Inherited analytic conventions

Inherited analytic conventions

A positive-plane plaquette uses the ordered Wilson product in the public statement. All lattices are four dimensional, with cover and period- lattice (. A positive link ( joins to; reverse reading takes its inverse. Signed coordinate permutations retain this convention. Distance is the maximum centered-coordinate distance. A support is a finite site set, and its internal links have both endpoints in it.

The physical Lie-algebra inner product and chord are

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⟨X, Y ⟩g = −m−1 Re tr(XY ),
chord(g) = ∥g −1∥F/√m.

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Fix the source’s orthonormal real color basis. Matrix Frobenius norm is √m times color norm. The
inner product is positive on the skew-Hermitian algebra and conjugation invariant. Inversion and
conjugation preserve the chord, and the triangle bound is chord(gh) ≤chord(g) + chord(h). The
color projection of (g −g−1)/2 is the source’s Lie-algebra curvature readout. These normalizations
are kept in the quadratic action, the coupling slope and the trace calculation.
At scale i, the spacing is ai > 0, the odd integer block ratio Li = ai/ai+1 ≥3, and ti = log(a−1
i ),
so ti+1 −ti = log Li. Block bounds and coupling-entry delays are chosen before cutoffand physical
period. The reference clock is

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X
bβi =  ̄β + sti,
s = 11CA
24π2 ,
CA = −
1
dim g
α
tr(ad Tα)2 > 0.

Represented couplings retain their distinct coordinates and explicit comparisons with this clock. The radius- halo of is the union of its radius- lattice balls. The cover number cov is the least number of such balls whose union is connected by unit lattice steps and contains their centers need not belong to. Periodic wrapping is treated by that same connected-union definition.

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For a fixed stage ratio L, retain

log (log temp(

Rate means that for every, eventually. Thresholds may depend on the fixed stage and ratio but not the period. Rate zero allows subpower growth and does not assert boundedness. The fine threshold is fiery() = temp()). Finite sums preserve the larger rate, products add rates, and a positive exponent gap absorbs fixed multiplicities and subpower factors. Later scale-path exponents retain their separately specified paths.

Every local weighted bound retains derivative order, allowed directions and exponential support weight. Countable sums are controlled absolutely, equivalently by a common bound on finite partial sums of the nonnegative majorant. This convention permits countable dictionaries without assuming a finite number of terms. Terms such as a bill, a payment or a paid bound in the source denote these explicit inequalities, not additional probabilistic assumptions. The lift carries these conventions as realized analytic data; conjugation of the finite frames preserves the color inner product, while arbitrary componentwise boxes still require their explicit coordinate transport.

5.1 Notation and the roles of recurring symbols

The construction uses several local alphabets. The following register fixes their roles; later local uses are qualified by their section and indices. In particular, a local parameter called a mass or a gap need not be a spectral gap of the reconstructed physical theory.

Accessible table transcription

Table transcription, source page 13

Notation	Meaning and convention
X, Y, K(Y )	Native admitted support, literal preparation, and raw realized sector.
YM
	Physical data occur in the third object, not in the source signature.
π, σ, ι	A readout and its specified sections. The finite frame section is σ(U) =
	(1, U).Other sections are named for their actual domains; none is a
	global inverse of every readout.
G ⊆U(m), g	The given closed matrix group in its faithful representation, its matrix
	size m, and its simple nonabelian real Lie algebra. The hypothesis does
	not add connectedness.
	−1
Ue, Ae, rx	Physical link, raw framed link and endpoint frame, Ue = rAery for
	x
	e : x →y. A reversed link is read by inversion.
N, p, i	Mesh cutoffindex, period index and scale level. The full N →∞limit
	is taken first at each suitable fixed p; the outer period limit may use the
	stated common refinement.
	−1
h, L, d	Physical mesh h d, physical integer period L
NpN	N =N p and positive denom-
	inator dN. The lattice side is LpdN.
a, L, t	Scale spacing, block ratio L a/a t
iii	i =ii+1 and logarithmic scalei =
	−1
	log(a). Some retained calendar passages call this last clock τi; it is
	i
	the same scale clock there. L L.
	i is not the physical periodp
b	b ̄
β, β	Actual calibrated action coefficient and reference clock ββ + st.
N,p,ii	i = i
	Their difference is estimated, not set to zero.
	2
s, b, C	The computed logarithmic slope s = 2bC/(24π). Later occur-
0A	0 = 11A
	rences of s as a positive radius are local scale variables, not this slope.

Accessible table transcription

Table transcription, source page 14

Notation	Meaning and convention
	−1/2
g, g	Regulator coupling g βand the Wilson and measured position
iW , gX	i =N,p,i
	couplings used in the final comparison. Their asymptotic identification
	is proved; it is not a notation convention.
SW , S(x, v)	Wilson action and stationary residual, respectively. The latter is the
	vertical derivative of the local score and is not an action. The unadorned
	S in the predictor Fourier identities is yet another locally defined symbol.
C, P	Covariances, precision/comparison matrices or coefficient maps accord-
	ing to their declared types. In a Gaussian change of variables C0 and CV
	denote distinct base and varied covariances. An auxiliary nonsymmetric
	inverse is not itself a covariance.
E, N, F	Degree-d coefficient space, null subspace for all separated spectator tests,
ddd
	and physical quotient Fd = Ed/Nd. Inclusions and derivatives on these
	quotients are constructed and proved compatible.
deg, |a|	Engineering degree of a label and total engineering degree |a|=
	P
	deg(a) of a cluster. Polynomial position degree and field count are
	i i
	separately specified in the radial-jet chapter.
χ
S, S, μ	Unselected and character-dependent separated correlations, and the full-
nnph
	R S
	χ
	support phase measure, so Sdμ(χ). The source often writes
	n =n ph
	this measure as μ; it is distinct from the finite Wilson laws.
Ω, D	Selected vacuum and common invariant word domain. The native prepa-
	ration subscript in RealPrepΩis a source convention, not a claim that a
	vacuum has already been derived.
Θ, GΘ, E(r)	Centered degree-four physical trace label, its selected Euclidean two-
	point function, and the positive-time Laplace reader identified with it in
	the spectral argument.
	4
A, c	The full trace amplitude is A/π. Other local scalar or color
GG	G = 3 dimR g
	projection constants called cG retain their local definitions.

Terms such as bill, payment, price and purchase denote the explicitly stated inequalities and the order in which their constants and thresholds are chosen. A calendar is the common sequence of scales, stage assignments and entry thresholds. A beacon is a specified finite reference view used to read a local coefficient, and a path book retains the path-series and insertion data of a chosen history. These words do not supply extra hypotheses beyond their mathematical definitions. Their exact weights, derivative orders and choices remain in the corresponding sections.