b) = es covB(X)b.
(AB6)

The factor is at least one. Every sum of envelopes below means a nonnegative summable family together with its asserted bound on the sum. Equivalently, every finite partial sum obeys that bound; taking increasing finite partial sums proves summability and the same total bound for countable dictionaries.

A regular address is () with. A regular family has anchored budget if it has envelopes0 satisfying (AB5) and

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X
i:oi=o
es covB(Xi)bi ≤δ.
(AB7)
sup
o∈Λn

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The representation used after the kinetic reset additionally has Ei(1) = 0. On a nonwrapping
support its identity Hessian
D2v 7→Ei(inc(1, v))

0[A, C]

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vanishes whenever A, C are affine link fields in the canonical lift about oi. Explicitly an affine field
has
Ax,a = aa +
X
b
lift(x −oi)b dab.
(AB8)

The reset chapter constructs this Hessian cancellation; it is not imposed on arbitrary primitive functions before projection. A forced address consists of a support, a nonempty inventory, and a fixed extra predicate). Its coefficient is globally Borel and gauge invariant and vanishes unless

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Yi ∩Fθ(U) = Ii,
Pi(U).
(AB9)

The predicate is Borel and gauge invariant; both coefficient and predicate may read links and final flags on, hence have the physical read enlargement (AB2). A forced budget consists of numbers 0 such that on every configuration and

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X
i:x∈Yi
es covB(Yi)ci ≤κ.
(AB10)
sup
x∈Λn

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This is a support-hit budget, in contrast to the single-anchor sum (AB7). Coefficients equal to zero
still retain their addresses and eligibility predicates.
For a finite set D of forced addresses, the gas term is zero unless their supports are pairwise
disjoint and their inventories have union exactly Fθ(U); otherwise it is Q
i∈D Ki(U). Summing
these terms over finite D defines the forced gas. The regular exponent is the sum of 1Hθ(Xi)Ei. At
a fixed finite torus these series are absolutely convergent: assign each regular term to its anchor
and each forced term to one site of its nonempty support. Equations (AB7)–(AB10) bound their
total envelope sums by |Λn|δ and |Λn|κ. The sum of absolute products over all finite subsets is at
P
most Q
i ci. Restricting to compatible subsets only decreases it. If there are no flags,
i(1 + ci) ≤e
only the empty forced set can contribute, since each inventory is nonempty; the gas is then exactly
one.

Signed translations are implemented by fixed permutations of the whole regular and forced address lists, carrying anchors, supports, inventories, predicates and coefficient functions together. Thus comparing two activity values in a single chart always means comparing the same lists and symmetry permutations. It does not mean identifying two representations merely because their total densities agree.

An early source budget uses + 3, rather than, in (AB7). This is the pre-reset reserve. The three additional support-exponent units pay the local splitting and conversion operations explicitly described in the reset and signed-expansion chapters. Reducing the exponent from decreases every nonnegative summand. The rotation of derivative directions costs the finite factor

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turnFee(p) = (2 2p)j.
(AB11)

Indeed the anchor itself is transported exactly. A reversed link is stored at its other endpoint, at distance at most one from the transported base, so 1 +) costs 2 in (AB4). Its right increment can be converted to a left increment by the adjoint action of the background link, which preserves color norm. Expanding each pulled-back left derivative into its positive-link and negative-link contributions gives at most 2 ordered lists at order; each has length and direction

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inflation at most 2p. Multilinearity of the iterated derivatives bounds their sum by (2 2p)qb, which
is at most (AB11) times b for q ≤j. This expansion follows by induction on the ordered list, using
the exact left/right increment identity and the disjoint link supports of the positive and negative
parts. Support enlargement is charged separately by its cover-number inequality; it is not included
in this factor.
At level i put xi = bβi and Mi = exp((log log xi)2). The normalizations used in (LCN1) are the
calendar’s actual allowances

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bPi = x−11/10
i
,
kPi = exp(−M1/5
i
).
(AB12)

Both are strictly positive. Thus means exactly that its regular and forced components obey (AB7) and (AB10) with budgets and, respectively, in their fixed address layout. The retained physical states have8. Triangle inequalities give the norm properties used later: sums of admissible nonnegative envelopes remain envelopes, derivatives are linear, and scalar multiplication multiplies all envelopes by the scalar modulus. Least envelopes, followed by finite partial-sum limits, give attainment of an infimum whenever it is used in (LCN1). The separated completion removes only differences of zero price in these tested germs and forced values. The definitions (AB1)–(AB12) are the ordinary mathematical content of RG/Masks, RG/Activities, the reset Quotes and Investor definitions, and the activity normalization in FullVolume/Sym/Shorn. They will be used with the same domains and directions in every subsequent reference to a regular, anchored, early or forced-hit budget.

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Source–readout formulation.
The target objects of this subsection are functions and events
on the finite configuration space Cn = GΛn×{0,1,2,3}, which is the readout space U of Theorem 7.1
for the torus Λn. The realized sector is Z = GV ×GE with readout π(r, A)e = r−1
x Aery and section
σ(U) = (1, U). An activity datum is lifted by composition: every regular representative, forced
coefficient and predicate F is replaced by F ◦π, while every piece of lattice bookkeeping (anchor,
support, inventory, address list, scope) is copied unchanged into the raw tuple. The operations
to be lifted are the flag map, the mask, the left increment and its ordered derivatives, the signed
lattice permutations, and the sums defining the budgets and the norm. The fibre of π is the frame
orbit; the statements below show that all of these operations commute with π and that every
number defined in (AB1)–(AB12) is constant on that fibre. The exactness class is an embedding
with section: nothing in the budgets is enlarged, sharpened or lost.

Proposition 9.1 (Frame-blind flags). For every and every plaquette based at

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π(r, A)p = r−1
x Aprx,
chord(π(r, A)p) = chord(Ap),
Fθ(π(r, A)) = Fθ(A).

Hence, a closed Borel set stable under the frame action of Theorem 7.1, and for every gauge-invariant one has (

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Proof. The path identity Uγ = r−1
x Aγry at a closed path gives the conjugation formula. The chord
∥g−1∥F/√m is invariant under conjugation by a unitary matrix, which gives the second equality at
every plaquette and therefore equality of the two flag sets. The preimage formula is a restatement.
The frame action sends Ap to kxApk−1
x , which again preserves every chord. Finally π(r, A) is the
gauge transform of A by gx = r−1
x , so a gauge-invariant F has F ◦π(r, A) = F(A).

Thus the frame column is a pure spectator for gauge-invariant activities. It is not a spectator for the charts, axial gauges and predictors of the nonlinear chapter, which are not gauge invariant; there the frame is retained and the covariance identities of the tangent chapter are used instead. A

flag-reading coefficient lifted by composition with reads the raw links internal to of (AB2) and, unless it is gauge invariant, the frames on For the increment and the derivative define, for and

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incZ(z, tB) =
r, (exp(t rxBer−1
x )Ae)e:x→y

,
∇Z
B::lΦ(z) = d
dt(∇Z
lΦ)(incZ(z, tB))

t=0,

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with ∇Z
∅Φ = Φ. The raw direction is the target direction conjugated into the stored frame; the
frame itself is not moved.

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Theorem 9.2 (Ordered-derivative recovery). For all z ∈Z, B ∈An and real t, π(incZ(z, tB)) =
inc(π(z), tB). If F has the ordered derivatives (AB3) through order j at π(z), then F ◦π has the
ordered derivatives ∇Z
lthrough order j at z and

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∇Z
l(F ◦π)(z) = (∇lF)(π(z))
(|l| ≤j),

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with the same ordered list on both sides. The raw-link direction rxBer−1
x
has the same color norm
as Be.

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Proof. For a link e : x →y, r−1
x exp(t rxBer−1
x )Aery = exp(tBe) r−1
x Aery, since conjugation com-
mutes with the exponential. This is the first identity. The derivative identity is proved by induction
on the list. It is trivial for the empty list. If it holds for lon a neighbourhood of the curve, then

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∇Z
B::l(F ◦π)(z) = d
dt(∇lF)
π(incZ(z, tB))


0 = d
dt(∇lF)(inc(πz, tB))

0 = ∇B::lF(πz).

The increment inc leaves the frame fixed, so the induction never leaves the fibre over the varying readout and the order of the list is not changed. The color inner product conjugation invariant, which is the last statement.

Proposition 9.3 (Budgets are fibre constants). Let be a regular representative on and forced data. For every frame

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1. b is an envelope (AB5) for F on (o, X) if and only if |∇Z
l(F ◦π)(r, A)| ≤b for all A ∈Hθ(X),
|l| ≤j and lists obeying (AB4);

2. the prices (AB6), the anchored budget (AB7), the forced bounds, the eligibility condition (AB9) and the forced budget (AB10) are the same numbers and the same events for the lifted family;

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3. the identity-value and affine-Hessian conditions lift to every flat raw configuration Ae = rxr−1
y :
there Ei ◦π = Ei(1) = 0 and the second derivative ∇Z
(A,C)(Ei ◦π) vanishes for affine A, C of
the form (AB8).

Proof. (1) By Proposition 9.1, () lies over the healthy domain exactly when does, and maps) onto itself by a gauge transformation. Theorem 9.2 equates the two derivative values with the same list, so the two suprema coincide. (2) Supports, anchors, cover numbers and inventories are lattice data untouched by; coefficients and predicates are gauge invariant, so their lifted values at () are their values at; flags agree by Proposition 9.1. (3) A flat raw configuration reads to = 1, and the Hessian is the order-two case of Theorem 9.2.

Gas and exponent. By Proposition 9.3 each term of the forced gas and of the regular exponent, lifted by composition, equals the corresponding original term at and has the same majorant. Consequently the absolute convergence statements derived above hold for the lifted series with the same bounds, and Gas) = Gas( Exp) = Exp(

termwise and before summation. The address lists are part of the realized tuple: two representations with the same total density but different address layouts are different source records, exactly as in the supplied construction, and the readout does not identify them.

Symmetries and the turn fee. On a signed lattice permutation acts geometrically on frames and raw links, with inverse raw links under orientation reversal; the path identity gives, so the lifted address lists are permuted by the same fixed permutations. In the derivation of (AB11) the conversion of a right increment into a left increment uses, in the realization, the raw background link, exp() = exp( , and the frame conjugations of Theorem 9.2 preserve the color norm. The fee is therefore the same number on both sides of and is not paid twice.

Theorem 9.4 (Isometric recovery of the activity chart). Let Act be the space of activities in a fixed address layout with finite norm andA = A, applied to each regular representative, forced coefficient and predicate while keeping the address lists. With the lifted envelopes of Proposition 9.3(1), is a linear isometry of (Act onto its image; restriction to the section is its inverse; descends to the separated completions and intertwines the signedtranslation permutations, the reserve reduction and the turn-fee estimate (AB11).

Proof. Linearity is clear. Equality of norms is Proposition 9.3 applied to each address, followed by the same suprema and sums. (A A because = id. An activity difference has zero price exactly when all its tested germs and forced values vanish, a condition preserved in both directions, so is well defined and isometric on the separated quotient and extends to the completion. The intertwining statements were proved in the two preceding paragraphs.

The status of Theorem 9.4 is exact recovery on the declared class. What the realization does not supply is a bound on derivatives in frame directions for functions that fail to be gauge invariant; none is used for activities. What is realization data rather than a consequence of admission is everything in (AB1)–(AB12) itself: the threshold, the stiffness and budget points, the envelopes and the calendar normalization (AB12) are chosen in the supplied construction and retained in the raw tuple.

9.2 The finite Wilson law is a probability measure

First the finite Wilson law is an actual probability measure. Every unitary matrix has Re tr: each diagonal entry has modulus at most one. Thus every plaquette price lies in [0 If there are plaquettes and0, its continuous density relative to the product Haar probability measure obeys

In particular normalization is legitimate and bounded measurable cylinders are integrable. A site gauge change multiplies a link on its two ends; by Haar invariance its change of variables preserves the product prior. Its plaquette holonomy is conjugated at the base site, so the trace and the action are unchanged. The normalized law is consequently gauge invariant. A signed lattice permutation merely reindexes plaquettes, possibly reversing their holonomies; inversion preserves their real

traces. It too preserves both the prior and the law. The local representatives in (PF12), and their finite products, therefore have precisely the finite-lattice expectations used below.

Source–readout formulation. In the realization the density of Theorem 7.2 is the same function of the raw links,, and the frame factors integrate to one. Therefore

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ZZ = Z ∈[e−2βF , 1],
π∗eμ = μW ,

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with the same two-sided density bounds. A physical gauge transformation g is lifted by rx 7→rxg−1
x
with A fixed, so it does not move the density at all; a signed lattice permutation acts on both
columns and preserves the product Haar prior and the raw action for the reasons just given. The
local representatives used in the reflection argument and their finite products have exactly the same
expectations before and after the lift. This is exact recovery of a normalized probability law, not
of an unnormalized weight modulo a gauge volume.