Convergence of interacting scale histories · Part 2

15.7 Finite-depth trajectory approximation

At finite horizon prescribe an admissible normalized scalar script. Activity updates preserve the norm ball 18 and satisfy

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∥Φi(λ, x) −Φi(λ′, x′)∥≤c0 max{|λ −λ′|, ∥x −x′∥},
c0 = 1/16.

The scalar correction has the corresponding coefficient 4. At fixed depth, set activity to zero and apply updates backwards, setting unused later levels to zero. Finite decreasing induction and ball invariance keep this comparison admissible. Against any trajectory with the same scalar script, its discrepancy at is at most , from top distance at most one (indeed 14). For histories 0 with the same fixed-level scalar limits, truncated activities at depth converge together by finite descending one-step comparison. Choose 3, then take histories late enough that their 0 truncations differ by less than3. Both truncation errors give level- difference below. Applying

to escaping subsequences gives the common limit in the complete activity space. Scalar corrections then converge by one-step comparison with the convergent next activity. Finite initial segments discarded to ensure all required horizons are present do not change these limits. Inputs are the same paid contraction, ball invariance and local Gaussian/cluster/reset comparison. At finite horizon correction is adjacent physical-coupling difference minus the committed path coefficient. Taking these scalar limits gives the banked flow identity without infinite telescoping. This supplies the fixed-scalar convergence used in the diagonal extraction and banked uniqueness argument.

15.8 Labelwise comparison yields reset budgets

At fixed period there are finitely many supports. Linearity of budget mass gives 0 with mass( for0. This constant may depend on period. For uniform-tail majorants and arbitrarily small fixed-label differences, set. Choose finite with tails at most2 and eventually bound each label in by + 1)]. These bounds on and original tails elsewhere sum to at most, giving weighted budget This applies to regular paid and forced all-field differences and recovers the activity norm. Each source has a finite paid total; diagonal projection yields ∆ 0. Give the source difference paid budget below 2. Projection linearity then yields∆∆ 2, so increments are bounded and compare. Canonical stationary path

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slots and atoms have fixed supports despite coefficient changes; low homes, high footprints and
reset listing coincide. Final dictionary consumption transfers comparisons on this listing; vacuum
and coupling scalars do not affect activity comparison. Remaining coefficient families are residuals,
freshness, loans and attached logs.
The lift retains scalar scripts, truncation depth and fixed
labelled supports.
Recovery preserves finite induction and the explicit finite-head, uniform-tail
budget construction, including its fixed-period scope.
Let x[b]
m denote the zero-terminal approximation at depth b. Realize the fixed scalar script and
every update in that finite backward recursion. Since one-step readout recovers Φi, finite induction
gives
Rm(bx[b]
m) = x[b]
∥xm −x[b]
m∥≤cb−m
m,
.
0

Here reads the activity presentation, whose recovered norm is the original one. Thus the same truncation depth works for the lifted three-error comparison, and section recovery preserves its completion limit. For the labelwise budget, the source object retains the chosen finite head and the nonnegative tail array. Their total bounds remain2 and2, respectively, under coefficient readout, so the same fixed-period constantgives budget. The diagonal projection reader then recovers the bounded scalar increment and its vanishing difference. These identities transport the

exact finite approximation and reset-budget construction with their original order of choosing depth, tolerance and sufficiently late histories.

15.9 Comparison of reset coefficient families

Regular reset terms are source rebates, low freshness and high freshness. Rebate and fault differences equal the rebate and fault of the difference; the fixed reset-loss constant preserves paid comparisons and uniform tails. For a fixed low label the slots and support are fixed and only mark varies. Choose once a reference mark= 0 (unit path coefficients suffice for this algebraic label test). Freshness is the fixed reference function times. Coefficient comparison gives fixed-label convergence; weighted low-mark budgets with spare exponential width give uniform tails, since bounded-length labels form a finite set. High freshness compares its bounded scalar parameters and actual cubic/quartic coefficients; spare-width atom budgets give tails. Finite seat replication preserves them. The projected increment is already a bounded compared scalar. For loans, common homes and kept tests give exactly

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loanR(α, ∆) = loanR∗((m/m∗)α, (m/m∗)∆).

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The direct mark is fixed. This follows from the literal loan definition: its live pair term is pro-
portional to the mark, its proxy is independent of the numerical path coefficients, and its home
and kept tests are common. Linearity makes reference-loan differences loans at scalar differences.
The all-field shape bound tends to zero uniformly, including failed-health boundaries. A common
majorant is
el/ρ|m|

2t(Bα + B∆) + 1
2B∆pairBill t2
.

Positive residual width after spending supplies uniform tails. Faults plus loans therefore compare attached logs; rebates plus freshness compare regular resets on the original canonical enumeration. Final consumption uses1. For, preserving summable tails. Exponential continuity on bounded compared values and the varying-configuration selection argument give uniform fixed-label differences. Forced outputs are finite products of old hard and increment labels with the same canonical eligibility test on both histories. Bounded-factor comparison survives this indicator even when it varies with history. Finite-subset product budgets preserve tails; canonical injective drawer placement, zero elsewhere, preserves totals and tails by finite preimage sums. The regular embedding does likewise. Thus output arrays have the quantitative weighted difference budgets, without differentiating hard indicators. The width hypothesis leaves a positive exponential margin after is spent. The lift retains reference marks, common predicates and canonical embeddings. Readout preserves their exact factorizations and the finite-head/spare-width tail proof, recovering the same norm convergence at health boundaries.

The final coefficient comparison has a useful quantitative recovery form. For two logarithm values with, integration along their line segment gives

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ea −eb = (a −b)
Z 1
0
eb+s(a−b) ds,
|ea −eb| ≤eB|a −b|.

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This estimate is uniform in the evaluated configuration. For a fixed finite product, telescoping gives

n
Y
n
Y
 ≤
n
X
k=1
|ak −bk|
Y
r<k
|ar|
Y
r>k
|br|.

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r=1
ar −
r=1
br

Multiplication by the common eligibility indicator preserves this bound. Compose every factor and indicator with readout: the same estimates and reference-mark loan identities hold, and the

section recovers each coefficient. The spare-width majorants and canonical injective placements are retained with their original labels, so finite-head comparison plus uniform tails gives the same weighted output difference budgets. This recovers the reset convergence at discontinuous health boundaries through function values and common predicates.

15.10 Closing one-step activity-norm convergence

For two incoming Cauchy activity presentations with vanishing norm difference and common coupling limit, canonical envelopes give regular and forced comparisons. Their common completion limit makes their original presentation sequences Cauchy and their difference tend to zero by the isometric embedding. Common stationary paths and kernels supply microscopic comparisons, placing actual source outputs on one listing and resetting with the same committed coefficients. The exact update is

Projection comparison and the common stationary coefficient limit give vanishing output-coupling difference. To recover the actual activity norm, let 0 be its budget normalizations. For target, choose simultaneous anchored regular and forced difference budgets below ( and (. Reset tail lemmas supply these eventually; common supports and formats make them budgets for the actual difference. The norm characterization yields at most. Thus quantitative tails, not solely labelwise limits, prove output-norm convergence. Isometric completion convergence supplies the incoming Cauchy hypotheses; continuity of the physical-coupling map supplies the coupling limit. The chosen physical step identifies its actual output with the update presentation, without selecting another representative. Its scalar correction is output minus input coupling minus committed path coefficient, proving the fixed-input convergence used in finite-depth approximation. The lift retains the common presentation, norm normalizations and exact update. Readout preserves both anchored budgets and their isometric norm characterization, transporting this one-step limit with the same reset tail dependencies.

15.11 Uniqueness of banked history limits

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For two banked limits with common bottom coupling at level zero, let scalar, activity and correction
coordinates be fi, Xi, gi and their primed versions. Put di = max{|fi −f′
i|, ∥Xi −X′
i∥} and γ0 =
1/128. Finite-trajectory comparison passed to limits gives

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∥Xi −X′
i∥≤8γ0di+1,
|gi −g′
i| ≤4sidi+1.

Scalar flow and base equality imply

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Ωi −Ω′
i = −
X
j<i
(gj −g′
j),
bi|fi −f′
i| ≤|Ωi −Ω′
i|,
X
j<i
sj ≤γ0bi,

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with bi > 0 and only finite sums at each i. Individual activity bounds and scalar-flow/correction
bounds give di ≤1/4. Hence D = supi di is finite and nonnegative. The activity difference is at most
8γ0D; scalar flow gives bi|fi −f′
i| ≤4D P
j<i sj ≤4γ0Dbi. Divide by bi and take the supremum
to obtain D ≤8γ0D = D/16, so D = 0. All scalar and activity limits agree. The inequalities
come from paid finite trajectories; the nearby finite trajectories are constructed by (FCA7) and
(FCB5), with positive comparison weights and cumulative bounds (FCB1)–(FCB3), and the fixed-
scalar comparison identifies their activity limits. This is the uniqueness input upgrading diagonal
subsequences. The lift retains base coupling, positive coordinate weights and finite-flow identities.

Recovery preserves those identities and the uniform bounds, transporting the same supremum contraction with its finite-trajectory dependencies.

On the recovered activity presentation, use the transported norm on its section. The activity readout and section are then inverse isometries there. They extend to the respective completions by taking Cauchy representatives: if represents a completion element, read the sequence termwise, and its equal norm differences show independence of the representative. The extended maps still compose to the identity. Consequently the quantitative regular and forced difference budgets above recover the actual output completion distance as well as each evaluated coefficient. The physical coupling update is retained as a joint scalar reader, so its three-term difference passes to the same limit. For banked limits these isometries recover each exactly. The bounded family therefore has the same supremum under readout, and the recovered finite-flow inequalities give

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0 ≤D ≤D/16,
D = 0.

Section recovery returns equality of the original scalar and activity limits. Thus the uniqueness step and the one-step norm convergence use the same completed presentation and the same quantitative budget normalizations, retaining their finite-trajectory comparison dependencies.

15.12 Extracting stationary path coefficients and their higher jets

There is a separate extraction behind the stationary paths just used. It concerns numerical path coefficients, not completed activity presentations. Combine all repeated occurrences of the same transported-curvature slot in each row by adding their coefficients. Do the same for each color component of each ordered pattern of two, three or four slots in the loop, cubic and quartic atoms. Denote this countable coefficient array by), where is the upper horizon and the fixed level. The length) is the slot length or the sum of the pattern’s slot lengths. There are finitely many keys of bounded length: a bounded-length walk has a bounded number of steps from a finite alphabet, and there are only finitely many plaquette and color types.

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The path bounds give, for each row or color component separately,
X

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κ
|aN,m(κ)|eσml(κ) ≤Cm
eventually in N.
(SH1)

Here is the row allowance, the loop allowance for two-slot atoms, or the common higher-atom allowance for three-and four-slot atoms, as appropriate. An individual coefficient is bounded by , because the exponential weight is at least one. Enlarge these bounds to include the finitely many exceptional’s. The product of the resulting compact intervals, over the countable set of pairs (), is compact and metrizable. Every escaping horizon sequence consequently has one subsequence on which all these coefficients converge simultaneously.

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Write its limit am(κ). For each finite set of keys, pass the inequality (SH1) to the limit. Taking
the supremum over finite sets gives (SH1) for am at the original width σm. More is true at every
strictly smaller width τ < σm:
X

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κ
|aN,m(κ) −am(κ)|eτl(κ) −→0.
(SH2)

Indeed the part with length at most is a finite sum tending to zero, whereas the remaining part is at most (SH3)

Choose first and then the horizon. This order proves (SH2) and permits pairing the coefficients with every function bounded by. It also preserves the zero and first loop moments, since

each displacement coordinate is bounded by path length and a polynomial in length is bounded by an exponential of any positive width. The row symbol at zero is such a pairing with a bounded indicator of the plaquette type. Its limit preserves the finite metric identity. Color invariance passes by the finite sum over color coordinates. Reconstructing the rows and atoms from this common coefficient array therefore retains all the original path hypotheses and allowances.

The convergence includes derivatives of the unmasked physical functions at every order. To justify this assertion, complexify matrix entries around a physical configuration and use inverse links as independent conjugate-entry coordinates. The extension of a slot of lengthis a holomorphic polynomial bounded on a sufficiently small common ball by. A product of slots has the same bound with and the sum of their lengths. Equations (SH2)–(SH3) give uniform convergence of the row and atom series on that ball. Cauchy’s integral formula on a smaller ball gives convergence of each derivative, with its fixed radius loss and factorial. Restricting to real smooth parameter curves then gives convergence of all real Fréchet jets. This argument applies to the actual group words; no finite cutoffof the limiting path is substituted for them.

For completeness, the stationary-graph derivative argument used to pass through the nonlinear minimization is as follows. Let) be the finite profile, let be its limit, and let ) be their analytic stationary branches through zero. Put, and

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HN = DyFN(0, 0),
H = DyF(0, 0).
(SH4)

The preceding all-order convergence implies convergence of every jet of, in particular . The stationary gap makes these vertical Hessians invertible. Inversion is continuous near the invertible . This is the inverse needed here, acting from vertical covectors to vertical vectors.

Differentiate)) = 0 repeatedly. With)), the exact diagonal chain rule is a finite sum over partitions of the derivative slots:

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0 =
X
π
D|π|FN(0)[ D|B|gN(0)[v|B|] : B ∈π ].
(SH5)

The single-block term contains] and the known horizontal derivative Every other block has size less than. Thus (SH5), solved by , proves convergence of the-th derivative by induction from the lower derivatives. In an ordered-partition convention identical single-block contributions are grouped with their positive multiplicity; dividing by that multiplicity gives the same recursion. Polarization of symmetric real multilinear forms supplies mixed derivatives. The case(0) = 0. Composing the convergent graph jets with the profile jets proves convergence of all stationary value jets.

There is also a precise cancellation at the first unmatched profile order. On a stationary graph (SH6)

The first equation is stationarity at zero; differentiate stationarity once to obtain the second, and use Hessian symmetry for the reversed order. Suppose agree through order + 2. Their stationary graph jets agree through order + 1, by the recursion above. In the chain rule at order + 3, any term involving a graph derivative of order at least + 2 has either one block or two blocks with the other block of size one. These are exactly the terms killed by (SH6). All remaining graph derivatives match. Moreover the difference has zero jets through order + 2, so its composition at order + 3 has only the all-singleton partition. Consequently

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Dk+3(P ◦g −Q ◦h)(0)[vk+3] = Dk+3(P −Q)(0)[(Dg(0)v)k+3].
(SH7)

This is the stationary value identity used for the cubic and quartic response comparisons ( higher unknown graph derivatives have not been discarded without compensation.

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In the lattice profile, the linear map in (SH7) is the derivative of the actual fine logarithmic
inclusion of the slice graph. Its derivative is the fine stretching map applied to (v, Du(0)v). A
different tree-gauge representative changes it by an exact lattice gradient. Gauge invariance, to-
gether with vanishing of all lower jets of the profile difference, makes its first nonzero derivative
insensitive to such a gradient: differentiate the gauge identity and observe that every correction in-
volving a nonlinear gauge derivative multiplies a lower, zero profile jet. Hence the stationary matrix
response and its tree-extracted version give the same first unmatched value. Passing an eventual
one-sided bound through the convergent branch derivatives proves the corresponding bound for the
limiting coarse profile. Interchanging the two profiles gives the reverse bound. This is the response
comparison used together with the decay and base matching estimates in the stationary-history
construction.

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The one-loop profile also has a literal decomposition before passage to the limit:
"
#

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loop action + 1
2 log det(∂2
yPN) −log
Y
e offtree
J(ye)
◦gN.
(SH8)

The determinant is nonzero at zero by the same positive vertical Hessian. Determinant is polynomial in its entries, logarithm is analytic near this positive value, and the Haar logarithms are analytic in the common chart. The jet convergence just proved therefore applies to (SH8), including its second jet. A finite inequality comparing this second jet to the adjacent draft (the coupling coefficient times the quadratic action plus its loop action) passes to the limit by continuity of absolute value. Field-independent scalar offsets are not identified by this argument.

Finally the extra physical covariance and response rows have their own quantitative, full-sequence convergence. In their weighted row and column norms the finite suppliers give, for

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∥UN,i −UM,i∥row, ∥UN,i −UM,i∥col ≤B clockk(i + 1)r
 aN
c
,
c > 0.
(SH9)
ai+1

For the covariance row the distance weight has exponent at unit output scale; for the response row it has exponent at output scale. Since0, (SH9) gives an entrywise Cauchy limit. Fix a finite set in a weighted row or column, pass its absolute sum to the limit, and then take the supremum over finite sets. This retains (SH9) with in place of, in both norms and at precisely these scales. The limits thus supply the physical rows on the same common witnesses, with quantitative tails; they are not an additional independently chosen stationary subsequence.

There is a further full-sequence assertion for the actual metric rows, which we separate from book extraction. At every fixed torus, the unamended diagonal matrix has the limit given by flat quadratic convergence. Its inverse matrices have entry bounds from one common positive strip allowance. A finite-dimensional compactness argument shows that the limiting diagonal matrix is invertible: extract any convergent subsequence of the bounded inverses, pass the identity = 1 to the limit, and obtain a right inverse of. A square matrix with a right inverse is invertible. Continuity of inversion then proves convergence of the entire sequence of inverses, independently of the extracted subsequence.

For the actual amended rows the prescribed construction, not merely agreement on curls, gives at that torus ) (SH10)

where is the fixed amendment coefficient at this level. All operations are continuous finite matrix operations. Thus converges. Its inverse converges by the same bounded-inverse argument, using the metric’s original strip allowance. This explains why the amendment must be retained from the same construction; agreement of potential Hessians alone would not imply (SH10).

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To recover cover coefficients from these all-period limits, take two possible coefficient limits
R, R′ of a uniformly exponentially bounded row. Their folded kernels agree on every scheduled
torus. Fix a cover displacement z and choose a torus whose centered box contains z. The folded
coefficient at its residue is R(z) plus the other congruent coefficients. Their sum is bounded by
a constant times the exponential tail outside that box, uniformly in the history index. As the
scheduled periods grow, this tail tends to zero, hence R(z) = R′(z). Compact coefficient extraction
therefore has only one possible limit; the full row converges coefficientwise. Splitting into a finite
displacement set and its exponentially bounded complement proves convergence in every strictly
smaller strip norm, exactly as in (SH2)–(SH3). The argument applies to the diagonal, its dual and
the amended metric. The physical covariance and response rows have in addition the rate (SH9).

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The extraction (SH1)–(SH3) for nonlinear path books alone remains subsequential. The auxil-
iary response comparison is used through exactly the orders it supplies: base matching through the
second jet, then cubic and quartic comparisons (SH7), and the loop second-jet comparison (SH8).
In particular any two extractions share the second profile jet, since it is the same quadratic matrix
limit just proved. The recursive first-unmatched-jet argument compares the next two jets with their
stated bounds. It does not assert equality of arbitrary all-order path books or field-independent
scalar offsets. Whenever the subsequent primitive-continuity argument uses common path coef-
ficients, its two sequences are taken from the already selected common book refinement.
Full
quadratic-row convergence and retained nonlinear-book convergence thus have different, explicit
scopes.

Source–readout formulation. Two different kinds of limit object are produced in this subsection, and the lift must keep them apart because they have different fibres. The coefficient array of a nonlinear path book is extracted from a compact set; its limit is a selected point, and different extractions may select different points. The quadratic rows have full-sequence limits and are determined. The following two statements say exactly which readers are insensitive to the selection.

Proposition 15.2 (First unmatched stationary value). On a stationary graph

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DP(0)[(0, w)] = 0,
D2P(0)[Dg(0)v, (0, w)] = 0.

If profiles with stationary graphs agree through order, then

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Dk+3(P ◦g −Q ◦h)(0)[vk+3] = Dk+3(P −Q)(0)[(Dg(0)v)k+3].

Proof. The first identity in (SH6) is stationarity at zero, the second its derivative along the graph with Hessian symmetry. If agree through order + 2 then agree through order and the recursion (SH5) gives equal graph jets through order + 1. In the chain rule at order + 3 a term containing a graph derivative of order at least + 2 has one block, or two blocks the other of size one; these are exactly the terms annihilated by (SH6). All remaining graph derivatives match, and, having zero jets through order + 2, contributes at order + 3 only through the all-singleton partition.

Proposition 15.3 (Two scopes of convergence).

1. Quadratic rows (diagonal, dual, amended metric, covariance, response): full-sequence convergence on the common witnesses, with the rate (SH9) for the covariance and response rows. The folding readout from cover coefficients to the family of all scheduled tori is injective on exponentially bounded rows.

2. Nonlinear path books (SH1)–(SH3): subsequential convergence only. Two extractions share the second profile jet (the quadratic matrix limit of (1)) and are compared at the next two jets by (SH7) and at the loop second jet by (SH8); equality of all-order books or of field-independent scalar offsets is not asserted.

Whenever a later argument uses common path coefficients, its two sequences are taken from the already selected common book refinement.

Proof. Both parts collect what was proved above; injectivity of folding is the tail argument for

The source record is the pair (book refinement, quadratic row limits). The first component is a selected point of a compact set (a non-canonical choice, recorded with its refinement map); the second is determined. The readers used later are exactly the jets listed in Proposition 15.3(2) and the rows of (1); a reader of higher book jets would not be constant on the fibre of extractions and is never applied. This is the method’s distinction between a recovered quantity and a retained witness, and it is the reason the common family is assembled from one composed refinement.

15.13 Retained coefficients recover physical row comparisons

At a fixed step the retained-history comparison data consist of the limits of the upper and lower path books, hence of their quadratic, cubic and quartic atom coefficients, along the single common horizon refinement selected in (SH1)–(SH3). They also include the three cover-kernel limits and the coupling and activity limits. Here two escaping subsequences mean two subsequences of that retained horizon refinement; both therefore have the same book limits. This is not an assertion that arbitrary subsequential path books extracted before the refinement are equal (Proposition 15.3). The coupling comparison is passed to the same tail ultrafilter, while the retained activity comparisons supply the regular-occurrence and forced-coefficient evaluations at arbitrary varying fine configurations. This constructs the comparison data for the local formulas. The full fixed-volume limit of the Wilson smeared integrals is a separate conclusion, proved in Subsection 18.4; it is not obtained by declaring path-book compactness to be full convergence. Cutoffs make physical linear expressions finite: bounded-length canonical paths have finitely many plane/displacement labels. Shortening is therefore a finite compared-coefficient sum of transported curvature slots; its direct branch is fixed curvature. Coarse kernels at cutoff retain bounded cover displacements. Fine-to-coarse route length islift(, with lift(, so retained displacements lie in radius . Common endpoint and cutoffpredicates leave finite sums along the same routes. Smooth word multiplication, inversion and adjoints compare all three kernels through derivatives. Coarse vision contracts these kernels with compared plotted fields. Linear direction rows are fixed curl, finite cutoffcurl sums or fixed coarse noise maps. Signed displacement and carrier transport preserve comparison; reversed links retain their negative adjoint and shifted base. Nonlinear rows retain their actual clipped fine parts, coarse-response half-squares and cross terms, wire deficit/noise square, negative coarse deficit with two departure terms, and tight-response half-square. Fine kept/clipped tests are bounded common factors on the same fine configuration; no limit through their indicators is required. Regular sources use occurrence comparisons; atom and tail sources retain compared coefficients and actual masked words. These give the row and inherited-source comparisons feeding