14 The analytic stationary graph, loop moments and the cali- brated coupling history · Part 10

where gold(x) = y−23/20. Every multiplier has rate zero. The fixed support exponential and turning
price preserve rate −23/20, proving the claimed rate of εj. Thus finite interpolation combines a
large-radius signed estimate and a sharper real individual estimate. For the primitive source, the
individual bound b is its regular price (FCC1) plus the beacon contribution. With x = bβi, it has
form
b = 3t Cjx−11/10 + εj(x),
εj has rate −23/20,
(LCN9)

a guide and a forest which the evaluated density does not see. The method’s descent criterion asks
whether the step is constant on the fibres of “choice of presentation”, so that it defines a map of
the normed activity data (λ, A) ∈R× bHP themselves. The derivation above proves this and proves
analyticity of the descended map.

Lemma 14.27 (Signed-combination bounds are series bounds). If f(t) = P
n(t/p)nan with
P
n ∥an∥≤C then f has comparison price C at radius p. Conversely, if f has comparison price C
at radius p, then for every p′ < p it is given on [−1, 1] by an absolutely convergent series in t/p′

Proof. The first statement interchanges a finite sum with an absolutely convergent series.
For
the converse consider the moment vectors m(t) = ((t/p)n)n ∈l∞.
The hypothesis says that
P
j cjm(tj) 7→P
j cjf(tj) is a well-defined linear map of norm at most C on their span, hence
on its closure. The coordinate vectors lie in that closure: subtract the already recovered lower
coordinates from m(t) at small t ̸= 0, divide by (t/p)n and let t →0.
Their images are the
coefficients an, and m(t) = P
n(t/p)nen converges in norm after restriction to radius p′ < p by the
geometric majorant.

Proof. (1) is as stated; it applies (LCN3) to the two-point measure. (2) Recover the line series at
radius q/4 by Lemma 14.27. Polarization makes the degree-n coefficient a symmetric multilinear
map: the finite-difference formula divides by n!, has 2n summands and each sum of n unit vectors
has norm at most n; nn/n! ≤en gives (LCN5). For increments z, z′ of norm at most 1/4 telescoping
each diagonal gives
X

Part (1) is the descent criterion in its exact form: the step maps are constant on the fibres of the
presentation, so they are well defined on the common normed presentation R × bHP . With si = 2fi
the constants are the scalar constant 4si and the activity constant 8γ used in the stability argument
below. The exponent 301 is the actual finite interpolation order of the supplied construction; ana-
lyticity of the descended maps is a consequence of the signed-moment bounds through Lemma 14.27
and is not an additional hypothesis. In the frame realization the activity space and its norm are
transported isometrically by Theorem 9.4, and the step maps are intertwined by Theorem 14.20;
the Lipschitz constants (LCN7) are therefore the same numbers for the lifted maps. The exactness
class is exact recovery of the maps together with their norms.

The retained forward recipe also gives quantitative stability of the complete finite trajectory under
changes of its bottom coupling. This is the finite statement behind the auxiliary trajectories in the
uniqueness argument. Write xi = bβi and define

The positive square root is used. By (FCA8), wiλi ∈[17/20, 23/20]; this wider interval contains
[
p
3/4,
p
5/4]. The activity coordinate Xi is its image in the common completed activity chart.
At a fixed finite horizon all these coordinates live in one finite product of their scalar and activity
spaces.
Let γ = 1/128. The actual local analytic comparison, including the initial state at N, bounds
differences of activity outputs by 8γ times the maximum input-coordinate difference. Its scalar
correction output has Lipschitz bound 4si times that maximum. The weights are explicitly

where v > 0 is the fixed comparison cost and Cj > 0 the stage’s local analytic price. The common
small-field threshold is θi = x−1/2
i
exp((log log xi)2). Thus si ≤2vCstage(i)x−1/10
i
. These comparison
bounds apply to the same retained primitive operation at both points, as ensured by its fixed layout
and choices.
The calendar pays the cumulative weight inequality
X

Since x−1/10/x−1/32 →0, eventually the upper bound in (FCB2) is at most γ times this increment.
Include this one eventual threshold in the coupling-entry delay for each of the two allowed adjacent-
stage possibilities. It follows at every realized step that si ≤γ(bi+1−bi). Summing gives P
j<i sj ≤
γ(bi −b0) ≤γbi. Positivity and monotonicity of xi give b0 ≤bi. Thus (FCB3) is a consequence of
the same schedule, not a supplementary assumption on a chosen pair of trajectories.

Compare two successful trajectories of the deterministic recipe, with bottom-coupling difference
δ. Let
U = max
0≤i≤N max{|λ′
i −λi|, ∥X′
i −Xi∥}.

The local comparisons give ∥X′
i −Xi∥≤8γU, including the finest-state comparison, and correction
differences at most 4siU. Subtracting the finite coupling recursions cancels their common path
coefficients:
β′
i −βi = δ −
X
j<i
(∆′
j −∆j).
(FCB4)

Theorem 14.29 (Injective, Lipschitz-invertible bottom reader). Let two successful trajectories of
the deterministic recipe have bottom-coupling difference δ and U = max0≤i≤N max{|λ′
i −λi|, ∥X′
i −
Xi∥}. Then

In particular the successful finite trajectory with a prescribed bottom value is unique within the
retained recipe: the bottom reader BN,p of Theorem 14.24 is injective, and b∗
N,p is the unique point
of the fibre over  ̄β.

Proof. Theorem 14.28 gives ∥X′
i−Xi∥≤8γU (including the finest-state comparison) and correction
differences at most 4siU. Subtracting the coupling recursions cancels the common path coefficients,

while (FCB1) factors β′
i −βi = bi(λ′
i −λi)(wiλ′
i + wiλi) with last factor at least 17/10 > 1. With
(FCB3), |λ′
i −λi| ≤|δ|/bi + 4γU ≤|δ|/b0 + 4γU.
Taking the maximum over levels and both
coordinates gives the first inequality, and 8γ = 1/16 the second. If δ = 0 the finest normalized
couplings, hence the finest couplings, agree; the startup states are equal by the fixed startup rule
and descending determinism gives equality of all represented states.

For the calibrated finite history, (FCA3), βN,p,0 =  ̄β, and the exact recursion for bβi imply

This is uniform control of the actual finite-cutoffcoupling, including m = N. It is stronger than
merely writing down the reference line bβm =  ̄β + sτm.
Dividing by τm > 0, and then absorbing the fixed intercept  ̄β/τm, yields

Explicitly apply (RUN4) with error ε/4, and take τm ≥4(| ̄β| + 1)/ε; the triangle inequality gives
an error smaller than ε. Taking m = N proves the bare coupling slope in the theorem, uniformly
in the physical period.
Set b0 = 11CA/(48π2), so 2b0 = s. For arbitrary cutoffand period sequences Ni ≥i eventually
and pi, let gi = β−1/2
Ni,pi,i. Positivity of the coupling and (RUN5), followed by continuity of the square
root and reciprocal at s > 0, give

The equality used here is β−1/2√τ = (β/τ)−1/2 for β, τ > 0. This regulator coupling is defined
from the inverse action coefficient. Its numerical leading logarithm does not by itself identify it
with a coupling defined by a physical position-space observable. That separate identification, and
its normalization, are proved in the trace and Ward calculation.

Source–readout formulation.
Equation (RUN5) recovers the first quantitative clause of the
supplied public conclusion (Section 4) as a property of the calibrated realized history: the readout
Readslope(N, p) = βN,p/ log(h−1
N ) converges to s uniformly in p, with witnesses the calendar c and
the trajectories of Theorems 14.24 and 14.29. It is stronger than writing down the reference line
bβm, since it controls the actual finite-cutoffcoupling.
The regulator coupling gi of (RUN6) is
defined from the inverse action coefficient. Its leading logarithm does not by itself identify it with a
coupling defined by a physical position-space observable; that identification and its normalization
are a separate result of the trace and Ward calculation (Section 23.1). The lift keeps the two notions
as two different readers of the same realized history, and no physical significance is attached to
(RUN6) beyond the stated limit.