# Morse handles from stopping flows and finite singular chains

Written by GPT-6.1 Sol (OpenAI), Ultra, October 2026. Independently authored exposition and original diagrams: CC0-1.0. Self-checked by the writing AI. Classical Morse and singular-homology methods are explained here; no novelty of the theorem is claimed and no external book prose or diagrams are reproduced.

The geometric argument below proves the missing handle statement H of TP041. It does not use a CW decomposition theorem. The argument states the elementary smooth inputs it uses and proves the relative sublevel and ordinary finite-chain deductions. Smooth inverse/flow and Morse-coordinate proofs are included; the existence of a strictly plurisubharmonic Morse perturbation uses the exact written SH03 provider relative to its own Sard, cutoff and other lower inputs. This distinction is retained at the end.

## MH0. Statement, coefficients and boundary cases

Let M be a Hausdorff second-countable smooth manifold without boundary of finite real dimension N. Let f:M→R be smooth, proper, bounded below and Morse: the real Hessian at every critical point is nonsingular. Fix a nonnegative integer d and assume every critical index is at most d. A coefficient ring k can be any associative unital ring; all chain maps below are integral maps extended to k, so commutativity, a field, torsion freeness and flatness assumptions are unnecessary. Write C_j(X;k)=C_j(X;Z)⊗_Z k with its natural left k-module structure. Every chain is a finite sum of actual continuous singular simplices. Ordinary negative chain groups vanish. Reduced homology is used only when explicitly marked.

**Theorem MH0.** There is an increasing sequence of open regular sublevels W_0=∅, W_1,W_2,… covering M. Each step crosses either no critical value or one value with finitely many critical points p. With λ(p) their indices,

\[
 H_j(W_i,W_{i-1};k)\cong
 \bigoplus_{p\text{ crossed at step }i}
 \begin{cases}k,&j=\lambda(p),\\0,&j\ne\lambda(p),\end{cases}
 \qquad j\geq0. \tag{MH1}
\]

The direct sum is zero at a regular step. Consequently

\[
 H_j(M;k)=0\quad(j>d). \tag{MH2}
\]

The local identification is natural for the inclusions used in the proof, after choosing an orientation of each negative coordinate disc. It does not assert that groups from several distinct critical values simply add: connecting homomorphisms can cancel generators. For arbitrary regular compact endpoints a<b, even with several values between them, the proved consequence is H_j(M^b,M^a;k)=0 for j>d, where M^t={f≤t}.

If M is empty all statements are vacuous. Dimension zero is allowed: a 0×0 Hessian is nonsingular and every point has index zero. A compact zero-dimensional stage is a finite set; its relative H_0 is one copy of k for every newly added point. If N<d the proof still applies. No orientation of M is required. If one instead specifies an integer d<0, a nonempty M is impossible under the index bound: properness and a lower bound make f achieve a minimum, whose index is zero. Thus that case reduces to the empty manifold.

## MH1. The finite-chain tools, proved at the level used here

A singular boundary is the alternating sum of deleted faces; twice-deleted faces cancel, so ∂²=0. For a homotopy F:X×[0,1]→Y the prism over an ordered j-simplex has the j+1 simplices

\[
 [(v_0,0),\ldots,(v_i,0),(v_i,1),\ldots,(v_j,1)],
 \quad0\leq i\leq j,
\]

with signs (-1)^i. Applying F gives a finite operator P satisfying ∂P+P∂=F_{1*}−F_{0*}; internal faces cancel, the top and bottom faces remain, and side faces give −P∂. If F carries a subspace A into B at all times this identity descends to relative chains. Thus homotopies of pairs induce equal homology maps over k.

The pair complex is \(C_*(X,A)\)=\(C_*(X)\)/\(C_*(A)\). It is free on simplices not wholly in A. The sequence 0→\(C_*(A)\)→\(C_*(X)\)→\(C_*(X,A)\)→0 splits degreewise on simplex bases, also over k. Its connecting map sends a relative cycle represented by c with ∂c in A to [∂c]. Changing the lift changes that class by a boundary. If ∂c=∂a in A, then c−a is an absolute cycle. Conversely an A-cycle bounds in X exactly when it is such a connecting image. These facts prove the pair long exact sequence and its naturality. For A⊂B⊂X the equally split sequence 0→\(C_*(B,A)\)→\(C_*(X,A)\)→\(C_*(X,B)\)→0 proves the triple sequence. Five-lemma uses here are elementary chases in these sequences: to lift a middle element, first lift its next image, then correct by the preceding image; for a kernel element, perform the same correction one position earlier.

We need **open excision**, not excision of an arbitrary closed piece. Here is its finite-chain proof. Barycentric subdivision S is a chain map. On affine standard simplices define it inductively by coning the already subdivided boundary to the barycenter; the cone identity ∂(b*c)=c−b*∂c proves ∂S=S∂. Coning the cycle σ−Sσ−T∂σ, inductively in dimension, gives a homotopy ∂T+T∂=1−S. In degree zero its augmentation is zero, so the same reduced cone identity works. All operations remain in the original simplex image. A barycentric subsimplex has diameter at most j/(j+1) times the old diameter: its vertices are barycenters of nested faces, and comparing face averages gives this bound. Iteration makes the mesh tend to zero. The inverse-image open cover of a compact standard simplex has a Lebesgue number, so every finite chain becomes small for any specified open cover after finitely many subdivisions.

For an open cover, inclusion of the complex of small simplices induces an isomorphism in homology, including relative homology. For surjectivity subdivide a finite cycle sufficiently; S^m is homotopic to identity by the finite sum of subdivision homotopies. For injectivity subdivide a finite bounding chain sufficiently; the homotopy keeps a chain that was already small small, so the original small cycle differs from its subdivision by a small boundary. For a relative cycle or relation, also subdivide the finitely many A-chains involved; all homotopies preserve A. These arguments prove both directions over any k without tensoring an unproved quasi-isomorphism.

If E⊂A⊂X and closure(E)⊂int_X A, apply this small-chain result to the open cover int_X A, X\closure(E). In the relative quotient the first cover member contributes zero, and the remaining complex is that for (X\closure(E),A\closure(E)). Restricting the cover to X\E gives exactly the same relative small-chain quotient. Hence

\[
 H_j(X\setminus E,A\setminus E;k)\xrightarrow{\sim}
 H_j(X,A;k). \tag{MH3}
\]

For an open cover X=U∪V, small chains also give the degreewise split sequence with maps c↦(c,−c) and (a,b)↦a+b from \(C_*(U\cap V)\) to \(C_*(U)\)⊕\(C_*(V)\) to the small-chain complex of X. Its exact sequence is Mayer–Vietoris, with the reduced version from augmented chains.

A point's chain complex has one generator in each nonnegative degree, with differential identity in positive even degrees and zero in odd degrees. Thus a contractible nonempty space has H_0=k and positive groups zero. Cover S^m, m≥1, by two slightly enlarged hemispheres. Each contracts, their intersection retracts to S^{m−1}, and reduced Mayer–Vietoris gives \(\widetilde H_j(S^m;k)\cong\widetilde H_{j-1}(S^{m-1};k)\). The initial S^0 has two components and reduced H_0=k. Therefore the ball pair sequence proves

\[
 H_j(D^\lambda,S^{\lambda-1};k)=
 \begin{cases}k,&j=\lambda,\\0,&j\ne\lambda\end{cases}
 \quad(\lambda\geq1). \tag{MH4}
\]

For λ=1 the reduced group of its two endpoints is used; the map H_0(S^0)→H_0(D^1) sums the two coefficients. A positive interval has boundary right minus left and represents the relative generator. For λ=0 the actual pair is (point,∅), with ordinary H_0=k, not the reduced homology of a point. Finite chains triangulating an oriented disc give the generator in MH4: identify a simplex with a ball by radial projection from interior points, orient it, and subdivide if desired; its boundary lies in the boundary sphere and has the positive sphere generator from the hemisphere calculation. No cellular-homology theorem is used.

## MH2. Smooth inputs and the exact Morse chart

A smooth metric can be made by a locally finite smooth chart partition and weighted chart metrics. The same partition supplies cutoffs equal to one near prescribed compact sets inside charts. We retain the elementary smooth-partition input of the programme.

For completeness, the local inverse and ODE inputs are as follows. If DF(0)=B is invertible, shrink a closed Euclidean ball until ||I−B^{-1}DF||≤1/2. For y near F(0), x↦x+B^{-1}(y−F(x)) preserves that ball and contracts by 1/2. Its geometric-series iteration gives a unique solution and a continuous inverse; inserting the inverse Lipschitz bound in F's differentiability remainder gives derivative DF(x)^{-1}. Repeated differentiation of the inverse identity gives smoothness. A square submatrix of an implicit equation gives the corresponding implicit-function conclusion by this inverse theorem.

For a smooth vector field G on a chart ball with |G|≤C and ||DG||≤L, take T with TC below the spatial margin and TL<1/2. The integral operator x(t)↦x_0+∫_0^tG(x(s))ds contracts the closed ball of paths, giving existence and uniqueness. The initial-point derivative solves J(t)=I+∫_0^tDG(x(s))J(s)ds. The same contraction and differentiability remainder prove that derivative exists and is continuous. Higher derivatives solve the same linear equation with already known inhomogeneous terms; induction gives smooth dependence. Uniqueness glues chart solutions. A trajectory that remains in a fixed compact subset can be continued through any finite time: a finite covering by chart balls gives a uniform positive local existence time, and successive extensions rule out a finite terminal time. This is the only continuation fact used below.

Here is the Morse coordinate proof, following the actual programme Lemma 4.1 in AN04. Translate a critical point to zero. A nonsingular symmetric Hessian has a vector on which its quadratic value is nonzero; otherwise polarization would make it zero. Make that vector the first coordinate. The implicit theorem solves ∂_1 f=0 as x_1=h(x'). With v=x_1−h(x'), Taylor's integral remainder gives

\[
 f(h+v,x')=g(x')+\tfrac12v^2 A(v,x'),\qquad
 A=2\int_0^1(1-t)f_{11}(h+tv,x')\,dt. \tag{MH5}
\]

Near zero A has constant nonzero sign. The change z_1=v|A|^{1/2} is locally invertible and replaces the last term by ±z_1²/2. The Hessian of g is the Schur complement of the first Hessian pivot. Block elimination gives its determinant as det(H)/H_{11}, so it remains nonsingular. Induction proves a diagonal signed quadratic normal form. Congruence preserves its negative index. For each negative signed coordinate set u_j=z_j/√2, and for each positive signed coordinate set v_j=z_j/√2; thus ±z_j²/2 becomes ±u_j² or ±v_j². Regroup them to obtain

\[
 f=c+q(u,v),\qquad q(u,v)=-|u|^2+|v|^2,
 \quad u\in\mathbb R^\lambda,\ v\in\mathbb R^\nu,
 \quad\lambda+\nu=N. \tag{MH6}
\]

The proof works also for N=0 with no coordinates. It gives a neighborhood for each point, not a universal radius. Nonsingularity also makes df=0 isolated by the inverse theorem applied to df in a chart. The critical set is closed. Each M^t is compact: choose a lower bound L of f and write M^t=f^{-1}([L,t]) (empty if t<L). A compact stage has finitely many critical points, since an infinite closed set of isolated critical points would have a critical accumulation point. Thus its critical values are finite. Critical values in any bounded range are locally finite, and regular values can be chosen between consecutive distinct critical values without invoking Sard for an already Morse function.

## MH3. A regular compact slab

Let a<b be regular and suppose there is no critical point with a≤f≤b. On a neighborhood of this compact band put

\[
 Z=-\nabla f/|\nabla f|^2,\qquad df(Z)=-1. \tag{MH7}
\]

There is a slightly larger critical-free band: otherwise critical values accumulate at a or b, contrary to MH2. Its field has a smooth local flow Φ. For x with a<f(x)≤b, continue Φ until time f(x)−a. Up to that time f(Φ(t,x))=f(x)−t, so the trajectory remains in the compact band and MH2 ensures continuation. It lands at f=a. Define D_s(x)=Φ(s max(f(x)−a,0),x) above a, and D_s(x)=x on M^a. Smooth dependence and time tending to zero at f=a make this continuous across the boundary. It stays in M^b, fixes M^a, and is a strong deformation retraction. In particular

\[
 H_j(M^b,M^a;k)=0\quad\text{for every }j\geq0. \tag{MH8}
\]

The formula applies if the band is empty as the identity. A cutoff can extend the field beyond its band if desired, but the stopped construction requires neither completeness on all of M nor a global lower bound for |∇f|.

## MH4. One critical value and disjoint compact blocks

Choose regular a<b with exactly one critical value c between them, containing finitely many points p_1,…,p_h. First take sufficiently close inner endpoints c−ε and c+b_0, and translate c to zero. The general endpoints are handled in MH8. At each p_i choose disjoint Morse chart neighborhoods. Choose compact blocks

\[
 B_i=D^{\lambda_i}(r_i)\times D^{\nu_i}(s_i),\qquad
 s_i^2<b_0,\qquad r_i^2>s_i^2+\epsilon\ (\lambda_i>0). \tag{MH9}
\]

All blocks and slightly enlarged blocks lie inside their charts. These choices are possible after reducing ε and b_0: start with chart-sized small radii r_i>0 for positive indices, choose s_i still smaller, then choose ε smaller than every r_i²−s_i² and b_0 above every s_i² while still below the next critical value. Equivalently choose all radii on the scale √ε and b_0 on the same small scale; finite many charts suffice. For index zero there is no u-exit condition; choose its positive radius with s_i²<b_0. For maximal index ν_i=0 take D^0={0}, s_i=0 and r_i²>ε. For N=0 the block is an open-and-closed singleton, with no faces.

Set X=M^{b_0}, A=M^{−ε}, and S=A∪⋃B_i. The whole B_i lies in X because q≤s_i²<b_0. Its u-exit face

\[
 E_i=S^{\lambda_i-1}(r_i)\times D^{\nu_i}(s_i)
 \quad(\lambda_i>0) \tag{MH10}
\]

lies strictly inside A, since q≤−r_i²+s_i²<−ε. That strict margin is essential for the stopping flow and collars. The block intersection C_i=B_i∩A equals {q≤−ε} within B_i. For λ_i=0 this set is empty. When ν_i=0 every boundary face of the block is already in int A.

## MH5. The global descending field and continuous stopping time

Use a smooth metric and start with G=−∇f. On each Morse chart use the exact field

\[
 W_i=(2u,-2v),\qquad dq(W_i)=-4(|u|^2+|v|^2). \tag{MH11}
\]

Choose chart cutoffs χ_i equal to one on neighborhoods of B_i, with disjoint supports. Define W=(1−∑χ_i)G+∑χ_iW_i, interpreting χ_iW_i by zero off its chart. This is smooth; df(W)<0 at every noncritical point, and it is exactly W_i near B_i. At a critical point it is zero. Normalization is needed in MH3; here this unnormalized smooth field is preferable across the critical points and a compactness bound supplies uniform decrease where we need it. Positive rescaling outside the boxes would give the same stopped argument.

Forward trajectories starting in X have nonincreasing f and remain in compact X, so exist at every finite time. All critical points in X are either strictly inside A or strictly inside one B_i. Thus the compact set K=X\int_M S contains no critical point. If K is nonempty there is η>0 with df(W)≤−η on K. For x∉S define T(x) as its first hitting time of S. If it had not hit by time (b_0+ε)/η+1, integration of df(W) would force f<−ε, a contradiction. Therefore T is finite and bounded uniformly. Put T=0 on S. If K is empty, X=S and the retraction is the identity.

We now prove continuity; a bare finite-time claim is insufficient. At a first hit outside int S, a point of ∂A has f=−ε and df(W)<0, so the implicit-function theorem gives a smoothly varying nearby crossing time of that regular level. A hit of a block outside int A cannot occur on a u-face, because that face has the strict margin in MH10. It can occur only on |v|=s_i with |u|<r_i and ν_i>0. There

\[
 W_i(|v|^2-s_i^2)=-4s_i^2<0, \tag{MH12}
\]

so its inward crossing time also varies smoothly. The same applies when a regular level and v-face meet: near the hit the first entry into their union is the minimum of their two smooth crossing times, hence continuous. In this intersection |u|²=s_i²+ε<r_i², so no u-corner is present. The only geometric u/v corners have already fallen inside int A. For index zero there is just the transverse v-entry; for maximal index there is no v-entry outside A; for a zero-dimensional singleton there is no exterior local boundary to analyze.

To justify that these local times describe the *first* hit, for x∉S and any δ>0 below T(x), the compact orbit segment [0,T(x)−δ] misses the closed S. Nearby initial points also miss it on that segment by smooth dependence and compactness. Near the hit, the transverse local descriptions give an entry by time T(x)+δ and exclude an earlier local entry except at the stated minimum. They prove lim T(x_n)=T(x) for x_n→x outside S. For x∈∂S, the same transverse descriptions show exterior points hit at times tending to zero; interior points of S have T=0 in a neighborhood. Every critical point of S lies in its interior. These observations also prove continuity across S.

Consequently

\[
 D_t(x)=\Phi_W(tT(x),x),\quad0\leq t\leq1,
 \qquad D_t|_S=1, \tag{MH13}
\]

is a strong deformation retraction X→S. Before the stop a trajectory remains in X by descent; at and after its fixed endpoint it stays there. Because A⊂S is fixed, this is also a homotopy equivalence of pairs (X,A) and (S,A). We stop the original smooth flow; we do not claim that a discontinuously set-to-zero vector field has a smooth flow.

## MH6. Collars and the good-pair quotient lemma

All spaces in the closed-union argument are compact metrizable Hausdorff spaces. We call (Z,C) a good pair here when C is closed and has an open neighborhood N in Z that strongly deformation retracts onto C fixing C.

For (X,A), take N={f<−ε+δ}∩X with δ>0 small enough that its band contains no critical point. The normalized descent of MH3, stopped at f=−ε, is the required deformation on N, fixing all of A.

For (B_i,C_i) with λ_i>0 choose

\[
 0<\delta<\epsilon/2,\qquad
 \delta<r_i^2-s_i^2-\epsilon. \tag{MH14}
\]

On the collar −ε<q<−ε+δ one has |u|²=|v|²+ε−(q+ε)≥ε−δ>0. The u-only vector field

\[
 Z_i=(u/(2|u|^2),0),\qquad dq(Z_i)=-1 \tag{MH15}
\]

fixes v and expands |u|. Its explicit time-t formula is u(t)=√(1+t/|u(0)|²)u(0). A trajectory stopped at time q+ε lands on q=−ε before the u-face, because that face has q≤s_i²−r_i²<−ε. It cannot exit through the v-face because v is fixed. Applying time s max(q+ε,0) above the lower boundary and identity below gives an open-neighborhood strong deformation onto C_i in B_i.

The pair (S,A) has the same collar. Near its regular level f=−ε patch Z_i, exact on each block's collar, with −∇f/|∇f|² elsewhere. Every patched term satisfies df=-1, so their partitioned sum does too. These fields are defined only near the level, away from all critical points. For points of S above A the field is Z_i in the relevant block and stays there until it meets A, by the preceding strict margin. For points of A it is stopped at time zero. Thus N_S=S∩{f<−ε+δ} strongly retracts onto A. Index-zero blocks do not meet this collar if δ<ε; their intersections C_i are empty and are treated separately below. Finite many blocks allow a common positive δ; no global chart radius is assumed.

**Lemma MH6 (good-pair quotient, from open excision).** If Z is compact Hausdorff, C is closed and nonempty, and (Z,C) is good as above, its quotient map induces

\[
 H_j(Z,C;k)\cong\widetilde H_j(Z/C;k),\qquad j\geq0. \tag{MH16}
\]

**Proof.** The pair (N,C) has zero relative homology by its deformation. The triple sequence therefore makes \(H_*(Z,C)\)→\(H_*(Z,N)\) an isomorphism. Since closure_Z C=C⊂int_Z N, the proved open-excision theorem MH3 removes C and gives \(H_*(Z,N)\)≅\(H_*(Z\setminus C,N\setminus C)\). Collapsing C gives a compact Hausdorff quotient: distinct points outside C can be separated from C and each other in the compact Hausdorff Z; equivalently its collapsed equivalence relation is closed. The image N/C is open, because N is saturated and open. It is contractible: the neighborhood deformation descends to one fixing the collapsed point. Its descent is continuous. Indeed q:Z→Z/C is a closed surjection between compact Hausdorff spaces, so q×1_[0,1] is a closed quotient map; restricting to the saturated open N×[0,1] proves the required homotopy quotient property.

Open excision on (Z/C,N/C), now removing its collapsed point \(*\), identifies its relative homology with \(H_*(Z\setminus C,N\setminus C)\), since q is a homeomorphism off C. Contractibility of N/C and the triple sequence identify \(H_*(Z/C,N/C)\) with \(H_*(Z/C,\{*\})=\widetilde H_*(Z/C)\). All comparisons commute with q, which proves MH16. This does not apply a closed-union excision theorem without hypotheses. ∎

For C=∅ use Z_+=Z⊔\(\{*\}\) and the natural identification \(H_j(Z;k)=\widetilde H_j(Z_+;k)\). At j=0, a component generator is identified with its difference from the added basepoint, so this is ordinary H_0(Z), not \(\widetilde H_0(Z)\). In positive degrees the identification follows directly from the split chain complexes. If Z also is empty both groups are zero. This convention will handle minima and an empty lower stage correctly.

## MH7. Closed union, finite wedge and the local relative group

If A≠∅, the inclusions induce a homeomorphism

\[
 S/A\ \cong\ \bigvee_{i=1}^h Q_i,
 \qquad Q_i=
 \begin{cases}B_i/C_i,&C_i\ne\varnothing,\\(B_i)_+,&C_i=\varnothing.
 \end{cases} \tag{MH17}
\]

If h=0 the right side is a point. This is a bijection because the boxes are disjoint and all identifications are precisely within A. It is continuous from the finite disjoint union with its basepoints identified, and compactness of that source and the Hausdorff quotient target make it a homeomorphism. For C_i empty, the added basepoint is needed: the entire unattached minimum block remains a separate component. If A=∅, instead apply the same statement to S_+=⋁(B_i)_+. In this case C_i all are empty. Never collapse an empty A as though it were a nonempty connected subset.

We spell out finite-wedge homology without cellular assumptions. Each Q_i has an open contractible neighborhood V_i of its basepoint whose contraction fixes that point: use its good collar quotient if C_i≠∅, and the isolated point if C_i=∅. In the finite wedge put U_i=Q_i∪⋃_{h≠i}V_h. It is open, contracts the V_h parts onto the basepoint and thus deformation retracts onto Q_i. Distinct U_i intersect in ⋃V_h, a contractible open wedge of those neighborhoods; the contractions paste continuously on their finite closed-in-the-union branches. The same intersection occurs between U_i and a union of preceding U_h. Reduced open Mayer–Vietoris, inductively, therefore identifies reduced homology of the whole wedge with the direct sum of the \(\widetilde H_*(Q_i)\). The maps are induced by branch inclusions.

By MH16, the empty-intersection convention and MH13 we have

\[
 H_j(X,A;k)\cong H_j(S,A;k)
 \cong\bigoplus_i H_j(B_i,C_i;k). \tag{MH18}
\]

It remains to compute one block. If λ=0, C=∅, the contractible block contributes ordinary H_0(B;k)=k and zero higher groups. If λ>0, every point of C has |u|²≥|v|²+ε, so u≠0. The radial expansion

\[
 (u,v)\longmapsto
 \left(\left[(1-t)+t\frac r{|u|}\right]u,v\right)
 \quad(0\leq t\leq1) \tag{MH19}
\]

strongly retracts C onto E=S^{λ−1}(r)×D^ν(s), fixing E. The u-radius increases, so q does not increase and the entire deformation stays in C. By the pair sequences and their naturality, the inclusion (B,E)→(B,C), identity on B, induces a relative homology isomorphism: E→C is a homotopy equivalence. We do not extend MH19 through u=0 in all of B. Finally the pair homotopy (u,v)↦(u,(1−t)v) retracts (B,E) onto (D^λ(r),S^{λ−1}(r)). It preserves E at every time, though it need not fix every point of E. By MH4,

\[
 H_j(B,C;k)=\begin{cases}k,&j=\lambda,\\0,&j\ne\lambda.\end{cases} \tag{MH20}
\]

This includes λ=N when ν=0. The oriented negative disc D^λ(r)×{0}, triangulated as a finite chain, represents its generator; its boundary lies in C. In dimension one that boundary is the difference of two lower-region points, which need not vanish in H_0(A). Thus a relative handle class is not automatically an absolute cycle.

## MH8. General finite endpoints and several levels

Return to any regular endpoints a<c<b with only the critical value c between them. Pick close inner a'<c<b' for MH4–MH7. The regular deformation M^b→M^{b'} fixes M^{b'} and hence M^a, identifying \(H_*(M^b,M^a)\) with \(H_*(M^{b'},M^a)\). The inclusion M^a→M^{a'} is a homotopy equivalence by MH3. The triple sequence or natural pair sequences therefore identify the latter group with \(H_*(M^{b'},M^{a'})\). This proves the direct-sum formula MH18–MH20 for those original endpoints. With no crossed critical value MH8 applies.

An arbitrary compact interval [a,b] has finitely many critical values. Insert finitely many regular separators so each step crosses at most one. If all indices are ≤d, every step's relative groups vanish above d. Apply the triple sequence successively for M^a⊂M^{t_h}⊂M^{t_{h+1}}: when both H_j(M^{t_h},M^a) and H_j(M^{t_{h+1}},M^{t_h}) vanish, exactness makes H_j(M^{t_{h+1}},M^a) vanish. Starting from the zero pair proves the general relative vanishing above d. This argument does not assert additivity of relative groups across distinct levels or overlook attaching-map differentials.

Likewise for absolute groups, begin below the lower bound, where the sublevel is empty. At every step and every j>d, the exact segment H_j(old)→H_j(new)→H_j(new,old) has zero outer groups and makes the new group zero. Consequently every compact regular sublevel has H_j=0 above d. All exact-sequence representatives and lifts are ordinary finite chains by MH1.

## MH9. Increasing OPEN stages with compatible pair maps

Choose t_0 below a lower bound and a cofinal increasing sequence of regular numbers t_i→+∞, with at most one distinct critical value in each interval (t_{i−1},t_i). Such a sequence exists because critical values are bounded below and finite in each bounded interval. Start with cofinal regular values near the positive integers, and insert finitely many separators between the finitely many critical values in every resulting bounded interval; the resulting locally finite sequence is countable and cofinal. Several critical points at one value are kept at that same step.

Set W_i={f<t_i}. These are increasing open sets, W_0=∅ and their union is M. Choose s_0<t_0 still below the lower bound. For i≥1 choose s_i∈(t_{i−1},t_i) sufficiently close to t_i that [s_i,t_i] is critical-free; then s_{i−1}<t_{i−1}<s_i. Put K_i=M^{s_i}. The explicit regular flow of MH3, for x∈W_i with f(x)>s_i run for time f(x)−s_i, gives a strong deformation W_i→K_i. Its trajectories lie in the compact closed band [s_i,t_i], and it fixes K_i. Even though W_i need not be compact, this particular collar is; if its band is empty the map is the identity. K_0 and W_0 are empty.

The inclusion maps form genuinely commuting diagrams

\[
 \begin{array}{ccc}K_{i-1}&\longrightarrow&K_i\\
 \downarrow&&\downarrow\\W_{i-1}&\longrightarrow&W_i.
 \end{array} \tag{MH21}
\]

Both vertical absolute maps are homotopy equivalences. Naturality of the pair long exact sequences and the five-lemma chase give an isomorphism H_j(K_i,K_{i−1};k)→H_j(W_i,W_{i−1};k) in all nonnegative degrees, including zero. The separate deformation retractions are not asserted to form a map of pairs; it is the actual inclusion diagram and exactness that prove the relative comparison. The strip from s_{i−1} to t_{i−1} has no critical value, and from s_i to t_i has none, so the compact pair crosses exactly the critical points intended for the open pair. MH8 supplies MH1 for these actual open stages.

## MH10. Passage to the whole manifold, with ordinary chains

Induction using MH1 and the absolute pair sequence proves H_j(W_i;k)=0 for j>d. Every ordinary finite chain has compact image, the finite union of compact simplex images. An increasing open cover contains that image in one stage: take a finite subcover and its largest stage. Every cycle on M therefore belongs to a W_i. If a class from a stage bounds in M, its actual finite bounding chain and the original cycle lie together in some later stage, where that relation already holds. Thus

\[
 \operatorname*{colim}_i H_j(W_i;k)\xrightarrow{\sim}H_j(M;k). \tag{MH22}
\]

These are explicit surjectivity and injectivity statements, without an inverse limit, dimension bound on individual groups or locally finite chains. The stage groups above d are zero, so MH22 proves MH2. A countable locally finite sum escaping every stage would not meet this proof's compactness condition and is deliberately excluded.

## MH11. Strict Levi positivity and the TP041 interface

At a critical point of a smooth strictly plurisubharmonic function φ on a complex d-manifold, let B be its real Hessian and J multiplication by i. With ∂/∂z_j=(∂_xj−i∂_yj)/2, direct expansion gives

\[
 B(v,v)+B(Jv,Jv)=4\mathcal L_\varphi(v)>0\quad(v\ne0). \tag{MH23}
\]

At a critical point the Hessian is a coordinate-independent bilinear form. If its negative eigenspace W had dimension q>d, then dim(W∩JW)≥2q−2d>0. For nonzero v=Jw in this intersection with w∈W, also Jv=−w∈W. Both terms on the left would be negative, contradicting MH23. Thus a nondegenerate strictly plurisubharmonic critical point has index ≤d. This proves the precise bound, including its factor four; strict Levi positivity by itself does not prove nondegeneracy.

The included SH03 source's sections “Small global perturbations can retain all three properties” and “A generic differential misses the bad cotangent locus”, specialized to the zero section Λ=Λ_0⊂T*X and empty bad locus, supply a bounded-below proper strictly plurisubharmonic Morse φ. Here is the exact consumed mechanism, relative to its named smooth/cutoff/Sard inputs. Compactly supported real coordinate functions w_i have differentials spanning cotangent fibres. Choose ε_i>0 summably small in absolute value, Levi operator norm relative to its positive local minimum, and successive compact C^i norms. For parameters a in the unit ball of ℓ² the smooth sum φ_a=ρ+1+∑a_i ε_iw_i stays uniformly bounded away from loss of Levi positivity and within 1/2 of ρ+1, so properness and a lower bound survive. Over a compact base set a finite set of parameters spans every cotangent fibre. The map (x,e)↦(x,dφ_{a+e}(x)) is then a submersion. Its inverse image of the zero section has the dimension of this finite parameter space. Sard applied to its projection gives dense regular parameters; linearizing the zero equations says regularity is exactly invertibility of the Hessian. On compact sets this property is open. A countable compact exhaustion and the nested-ball Baire argument in complete ℓ² select one parameter regular on every stage. Thus φ is Morse.

Apply this to the exact proper strict-psh projective exhaustion in TP041 TP3, under those retained lower inputs. MH23 bounds all indices by the complex dimension d. MH9 gives the *actual increasing open stages* and their pairs that TP041 H requests, now with their finite-chain handle proof rather than a CW assertion. MH10 then gives ordinary H_{d+1}(U;Q)=0. Together with the normal Thom/tubular construction in TP041, the ordinary pair exact sequence makes the defined homological normal-first tube injective. This is a geometric receiving implication relative to those exact providers, not an identification of an arbitrary affine C8 cycle with that tube.

## MH12. Credits, exact inputs and remaining limits

The coordinate proof MH2 follows the independently authored AN04 “Stationary phase and critical manifolds”, Lemma 4.1: implicit critical coordinate, Taylor pivot square and Schur-complement induction. Its stated IFT and smooth prerequisites are retained. The inverse/ODE proof and smooth metric/partition interface agree with the included DGCHAR “Manifold duality and tubular sections”, §3 Lemma 3.1; its own authored source is CC0 and its current source credits distinguish the verified Spring 1957 Milnor manuscript, notes by James Stasheff, from Hatcher's compact-support comparison. The finite-chain prism, subdivision, excision, sphere and exact-sequence arguments in MH1 agree with the included complete DGCHAR “Thom classes and Euler classes”, §§1–3, authored by GPT-6.1 Sol (OpenAI), Ultra, CC0. Its valid Hatcher and Spring 1957 Milnor/Stasheff background credits are method context; its citation and scope paragraphs are revised while every mathematical proof, formula and exercise body remains complete. Cited source expression retains its own rights. SH03 “Holomorphic Morse exhaustions on Stein manifolds”, authored by GPT-6.1 Sol (OpenAI), Ultra, CC0, supplies the exact perturbation interface described in MH11 and retains its smooth-coordinate, Sard and analytic lower obligations.

Classical Morse handle theory and ordinary singular excision are the historical mathematical methods. The stopping block, explicit corner/hitting-time proof, collar quotient comparison and figures here are independently written. No external textbook body was read or reproduced in this task, and no exact original-book proof match or book error is claimed. Local programme bodies were read only in the needed scopes and retain their stated origins and terms. A Blender scene would not explain the finite-chain relative pair better than the exact planar block/flow and quotient diagrams, so the reproducible figures use planar coordinates and exact algebraic labels.

The candidate supplies the geometric ordinary finite-chain handle theorem H, including zero and maximal indices, simultaneous points, finite compact stages and compatible open stages. It does not close the still lower Sard/weak/analytic foundations of the perturbation provider, certify all its sheaf results, or admit whole provider courses. General rational-form spanning, the actual affine/projective C8 representative and multiplicity/sign/scalar identification, the alternative compact-support closed–open comparison, historical coefficients, other C6–C8 hypotheses, full component constancy and recursive course closure remain separate. Public visibility and publication are outside this writer task.