Lemma 62 ([Boundary corners of the chosen domains]). The independent-argument domains in this section can be chosen so that the following holds for every sequence in any one of them, including a sequence on which imposed strict margins …

There are two different kinds of margin. A working-box margin restricts data to a smaller part of a region on which the analytic formula and its existence estimates are already valid. A structural margin chooses a positive coordinate, a nonzero algebraic divisor, an ordering, or a clock regime. We check their boundaries separately. This is a property of the concrete domains, stronger than completeness for sequences whose margins stay positive. Analytic boxes, amplitudes, and state room. Choose three nested closed boxes for each bounded argument: an admitted box, a larger working box, and an analytic-extension box, each contained in the interior of the next. The finite preparation and root charts provide the outer boxes before the admitted ones are chosen. Similarly choose an admitted input-state set strictly inside a working state disk and the working disk strictly inside the field's analytic state disk. Choose the admitted amplitude bound η , coefficient bounds, and any integrated prefactor bounds so that the real existence estimate holds on a slightly larger closed range than the admitted one. These choices are made once. For example a zero-base passage with integrated forcing and Lipschitz bounds Cη sends |z_ in |≤r_0 into |z(t)|≤(r_0+Cη)e^ Cη ≤r_1<r_2, where r_2 is the analytic state radius. Stable damping permits the same bound; the regular base is first removed on its specified larger flow box. Thus a working-box margin tending to zero reaches the edge of the admitted box, which still has uniform analytic room. It does not mean that a trajectory reaches the singular boundary of the field. Here all needed integrated estimates hold for the independent ambient families. In, the base has the small coefficient b , while the layer integral of the other term is bounded by C|b'|η , independently of L . The physical equality b=Lb' is not used for that estimate. In the mixed simple model, ∫|g|\, t≤Cη and the homogeneous term is damped for every W,L>0 . The mixed annular fields have a designated fast factor with a fixed rate margin, giving the bound preceding for every independent bounded prefactor. The additive bounds integrate nonconstant powers against w/w in (A), and power-corrected fast layers against w in (B); they hold uniformly on the closures of the smaller amplitude ranges and for q≥q_* . In the one-rate stable model the fixed lower large-rate threshold may be increased before defining its domain; the bounded slow field is then controlled by damping even at that threshold. The bounded- w_0 ordinary annular kernel has the guard in; its forcing and Lipschitz integrals are at most Ch on the closed admitted range, even for independent normalized profile arguments. Bounded-clock factors have analytic ODE dependence; if a finite limiting slow interval is too long for one base-flow chart, take one fixed finite subdivision and smaller flow charts. The original trajectory estimate supplies the same state room at its internal states. If it is the damping of an old simple kernel that tends to zero, use its inherited forcing estimate, not a claim about an arbitrary bounded stable remainder. In, ∫_0^1|Lg|\, s≤Cη independently of W≥0 . In, (M)/(y+δ)\, s=( y)/(y), so the nonconstant-power estimate bounds its forcing and Lipschitz integrals independently of S≥0 , including S=0 by continuity. Thus zero damping is harmless for these particular old kernels; no such assertion is needed for a general stable equation at zero rate. Consequently input, output, amplitude, and artificial state-range inequalities can vanish without loss of these larger existence bounds. Rate, sign, and unit margins. All nonzero rational one-rate or additive rates are fixed positive numbers. Mixed (I) has fixed positive layer rates and nonnegative damping at its limiting boundary. In the away-from-resonance mixed (II) rule, let r_*>0 be a lower bound for the finitely many actual sign-adjusted rates on the chosen coefficient range after shrinking the generator sizes. Impose in the independent domain, with slack, the inequalities r_i>r_*/2 . At their boundary the rates are still positive. Here r_i=r_i(W/L) is the actual sign-adjusted rate of the independent kernel, not a rate evaluated at a separately copied ordinary coefficient. Thus no rate tending to zero at a positive limiting slope is admitted, even off the physical coefficient graph. The same choice applies to the designated fast rates in the small-slope rule. Its pure-state rates |β_i|c can tend to zero, precisely when c→0 ; the field then still vanishes without the designated fast layers. The both-large case is the zero-slope alternative of the mixed theorem, the bounded W case is--, and bounded clocks are treated below. Pure slow layers are never used alone to bound the forcing integral. Likewise fix the domains of prepared units, root linearizers, and profile inversions on the larger working boxes where their required Jacobians and unit divisors are bounded away from zero. This applies to F,F_s,b,b(0),h , the denominator in , and the root-chart transversality constant. Choose |δ|≤δ_*<1 on the working box, so 1+δ/y and y+δ , y≥1 , are units even on the admitted boundary. A margin choosing the sign of an algebraic prefactor, such as B or p , may instead tend to zero. The kernel formulas are analytic in that prefactor at zero; if a normalization divided by it, that division is an auxiliary graph of the next paragraph, not an uncontrolled analytic argument. Overlapping magnitude rules permit the same conclusion at their finite cutoff boundaries. Algebraic divisors and positive endpoint variables. A physical positive variable or an algebraic normalization divisor may approach zero. Keep every bounded normalized value as its own graph coordinate; for instance y=a/d uses dy=a , with d≠0 retained as a domain condition. Positive rational powers use polynomial relations with the designated positive branches. These relations are analytic also at zero, although their graph Jacobians may then become singular. At each point of the sequence the original nonzero or positive condition gives the same locally unique graph as before. A fixed finite number of differentiations produces only finite powers of these divisors, which can be cleared without changing the local solution set there. The coefficients of the resulting equations are analytic at a bounded graph limit. If a required normalized value is unbounded, it is instead a new recoverable large coordinate. A vanishing determinant imposed by a later rank restriction behaves in the same way: clear its reciprocal powers in the finite equations and retain its strict domain condition. No uniform inverse bound at the limiting boundary point is asserted or needed for the local graph equivalence at the sequence points. Zero clocks, ordering boundaries, and ratios. The ordinary normalized formulas explicitly include zero length. The nozero short field uses its bounded integrated coefficient. The nozero long field, if it reaches a bounded-clock corner, retains independent b,b' in. The simple mixed field is -Wv+Lg , whose layers have exponents affine in L . Mixed (II) has field Lp G and layers whose exponents are the fixed affine combinations αL+βW . All are analytic at bounded (L,W) , including a zero value of either clock, without evaluating W/L in a field coefficient. A duplicated bounded coefficient remains independent, with its product tie retained. If a slope interval itself is imposed using a ratio, its inequalities and tie may be cleared by L>0 ; no ratio is necessary inside these bounded-clock kernels. For example even L,W→0 and W/L→∞ cause no singularity in the mixed (I) normalized field. A diverging ratio already present as an auxiliary coordinate may still be retained as a large algebraic input, but does not invoke the unbalanced mixed primitive when the two primary clocks are bounded. In particular no auxiliary ratio is introduced merely to express a slope which these bounded-clock formulas have already eliminated. The mixed auxiliary ratio is required only in the specified regimes in which both primary clocks tend to infinity. For additive kernels the fixed bound q≥q_*>0 prevents a zero denominator. If Q stays bounded, then so does M=(Q-q)/q , and is regular, including Q-q=0 . For a finite large limit of M , the single fixed K in provides analytic fraction charts. If Q is unbounded, the divergent-ratio or bounded-ratio packet alternatives already proved apply. If two positive physical endpoints both approach zero, their ratio is handled by its graph and: bounded log length has the normalized extension just described, while a divergent log length is an explicit large coordinate. Vanishing physical clock speed similarly affects its algebraic endpoint normalization; the normalized field's unit denominator remains in its larger working box. This list exhausts the imposed terminal margins. Fractions and other rewritings use an exact identity of the old independent kernel on a neighborhood of each sequence point, with the same exogenous arguments. The fixed finite state and jet graphs therefore preserve isolation there, even if their neighborhoods shrink along the sequence. If all enlarged coordinates are bounded, the preceding checks give analytic kernel factors and analytic divisor-cleared graph equations at their limits. Otherwise the divergent enlarged coordinates are available for the flag and packet alternative. The two conclusions are not conflated with positive-margin completeness.