Convergence of the interacting scale histories · Part 10

What the path extraction does and does not select The already established contraction fixes the banked activity limits. The path coefficients and their higher jets are additional presentation records and may require compact extraction. Their selected limits are used to represent those same activities and insertions. The extraction does not change the Wilson family or turn its already unique banked limit into a choice between different physical values. Conversely, uniqueness of the banked reader alone does not assert convergence of every unreduced path presentation.

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.

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 ( different tree-gauge representative changes it by an exact lattice gradient. Gauge invariance, together 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 involving 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

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comparison used together with the decay and base matching estimates in the stationary-history
construction.
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).

To recover cover coefficients from these all-period limits, take two possible coefficient limits of a uniformly exponentially bounded row. Their folded kernels agree on every scheduled torus. Fix a cover displacement and choose a torus whose centered box contains. The folded coefficient at its residue is) 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). 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).

The extraction (SH1)–(SH3) for nonlinear path books alone remains subsequential. The auxiliary 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 coefficients, 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.

From retained coefficients to 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/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. In this paragraph, 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. 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 “Extending the full fixed-volume mesh limit”; it is not obtained by silently declaring path-book compactness to be full convergence.

The cutoffmakes each physical linear expression finite. There are only finitely many canonical path labels of length at most: their plane type and integer displacement lie in a finite coordinate box. In shortening with length multiplier 0, every retained label has length at most. Thus the cutoffshortening is a finite sum of its compared scalar coefficients times the corresponding transported curvature slots. The direct plotted branch is the fixed curvature function and needs no coefficient limit.

Likewise a coarse routed kernel at cutoff uses only cover displacements of coordinate length at most. For a fine-to-coarse route from a lifted residue to, route length islift( Sincelift(, the retained displacements lie in the box of radius. Its endpoint and cutoffpredicates are the same in both histories. Each nonzero summand is its compared kernel coefficient times adjoint transport along the same finite route. Smooth finite word multiplication, inverse and adjoint action therefore compare these summands and their derivatives. Summing proves comparison of all three physical kernels.

Coarse vision is a finite contraction of these kernels with the compared plotted coarse fields. The linear field on a logarithmic fine direction is either the fixed discrete curl, a finite cutoffpath sum of such curls with the compared path coefficients, or the fixed coarse noise linear map. An oriented row additionally applies the common signed displacement rule and carrier adjoint transport. A reversed link uses the negative adjoint-transformed displacement; its base shift is the one already fixed by the oriented link convention. These operations are smooth finite sums and products and hence preserve comparison.

The nonlinear energy rows are then their original finite scalar expressions: the direct clipped

fine part plus the two coarse-response half-squares and cross terms; the wire curvature deficit and effective noise square; the negative coarse curvature deficit together with the two departure terms; and the tight-response half-square. The clipped or kept fine tests act on the same fine configuration in the two histories. Their varying numerical values can therefore be retained as bounded common factors, without taking a continuity limit through the test. The inherited regular source uses the regular-occurrence comparison directly; the atom and path-tail sources use their retained scalar coefficients and the same actual masked words. These prove the physical row and inherited-source comparisons feeding the packet and mean calculations above. The underlying retained stationary and activity convergence is the history input, whose extraction must use the common choices described in the construction.

Mean, local source and chain expressions

The bill vector comparison can be made explicit. A selected row’s strain is its free residual vision plus the physical row operator applied to the saved log field and the hot part of the varying fluctuation. Selection and hot-coordinate sets are fixed by the draft. The saved log is the same all-argument local function on both sides; its uniform derivative bounds give its comparison. On hot coordinates use the compared fluctuation; on dormant coordinates its lens contribution is identically zero. Physical row comparison therefore compares the strain.

For the fine free residual, compare the actual base-stroke map and subtract the compared coarse vision. For a noise residual, the base-stroke part is common; its derivative bounds hold in the clean hull ensured by viability and selection. Adding and subtracting the same compared vision then gives the residual comparison. Thus the clean-hull condition is used exactly on selected noise rows, rather than postulating clean fields everywhere.

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The bill vector is the transpose form applied to the selected zero-fluctuation strains, with their
original scaling. Finite matrix-vector multiplication compares it. The protected mean is minus
the transpose window-column matrix applied to this bill; hence it compares as asserted above. Its
scaled stationary defect is the plate times this mean plus the bill, multiplied by −√β, and therefore
compares as well. The positive common coupling limit is guaranteed by the lower coupling-window
endpoint; taking square roots and reciprocal scales is consequently valid.

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The chain terms are now finite expressions in compared matrices and vectors. Their successive
factors are the appropriate negative plate–covariance defect while the index is within the prescribed
itinerary, and the plate factor at its terminal position. Matrix products compare these factors,
with their order unchanged. The quadratic fluctuation contraction and the linear stationary-defect
contraction compare by finite coordinate sums; the local cone cutoffs multiplying them are smooth
functions of squared fluctuation norm. The blank-reference chain uses the empty store and identity
output field, whose viability is automatic. Its plate has the same common lower bound, so the same
inverse and determinant comparisons apply. The condensation source is one half the difference of
the blank and actual local determinant expressions.

The Haar source is smooth for all fluctuation arguments after its cutoff. Inside the cutoff support its argument lies strictly within the positive Haar chart radius, so the chart logarithm is smooth. Outside that support the source is locally zero. The strict chart margin covers the cutoff boundary and proves the all-argument claim. This justifies its use in products without extending an uncut logarithm beyond its domain. Finally the packet logarithm is the original finite combination

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−β(chalky + crispy) +
X
visited
condensation +
X
plaza
HaarSource −|pens| log(chartMass(β)).

Here chalky is the posted root energy minus its selected quadratic costs; crispy is the selected quadratic cost at the protected mean. Each cost is a finite sum of squared strains over the actual parent rows. The root-energy comparison is the corresponding nonlinear energy comparison minus that same cost. The chart mass is positive and continuous at the positive coupling limit, so its logarithm compares. All posted, visited, plaza and pen sets are fixed finite sets in the draft. This proves the packet-log comparison from the strain, root-energy and determinant comparisons, with the original normalization retained.

The four vertex comparisons

After fixing the draft, the ordinary integrand has the same four vertex types as its absolute activity bound. The pointwise comparison treats their predicates explicitly.

For a connected block packet, connectedness and the containment/empty-flag test are fixed discrete predicates. If a predicate fails the corresponding factor is identically zero in both histories. Otherwise the packet is the exponential of its compared energy expression, multiplied by the common contract factor, tickets and profiles, and the compared smooth local cutoffon its actual enlarged plaza. The fine-configuration ticket and profile factors may vary with the history index but are the same on both sides and uniformly bounded. Their output-field dependence uses the already chosen smooth wire profile. Thus the finite product comparison applies without requiring a limit of the ticket indicator.

For an old forced vertex, compare its actual forced coefficient evaluated on the same fine configuration, as supplied by the retained forced-activity comparison, and multiply by the compared cutoffon its true footprint. For a regular exponential vertex, compare its logarithmic expression first and then apply the smooth exponential and subtract one. Its branches are inherited regular terms, the short local energy source, the undercurrent condensation term, the Haar term and the chain-row terms. The tests for being inside the posted region, for voyage membership and for hot exterior coordinates are fixed by the draft; their zero branches agree exactly. The nonzero branches use the protected mean, covariance/window, local logarithm and row comparisons already needed for these expressions, with the actual signs and factors of retained.

For a warning vertex, its mandatory/hot-region predicate is also fixed. The factor is a compared cutofftimes the actual fine profile or contract charge minus its constant reference value, or an exterior cone minus one. Fine profiles and finite label charges are bounded common factors; the exterior cone is a smooth function of the compared fluctuation direction. Its mandatory branch is identically the stated constant in both histories. Thus all warning comparisons retain their original hard predicates.

Finite products over the molecule give the integrand comparison, and multiplication by the compared density gives the transformed integrand comparison. The derivative version uses the same four factors in their local smooth extended formulas, the same fixed fine configuration, and the compared steered fluctuation and corrected Gaussian coordinates. Its full finite product is identified with the original differentiated integrand by the exact local formula on a neighborhood of the healthy base. These statements compare the true vertices rather than replacing hard indicators by smooth approximations. The underlying energy, row, cutoffand Haar comparisons are the finite-coordinate ingredients of these four cases.

The marked primitive comparison

The marked premium contains a test that its distinguished badge occurs in the molecule. The badge is fixed by the common mark key, so this test agrees in the two histories. If it fails, both premiums

vanish. If it holds, the integrand replaces precisely that vertex by the paid marked ingredient and retains the product over the molecule with that badge erased.

At fixed physical fluctuation and common draft, the reconstructed fine configuration is the same on both sides. The ingredient comparison is therefore the comparison of the two mark entries evaluated on this common, possibly history-dependent, fine configuration. Multiply it by the common finite enforcement cutoff, the compared ordinary product over the erased-badge molecule, and the compared Gaussian density. This proves the pointwise comparison of the marked amplitude. The exact reconstruction identity between the paid ingredient and this expression uses viability; it is applied on the eventual viable set.

For the derivative integrand the marked entry remains evaluated on the fixed reconstructed fine input. Output-field derivatives act on its smooth steered cutoffand on the remaining ordinary vertices, as prescribed by the local reweighting formula. The local chart identity identifies the actual marked ingredient with this expression near the healthy base. The compared steering map gives comparison of the cutoff; the four ordinary vertex comparisons give comparison of the remaining product; the local Gaussian ratio and Jacobian give comparison of the reweighting factor. Their product is exactly the derivative integrand, not a derivative of a newly substituted marked function. This explains why convergence of mark entries under varying-configuration evaluation is sufficient here.

The same Gaussian-polynomial majorant bounds these marked derivatives and permits the integral passage already proved. Finite boundary and forest averages then compare the actual marked grove. Its all-field bound also gives convergence of its uncut amplitude under arbitrary varying output configurations, using the fixed-draft argument separately at each gauge-slice point. This supplies both the regular marked factor in the one-mark cluster stream and the forced marked comparison used in the later reset.

Uniform derivative comparison of a single cluster stream

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A one-mark connected stream with r occurrences is its fixed Ursell coefficient divided by r!, multi-
plied by
r
X
ν=1
lν
Y
μ̸=ν
uμ,

where is an ordinary primitive entry and its marked replacement. Its actual support is the occurrence union when the mark speaks to that label, and its zero alternative otherwise. The speaking condition depends on the fixed mark key and label inventory, so corresponding histories have exactly the same support. A silent stream is identically zero.

The stream’s derivative bill is summable uniformly in the history index. To see the essential bound, let bound ordinary and marked entries through derivative order The ordered product rule bounds the term with mark by

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(1 + r)jbν
Y
μ̸=ν
aμ.

Replace. Since 1 +, this absorbs that derivative factor. The ordinary bounds may be taken as the smaller of their uniform primitive envelope and their original purchased budget; taking the smaller preserves summable tails. A positive reserve, selected by the one-mark budget argument, makes the inflated array fit the connected-cluster reserve.

Expanding its product with nonnegative entries gives

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

d
X
Y
μ̸=ν
2jaμ
2jbν ≤
Y
μ
fμ.
ν

Thus a valid stream bill is times the ordinary absolute connected-cluster weight for. The rooted cluster estimate makes this family uniformly summable with small tails. Multiplication by the fixed Ursell factor and! is retained exactly.

For a fixed speaking stream, a healthy union makes every occurrence support healthy. The primitive comparison along any allowed configuration family then compares all its ordinary factors and its single marked factor through derivative order. Finite product and sum differentiation compares the stream itself. For a silent stream the assertion is immediate. Passing through every ultrafilter above the tail filter turns this into ordinary convergence of the derivatives of any fixed-order allowed test family.

Here is why such test-family convergence gives a uniform paid derivative bound, including on the closed healthy set. If an order- derivative bound failed uniformly over the allowed maps from and healthy base configurations, select a violating map at each bad history index. The assumed convergence for that sequence of maps contradicts the violations. Take the finite intersection for . This yields a single late cutofffor all these orders.

To extend the test to an arbitrary finite-dimensional parameter space and directions define

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(
vν/∥vν∥,
vν ̸= 0,
0,
vν = 0.
k
X
Lz =
1
k + 1
ν=1
zν bvν,
bvν =

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With the coordinate maximum norm, ∥L∥≤1. The vectors aν = (k + 1)∥vν∥eν satisfy Laν = vν.
Pulling an allowed map back by z 7→Lz + p preserves its allowed derivative bounds. A test bound
b/(k + 1)k then bounds its derivative on the aν by b Q
ν ∥vν∥, proving the full operator bound b.
Smoothness for this subtraction is furnished by the separate primitive derivative bills at the healthy
base; the mask itself is never differentiated.

Finally convert these arbitrary-map derivative bounds to the anchored paid direction convention. At a fixed period 1 + dist() is bounded by one finite . The conversion costs at most independent of the history index. Divide the desired tolerance by this constant before applying the uniform comparison. The identity configuration makes the healthy test set nonempty. This proves the regular comparison for a stream at its actual fixed support. Real and imaginary streams are treated separately, multiplied by their common normalization and recombined, then embedded by zero into the common enumeration. This supplies the microscopic regular estimate used above.

Passing microscopic comparisons through projection

The microscopic regular comparison means that both arrays are regular on one fixed family of supports, have uniformly summable paid bounds with uniformly small tails, and, at each fixed label, their difference eventually has paid derivative bound below every positive tolerance. The forced comparison means both arrays have absolute bounds with uniformly small summable tails and, for each fixed label and every sequence of configurations, the difference of their values at tends to zero. The latter implies eventual uniform smallness over configurations: otherwise choose a violating configuration on each bad index to contradict that sequence test. Neither notion assumes continuity of the masks.

The tail hypothesis gives a useful dominated-series principle. If two arrays have uniformly summable absolute tails and their fixed-label differences tend to zero, split the sum into a finite set