the store and boundary choice remain variable, and they range over a finite set at a fixed lattice step. For any ultrafilter one pair is therefore constant eventually: otherwise the complements of all its finitely many fibers would have an empty intersection in the ultrafilter. Replacing the gauge-slice coordinate by the fixed integration point produces one common draft. Viability transfers to it on that eventual set. No continuous gauge-slice variable is claimed to become constant along the original histories.
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In this common draft, the atmospheric physical fluctuation w is the same in the two histories. The restart map is p zβ(w) = β (w −tX).
The protected-mean comparison and coupling comparison imply comparison of this map. More generally its corrected version along an allowed output-field family replaces
by the common smooth steered field before subtracting the protected mean. The exact identity
ensures that the original real integrand is recovered; the analogous steered identity is used for its derivative integrand.
The Gaussian density formula is
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ρC(z) = (
√
2π)−D exp{−1
2 log det C −1
2zTC−1z}.The covariance comparison, its uniform positive floor, and the finite-dimensional inverse and determinant formulas compare every term in this expression. The floor also ensures that any subsequential density limit at a fixed finite
is strictly positive: the logarithmic exponent has a finite limit and its exponential is positive. Thus division by the corresponding marginal density is legitimate in the local reweighting formula.
The same argument applies to every principal covariance restriction on a fixed finite set of tested bond/color coordinates. Its positive floor is inherited from the full covariance, and its inverse and log determinant depend smoothly on that restricted matrix. The local correction factor is exactly its finite Jacobian factor times the density of the corrected local fluctuation divided by the density of the uncorrected local fluctuation. The Jacobian is a finite product of the smooth link steering factors. Each steering factor is evaluated near the identity in the common draft; the chosen local chart supplies its smoothness through the required derivative order. Consequently this entire correction factor compares through that order. This establishes the density and localchange-of-variable factors used in the ordinary and marked integral comparisons; comparison of their remaining vertex factors is treated separately.
The protected change of variables and its determinant
The steering factors in the preceding ratio have the following construction. Fix a small positive radius
, independent of the later scale delay, on which the logarithm is a left inverse of the exponential through radius 2
. Shrink a matrix neighborhood of 1 so that log(exp
) is analytic for
, and so that for group elements
it exponentiates back to exp
when
These choices follow by continuity on a closed ball strictly inside the original chart. Let
the smooth innershare cutoff, one on
, zero for
. Define, with this multiplication order,
log(exp
Φ
) = det(1+
Outside the cutoffsupport the first expression is identically zero in a neighborhood, so the map is smooth there regardless of the exterior values of the logarithm. On the support, smoothness follows from the chosen analytic chart. At
and its derivative vanish, Φ
= id, and
Compactness of
, joint continuity, and the exterior zero identity supply one neighborhood of
= 1 on which, uniformly for every
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∥pM(a)∥< d/2, ∥DapM(a)∥< 1/2, jM(a) > 0. (PV2)
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For the determinant assertion, use its continuity and its value one on the compact ball; outside the ball it is exactly one. The derivative bound makes pM Lipschitz with constant 1/2. Thus ΦM is injective, since equality of two images implies ∥a −b∥≤1 2∥a −b∥, and surjective, since for each target b the contraction a 7→b−pM(a) has a fixed point. Its derivative is invertible by the Neumann series. Consequently it is a global smooth change of real coordinates with positive Jacobian jM. For an original coarse background V0 and a nearby background V , the correction at a fine link is Me = (replica V0)e(replica V )−1 e , u′ e = ΦMe(ue). (PV3)
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The order matters: for ∥ue∥≤d,
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exp u′ e (replica V )e = exp ue (replica V0)e. (PV4)
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On inactive links the fine slice input remains unchanged. On active links the identity is needed
only where their protecting cutoffs are nonzero.
Those cutoffs imply ∥u′
e∥≤8a tfield.
Choose
the scale delay so that 8a tfield ≤d/2.
Then (PV2) implies ∥ue∥≤∥u′
e∥+ ∥pMe(ue)∥≤d, so
(PV4) applies there. This proves equality of the old and new reconstructed configurations on every
protected read. A cutoffproduct vanishing at the read makes the corresponding term zero. A
nonzero product forces equality on its whole actual footprint. This argument proves the equality
for old activities, atom words, row predictions, sector profiles and their complements individually,
including the hard decisions inside them. It does not differentiate a hard decision.
Writing the physical fluctuation as u = μV0 + gz, g = β−1/2 > 0, the corresponding Gaussian
coordinate change is
z′ = g−1{ΦM(μV0 + gz) −μV }.
(PV5)It obeys
) exactly, and becomes the identity at
. Restrict to the finite set
of bond/color coordinates required by the ground. Each link block of (PV5) has derivative 
Φ
Φ
; the scalar dilation factors cancel. The coordinate permutation between link/color arrays and Euclidean vectors contributes no absolute determinant. Therefore
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det Dz′ = Y e∈Ilinks jMe(ue) > 0. (PV6)
This is a product over tested links, with one color-space determinant per link, rather than one determinant per color component. Let
be the principal covariance matrices on
, with Gaussian densities
. Finite-dimensional change of variables gives for an integrable local function
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Y
!
ρV (z′(z))
Z
f(w)ρV (w) dw =
Z
f(z′(z))
e
jMe(ue)
ρ0(z) dz.
(PV7)
ρ0(z)
|
{z
}
LV (z)The denominator is strictly positive at every finite
. Both integrals can equivalently be written against the original full Gaussian, since the functions depend only on the indicated marginal and the unused coordinates integrate to one. This proves the local likelihood formula, including its direction and normalization.
Along a smooth coarse field family based at
, its finitely many matrices
lie in the common neighborhood from (PV2). Differentiation of (PV1) on the compact cutoffball gives uniform finite-order bounds on the displacement and determinant; exterior derivatives vanish or equal those of the constant one. Composition with the coarse family gives a displacement price of the same small order as that family’s motion and a determinant price of order zero. Equations (PV5)–(PV7) then supply the Gaussian score derivatives. The covariance floor and the finite inverse/determinant formulas control the score coefficients. The integrable Gaussian envelopes established above justify differentiation under the integral. Thus a marked entry can remain evaluated on the fixed original fine input: the derivatives fall on its protecting cutoff, on the smooth ordinary vertices, and on
The protected equality (PV4) identifies this formula with the original integral on a neighborhood of the base point. This is the mechanism used in the marked primitive comparisons and their product estimates.
Finite-window covariance comparison
For clarity, covariance comparison here is obtained from its actual construction. The form matrix consists of the selected physical row operators evaluated on coordinate impulses. Selection is determined by whether the row base is outside the store’s fixed six-radius interior. In an unselected row both histories have zero form entries. In a selected row the physical row comparison gives entrywise comparison through the required derivatives, with the fixed impulse held the same. The plate is the form Gram matrix plus the fixed dormant diagonal. Finite matrix multiplication and addition therefore compare its entries.
For each covariance column, its window is the fixed finite set prescribed by the reference frame. The principal plate restriction inherits the uniform positive floor. Inversion on this set of positive definite finite matrices is smooth, with derivatives controlled by the same inverse bound; hence the two restricted inverse matrices compare through the required order. Entries outside the window are identically zero in both histories. Placing the inverse column back in the full coordinates and symmetrizing gives the actual reference covariance, so it too compares. This retains the columnwindow construction and does not replace it by the full plate inverse.
The protected mean is obtained from these same window columns and its local bill vector. Their comparison gives the mean comparison used by restart. Together with the coupling-window comparison it supplies the affine physical-variable comparison. The scalar density proof above consequently rests on finite inverse matrices with a common lower eigenvalue bound. Its remaining physical-row and bill-vector inputs are precisely the stationary and local nonlinear comparisons, rather than an independent covariance convergence assumption.
Extracting the common activity comparisons
At each fixed level and physical period, the activity data lie in one common normed presentation space. Its dictionary, supports and threshold are fixed, and its norm controls anchored regular derivative mass and forced absolute mass. Its completion is used only for limits. The actual retained finite histories remain elements of the original space.
We spell out how a Cauchy sequence in this norm gives the evaluated comparisons needed above. For either the regular-occurrence map or the forced-coefficient map, the activity norm implies a nonnegative absolute budget of total at most
, uniformly over configurations. At a fixed period one may take
to be the number of sites times the corresponding anchored budget constant. For regular terms sum the anchored masses over their unique anchors; for forced terms choose a site from each nonempty forced inventory contained in its support. The local mass dominates the
unweighted bound. This yields the stated global budget. Choose its canonical least envelope
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q(a, i) = sup U |F(a, i, U)|.
The existing absolute budget makes this supremum finite, nonnegative and summable with total at most
. Linearity of
gives
). For a Cauchy sequence
, these envelopes have uniformly small tails. Indeed choose
with
2 for
, and a finite set controlling the tails of the finitely many
by
2. On its complement the displayed subadditivity bounds every later tail by
; the earlier tails are already controlled. If
are Cauchy and
0, then
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|F(an, i, U) −F(bn, i, U)| ≤q(an −bn, i) ≤C∥an −bn∥
uniformly in
for each label. The two envelope families also have the uniform tail property just proved. Thus regular occurrences with their actual fixed healthy masks, and forced entries with their actual inventories, have exactly the evaluated comparison and tail bounds used in the local construction. Masked regular evaluation is linear because the threshold and each labelled support are fixed in this presentation.
For any two escaping subsequences of a single Cauchy activity sequence, their norm difference tends to zero by the Cauchy criterion. The coupling sequence has the same property and converges in
. The retained history’s actual activities are eventually these normed presentation elements at every fixed level. Hence all incoming regular, incoming forced and outgoing forced comparisons required by the two-history transfer argument follow from the common activity Cauchy sequence. Completion convergence suffices: the completion embedding is isometric, so convergence there makes the original presentation sequence Cauchy.
The scalar and activity convergence themselves are obtained by the retained-history construction as follows. At each level its normalized scalar coordinate is bounded in a fixed finite interval, by the positive width and the uniform normalized-coupling bound. Compactness of the countable product of these intervals gives one diagonal escaping subsequence on which all scalar coordinates converge. The fixed-scalar construction then gives convergent activity data and the associated intermediate quantities at all levels along that subsequence. To promote this to the full sequence, repeat the same extraction on any escaping subsequence. The comparison of two such banked limits with the same limiting terminal coupling identifies their scalar limits. The subsequence criterion for convergence therefore gives full scalar convergence, after which the fixed-scalar construction gives the full activity convergence. The section on uniqueness of banked history limits proves the comparison identifying these limits. Compactness is used only to obtain the candidate limits.
For the winning trajectories the bottom coupling at level zero is the same prescribed center ̄
so its convergence hypothesis holds. The finite-horizon alignment identifies their paths with the original stationary paths and their normed activity presentations with the actual retained activities. This provides the Cauchy sequences used throughout the fixed-level insertion proof, on the original common choices.
Finite-depth approximation of the activity trajectory
At a fixed finite horizon, regard the normalized scalar coordinates as an externally specified admissible sequence. The activity update maps the next-level activity to the current one and keeps its norm at most 1
8. On this ball its paid comparison is
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∥Φi(λ, x) −Φi(λ′, x′)∥≤c0 max{|λ −λ′|, ∥x −x′∥},
c0 = 8/128 = 1/16.The accompanying scalar correction satisfies the corresponding bound with coefficient 4
. These are the finite-history estimates used in the banked-limit comparison.
For any intermediate depth
, construct a comparison activity trajectory by setting its activity at depth
to zero and applying the same updates backwards below
; set unused later levels to zero. Finite decreasing induction constructs this trajectory. The paid ball-invariance estimate keeps every value admissible. Compare it with any full trajectory having the same scalar script. At depth 
the distance is at most one (in fact at most 1
4); each backward step multiplies that distance by at most
. Thus at
the discrepancy is at most
. 0
Now compare two sequences of histories whose scalar scripts have the same limit at every fixed level. For the trajectories truncated at the common fixed depth
, the top activity is identically zero in both histories. Descending induction, using the fixed-input one-step comparison at each of the finitely many levels, gives a common limit of their truncated activities. Given
0, choose
so
3. For sufficiently late histories both horizons exceed
and their truncated activities 0 differ by less than
3. The two truncation errors prove that the original level-
activities differ by less than
Applying this argument to every pair of escaping subsequences proves that the activities have a common limit in their complete activity space. The scalar correction limit follows by the one-step comparison using the already convergent next-level activity. Finite initial segments discarded to ensure all required horizons are present do not change these limits. This supplies the fixed-scalar convergence used in the diagonal extraction and banked uniqueness argument. Its fixed-input one-step comparison is the local Gaussian/cluster/reset comparison developed above; the ball-invariance and contraction constants are the paid one-step estimates on that same update.
The physical scalar identity also survives this passage. At finite horizon the correction equals the difference of adjacent physical couplings minus the committed path coefficient. The latter has its stationary limit. Taking the three scalar limits therefore gives the displayed banked flow identity, with no infinite telescoping sum.
From labelwise comparison to reset budgets
At a fixed physical period there are finitely many site supports. The budget weight is linear in the nonnegative scalar bill, so there is a finite constant
0 with
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p.mass(D, t) ≤P∗t (t ≥0)
for every support
. This constant may depend on the fixed period; no uniform volume assertion is used in this convergence argument.
Suppose a labelled difference has a nonnegative uniform-tail majorant
, and each fixed label eventually admits every positive tolerance. Given a desired total weighted budget
, put
Choose a finite label set
with every majorant tail at most
2, and eventually bound each label in
by
+ 1)]. Use these small bounds on
and the original majorant elsewhere. The resulting nonnegative array has total at most
, hence weighted budget at most
. This applies both to the paid regular difference bounds and to the all-configuration forced difference bounds. It proves the quantitative difference-budget statements used to recover the activity norm.
The same construction controls the projected coupling increment. Each source separately has a common finite paid total, obtained by multiplying its uniform-tail bound by
. The diagonal projection estimate gives 
∆
threshold
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For the difference of two sources, choose its paid deviation budget below t2 0ε/2. Linearity of projec- tion and its diagonal drift estimate then give |∆n −∆′ n| ≤ε/2 eventually. Thus the two increment
sequences are uniformly bounded and their difference tends to zero. This proves the increment comparison used in the exact coupling update, rather than assuming a separate scalar continuity principle.
The reset dictionaries themselves are on common supports. The canonical stationary path presentations retain their fixed slot and atom supports, so the low homes and high footprints do not change with their numerical coefficients. The reset listing is therefore identical for both path families. Its regular and forced comparisons can be transferred through the final dictionary consumption on this same listing. The final vacuum and coupling scalars are irrelevant to that activity comparison. These observations isolate the remaining task in reset convergence: compare the actual residual, freshness, loan and attached-log coefficient functions with uniform tail budgets.
Comparison of the reset coefficient families
The regular reset family is the disjoint union of the source rebates, low freshness terms and high freshness terms. For a rebate, the difference identity is the rebate of the source difference. Its paid bound is multiplied by the fixed reset-loss constant, so both the uniform tail bound and each label’s arbitrarily small comparison bound pass through. The fault difference has the same linear identity and the same all-field reset-loss bound, yielding the forced comparison.
For a fixed low label, the actual slots and support are independent of the stationary numerical coefficients. Only its mark
varies. Choose once a reference stationary coefficient array for which this label’s mark
= 0 (all path coefficients equal to one suffice for this single-label algebraic test, and the direct mark is positive). Its freshness is then the fixed reference freshness multiplied by 
. The coefficientwise path comparison makes these scalar ratios compare, so the fixed-label paid freshness difference tends to zero. Its uniform majorant is the original weighted low-mark budget times its fixed low-cost polynomial and the common threshold factor. Bounded-length low labels form a finite set, since each pair of path labels has total length bounded. The spare positive exponential width therefore makes these majorants have uniformly small tails.
High freshness is minus the sum of the two bounded compared scalar parameters times the actual cubic or quartic atom. The atom coefficient comparison gives fixed-label convergence; the weighted atom budget with spare width gives uniform tails. Finite signed-seat replication and the fixed orientation multiplier preserve all these bounds. The projected source increment is a bounded compared scalar by the previous argument, so it may be used as one of those two parameters.
For the low loans, the same reference-mark factorization gives exactly
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loanR(α, ∆) = loanR∗ (m/m∗)α, (m/m∗)∆ .
This follows from the literal loan definition: its live pair term is proportional to the mark, its proxy is independent of the numerical path coefficients, and its home and kept tests are common. In the direct case the mark is fixed. Subtracting the two reference loans is the reference loan at the differences of their two scalar arguments. Its all-field shape bound tends to zero as those arguments tend to zero. Thus the loan differences converge uniformly over configurations, even on the boundary of a failed health test.
The common absolute majorant for these loans is
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el/ρ|m| 2t(Bα + B∆) + 1 2B∆pairBill t2 .
The width hypothesis leaves a positive exponential margin after
is spent. The weighted low-pair budget and the finiteness of bounded-length labels therefore give uniformly small tails. Combining faults and loans proves the attached-log comparison. Combining rebates and the two freshness
families proves the regular reset comparison, all on the same canonical listing and with its original countable enumeration.
The final consumption first turns every attached logarithm
into
the mean value estimate gives
. A common total absolute budget supplies such a 
, so the increment family has uniformly small summable tails. At each fixed label exponential continuity, applied to the bounded compared values under arbitrary varying configurations, gives convergence of the increment difference, and the varying-configuration selection argument makes this uniform in configurations.
The resulting forced family is indexed by finite subsets of the union of the old hard labels and these exponential-increment labels. Its coefficient is their finite product when the canonical support/eligibility test holds, and zero otherwise. The same test is used on both sides. Finite products of bounded compared values compare, and multiplication by this common indicator preserves the comparison even if the indicator varies with the history index. The finite-subset product budget preserves uniform tails. Canonical injective reindexing places these coefficients in the drawer, with zero outside its image; finite sums over the preimage establish preservation of both totals and tails. The regular family is likewise placed by its canonical support embedding. Hence the completed output arrays have the quantitative weighted difference budgets used in the activity-norm closure. This closes reset convergence without differentiating a hard indicator.
Closing the one-step convergence in the activity norm
Suppose two incoming activity presentations are Cauchy with norm difference tending to zero, and their couplings have a common limit. The canonical-envelope argument above gives the ordinary regular-occurrence and forced comparisons. The common stationary path and kernel limits then supply the microscopic comparisons for the two physical steps. Their actual output source dictionaries are expressed on one common listing by the local construction. The path reset is performed with those same dictionaries and the same committed coefficients.
The coupling after this step is exactly
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βout = βin + αpath + ∆source.
The projection comparison gives a vanishing difference of the two source increments ∆
, and the stationary path coefficients have the same limit. Thus the two output couplings differ by a quantity tending to zero. The regular and forced reset comparisons give the analogous convergence of the output activity presentations.
To see the last implication in the actual norm, its budget characterization has two positive normalizing constants,
and
. For a target
0, choose the regular difference budget below 
and the forced difference budget below (
, simultaneously at every anchor. The regular and forced comparison tail lemmas supply these bounds eventually. The common support and format data permit their use for the actual difference presentation. The norm characterization then bounds its norm by
. Consequently the output norm difference tends to zero. This step requires the quantitative tail comparisons after reset, not just pointwise convergence of individual output coefficients.
The incoming completion data used by the finite-history update satisfy exactly these hypotheses. Their common completion limit makes their original presentation sequences Cauchy and their difference tend to zero by the isometric embedding. The continuous physical-coupling map sends their common normalized-scalar limit to a common physical limit. The chosen physical step identifies its actual output activities with the update’s original presentation element; no new output
representative is selected for the comparison. The displayed one-step coupling identity also identifies the scalar correction as output coupling minus input coupling minus the committed path coefficient. This proves the fixed-input update convergence invoked in the finite-depth argument, once the regular and forced reset tail comparisons are established.
Uniqueness of banked history limits
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The banked contraction compares histories at the same retained normalization. Its tail error tends to zero after the common finite-depth approximation is fixed. The double-tail Cauchy argument proves convergence of the whole relevant inner family, not merely convergence of an extracted subsequence. This is the convergence input used for the eventual full fixed-volume mesh limit. Here is the comparison that makes the diagonal extraction unique. Consider two banked limits with normalized scalar coordinates fi, f′ i, completed activity data Xi, X′ i, and local scalar correction limits gi, g′ i. Their bottom coupling limits at level zero agree. Write
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di = max{|fi −f′
i|, ∥Xi −X′
i∥},
γ0 = 1/128.
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The one-step comparison, applied to the two limits by approximation with actual finite trajectories, gives ∥Xi −X′ i∥≤8γ0di+1, |gi −g′ i| ≤4sidi+1.
The scalar flow identity and equality at the base give, in physical coupling coordinates,
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Ωi −Ω′ i = − X j<i (gj −g′ j).
The normalized-coordinate comparison has a positive weight
with
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bi|fi −f′ i| ≤|Ωi −Ω′ i|, X j<i sj ≤γ0bi.
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All these inequalities use finite sums at each i. The original small activity and scalar bounds first give di ≤1/4 uniformly: for the scalar part apply the finite scalar-flow identity and the coarse individual correction bounds; for the activity part add the two individual norm bounds. Let D = supi di, which is therefore finite and nonnegative. The one-step activity inequality yields ∥Xi −X′ i∥≤8γ0D. The scalar identity and correction inequality yield bi|fi −f′ i| ≤4D X j<i sj ≤4γ0Dbi.
Since
0, the scalar difference is at most 4
. Hence
for every
, and taking the supremum gives
16. Thus
= 0, proving equality of every scalar and activity limit. The one-step inequalities arise by taking limits in the paid finite-trajectory comparison; the nearby finite trajectories are constructed by (FCA7) and (FCB5), with positive comparison weights and cumulative bounds (FCB1)–(FCB3); the fixed-scalar comparison identifies their activity limits. This is the precise contraction used to upgrade the subsequence construction above.
Extracting stationary path coefficients and their higher jets
The path book retains stationary coefficients, their higher jets and the true supports of the corresponding rows. Absolute convergence at a smaller analytic width permits termwise differentiation. Insertion transfer then consumes those same coefficient comparisons, so that changing the path presentation does not amount to changing the represented local insertion.
Source: A Wilson Yang–Mills Construction: A Natural-Language Mathematical Exposition. Both papers belong to one Zenodo record. No external funding.