Relative descending paths with finitely many same-index passages

A nonempty space of admissible families. Near a fold write the already supplied function, after rescaling the center variable and parameter by nonzero constants, as $f_s=a(s)+x^3-\mu (s)x-\mathrm{\mid }u{\mathrm{\mid }}^2+\mathrm{\mid }v{\mathrm{\mid }}^2,$ where $u\in {\mathbb{R}}^k$, $v\in {\mathbb{R}}^{d-k-1}$, ${\mu }^{\mathrm{\prime }}(s_{\ast })\mathrm{\ne }0$. Fix the exact field $X_s=(\mu (s)-3x^2){\mathrm{\partial }}_x+2u{\mathrm{\partial }}_u-2v{\mathrm{\partial }}_v$ on a smaller fixed box. The two Morse points, when $\mu >0$, have $x=-\rho ,+\rho$, $\rho =\sqrt{\mu \mathrm{/}3}$, and indices $k+1,k$ respectively. Away from protected birth neighborhoods choose smooth parameter Morse coordinates along each regular critical arc, agreeing with the prescribed endpoint field chart at an endpoint. Here is the chart patching issue: in overlapping parameter Morse charts the transition preserves the fixed signed quadratic form. Its germ is isotopic to its linear part by $h_t(z)=t^{-1}h(tz)$, continued at $t=0$ by $Dh(0)$. The linear part can be joined within its component of the quadratic-form group; reverse coordinate frame components first if needed, which leaves the normalized model field unchanged. A finite subdivision along the interval thus patches chart germs on smaller neighborhoods. There is no critical-arc monodromy issue: between folds and parameter ends each arc projects diffeomorphically to an interval. Use $X=(2u,-2v)$ away from the birth neighborhoods and the exact cubic gradient near each fold. The transition between these two choices can be made explicitly without mixing the local invariant disks. On a compact parameter annulus away from the fold, scalar square completion at either root supplies a smooth center coordinate $z$ with $g_{\mu }(x)=g_{\mu }(r)\pm z^2$. Leave the transverse $u,v$ coordinates unchanged. In these coordinates the cubic gradient has center component $a_s(z)z{\mathrm{\partial }}_z$, with $a_s$ having the appropriate fixed nonzero sign, and transverse components $2u,-2v$. Interpolate $a_s(z)$ to the signed constant $\pm 2$ with cutoffs supported on this compact regular annulus and inside the chart. Descent persists because each center summand has the correct sign, and the stable/unstable coordinate planes remain unchanged. Match the normalized end of this annulus to the regular Morse charts. No linear eigenvalue normalization of the cubic field itself is assumed. A background negative gradient, the finitely many local fields, the prescribed endpoint fields, and the vertical collar field can be patched with a smooth partition of unity. Away from critical points the inequality $d{f}_s(X_s)<0$ is convex, and all local choices coincide on smaller prescribed model regions. Keep the just constructed critical-point fields on those smaller regions. Endpoint constancy and spatial-collar equality are retained. This constructs an initial admissible family. Allow perturbations supported away from the smaller model regions, endpoint parameter intervals, and spatial collars. On every compact permitted support the strict descending inequality has a positive margin. Consequently sufficiently small smooth perturbations remain admissible. All subsequent perturbations are of this form. Regular local unstable and stable sections, including their normal coordinates, depend smoothly on $s$ and are fixed with respect to these perturbations: use the fixed model or the explicitly cited parameter-disk contraction. Their finite flow extensions use the actual flow supplier. Thus no parameter stable-manifold theorem beyond the supplied elementary proof is being silently imported.

The incidence maps and their actual adjustable directions. For regular critical branches $p_i(s),q_j(s)$ choose small outgoing unstable and incoming stable sphere sections $A_p(s,\xi ),B_q(s,\eta )$ inside their fixed local model regions. Their dimensions are $i-1$ and $d-j-1$. A connecting orbit is uniquely represented by ${\mathrm{\Phi }}_{X_s}^t(A_p(s,\xi ))=B_q(s,\eta ),t>0.$ The source variables $(s,\xi ,t,\eta )$ have dimension $d+i-j$. Equality with the diagonal in $W\times W$ has codimension $d$. The expected dimension is therefore $i-j$. All domains are open finite-flow domains; in particular no infinite-time evaluation is used. Cover sphere sections by finitely many charts and these domains by countably many relatively compact coordinate pieces, with time bounds $1\mathrm{/}m\le t\le m$. Parameter subintervals stay away from folds and from parameter ends. A countable exhaustion covers every ordinary connecting orbit. At every required incidence the orbit has a subsegment outside all protected critical charts. Otherwise it could not join distinct critical charts. The only two critical points allowed inside one protected birth chart are the adjacent-index newborn pair; their local connection has positive drop and is not one of the required nonpositive-drop incidence problems. Choose a sufficiently short flowbox on the external subsegment, with coordinates $(z,r)$ in which $X_s={\mathrm{\partial }}_r$ at the parameter in question. A smooth parameter-dependent version follows from smooth dependence of the finite flow and the inverse function theorem. Changing the field there to ${\mathrm{\partial }}_r+\beta (r)\chi (z)w\cdot {\mathrm{\partial }}_z$ with $\int \beta =1$ changes the exit point to first order by arbitrary $w\in {\mathbb{R}}^{d-1}$ in the transverse directions. The cutoff is one along the chosen segment and zero near the box boundary. The time variable supplies the missing flow direction. Transport by the subsequent flow is an invertible differential. Thus the derivative of the incidence equation in these $d-1$ shear parameters together with $t$ surjects onto $TW$. Equivalently, one may use the exit level as target and omit $t$; then the shear alone is onto the $(d-1)$-dimensional target. The descending inequality survives small shears, and all protected regions are untouched. For two distinct trajectories at one parameter, their interiors are disjoint by uniqueness. Choose one small flowbox on each, avoiding the other compact segment. The shear parameters independently prescribe their transverse exits. This remains true if the trajectories share an initial or terminal critical point: their chosen boxes lie away from those points. Smooth parameter cutoffs supported near the parameter make these legitimate perturbations of the whole family. The product two-incidence map is therefore transverse to the product diagonal. The same argument applies to any finite collection of distinct trajectory segments, though only two are needed here. For one equal-index incidence the transverse inverse image has dimension zero. For two distinct equal-index incidences at the SAME parameter the source has one shared $s$ rather than two, so its dimension minus the two diagonal codimensions is $-1$. For a single incidence at a fixed critical-value-crossing parameter the dimension is $i-j-1$, negative when $i\le j$. These are explicit finite-dimensional incidence calculations, with independently adjustable derivatives just established.

Finite Sard on compact pieces, then an actual convergent exhaustion. Here are details converting the preceding pointwise directions into one smooth family. They replace an anonymous infinite-dimensional residuality assertion. Fix a relatively compact piece of one of the above incidence domains and an admissible family $X$. Its zero set in the compact closure is compact. At each zero choose the finite shear directions from phase 2, and use a finite subcover of that zero set. Taking the union of these finitely many directions produces ONE finite-dimensional perturbation family whose universal incidence is transverse near that compact zero set. After shrinking its parameter ball, all zeros on the compact piece remain in that neighborhood; the complement was compact and had positive distance from the diagonal. Apply finite-dimensional parametric transversality [F5] to this family on the open neighborhood. Its good parameters meet every ball. Thus transversality can be achieved arbitrarily closely on the compact piece. The same argument for product maps excludes two distinct equal-index incidences at a common parameter. Exhaust the distinct-segment configuration domain away from its diagonal; repetitions of one trajectory are not counted as two events. Transversality on a compact piece, and avoidance of a closed diagonal there when the expected dimension is negative, are open conditions in a sufficiently high finite $C^r$ topology. Enumerate all compact pieces of all required single, double, and fixed-parameter incidence problems, with slightly larger pieces whose interiors cover their closures. Starting from $X^{(0)}$, at stage $n$ use finite Sard to achieve condition $n$ and preserve conditions $1,\dots ,n-1$; choose the change smaller than half a previously reserved openness radius for each of these finitely many conditions. Also choose its $C^n$ norm below $2^{-n}$ and small enough to preserve descending margins on its compact support. Reserve positive tail budgets for all achieved conditions: after stage $n$, require the sum of all later $C^{r_n}$ changes to be smaller than half its openness radius. This can be enforced by choosing every later change below the minimum of finitely many geometric budgets. The sequence converges in every $C^r$ norm to one smooth field family. Every achieved condition survives its total tail. Protected regions stay exactly fixed. A point outside the protected critical regions lies in some compact exhaustion piece with a descending margin; its margin is protected once that piece enters the list. Equivalently one can bound each field change pointwise by a fixed small positive multiple of the original $-df(X)$ on the permitted region. Hence the limit remains strictly descending everywhere away from the critical set. The list comprises: ordinary single incidences with $i\le j$; all ordinary double same-index incidences with distinct trajectories; all nonpositive-drop incidences at each fixed critical-value-crossing parameter; and the fixed-birth incidence problems in phase 6. Parameter-endpoint problems are already satisfied by the prescribed endpoint fields and need no change. Positive-drop single incidences are not imposed, so even non-Morse–Smale endpoint fields satisfying the stated nonpositive-incidence hypothesis cause no difficulty. No claim of a single bounded polynomial parameter family for arbitrarily long trajectories is made: each compact finite-time problem has a finite family, and the displayed convergent construction handles the countable exhaustion. One can make the domain exhaustion independent of evolving finite-flow domains as follows. Extend the fixed collar field smoothly across a double of $W$, and extend every permitted perturbation by zero, so bounded-time evaluations are globally defined on a compact ambient manifold. In the compact source pieces impose also that the entire orbit segment stays at distance at least $1\mathrm{/}m$ from $\mathrm{\partial }W$. This is a closed condition. Every relevant connecting segment eventually satisfies it, since its endpoints and its compact finite-time image lie in the interior. Only zeros satisfying this condition require the shears above; their neighborhoods can be chosen in the interior of $W$. Compactness controls all such zeros for nearby fields too. Possible ambient trajectories outside $W$ are not included and acquire no genericity claim. This resolves domain changes during the successive small perturbations. For the resulting family, dimension gives no index-increasing ordinary trajectory and locally isolated equal-index incidences. It also excludes simultaneous distinct equal-index trajectories, including pairs joined end-to-end at the same parameter. The derivative at an equal-index incidence is an isomorphism after quotienting the flow direction. On an incoming stable section this says exactly that its $i-1$ source-sphere directions and its one parameter direction span the $i$ normal unstable directions. This is the transversality hypothesis of the local framed-passage comparison, not full slice-wise Morse–Smale transversality.

Uniform broken-orbit compactness away from folds. For the final smooth family constructed in step 3.1, take parameters $s_n\to s_{\ast }$ and ordinary connecting trajectories with critical endpoints converging to distinct regular critical points at $s_{\ast }$. In local Morse coordinates the height-parametrized Euclidean speed is at most $C\mathrm{\mid }a-c(s_n){\mathrm{\mid }}^{-1\mathrm{/}2}$. Smooth parameter charts have uniformly bounded coordinate derivatives on compact closures. Outside smaller critical charts, $\mathrm{\mid }X\mathrm{\mid }\mathrm{/}\mathrm{\mid }df(X)\mathrm{\mid }$ is uniformly bounded. Summing the finitely many chart estimates gives, for any height interval of length $\delta$, a uniform length bound $L\delta +C{\sum }_{\text{critical branches}}\sqrt{\delta }\mathrm{.}$ The same estimate works for general negative-gradient metric representatives with uniform local constants: $\mathrm{\mid }X\mathrm{\mid }\le C\mathrm{\mid }z\mathrm{\mid }$, $-df(X)\ge c\mathrm{\mid }z{\mathrm{\mid }}^2$, and $\mathrm{\mid }f-c\mathrm{\mid }\le C\mathrm{\mid }z{\mathrm{\mid }}^2$. Our exact local fields are a special case. Choose an auxiliary Riemannian distance and extract a uniformly convergent subsequence of height curves by compactness and this equicontinuity estimate. Moving endpoint heights can be identified by affine maps when their limiting difference is positive. On a compact interval of regular limiting heights the curves solve the height ODE $h^{\mathrm{\prime }}=X_s\mathrm{/}d{f}_s(X_s)$. Uniform convergence and smooth convergence of the coefficients pass this equation to the limit. Uniqueness makes the limiting segment one $X_{s_{\ast }}$ orbit. Its endpoints are critical; an orbit cannot reach an equilibrium in finite time. There are only finitely many critical points, and height strictly decreases along every nonconstant component, so the result is a finite broken trajectory. No transversality, gluing, or index-drop bound is used in this extraction. Length is bounded by the number of distinct critical heights traversed. Equal critical values cannot be linked by a nonconstant orbit. If endpoints approach two distinct critical points whose critical values coincide at a value crossing, no connecting orbit exists for all sufficiently large $n$. Indeed choose disjoint fixed smaller critical charts. A joining trajectory must cross a compact annulus outside one such chart; on it the field and the descending inequality have uniform bounds. Along arclength the loss is $-df(X)\mathrm{/}\mathrm{\mid }X\mathrm{\mid }\ge c>0$. Traversing the annulus therefore loses a fixed positive height. This contradicts the endpoint height gap tending to zero. This also prevents accumulation at a value-crossing parameter by disappearing connecting segments of zero height.

Uniform compactness through a cubic fold. The Morse estimate is not uniform when a birth eigenvalue tends to zero. Here is the replacement, rather than an assertion that Morse compactness automatically covers the fold. In the fixed fold box put $g_{\mu }(x)=x^3-\mu x$. If $\mu \ge 0$, choose $r=\mathrm{sign}(x)\sqrt{\mu \mathrm{/}3}$ (either sign at $x=0$). Direct factorization gives $g_{\mu }(x)-g_{\mu }(r)=(x-r{)}^2(x+2r),g_{\mu }^{\mathrm{\prime }}(x)=3(x-r)(x+r)\mathrm{.}$ Because $x,r$ have the same sign, $\mathrm{\mid }x+2r\mathrm{\mid }\le 2\mathrm{\mid }x+r\mathrm{\mid }$ and $\mathrm{\mid }x-r\mathrm{\mid }\le \mathrm{\mid }x+r\mathrm{\mid }$. Therefore $\mathrm{\mid }{g}_{\mu }(x)-g_{\mu }(r)\mathrm{\mid }\le C\mathrm{\mid }{g}_{\mu }^{\mathrm{\prime }}(x){\mathrm{\mid }}^{3\mathrm{/}2}\mathrm{.}$ For $\mu <0$ use $r=0$ and the elementary bound $\mathrm{\mid }{x}^3-\mu x\mathrm{\mid }\le C(x^2+\mathrm{\mid }\mu \mathrm{\mid }{)}^{3\mathrm{/}2}=C^{\mathrm{\prime }}\mathrm{\mid }{g}_{\mu }^{\mathrm{\prime }}(x){\mathrm{\mid }}^{3\mathrm{/}2}$. On a sufficiently small fixed box, $\mathrm{\mid }Q(u,v)\mathrm{\mid }\le C(\mathrm{\mid }u\mathrm{\mid }+\mathrm{\mid }v\mathrm{\mid }{)}^{3\mathrm{/}2}$. Thus, writing $D_s$ for the two local critical values when $\mu \ge 0$, and for the single distinguished value $a(s)$ when $\mu <0$, one obtains $\mathrm{dist}(f_s(x,u,v),D_s)\le C\mathrm{\mid }\mathrm{\nabla }{f}_s(x,u,v){\mathrm{\mid }}^{3\mathrm{/}2}\mathrm{.}$ For the exact negative gradient, height speed is $1\mathrm{/}\mathrm{\mid }\mathrm{\nabla }{f}_s\mathrm{\mid }$. Away from the finitely many heights in $D_s$ it is consequently at most $C\mathrm{dist}(a,D_s{)}^{-2\mathrm{/}3}\le C{\sum }_{c\in D_s}\mathrm{\mid }a-c{\mathrm{\mid }}^{-2\mathrm{/}3}\mathrm{.}$ Integration over ANY height interval of length $\delta$ is bounded by $C^{\mathrm{\prime }}{\delta }^{1\mathrm{/}3}$, uniformly in the parameter. A singular displayed bound at an isolated distinguished height is harmless to the integral. Together with the regular-chart and outside-chart estimates, this gives uniform height-curve equicontinuity even for trajectories spending arbitrarily long time in the birth bottleneck. The extraction in phase 4 now applies at the birth parameter too. The limiting function has only finitely many critical points including the one degenerate point; outside its critical heights the height ODE is nonsingular. Thus every limiting chain consists of ordinary trajectories and, possibly, incoming/outgoing trajectories at the fold. Its component heights strictly decrease. For completeness the limiting fold half-manifolds are explicit. At $\mu =0$, $\dot{x}=-3x^2$ has solution $x(t)=x_0\mathrm{/}(1+3x_{0}t)$. A point approaching the fold in positive time must have $u=0,x\ge 0$; one approaching it in negative time must have $v=0,x\le 0$. The strong versions have $x=0$. The other coordinates solve $u(t)=e^{2t}{u}_0$, $v(t)=e^{-2t}{v}_0$. Intersect these four sets with the entry/exit faces of a small rectangular block. They are compact disks or unions of disk faces with corners. Their dimensions and boundary strata are stated in phase 6. This also identifies limits of trajectories beginning/ending at a newborn critical point: the upper point's outgoing section approaches the weak unstable section, the lower point's outgoing section approaches the strong unstable section; the lower point's incoming section approaches the weak stable section, the upper point's incoming section approaches the strong stable section. These facts follow directly from the invariant equations $v=0$ or $u=0$ and the scalar phase line with roots $-\rho ,+\rho$. For example the upper unstable center direction occupies $x<+\rho$ and limits to $x\le 0$, whereas the lower unstable section has exactly $x=+\rho$ and limits to $x=0$. There is no extra trajectory entirely in the fold box joining the two newborn points: the transverse linear equations force $u=v=0$, and the scalar phase line gives the one standard connection. The energy-annulus argument in the actual elementary supplier excludes leaving the box and returning to join them when their value gap is small.

Fixed-birth conditions and exclusion of fold accumulation. At a fixed birth parameter the weak/strong unstable manifolds have dimensions $k+1,k$; their outgoing sections have dimensions $k,k-1$. The weak/strong stable manifolds have dimensions $d-k,d-k-1$; their incoming sections have dimensions $d-k-1,d-k-2$. Use all smooth face interiors and corner strata of the compact sections. Lower-dimensional corner strata only improve the inequalities below. Empty strong sections at extreme indices are omitted. Add the following fixed-parameter incidence maps to the exhaustion of phase 3, using the same finite-flow equality and the same regular-interior shear. All source dimensions below already subtract the target diagonal codimension: The dimensions are: ordinary $p_i\to q_j$, $i-j-1$; ordinary $p_i\to$ weak stable fold section, $i-k-1$; ordinary $p_i\to$ strong stable fold section, $i-k-2$; weak unstable fold section $\to q_j$, $k-j$; strong unstable fold section $\to q_j$, $k-j-1$. Consequently at the birth parameter: ordinary connections strictly decrease index; a connection INTO the weak fold can only come from index $i\ge k+1$; a connection INTO the strong fold can only come from $i\ge k+2$; a connection OUT OF the weak fold can only end at $j\le k$; a connection OUT OF the strong fold can only end at $j\le k-1$. These are not assumed compact-transversality statements on infinite-time maps: their finite-time maps and compact exhaustion were supplied in phases 2–3, and their fold sections are explicitly the compact faces just computed. Suppose equal-index connections accumulated at a birth. Apply phase 5. If both endpoint branches survive independently of the birth, their limiting broken chain either avoids the fold, in which case every component strictly lowers index at this fixed parameter, or enters and leaves it. In the latter case its incoming ordinary index is at least $k+1$ and its outgoing ordinary index at most $k$, so traversing the fold produces a strict net drop too. Every other component strictly drops. Neither chain can have equal endpoint indices. If the source is the newborn upper point (index $k+1$), its first limiting nonconstant component leaves the weak fold, so reaches an ordinary index at most $k$; later components strictly decrease, including any allowable further chain. If it is the newborn lower point (index $k$), the first component leaves the strong fold and reaches index at most $k-1$. These exclude equal-index target endpoints. Dually, a target which is the newborn lower point requires arrival at the weak fold from index at least $k+1$; an upper target requires arrival at the strong fold from index at least $k+2$. Subsequent/preceding strictly decreasing components again exclude equal endpoints. A chain cannot revisit the same fold: its nonconstant components strictly lower function height. If both endpoints belong to the newborn pair, their indices are different; one branch cannot connect to itself by strict descent. These exhaust the possibilities. Hence a neighborhood of every birth is free of equal-index events. The same argument excludes accumulation of index-increasing connections; those have already been excluded at every regular parameter by phase 3.

Global finiteness and no simultaneous events. Assume infinitely many equal-index connections exist. By compactness of parameter space and finiteness of the critical-arc decomposition, choose a subsequence with fixed source/target branch labels and $s_n\to s_{\ast }$. Parameter endpoints are stationary and have no such connections. Birth limits were excluded by phase 6. At an ordinary parameter, phase 4 supplies a broken chain. Each component is non-index-increasing by phase 3; since the total drop is zero, EVERY component has zero drop. A chain of two or more components would give two distinct same-index trajectories at the SAME parameter, excluded by the product-incidence condition of phase 3. Therefore the limit is unbroken. An unbroken limiting trajectory has positive finite transit time between the protected source/target sphere sections. Smooth dependence and transverse section crossings imply convergence of its finite-time incidence coordinates, including that time. The transverse zero-dimensional incidence set is locally a single point. Distinct events cannot accumulate there. This contradiction proves finiteness. Critical-value crossings cause no exception: phase 3 excludes a nonpositive-drop connection at the crossing parameter, and phase 4 excludes collapse into the two equal-value endpoints. The product-incidence calculation already excludes two distinct equal events sharing a parameter. This proof used neither gluing of broken trajectories nor an index bound as an assumption in the compactness extraction. Strict index behavior was invoked only AFTER extraction, to rule out the possible broken equal-index limit.

Local normalization without preserving transit maps. Choose pairwise disjoint closed parameter intervals about the finitely many events of step 7.1, containing no fold, value crossing or other equal-index event. At one such event let $q_s$ be its destination. In its scalar square coordinates change $a_s(z)$ to the signed constant $\pm 2$ inside a radius $ϵ$ neighborhood, using spatial and parameter cutoffs, retaining $2u,-2v$. The center coefficient keeps its sign and is uniformly bounded away from zero on this compact regular parameter interval. Hence the modified fields remain descending and satisfy $\mathrm{\mid }X\mathrm{\mid }\le C\mathrm{\mid }w\mathrm{\mid }$, $-df(X)\ge c\mathrm{\mid }w{\mathrm{\mid }}^2$ near $q_s$, with constants independent of $ϵ$. Their local stable and unstable coordinate disks are exactly the old ones; on a fixed outer section their parametrizations and parameter derivatives agree. The exterior segment of the original source-to-$q_s$ incidence is unchanged, as are its normal derivative and transverse parameter jet. This assertion concerns trajectories ending at $q_s$, not trajectories passing through its chart. To justify preservation of the entire nonpositive-index event list, suppose arbitrarily small radii produced a new nonpositive-index connection. Such a new trajectory must enter the shrinking support. Extract a height-parametrized limit as in step 4.1: its uniform local estimates apply to the modified fields, which agree with the original family off the shrinking chart. The limit is a broken trajectory of the original field, with a break at $q_{s_{\ast }}$ if the shrinking support is visited away from its endpoints. Every component is non-index-increasing. A chain with total drop at most zero must therefore have drop zero on every component. An interior break would require at least two distinct equal-index connections at the same parameter, excluded by step 3.1. If $q_{s_{\ast }}$ is an endpoint, the local stable or unstable section and its exterior flow are unchanged, so the incidence is the original one, with the same jet; local transverse uniqueness excludes an additional incidence converging to it. One may take the outer block with equal stable/unstable radii: its entry face has $f\ge f(q_s)$ and its exit face has $f\le f(q_s)$, so a trajectory ending at $q_s$ cannot pass through that block and then return. This verifies the endpoint assertion even for trajectories with long transit times. If the limiting trajectory misses $q_{s_{\ast }}$, its compact exterior segment misses the support for all sufficiently small radii and likewise cannot be new. These contradictions show that a sufficiently small radius introduces no new index-increasing or equal-index trajectory. Thus the unique transverse event and its parameter persist, while transit maps for positive-drop trajectories are allowed to change. Perform this construction on the finitely many disjoint intervals. Birth models, parameter ends and collars are untouched. The resulting family has the same finite nonpositive-index event list and is normalized at every passage destination.

Full framed elementary data, including positive-drop tangencies. Fix a compact parameter subinterval containing no function birth or value crossing. The actual elementary-band supplier constructs a relative left-right trivialization $f_s\circ {\mathrm{\Psi }}_s={\eta }_s\circ f_{s_0}$, fixing the prescribed spatial collars. Transport all intermediate regular levels by this trivialization. Between successive transported levels there is exactly one critical point. Its fixed small unstable disk and complementary normal coordinates depend smoothly on $s$ (explicit in the chosen model; equivalently use the supplied parameter-disk construction). Intersect with its lower local sphere section and transport a sufficiently thin FULL normal thickening to the lower boundary of that elementary band. There is no intervening critical point. The compact union of these trajectories outside the protected chart has a uniform strict descending bound, so their hitting times are uniformly finite and smooth by the implicit function theorem. Local coordinates supply the framing on the small sphere; the differential of the actual finite flow transports those normal coordinates. Compactness of the parameter interval and attaching source gives one positive radius after taking a finite subdivision. The resulting maps $S^{i-1}\times D^{d-i}\to L_{\mathrm{i}\mathrm{n},s}$ are embeddings and constitute a smooth isotopy after the boundary levels are identified. The corresponding outgoing maps are formed by the same local handle model and outgoing regular flow, hence vary smoothly as well. Extend a full-thickening isotopy by the read compact-source isotopy extension lemma when ambient maps are wanted. A positive-drop tangency involves global trajectories crossing one or more critical-height bands. It does not make the one-critical-band local disk singular and does not insert another critical point between that disk and its chosen lower band boundary. None of the bounded-time construction in the preceding paragraph uses a global stable/unstable transversality condition. It therefore stays smooth THROUGH such a tangency, with its entire framed attaching embedding and outgoing boundary identification. This proves, at the exact elementary-band interface claimed here, that these tangencies require no additional handle move. It does not claim that the globally flowed unstable disk through all lower critical points is a smooth attaching sphere in the original incoming boundary. At a same-index event, step 8.1 retains the transverse normal-isomorphism from phase 3 and supplies the normalized destination model before the elementary-band construction is applied. Birth and value-crossing changes are exactly the supplied elementary comparisons, using the fixed local models already imposed. At every comparison, if $h_s$ is the incoming boundary identification and $d_s$ the elementary outgoing map, carry a later framed embedding $A$ by $d_s\circ A$, and carry the outgoing parametrization by composition with that same actual map. Changing level identifications conjugates the elementary map by the actual incoming/outgoing identifications. Compositions over the finite subdivision are actual maps, so the terminal boundary parametrization is retained. No terminal-map cancellation or suppression is asserted.