Proof. Embed the sphere in \(\mathbb R^{n+1}\). Pointwise exponential charts, the embedding \(H^1(S^1)\subset C^0(S^1)\), and the one-dimensional \(H^1\) product rule make \(\Lambda\) a smooth Hilbert submanifold of \(H^1(S^1,\mathbb R^{n+1})\). Its induced strong metric is complete. Indeed, an intrinsic Cauchy sequence is ambient Cauchy; its limit still takes values in the closed sphere, and a local Hilbert chart near that limit bounds intrinsic path distances by ambient distances. Time translations and reflection are isometries. Each acts smoothly, and their action is jointly continuous in the strong \(H^1\) topology.

Extend \(g\) to a smooth positive symmetric matrix \(A(x)\) along the ambient tangent bundle, preserving the tangent–normal splitting. Then \[E(\gamma)=\int_0^1 \langle A(\gamma)\dot\gamma,\dot\gamma\rangle\,\mathop{}\!\mathrm dt\] is smooth. To check compactness, suppose that \(E(\gamma_r)\) is bounded and \(\| \mathop{}\!\mathrm dE(\gamma_r)\|\to0\). Pass to a subsequence converging weakly in \(H^1\) and uniformly to \(\gamma\). Let \(P_x\) denote Euclidean tangent projection. The tangent fields \[\eta_r=P_{\gamma_r}(\gamma_r-\gamma)\] are bounded in \(H^1\). Testing \(\mathop{}\!\mathrm dE(\gamma_r)\) on \(\eta_r\), every term containing \(\gamma_r-\gamma\) without a derivative tends to zero: its uniform norm tends to zero, while the squared velocities have bounded integrals. This includes the terms from differentiating \(A(\gamma_r)\) and \(P_{\gamma_r}\). Consequently \[\int_0^1\! \langle A(\gamma_r)\dot\gamma_r, \dot\gamma_r-\dot\gamma\rangle\,\mathop{}\!\mathrm dt\longrightarrow0.\] Uniform convergence of the coefficients and weak convergence of the velocities now give convergence of their energy norms. Uniform positivity of \(A\) proves strong \(H^1\) convergence.

At a critical point the weak Euler–Lagrange equation says, in a coordinate interval, that the distributional derivative of the metric momentum is an \(L^1\) quadratic expression in the velocity. The momentum, hence the velocity, is continuous. The geodesic equation then gives smoothness. Its periodic boundary condition follows from the first variation. Conversely, geodesics and constants are critical.

Normalize the negative gradient by \[Z=-\frac{\nabla E}{\sqrt{1+\|\nabla E\|^2}}.\] This is a smooth invariant vector field of norm at most one, strict away from the critical set. A bounded vector field on a complete strong Riemannian manifold has a complete flow: a trajectory of finite time has finite length and a limit, through which the local flow extends it.

Finally, near the constants use coordinates \(\gamma(t)=\exp_x \xi(t)\), where \(\xi\in H^1(S^1,T_xM)\) has mean zero. The inverse function theorem gives a uniform tube over the compact zero section. The normal Hessian of energy is positive coercive by the mean-zero Poincaré inequality. Taylor integration and the positive square-root change of normal coordinates make the energy an exact positive quadratic form there. These constructions are equivariant. Small energy forces both the diameter and the ambient \(L^2\) derivative norm of a loop to be small, so all sufficiently small sublevels lie in this tube. Radial contraction gives the claim. ◻

Proof. Palais–Smale implies a positive lower bound on \(\|\nabla E\|\) in such an interval. The normalized flow therefore decreases energy at a uniformly positive rate there. A fixed sufficiently long time moves its upper sublevel into its lower sublevel, and the intervening flow gives the required homotopies. A small regular collar at an endpoint similarly compares the open and closed sublevels. Apply these constructions simultaneously at the endpoints of a pair. All maps are equivariant because they use the same invariant flow. ◻

Proof. Choose \(\rho>0\) below the injectivity radius. The open manifold \[P_N=\{(x_0,\ldots,x_{N-1})\in M^N: d_g(x_j,x_{j+1})<\rho\}\] is identified with the polygons formed by minimizing constant-speed segments; subscripts are cyclic. Its energy is \[f_N=N\sum_{j=0}^{N-1}d_g(x_j,x_{j+1})^2.\] Choose \(N\) so large that \(B/\sqrt N<\rho\). On each time cell, replace the central subinterval of proportion \(s\), \(0\le s\le1\), by its minimizing segment. Cauchy–Schwarz makes all endpoint distances shorter than \(\rho\), and shows that this replacement does not increase energy. At \(s=1\) the loop is a polygon, and polygons are fixed throughout.

For positive subinterval width, strong \(H^1\) continuity follows from smooth dependence of short geodesics on endpoints and continuity of restriction and rescaling in \(L^2\). As the width tends to zero, the replacement energy is bounded by the original energy on the disappearing intervals. Those integrals tend uniformly to zero along any strongly \(H^1\)-convergent sequence. This proves continuity also at \(s=0\). Centering the replacements proves the stated symmetries.

On the closed sublevel \(f_N\le B^2\), every edge has length at most \(B/\sqrt N<\rho\), so this sublevel is compactly contained in \(P_N\). Its boundary is smooth at a regular value. The polygon first variation imposes matching velocities at every vertex, so its critical points are exactly the smooth closed geodesics in the band.

At such a geodesic, decompose an \(H^1\) variation field into its piecewise Jacobi interpolation and a field vanishing at every vertex. Integration by parts makes the two summands orthogonal for the index form. On a cell the second summand satisfies the Dirichlet Poincaré inequality. A curvature upper bound, the speed bound \(B\), and sufficiently small cell width make its index form positive definite. The polygon Hessian is the first summand, which therefore has the full Hilbert index and nullity. Sard’s theorem for \(f_N\) gives dense regular cuts, since its critical values in the band are exactly those of \(E\). Compact smooth manifolds with boundary have finite CW models, completing the claim. ◻

Proof. A nonconstant geodesic has constant positive speed and is an immersion. Its periods form a closed discrete subgroup of \(\mathbb R\), so there is a least positive period. Dividing by that period gives its prime root.

To compare two primes with the same image, first use unit-speed parameters. Cover the parameter circle of the first by finitely many compact intervals on which it is an embedded arc. The inverse images of those arcs under the second geodesic form a finite closed cover of its parameter circle. By Baire’s theorem, one contains an interval. Within this interval choose a point whose image is not an endpoint of the chosen arc; an immersion cannot take an interval into that finite endpoint set. Near this point both geodesics lie in the same embedded arc and have the same tangent line. Their unit velocities agree up to sign. Uniqueness for the geodesic equation identifies them globally up to translation and reversal. Their least unit-speed periods agree, giving the assertion for period-one parametrizations as well. This proof allows self-intersections of either prime.

If reversal preserved an oriented phase orbit, one would have \(\gamma(t)=\gamma(a-t)\). Differentiation at a fixed point of this parameter reflection gives \(\dot\gamma=-\dot\gamma\), impossible for a nonconstant geodesic. ◻

Proof. Use the periodic index form over integer intervals at prime speed. In the form domain for the \((m+k)\)-period problem, require that the value at time \(m\) equal the initial value. In the direct sum of the \(m\)- and \(k\)-period domains, require that the two initial values agree. Both are restrictions of codimension at most \(n\), and the restricted forms identify. If the two indices drop by \(a,b\in[0,n]\), their original difference is \(a-b\). This proves the first estimate. The indices are finite since the index form is a positive derivative form plus a compact zeroth-order perturbation.

The sequences \(I_{i,m}+n\) and \(I_{i,m}-n\) are respectively subadditive and superadditive. Their normalized limits agree; applying the inequalities along multiples of each fixed \(m\) gives \[\frac{I_{i,m}-n}{m}\le\Delta_i \le\frac{I_{i,m}+n}{m}.\] Nonnegative indices give \(\Delta_i\ge0\). Equation (4) is the first estimate with two equal summands.

Let \(P_i\) be the real Jacobi Cauchy-data monodromy over one prime period. The periodic kernel is \(\ker(P_i^m-1)\). Take \(D_i\) to be the least common multiple of the orders of its root-of-unity eigenvalues, with value one if there are none. For \(s=\gcd(m,D_i)\), the same eigenvalues satisfy \(\lambda^m=1\) and \(\lambda^s=1\). On a contributing complex Jordan block, \(z^m-1\) has a simple root, so its kernel is the ordinary eigenspace. All other blocks contribute zero. Hence \[\ker(P_i^m-1)=\ker(P_i^s-1).\] ODE uniqueness identifies the corresponding fields as repeated root fields. This computation is over \(\mathbb R\) or \(\mathbb C\); it makes no division by multiplicity in \(\mathbb F_2\)-cohomology. ◻

Proof. Choose disjoint neighborhoods \(W_\alpha\) of the compact critical sets, and smaller \(V_\alpha\) with positive distance from the complement of \(W_\alpha\); the choices are equivariantly permuted. Work first in a fixed isolating band. Uniform drop outside the \(V_\alpha\) makes a sufficiently long segment remaining in that band visit one of them. Traversing the annulus between \(V_\alpha\) and the complement of \(W_\alpha\) takes positive time, since the speed is bounded and its width is positive. The traversal costs a uniformly positive amount of energy. Choose the final width \(b-a\) smaller than all these costs. Every segment of duration \(T\) with initial energy at most \(b\) and terminal energy at least \(a\) is then trapped in a single \(W_\alpha\), before and after its visit to \(V_\alpha\).

The first map in (6) is a pair homotopy equivalence: its inverse is the map induced by \(\phi_T\). The intermediate flow preserves the relevant subspaces. The finite closed cover \(X=N\cup Y\) gives \[N/Q\cong X/Y\] by closed pasting. This does not require \(X\) to be compact. At \(f_T=a\), the derivative of \(f_T\) along the flow is strictly negative. Transverse first hitting of this level is continuous, with hitting time zero on the lower side. Flowing for a fraction of that time gives a neighborhood retraction onto \(Y\) in \(X\). On the upper side, stop at equality to obtain the corresponding retraction onto \(Q\) in \(N\). The condition \(f\le b\) is preserved by descent. These are good pairs, so their quotient homeomorphism gives the second relative isomorphism. If a collapse set is empty, adjoin a disjoint basepoint.

The constructions are equivariant. Finite invariant closed covers remain closed covers after Borel construction, and the neighborhood homotopies descend. This proves the equivariant version without asserting compactness of a Borel space. Trapping makes \(N\cap W_\alpha\) and \(Q\cap W_\alpha\) clopen label pairs. Restriction to a fixed manifold intersects every space in (6) with that manifold, using the original flow. Finally, the maps in that diagram are inclusions and hence respect ambient cup products and connecting homomorphisms. ◻

Proof of the decomposition, naturality, and cup assertion. Choose an even polygon order adequate for the event. Start with one complete \(K\)-invariant strict descending gradient flow for the original \(f_N\). Its gradient is tangent to \(\operatorname{Fix}(\tau)\). Thus a critical point of the restriction is exactly a full critical point there, and the restricted flow is strict away from them.

By Lemma 7, the critical circles come in disjoint reversed pairs. The half-shift preserves each orientation circle and fixes it pointwise exactly when \(m\) is even; for odd \(m\), choose its neighborhood disjoint from the fixed locus. Use Lemma 9 with disjoint equivariantly permuted neighborhoods of these oriented circles. Reflection induction identifies a block with the ordinary pair of one orientation, while \(K\) induction retains its half-shift action. Literal intersection of the original lenses with the fixed manifold gives the full-to-fixed map. In particular, the actual diagrams are \[\begin{tikzcd} H_K^j(X,X_-) \arrow[r,"\cong"] \arrow[d,"\mathop{\mathrm{res}}"'] & H_K^j(N,Q) \arrow[d,"\mathop{\mathrm{res}}"]\\ H_K^j(X^\tau,X_-^\tau) \arrow[r,"\cong"] & H_K^j(N^\tau,Q^\tau), \end{tikzcd}\] with \(N^\tau=N\cap\operatorname{Fix}(\tau)\) and likewise for \(Q\). They commute with ambient relative maps and the associated exact sequences. No alternative local normal form is being used to identify these maps.

The finite union of prime images does not fill \(S^n\), for \(n\ge2\). Choose \(q\) outside it, and choose all critical neighborhoods inside the invariant open set \[\bigcap_{g\in K}g^{-1} \bigl(\mathop{\mathrm{ev}}_0^{-1}(S^n\setminus\{q\})\bigr).\] The entire lens therefore evaluates into the punctured sphere. Evaluation at zero is reflection invariant, so it induces \(ER\times_R N\to S^n\setminus\{q\}\). The top sphere class pulls back to zero. Hence its cup product on \(H_R^*(N,Q)\) is zero. The inclusion isomorphisms of Lemma 9 transfer this vanishing to the actual crossing. This uses only reflection invariance of evaluation; half-translation invariance is not needed. All choices are common to the finitely many labels at a simultaneous level. The uniform rank statements are proved next. ◻

Proof. Normal coordinates. Take \(N\) divisible by \(k\) and sufficiently large. The fixed manifold \(Z=P_{N/k}\subset P_N\) of the \(1/k\)-shift has restricted energy \(h=k^2f_{N/k}\). Near the selected orbit the bump vanishes. Use a cyclically invariant tubular metric. Symmetry makes the normal first derivative vanish along \(Z\). The Hessian has reducing fixed and normal summands, and Lemma 8 places its full kernel inside the fixed tangent space. The normal block is therefore nonsingular, of negative rank \[b_-=I_{i,m}-I_{i,s}\ge0.\]

We record the normal coordinate construction to keep the degenerate base unchanged. Taylor integration writes \[f_i(z,\nu)=h(z)+\langle A(z,\nu)\nu,\nu\rangle,\] where \(A_0(z)=A(z,0)\) is symmetric and invertible. Normalize by \(|A_0|^{1/2}\), putting \[B=|A_0|^{-1/2}A|A_0|^{-1/2},\qquad J=\operatorname{sign}(A_0).\] Near the zero section, \(B\) is close to \(J\). The near-identity series \(D=(JB)^{1/2}\) is \(J\)-self-adjoint, and \(D^*JD=B\). Thus the fiber inverse function theorem applied to \(D|A_0|^{1/2}\nu\) gives \[ f_i=h(z)-|u|^2+|y|^2. \tag{7}\] The construction is uniform near the compact orbit. The positive and negative bundles can be nontrivial; no trivialization of them is required.

The local quotient. Let \(\psi_t\) be a complete strict descending flow for the unmodified \(h\). In (7), lift it using metric connections, expand \(u\) exponentially, and contract \(y\) exponentially. The energy derivative is \[\mathop{}\!\mathrm dh(\dot\psi)-2|u|^2-2|y|^2<0\] off the critical set. Convex interpolation with a normalized negative gradient gives a complete strict full field agreeing with this one near the orbit.

Choose a full trap \(W_f\) whose base projection has compact closure inside the base chart and avoids all other \(h\)-critical points. The exact-field region contains uniform fiber balls over this closure. A smaller selected base trap \(W_h\) lies under uniform small fiber balls contained in \(W_f\); all other base traps lie outside its projection. First choose a common sufficiently large time \(T\) for full and base trapping, then sufficiently tight regular cuts \(a<c<b\). In particular, \(\sqrt{b-a}\) is smaller than the available fiber radius.

Apply Lemma 9 to the isolated full crossing. On its quotient \(N/Q=X/Y\), shrink \(y\) to zero. The new first \(T\)-paths have the same base and negative coordinate, and a scaled positive coordinate, so they stay in the exact region. Both initial and terminal energies decrease. Thus the maps take \(N\) into \(X\) and \(Q\) into \(Y\), and descend continuously to the quotient; some points may collapse. This retracts onto the \(y=0\) subset together with the basepoint.

Apply the full forward flow to this subset. While a point remains uncollapsed, every shifted \(T\)-segment is trapped in the exact region, so \(y=0\) persists and the base follows \(\psi\). On the compact base projection where \(h\ge b>c\), its drop has a positive lower bound. A single further time therefore sends all survivors into \(h\le b\). This last subset is preserved up to collapse, so its inclusion is a homotopy equivalence. The homotopies are continuous on the pointed quotient; products of a quotient map with the compact interval remain quotient maps.

Write \(h_T=h\circ\psi_T\). The invariant subset \(h\le b\) used in this homotopy equivalence, with its equality boundary before collapse, is described by \[ P=\{h\le b,\ h_T\ge a\}_{\mathrm{selected}},\qquad A_P=\{h_T=a\}\cap P,\qquad |u|\le e^{-T}\sqrt{h_T(z)-a}. \tag{8}\] For one direction, the projected segment stays in the base band and the projection of the full trap, so it belongs to the selected base label. Conversely, a selected base path is trapped in \(W_h\), and \(e^t|u|\le\sqrt{b-a}\) for \(0\le t\le T\). Its lift therefore stays in the exact region and satisfies the full lens inequalities. Equality in the radius bound is precisely its exit.

Let \(D_-\) and \(S_-\) be the unit disk and sphere bundles of the negative bundle over \(P\). Scaling by the continuous radius in (8) gives a homeomorphism from \[ D_-/(S_-\cup(D_-)_{A_P}) \tag{9}\] onto the final pointed subset. Indeed it is bijective away from the collapsed union; its domain is compact and its target is Hausdorff. The radius vanishes exactly over \(A_P\). This also handles rank zero and the disjoint-basepoint convention.

Relative Thom calculation. For completeness, we prove the relative Thom calculation for this lens, without requiring a manifold structure on its base. On \(P\), let \(t_p\) be the first forward time until \(h_T\le a\), truncated at a fixed \(T_0>0\). It is continuous. Finite hits are transverse since \(a\) is regular; at a point that has not hit by \(T_0\), either the hit occurs transversely at \(T_0\), or the strict terminal inequality persists in a neighborhood. The base flow stays in its clopen label until equality.

For a unit-disk coordinate put \[t(z,u)=\min\{t_p(z),-\log|u|\},\qquad -\log0=+\infty.\] This is continuous also at \(u=0\), where the radial entry is inactive for \(|u|<e^{-T_0}\). Its zero set is \(\mathcal A=S_-\cup(D_-)_{A_P}\). On the open neighborhood \(t<T_0\), flow the base for time \(\lambda t\), parallel transport \(u\), and multiply its norm by \(e^{\lambda t}\). Neither the base exit nor the sphere is passed before the stopping time. The residual minimum time is \((1-\lambda)t\), so the neighborhood is preserved, the union is fixed, and at \(\lambda=1\) it is reached. This proves that \((D_-,\mathcal A)\) is a good pair. Rank zero simply removes the radial alternative.

The sphere pairs are good by radial collars. Closed pasting identifies \[H^*(\mathcal A,S_-) \cong H^*((D_-)_{A_P},(S_-)_{A_P}).\] Under the natural absolute mod-2 Thom isomorphisms, restriction from \(H^*(D_-,S_-)\) to this group is the restriction \(H^{*-b_-}(P)\to H^{*-b_-}(A_P)\). The exact sequence of the triple \(S_-\subset\mathcal A\subset D_-\) gives \[\begin{split} 0\longrightarrow& \mathop{\mathrm{coker}}\bigl(H^{j-b_--1}(P)\to H^{j-b_--1}(A_P)\bigr) \longrightarrow H^j(D_-,\mathcal A)\\ \longrightarrow& \ker\bigl(H^{j-b_-}(P)\to H^{j-b_-}(A_P)\bigr) \longrightarrow0. \end{split}\] The base-pair exact sequence has the same kernel and cokernel. Thus \[ \dim H^j(D_-,\mathcal A) =\dim H^{j-b_-}(P,A_P). \tag{10}\] This argument compares vector spaces, rather than subtracting possibly infinite dimensions.

Absolute Thom applies to these compact metrizable bases as follows. Pull the bundle back along the CW weak equivalence \(|\operatorname{Sing}P|\to P\). The disk and sphere bundle projections are fibrations, so their induced total-space maps are weak equivalences. They give singular cohomology isomorphisms, including the relative disk/sphere groups and their restriction maps. Apply the usual mod-2 Thom isomorphism on the CW model, naturally under the same construction for \(A_P\). The rank-zero case is the base pair itself.

Finally, divide the base cuts by \(k^2\) in (1). The selected base label is the root label at \(c_i^s\), so (10) proves the asserted normal shift. Iteration is a homeomorphism onto the repeated-loop subspace, including its oriented phase circles; this identification uses no covering transfer or division by multiplicity. ◻

Completion of Theorem 10. There are only finitely many root levels \((s\ell_i)^2\), with \(s\mid D_i\). Around each distinct level choose a fixed isolated interval with regular endpoints. Choose one coarse polygon order \(N_0\) adequate up to every upper endpoint.

The total crossing pairs in \(P_{N_0}\) have finite total cohomological rank: their regular sublevels are compact smooth manifolds with boundary, and the pair exact sequence applies. Their relative cohomology vanishes outside \(0,\ldots,nN_0+1\). Every selected root label is a direct summand of such a total crossing by Lemma 9. This includes coincident root levels.

At an arbitrary event, choose the fine polygon order divisible by \(kN_0\) for every label, where \(k=m/\gcd(m,D_i)\). For every even label also impose divisibility by \(2k'N_0\), where \[k'=\frac{m/2}{\gcd(m/2,D_i)}.\] These are finitely many conditions. After the normal shift, polygon retractions through the Hilbert pair identify each finer root crossing with its coarse pair. Tight cuts within the fixed isolating intervals give the same groups by regular-strip deformations. The finite maximum of their total ranks is therefore a bound \(B\) for every iterate label.

If the root group in degree \(a\) is nonzero in the shift calculation, then \[j-I_{i,m}=a-I_{i,s},\qquad 0\le a\le nN_0+1.\] For example, \[A=nN_0+1+\max_{i,\ s\mid D_i}I_{i,s}\] is a uniform support constant. Enlarge the constants if necessary for the fixed computations: on \(\operatorname{Fix}(\tau)\) start with \(4f_{N/2}\), apply the same argument to \(m/2\), and divide cuts by \(4(k')^2\). The support is centered at \(I_{i,m/2}\).

The normal shift changes no total rank. All degeneracy remains in finitely many root crossings, so the proof assumes neither nondegeneracy nor analyticity. It never divides by \(\Delta_i\); zero-mean-index primes satisfy the same conclusions. ◻

Proof. Write \(v(t)=|\dot\gamma(t)|\) and \(L=\int_0^1v(t)\,\mathop{}\!\mathrm dt\). When \(L>0\), put \[\sigma(t)=\frac1L\int_0^t v(s)\,\mathop{}\!\mathrm ds, \qquad \phi_u(t)=(1-u)t+u\sigma(t).\] For \(u<1\), the absolutely continuous map \(\phi_u\) is an increasing homeomorphism with Lipschitz inverse. Set \(\Psi_u\gamma=\gamma\circ\phi_u^{-1}\). At \(u=1\) the inverse need not exist. Instead, define \(\Psi_1\gamma(s)\) to be the value of \(\gamma\) on the fiber \(\sigma^{-1}(s)\). Each fiber is an interval on which \(v=0\) almost everywhere, so absolute continuity makes this definition unambiguous. For \(L=0\) leave the constant path fixed for every \(u\).

For \(s_1<s_2\), choose ordered preimages \(t_1<t_2\) under \(\sigma\). Then \[d\bigl(\Psi_1\gamma(s_1),\Psi_1\gamma(s_2)\bigr) \le\int_{t_1}^{t_2}v(t)\,\mathop{}\!\mathrm dt=L(s_2-s_1).\] Thus \(\Psi_1\gamma\) is \(L\)-Lipschitz. Since \(\gamma=(\Psi_1\gamma)\circ\sigma\) and \(\sigma\) is a nondecreasing surjection, comparison of partitions gives \(\ell(\gamma)\le\ell(\Psi_1\gamma)\). The Lipschitz estimate gives the reverse inequality. Consequently \(\Psi_1\gamma\) has length \(L\) and speed \(L\) almost everywhere, and its energy is \(L^2\).

For \(u<1\), change of variables by \(\phi_u\) gives \[E(\Psi_u\gamma) =\int_0^1\frac{v(t)^2}{(1-u)+uv(t)/L}\,\mathop{}\!\mathrm dt.\] On \(\{v>0\}\), convexity of the reciprocal bounds the integrand by \((1-u)v^2+uLv\); on \(\{v=0\}\) it is zero. This proves (11), including \(u=1\) by the preceding argument. The same change of variables preserves length for \(u<1\). Endpoints are unchanged, and constant-speed paths have \(\sigma(t)=t\).

We verify joint strong continuity, including the possibly flat fibers at \(u=1\). Fix a smooth embedding of \(M\) in a Euclidean space and extend its metric to a smooth positive definite matrix on a neighborhood of \(M\). Suppose \(\gamma_i\to\gamma\) strongly in \(H^1\) and \(u_i\to u\). The one-dimensional Sobolev embedding gives uniform convergence of the paths. Smoothness of the metric gives \(v_i\to v\) in \(L^2\), and hence \(L_i\to L\).

If \(L>0\), the maps \(\sigma_i\) and \(\phi_{u_i}\) converge uniformly to \(\sigma\) and \(\phi_u\). Given \(s_i\to s\), choose \(t_i\) in the inverse fiber defining \(\Psi_{u_i}\gamma_i(s_i)\). Every subsequential limit \(t\) satisfies \(\phi_u(t)=s\). The path \(\gamma\) has one value on that fiber: for \(u<1\) the fiber is a singleton, and for \(u=1\) this was proved above. Therefore \[\Psi_{u_i}\gamma_i(s_i)=\gamma_i(t_i) \longrightarrow\Psi_u\gamma(s).\] A compactness argument makes this convergence uniform in \(s\). If \(L=0\), the image of each output is contained in the image of its input, so uniform convergence to the constant path follows directly.

The energies also converge. If \(L>0\) and \(u<1\), the reciprocals in the integral formula are uniformly bounded and converge in measure; combining this with \(v_i^2\to v^2\) in \(L^1\) proves convergence of the integrals. If \(u=1\), Cauchy–Schwarz and (11) give \[L_i^2\le E(\Psi_{u_i}\gamma_i) \le(1-u_i)E(\gamma_i)+u_iL_i^2\longrightarrow L^2.\] If \(L=0\), the upper bound \(E(\Psi_{u_i}\gamma_i)\le E(\gamma_i)\) tends to zero.

For completeness, energy convergence here implies strong, rather than merely weak, convergence of derivatives. Uniform convergence and the energy bound give weak \(H^1\) convergence to the already identified output. In the squared derivative norms, replace the metric matrix at the varying outputs by its value at the limiting output. The resulting error tends to zero, because the matrices converge uniformly and the derivatives are bounded in \(L^2\). The derivative norms thus converge in one fixed positive weighted \(L^2\) Hilbert norm. Weak convergence together with convergence of these norms gives strong convergence. This proves the asserted joint continuity.

Finally, for the reversed path the arclength map is \(1-\sigma(1-t)\). This identity proves commutation with reversal, also at the quotient endpoint \(u=1\). ◻

Proof. The space of such pairs is compact in \(C^0\) by the Arzelà–Ascoli theorem. Consider one pair \((\alpha_0,\beta_0)\) from this space, with endpoints \(x_0,y_0\). Simple connectivity gives a continuous homotopy between them, fixing the endpoints. Replace its paths by polygons on one sufficiently fine, fixed subdivision of the path parameter. Uniform continuity of the original homotopy ensures that all consecutive vertices are within the injectivity radius. Polygonal interpolation then gives an \(H^1\)-continuous homotopy with a finite length bound. Short pointwise geodesic interpolation joins its two endpoint polygons to \(\alpha_0,\beta_0\). These interpolations also have a finite length bound, since the original endpoint paths are Lipschitz.

This construction extends to nearby pairs with varying common endpoints \(x,y\). Attach the short geodesics from \(x\) to \(x_0\) and from \(y_0\) to \(y\) to each prototype path, using fixed, small positive time intervals at its ends. First choose those intervals small enough that the retimed prototype endpoints are uniformly close to the original endpoints; this uses their Lipschitz bounds. Then restrict \(x,y\) and the input paths to a sufficiently small \(C^0\) neighborhood. Short pointwise geodesic interpolation joins the actual first path to the adjusted prototype first path, and similarly at the other end. Both interpolations fix \(x,y\). Their derivatives are bounded by a fixed multiple of the derivatives of the two interpolated paths. Consequently their lengths have a common bound on this neighborhood. Every operation just described is continuous in the strong \(H^1\) topology. A finite subcover of the compact pair space, and the maximum of its finitely many bounds, give \(K\). ◻

Proof. Set \(D=\operatorname{diam}(M)+1\). For a fixed prototype choose shortest paths, called spokes, from its initial point to all its piece vertices. For nearby inputs extend these choices by short endpoint connectors, using fixed thirds of the unit interval. Shrink the neighborhood so that each spoke has length at most \(D\) and Lipschitz constant at most \(3D\). At the initial vertex use the constant spoke.

Proceed through the pieces in order. Assume the preceding prefix has already been replaced by its spoke. Retime that spoke and the next piece to halves of the enlarged prefix. This retiming preserves length and is continuous in \(H^1\): every nonconstant path has a positive time interval throughout the retiming. In the first step, where the prefix is empty, insert a constant wait at the initial point and retime the first piece. The two paths to be compared are now the spoke followed by the next piece, and the spoke to the next vertex. On a unit interval both have Lipschitz constant at most \[L_0=2\max\{1,3D\}.\] Apply Lemma 13 to replace the first by the second, and leave the remaining suffix unchanged.

During a replacement the active prefix has length at most \(K(M,L_0)\); during a retiming it has length at most \(D+1\). The suffix still consists of original pieces. Thus \(C=\max\{K(M,L_0),D+1\}\) bounds the increase over the length of the entire original path at every stage. At the end the path is its final spoke, of length at most \(D\). For a particular input there are finitely many pieces, hence finitely many local choices. Intersecting their neighborhoods makes the entire construction continuous on one neighborhood. This affects the neighborhood, but neither \(C\) nor \(D\). ◻

Proof. Normalize the family by Lemma 12. A loop of length \(L<a\) now has constant speed \(L\). Let \(C,D\) be the constants of Lemma 14. Choose \(B\) so large that \[ B/2>D+jC+2. \tag{12}\] For sufficiently large \(a\), the integer \(k=\lfloor a/B\rfloor\) is at least \(j+1\). Divide the loop into \(k\) equal time blocks, and each block into \(\lceil a/k\rceil\) equal pieces. Every piece, rescaled to a unit interval, has Lipschitz constant at most one. Every block has length \(L/k\).

For each parameter point, Lemma 14 supplies local recipes for the first \(j+1\) blocks. There are only finitely many blocks and pieces in this particular family. Choose a common neighborhood for these recipes, and cover \(P\) by finitely many such neighborhoods. Subdivide a triangulation finely enough that the closed stars in a final barycentric subdivision are subordinate to this cover. Color each vertex by the dimension of the simplex of the preceding triangulation whose barycenter it is. The colors lie in \(\{0,\ldots,j\}\) and are distinct on each simplex.

At a vertex \(v\) of color \(h\), use its local recipe on block \(h+1\). If \(\lambda_v\) is its barycentric coordinate, at homotopy time \(s\) run that recipe for the fraction \[s\min\{1,(j+1)\lambda_v\}.\] Two positive-coordinate stars of the same color are disjoint; at the boundary of either star the fraction is zero and the recipe is the identity. Recipes of different colors affect different blocks, keeping their endpoints fixed. They therefore paste to one continuous \(H^1\) homotopy. At each parameter at most \(j+1\) blocks are changed, and the length throughout is less than \(a+(j+1)C\).

At the terminal time some barycentric coordinate is at least \(1/(j+1)\), so one block has been completely shortened. The final length is consequently at most \[ L-L/k+D+jC. \tag{13}\] If \(L\ge a/2\), then \(k\le a/B\) gives \(L/k\ge B/2\), and (12) makes (13) less than \(a-1\). If \(L<a/2\), the rough bound \(L+(j+1)C<a/2+(j+1)C\) has the same consequence once \(a>2((j+1)C+1)\). Choose \(a_j\) larger than this threshold and large enough that \(k\ge j+1\). We may take \(H_j=(j+1)C\). ◻

Proof. Normalize by Lemma 12; the resulting cycle has \(F=\ell<b\). We describe a single finite model for all such normalized cycles. Choose an integer \(N\) large enough that paths of energy at most \((b+1)^2\) move less than a fixed small injectivity radius on intervals of length \(1/N\). Sampling at these times and joining successive vertices by minimizing geodesics gives a continuous polygonal map. For the discrete energy \[E_N(x_0,\ldots,x_{N-1}) =N\sum_{i=0}^{N-1}d(x_i,x_{i+1})^2, \qquad x_N=x_0,\] Cauchy–Schwarz gives \(E_N\le E\) on sampled loops. Choose \(b'\in(b,b+1)\) such that \((b')^2\) is a regular value of \(E_N\). The set \[Q=\{E_N\le(b')^2\}\] is contained strictly inside the open set of vertex tuples whose successive distances are smaller than the injectivity radius. It is a compact smooth manifold with boundary and therefore has finite-dimensional homology in every degree. Polygonal interpolation maps \(Q\) to loops of length at most \(b'\).

There is also a homotopy from each normalized loop to its sampled polygon, with one uniform energy bound depending only on \(b\) and \(M\). Indeed, on each interval both paths stay in a small normal neighborhood of its initial vertex. Pointwise short-geodesic interpolation is \(H^1\)-continuous, and the derivative of this interpolation is bounded by a fixed multiple of the two path derivatives. Its square-root energy is thus at most \(2C_0b\) for a fixed metric constant \(C_0\). This also gives a uniform length bound for the homotopy.

The kernel of \(H_j(Q;\mathbb F_2)\to H_j(\Lambda;\mathbb F_2)\) is finite-dimensional. Choose finite cycles representing a basis and finite global fillings of their polygonal images. Their supports are compact, so their lengths have a common finite bound. The sampled image of any cycle in the lemma represents a linear combination of this basis. The difference bounds in \(Q\), whose entire image has length at most \(b'\). Adding that comparison chain, the finitely many chosen fillings, and the two normalization and polygonal homotopies proves the claim with one \(R_{b,j}\) strictly larger than all these bounds. No bound on the number of simplices in the original cycle is needed. ◻

Proof. A finite singular cycle over \(\mathbb F_2\) can be regarded as the image of a cycle on a finite polyhedron of dimension at most \(j\). One explicit construction takes the finitely many singular simplices, pairs equal faces that cancel in their boundary, and glues these faces by their ordered affine identifications. The first barycentric subdivision has embedded closed simplices and vertices distinguished by their original face dimensions; a second subdivision removes multiple simplices with the same vertex set. This gives a finite polyhedron with its top-dimensional cycle and the original map to \(\Lambda\). The usual subdivision and prism chains identify its image with the original singular cycle. All these chains have image in the union of the original simplices. Thus the number of parameter simplices plays no role in a length estimate.

Apply Lemma 15 repeatedly, normalizing between successive applications. Starting at level \(a\), the successive starting levels are \(a,a-1,a-2,\ldots\), until the family lies below one fixed level \(b_j>a_j+1\). The prism chains at the successive stages all lie in \(\mathcal A_{a+H_j}\): the peak at stage \(r\) is bounded by \(a-r+H_j\), so these bounds are not added over the stages. Only finitely many stages are needed for each input cycle.

The resulting cycle is still globally null-homologous. Apply Lemma 16 at the fixed level \(b_j\). Taking \(D_j>\max\{H_j,R_{b_j,j}\}\) gives the required filling after adjoining the shortening prisms. If the original \(a\) is already at most \(b_j\), the bounded-level lemma applies immediately. This proves the assertion for every positive level, without a regular-value assumption. ◻

Proof. The resonance statement is Theorem 1.1 of (Hingston and Rademacher 2013). Its cohomological critical levels may be computed in singular cohomology: the open energy sublevels and \(\Lambda\) are paracompact, locally contractible Hilbert manifolds, where singular and Čech cohomology agree. Their Lemma 5.3 gives the rank bound; Lemma 5.4 and Equation (12) give the powers and the pairing, with the index in that equation replaced by \(m+1\). These statements apply for every \(n>2\) and every field. The indicated powers and pairing may also be taken integrally and reduced modulo two: the integral pairing is one, so its reduction is one in both parities of \(n\). The absolute class of degree \(3n-2\) has a unique relative lift because \(3n-2>n+1\). Finally evaluation retracts the inclusion of constant loops, so \([M]\ne0\); the rank bound gives the last assertion. ◻

Proof. Extension by zero defines a cochain, although generally not a cocycle. The ordinary cochain identity nevertheless gives \[\delta b=U\smile\delta\widetilde{h^*T}.\] If its front \(U\)-face is zero, the evaluation is zero. Otherwise the entire parent lies in \(O\), and every face entering the remaining coboundary is an actual pullback of \(T\). They cancel because \(\delta T=0\). This proves the assertion without discarding any extension-by-zero terms separately. Each block or skip face is contained in its parent image and has the same property.

In a contributing term the \(U\)-face first puts the whole rectangle in the closed intermediate tube, where \(h\) is defined and uniformly continuous on the compact parameter set. Its nonzero \(T\)-face cannot have its entire image in \(\{\ell<L\}\), and hence contains a point with \(\ell(h)\ge L\). Refining the factors so that \(\ell\circ h\) oscillates by less than \(\varepsilon\) on each such rectangle proves the estimate. The tube conditions follow in the same way from the gap between support and intermediate radii. Include all homotopy intervals among the factors before refining. The finitely many conditions then hold simultaneously. All later subdivision prisms and block or skip faces stay in the original rectangles, so both the domain and length conditions persist. ◻

Proof. The cut degree is \(1-n-p=-q\). The cap boundary formula applies by Lemma 19; its time-end terms vanish because \((x(0),x(t))\) is exactly diagonal at \(t=0,1\). Applying the same construction to a finite filling proves null-homology preservation. On a surviving parent the tested first loop has length at least \(L-\varepsilon\), while \[\ell(C_1(x,t))+\ell(C_2(x,t)) \le\ell(x)+2\varepsilon.\] This gives (23). If its upper level is nonpositive, no term survives. The join boundary formula uses the closed cocycle \(U_j\). Its output length is at most \((1+\kappa)\ell(x)+\ell(y)\), and \(\eta_j\) may be chosen after all finite length bounds have been specified.

For the unit, consider a shuffle of \(c\times x\). A constant-factor step remaining after the \(n\)-cap makes the output degenerate: the rotated constant is exactly \(x(0)\), so the output is independent of that factor. The remaining terms have all \(n\) constant-factor steps first. Their coefficient is the Thom evaluation on \([M]\times\{x(0)\}\), at the initial parameter vertex of the \(x\)-simplex, and equals one. Thus the output is a constant half followed by \(x\). Removing the waiting half by reparametrization gives (24) without changing length. ◻

Proof. Write \(J=J_{x(0),y(0)}\) and \(Z=(Jx)\#y\). On \(x\times I_s\times y\), first cap with \(\sigma=U_j(x(0),y(0))\), then with \(U_c(Z(0),Z(s))\), then with \(T(C_1(Z,s))\), and output \(C_2(Z,s)\). Call this cycle \(P(x,y)\). Its time endpoints vanish by (19). We prove \[ \mu(D_Tx,y)\simeq P(x,y)\simeq P(y,x) \simeq\mu(D_Ty,x). \tag{26}\] All homologies will have the bound in (25).

Proof. Hingston–Wahl give singular chain models on \(H^1\) loops by capping with the evaluation pullback of a diagonal Thom cocycle, then adding short arcs and concatenating or cutting (Hingston and Wahl 2023, Theorem A, Definitions 1.2 and 1.4). The dual relative operation is the Goresky–Hingston product (Hingston and Wahl 2023, Theorem 2.13); all sign differences disappear over \(\mathbb F_2\). We use these models with the auxiliary round metric and compare the actual maps and cochains. Choose their support radius to equal ours and their map domain to contain our larger tube; both choices can be made within their injectivity-radius requirement.

For joining, partially rotate the first input by \(J^u\) and apply their short-arc retraction, whose values have equal starts throughout. Writing \(z=x(0)\) and \(z'=y(0)\), the residual mismatch satisfies \(d_{\rm rd}(J^u z,z')=(1-u)d_{\rm rd}(z,z')\), so all inputs remain in its domain. At \(u=0\) this is their retraction; at \(u=1\) the inputs already have equal starts, so their retraction is the identity and gives our rotation retraction. This is a homotopy inside equal-start pairs. Interpolate their energy-optimal concatenation times to halves. For path pieces of square-root energies \(P,Q\), the optimal time for the first piece is \(P/(P+Q)\); during this interpolation its energy contribution is bounded by \[\max\{2P^2,P(P+Q)\}.\] This tends to zero if that piece collapses. The other piece is handled in the same way, and if both collapse both contributions vanish. Thus the timing homotopy is continuous also at constant pieces.

For cutting, their short-arc retraction followed by cutting gives exactly the two paths in (20), with optimal instead of half timings. The same interpolation compares them. At either time endpoint one of the two paths remains constant. Pairing with \(T\) and \(b\) evaluates on \(z\times I\) the cochain \[U_c(x(0),x(t))\smile\chi,\qquad \chi=T(C_1(x,t))\smile b(C_2(x,t)).\] Both tests are relative to constants. Hence \(\chi\) vanishes at \(t=0\) through its first factor and at \(t=1\) through its second factor.

Replace \(U_c\) by the reduced ordinary Thom representative used in that model. They differ by a coboundary relative to the far set. Although its primitive need not vanish on the diagonal, all of its time-end contributions are multiplied by \(\chi\) and vanish. If a cap convention reverses the cup order, the cup-one formula supplies the comparison. On a small simplex in the tube, \(\chi\) is a closed pullback; on a simplex in the far set, every face of the accompanying Thom cochain or relative primitive vanishes. Thus extension-by-zero errors contribute nothing, exactly as in Lemma 19. The same argument permits replacement of either relative test by a cohomologous representative. Join Thom representatives are compared in the same way, without interval-end terms.

Finally the Alexander–Whitney map in the coproduct evaluates \(T\otimes b\) as the displayed cup of the two pullbacks. Their small-chain replacement and our refined products represent the same relative chain class: both are homologous to the input, and small-chain inclusion is a quasi-isomorphism, also relative to the time ends, by subdivision. This proves (28) and the asserted join identification. The argument compares two relative tests; it makes no assertion about a canonical absolute extension of the coproduct. ◻

Proof. Represent \(\Theta^{\bullet(m+1)}\) by a finite cycle \(z\). Its cut has degree \(n+2(m+1)d-q=3n-2\). Lemma 22, commutativity of the relative product over \(\mathbb F_2\), and (16) give \(\langle\theta,D_Tz\rangle=1\). The rank-one calculation therefore yields \([D_Tz]=\Theta\).

Both \(|z|\) and \(|S|=n+q\) exceed \(n+p=q+1\). Apply Theorem 21 to these cycles, now forgetting the finite output length bound. Using the join identification, \[0\ne\Theta^{\bullet(m+1)} =[\mu(D_Tz,S)]=[\mu(D_TS,z)].\] If \(D_TS\) were null-homologous, its join with \(z\) would bound, by Lemma 20. Thus \(D_TS\ne0\) in \(H_n(\Lambda)=\mathbb F_2[M]\), proving (29). This determines the absolute constant component by interchange and nonvanishing, rather than by the relative scalar test alone.

The cut bound gives support in \(\mathcal A_{l-L+1}\). Its difference from a constant fundamental cycle is globally null in degree \(n\); Proposition 17 fills that difference in \(\mathcal A_{l-L+1+D_n}\). By (18), this is one fixed level \(B\). ◻

Proof. For a real rank-\(r\) representation of \(H\), its associated bundle over \(Y_H\) restricts to a bundle over \(A_H\). Suppose that its unit sphere is \(H/H'\). The sphere-bundle pair is then the Borel pair for \(H'\). The relative Thom sequence gives \[ \cdots\longrightarrow H_H^{j-r}(Y,A) \xrightarrow{\,w_r\,}H_H^j(Y,A) \longrightarrow H_{H'}^j(Y,A) \xrightarrow{\int}H_H^{j-r+1}(Y,A)\longrightarrow\cdots. \tag{44}\]

Here is a cochain construction that also justifies (44) when the inclusion \(A_H\subset Y_H\) is not a cofibration. For a vector bundle over \(Z\supset A_0\), let \(D,S\) denote its disk and sphere bundles, and let \(D_{A_0},S_{A_0}\) be their restrictions. In the singular chain complex of \(D\), put \[\mathcal L=C_*(S)+C_*(D_{A_0}).\] The two summands intersect in \(C_*(S_{A_0})\), since singular chains have the singular simplices as a basis. Restriction of cochains gives degreewise exact sequences \[\begin{align*} 0&\longrightarrow\operatorname{Ann}\mathcal L \longrightarrow C^*(D,S) \longrightarrow C^*(D_{A_0},S_{A_0})\longrightarrow0,\\ 0&\longrightarrow\operatorname{Ann}\mathcal L \longrightarrow C^*(D,D_{A_0}) \longrightarrow C^*(S,S_{A_0})\longrightarrow0. \end{align*}\] Surjectivity follows by extending linear functionals on chain subspaces. Cup with a Thom cocycle sends a relative base cochain into \(\operatorname{Ann}\mathcal L\): on \(S\) the Thom factor vanishes, and on \(D_{A_0}\) the base factor vanishes. Comparison of the first sequence with the base-pair sequence, using the absolute Thom isomorphisms, proves \[H^{j-r}(Z,A_0)\cong H^j(\operatorname{Ann}\mathcal L).\] The second sequence and the disk contraction give (44). The map back to the base is cup with the zero-section pullback of the Thom class, namely \(w_r\). Pullback of Thom classes gives naturality and the projection formula. The absolute Thom isomorphism used here is the ordinary mod-two one (Hatcher 2017, sec. 3.2, pp. 88–89); for a general base it follows by pulling back to its singular CW replacement and using the disk and sphere fibrations. This argument uses the sum of two chain subspaces and does not require an excision assertion for their union.

The standard \(G\)-plane has sphere stabilizer \(R\) and Euler class \(U\), proving (40). Let \(\rho_2\) be the representation that doubles rotation angles and sends the chosen reflection to \(\operatorname{diag}(1,-1)\). Its sphere stabilizer is \(K\). On \(K\) it has a fixed line, so its Euler class restricts to zero. The injectivity in (39) makes that Euler class zero already on \(BG\), giving (41). Finally, the sign representations with kernels \(1\subset R\) and \(R\subset K\) have Euler classes \(w\) and \(v\), respectively. ◻

Proof. Along (45), put \[e_t=-x\sin(2\pi mt)+u\cos(2\pi mt).\] A tangent field is \(a(t)e_t+b(t)\), where \(b(t)\perp\operatorname{span}\{x,u\}\). Half the Hessian is \[ \int_0^1\bigl(|a'|^2+|b'|^2-(2\pi m)^2|b|^2\bigr)\,dt. \tag{47}\] In each of the \(d\) normal directions, the negative Fourier frequencies are \(|k|<m\). Their real dimension is \(2m-1\). The null modes are the two normal modes at frequency \(m\), together with the constant tangential mode. They have dimension \(2d+1\) and are precisely the variations of the frame \((x,u)\). The nonzero Fourier eigenvalues, divided by the corresponding squared \(H^1\)-norms, are bounded away from zero. Thus the Hessian is invertible on the orthogonal complement of its kernel. At constants, the Hessian is \(2\int|V'|^2\), positive and invertible off its constant kernel.

We describe why the resulting attachment respects the full group. Use the strong metric induced by the Euclidean \(H^1\)-norm. Each parameter symmetry is a smooth isometry, and the action is continuous. Pointwise exponential gives a chart on the Hilbert normal bundle of the compact manifold \(C_m\), at a uniform small radius. In these coordinates Taylor’s formula writes the energy difference as \(\langle A(z,y)y,y\rangle\), with \(A(z,0)\) an invertible symmetric operator on the normal fiber. Normalize first by \(|A(z,0)|^{1/2}\), and let \(J=\operatorname{sign}A(z,0)\). For the resulting operator \(B\) near \(J\), the norm-convergent power series \(C=(JB)^{1/2}\) satisfies \(C^*JC=B\). The change \(y\mapsto C(z,y)y\) is a diffeomorphism after shrinking the tube. All these constructions commute with the parameter isometries. They give the exact normal form \[ E_0=c_m-|u_-|^2+|u_+|^2 \tag{48}\] on the negative and positive bundles.

For completeness, the sublevel attachment uses the original Hilbert space, rather than a polygon model with only finite rotational symmetry. Patch a complete normalized descending gradient field to the split field \[\dot z=0,\qquad \dot u_-=u_-,\qquad \dot u_+=-u_+\] inside the normal tube, using an invariant squared-radius cutoff. Both fields strictly decrease \(E_0\) away from \(C_m\) within the isolating energy band. The coordinate derivatives are uniformly bounded in a small tube over the compact zero section, so the patched field remains bounded. The ambient strong metric is complete; hence this field is complete. Palais–Smale and regular-band deformation (Lemmas 4 and 5) give a uniform decrease outside any smaller tube in an isolating energy band. The flow-lens construction of Lemma 9 therefore computes the crossing in that smaller tube. Use its notation \(X,Y,N,Q\), with lower energy cut \(t_-<c_m\). In split coordinates, contraction of \(u_+\) decreases both endpoint energies of a trapped flow segment. It defines maps \(N\to X\) taking \(Q\) into \(Y\), even when a contracted point leaves \(N\) through the lower endpoint. Via \(N/Q\cong X/Y\), these maps induce a contraction to the negative disk with its boundary collapsed. For flow time \(T\), the surviving disk before applying the flow has radius \(e^{-T}\sqrt{c_m-t_-}\); its boundary flows to the lower cut. The flow multiplies \(u_-\) by \(e^T\), identifying this disk and sphere with the usual negative-bundle disk and sphere. The disk is closed in the lens and meets \(Q\) exactly in its sphere. These statements remain true after Borel construction. The disk/sphere quotient consequently embeds in the lens quotient: the saturation of a closed subset of the disk is that subset, possibly together with the closed collapse set. The contraction fixes its image. The disk/sphere inclusion is a good pair. This proves (46) by the mod-two Thom isomorphism.

All pair maps just used are \(G\)-equivariant. The negative bundle carries its actual derivative action; no triviality of that action is assumed. Restriction to a subgroup and cup by an absolute class therefore commute with the attachment isomorphism. Wider regular cuts give the same assertion by regular-band deformation. Finally, sufficiently small energy places a loop in the positive normal chart of the constants, where contraction of the normal coordinate proves the last assertion. ◻

Proof. The \(G\)-action on frames is the right action through \(\rho_m\), up to the convention for inverse precomposition. For odd \(m\), the map from the critical Borel space to \(\mathrm{Gr}_2(\mathbb R^{n+1})\) has fiber the classifying space of the cyclic group of order \(m\). Local choices of frames give local trivializations. That fiber has no positive-degree mod-two cohomology, by transfer, so the map induces a cohomology isomorphism. The tautological plane pulls back to the plane of \(\rho_m\); its classes are \(w,U\) by (51).

These two classes generate the Grassmann cohomology. Indeed, projectivize the tautological plane. On the resulting ordered-line flag space, let \(s,t\) be the first Stiefel–Whitney classes of its two lines. Over the Grassmannian, the projective-bundle formula gives a free module with basis \(1,s\). Over \(\mathbb{RP}^n\), the same flag space is the projectivization of the first line’s perpendicular bundle, so its ring is generated by \(s,t\). Since \(s+t=w\), \(st=U\), and \(s^2=ws+U\), every polynomial in \(s,t\) reduces to \(P(w,U)+sQ(w,U)\). Uniqueness of the free-module expansion implies that every class pulled back from the Grassmannian is a polynomial in \(w,U\). The complementary bundle has rank \(d\), with total Stiefel–Whitney class \((1+w+U)^{-1}\). Its components are the polynomials \(A_i\), proving (53) and the odd-stage assertion.

If \(m\) is even, \(\tau\) acts trivially on the critical manifold. Consequently \[H_K^*(C_m)=H_R^*(C_m)[v].\] The same statement holds for the crossing modules after the Thom shift, with the actual equivariant negative bundle retained. The polynomial \(U=v(v+w)\) is monic in \(v\), hence acts injectively even though the coefficient ring in (49) has zero divisors. The injection in (41) makes \(U\) injective on the \(G\)-crossing as well. Sequence (40) therefore identifies the quotient of the \(G\)-crossing by \(U\) with the \(R\)-crossing. Choose a lift \(y_m\) of \(ex_m\); the bottom \(x_m\) already lifts. Their \(w\)-multiples generate modulo \(U\), by (50). Subtracting such a combination and dividing the remainder by \(U\) reduces degree by two, so induction from the bottom degree proves generation. ◻

Proof. First use \(H=R\), inducting on \(p\). By (50), the earlier interval module is generated over the ambient classes \(w,e\) by chosen extensions of its bottoms \(x_j\), \(m_0\le j<p\). The connecting homomorphism sends such a bottom into the new crossing in degree \(a_j+1<a_p\), where the crossing is zero. It is linear over \(w,e\), hence vanishes on the whole earlier module. This proves (54) for \(R\), and also justifies the inductive generation assertion.

For \(G\), Lemma 28 gives the same induction with the bottom generators and, at even stages \(j\), the chosen top generators of degree \(a_j+n\). Bottoms again have zero connecting image. For a top, the only potentially relevant degree is \(a_j+n+1\). If \(d>2\), then \(n+1=d+2<2d\), so this degree is less than \(a_p\). If \(d=2\), equality can occur only when \(j=p-1\) is even. In this case \(p\) is odd, and the target is its one-dimensional bottom \(G\)-crossing. Restriction to the corresponding reflection bottom is injective: both bottom Thom classes restrict to the nonzero bottom class. The reflection connecting map has already been proved zero. Naturality therefore kills the \(G\)-connecting map in this last case as well. This proves (54) in every dimension \(n\ge3\).

We pass to the global pair degree by degree. Every singular chain in the Borel construction is finite and has bounded round energy, since energy descends to a continuous invariant function. Thus \[H_j^H(\Lambda,Z_{m-1}) =\varinjlim_{p\ge m}H_j^H(Z_p,Z_{m-1}).\] Universal coefficients over the field \(\mathbb F_2\) identify its dual with the inverse limit of the degree-\(j\) cohomology groups. The local crossing groups vanish below \(a_p\), which tends to infinity. Sequence (54) consequently stabilizes in every degree. For \(j<a_{m+1}\), no subsequent crossing contributes, proving (55). All limits in this argument are taken in a fixed degree; the total cohomology module is the graded direct sum. ◻

Proof. The recurrence (52) implies \[ A_i^2+A_{i-1}A_{i+1}=U^i\qquad(i\ge1). \tag{59}\] Indeed, substitution of the recurrence makes the left side for \(i\ge2\) equal to \(U\) times the left side for \(i-1\); the initial value is \(w^2+A_2=U\).

At an odd crossing, (53) gives \[UA_dx_m=(A_{d+2}+wA_{d+1})x_m=0.\] The class \(A_dx_m\) is nonzero: on restriction to \(R\), it becomes \(w^dx_m\), which is nonzero in (50). The relevant part of the standard-plane Gysin sequence is \[H_R^{a_m+d+1}(Z_m,Z_{m-1}) \xrightarrow{\int} H_G^{a_m+d}(Z_m,Z_{m-1}) \xrightarrow{U} H_G^{a_m+d+2}(Z_m,Z_{m-1}).\] Its source is one-dimensional, spanned by \(ex_m\). Since its image contains the nonzero class \(A_dx_m\), exactness over \(\mathbb F_2\) forces \[ \int ex_m=A_dx_m. \tag{60}\]

Apply the same integration map to \(e\,\mathop{\mathrm{res}}_R\widetilde\zeta_m\) in the pair \((\Lambda,Z_{m-1})\). Naturality and (60) identify its restriction to the crossing with \(A_dx_m\). Both sides have offset \(d<2d\), so (55) gives \[\int\bigl(e\,\mathop{\mathrm{res}}_R\widetilde\zeta_m\bigr) =A_d\widetilde\zeta_m.\] The next map in Gysin is multiplication by \(U\), hence \[ UA_d\widetilde\zeta_m=0. \tag{61}\] Also \(A_{d+1}\widetilde\zeta_m=0\), because its local value is zero and \(d+1<2d\). Multiplying (59) at \(i=d\) by \(U\widetilde\zeta_m\) therefore gives \[U^{d+1}\widetilde\zeta_m =UA_d^2\widetilde\zeta_m+ UA_{d-1}A_{d+1}\widetilde\zeta_m=0.\] Its image in \(H_G^*(\Lambda,M)\) is \(U^n\zeta_m=0\).

For \(n=3\), one has \(d=2\), \(A_2=w^2+U\), and \(A_3=w^3\). The two intermediate global relations are exactly \[w^3\widetilde\zeta_m=0,\qquad U(w^2+U)\widetilde\zeta_m=0.\] Their proof used uniqueness only at offsets two and three, both strictly less than the next-bottom gap four. It did not extend an offset-four relation across that gap. ◻

Proof. We first prove the ordinary comparison away from fixed-width bands. Choose a regular endpoint \(a_+\in(a+D,a+D+1)\). A local group at an event \(m\ell_i\in[a,a_+]\) can contribute in degree \(j\) or its neighboring degrees only if \[|j-I_{i,m}|\le C_{\rm loc}.\] By (70), this implies \[ |j-\beta_i a| \le C_{\rm loc}+n+\beta_i(D+1). \tag{72}\] The right-hand side is bounded uniformly over the finite prime list. Thus, outside fixed-width bands about the \(\beta_i a\), the ordinary map \[H_j(X_a,M)\longrightarrow H_j(X_{a_+},M)\] is injective, by the crossing exact sequences. A class in its source that is zero globally is already zero in \(X_{a+D}\) by Theorem 24. Injectivity therefore makes the map \(H_j(X_a,M)\to H_j(\Lambda,M)\) injective. Field duality gives surjectivity of ordinary cohomology restriction.

In degrees above a fixed low threshold, the ordinary groups relative to \(M\) identify with the absolute groups, by the evaluation retraction and the cohomology of the constant sphere. In these degrees Proposition 18 gives global rank at most one. Its resonance estimate implies that every nonzero global class restricts nontrivially to \(X_a\) when \(j<\alpha a-c_0\), and restricts to zero when \(j>\alpha a+c_0\), for a fixed \(c_0\). Together with the preceding surjectivity, this proves \[ \begin{cases} H^j(\Lambda,M)\longrightarrow H^j(X_a,M) \text{ is an isomorphism},&j<\alpha a-c_0,\\ H^j(X_a,M)=0,&j>\alpha a+c_0, \end{cases} \tag{73}\] provided \(j\) also lies outside the fixed low-degree interval and the bands in (72). When the global group is zero, surjectivity gives the first assertion as an isomorphism between zero groups. Increase \(c_0\) to absorb all these restrictions.

Consider a present gap and put \(P_a^j=H_R^j(\Lambda,X_a)\) for this proof. Ordinary triple exactness and (73) give \(H^j(\Lambda,X_a)=0\) whenever the needed two neighboring ordinary restrictions are isomorphisms. The sign-line Gysin sequence of Proposition 25 consequently makes \(w:P_a^j\to P_a^{j+1}\) an isomorphism in the corresponding interior. After allowing \(n\) further degrees below, every element of \(P_a^j\) is divisible by \(w^n\) there. Its image in \(B_\infty^j\) is zero, since \(w^nB_\infty=0\). Applying this at \(j\) and \(j+1\) to the triple \((\Lambda,X_a,M)\) gives a natural short exact sequence \[ 0\longrightarrow B_\infty^j\longrightarrow B_a^j \longrightarrow P_a^{j+1}\longrightarrow0. \tag{74}\] The quotient identification commutes with \(w\). It gives both the claimed injection and the isomorphisms on \(Q_a\) in a present gap.

In an absent gap, (73) and sign Gysin make \(w:B_a^j\to B_a^{j+1}\) an isomorphism in the interior. Allowing \(n\) further degrees above makes \(w^n\) injective there. The image of \(B_\infty^j\) is killed by that power, so it is zero. Hence \(Q_a^j=B_a^j\) and the claimed isomorphisms also hold in an absent gap.

Choose the margin \(c\) larger than \(c_0+2n+6\) and the fixed low-degree threshold. Then all neighboring degrees and all powers just used lie in the required untrimmed ordinary ranges whenever the statements of the lemma apply. This is one fixed choice for all cuts and gaps. Finally, any reflection local contribution below cut \(a\) has \[j\le I_{i,m}+C_{\rm loc} \le \beta_i a+n+C_{\rm loc}\qquad(m\ell_i<a).\] The relative groups start at zero below the first event. The local crossing sequences therefore give \(B_a^j=0\) above \(s_Na+c\), after one final increase of \(c\). ◻

Proof. Fix consecutive slopes \(s<s'\) and write \[A=sa,\qquad r=(s'-s)a.\] There is a fixed \(\delta>0\) such that \(r\ge\delta a\) for every such gap, since the list of distinct slopes is finite. Suppose that some \(Q_a^j\) in this gap is nonzero. The isomorphisms in Lemma 34 make it nonzero throughout the integer gap. For large \(a\), choose \[l_0=\lceil A+0.10r\rceil\] and a class \(y_{l_0}\in B_a^{l_0}\) with nonzero image in \(Q_a^{l_0}\). Set \(y_l=w^{l-l_0}y_{l_0}\) through \(l\le\lfloor A+0.90r\rfloor\). Every resulting quotient class is nonzero. In particular, with \[\mathcal L=\mathbb Z\cap[A+0.15r,A+0.20r], \qquad q=\lfloor0.50r\rfloor,\] the classes \(y_l\), \(l\in\mathcal L\), are independent in \(\bigoplus_j B_a^j\), as are their powers \(w^q y_l\). Indeed they occupy distinct root degrees, and their quotient classes remain nonzero between degrees \(A+0.65r-O(1)\) and \(A+0.70r\). Figure 2 displays the intervals used below.

At the ambient cut \(2a\), let \(E_J\) and \(P_k\) be the actual normalized restriction image and root-degree projection from Theorem 32. Recall its nonnegative terms \[D_{k,J}=\dim P_k\Bigl(\bigoplus_l B_a^l\Bigr)-\dim P_kE_J, \qquad H_{k,J}=\dim(E_J\cap\ker P_k).\] In particular that theorem gives \[ \sum_{k\ge1}\bigl(D_{k,2k}+H_{k,2k}\bigr)\le C(1+a). \tag{75}\] Average \(D_{k,2k}\) over the integers \(k\in[A+0.30r,A+0.35r]\). There are a fixed positive multiple of \(r\) such integers for large \(r\), so some \(k\) satisfies \(D_{k,2k}\le C_1\), independently of \(a\), because \(r\ge\delta a\). The space \[W_0=\operatorname{span}\{y_l:l\in\mathcal L\}\] lies in the root degrees at most \(k\). Thus \[\dim(W_0\cap P_kE_{2k}) \ge\dim W_0-D_{k,2k}\ge c_1r\] for a fixed \(c_1>0\) and sufficiently large \(r\).

Choose a basis \(z_1,\ldots,z_\nu\) of this intersection and actual ambient lifts \[u_\mu\in H_K^{2k}(X_{2a},M).\] The normalized fixed restriction of \(u_\mu\) has the form \(z_\mu+c_\mu\), where every root label in \(c_\mu\) is greater than \(k\). Multiply the actual classes by \(w^q\). The selected terms \(w^q z_\mu\) remain independent, since \(w^q\) is injective on \(W_0\). Their labels are at most \(A+0.70r\), whereas the other terms have labels greater than \[k+q\ge A+0.80r-1\] or vanish. The latter terms cannot cancel any of the selected ones. Moreover all labels of these normalized images exceed \(A+0.64r\) for sufficiently large \(r\).

For any integer \(k'\in[A+0.61r,A+0.64r]\), there is enough actual degree to multiply further by a nonnegative power of \(v\), since \[ 2k+q\le2A+1.20r<2A+1.22r\le2k'. \tag{76}\]

The classes \[v^{\,2k'-(2k+q)}w^q u_\mu \in H_K^{2k'}(X_{2a},M)\] have unchanged normalized fixed images: multiplication by \(v\) changes the actual degree and leaves the root coefficients unchanged. Multiplication by \(w\) raises each root label by one, as used above. Thus the images are independent and lie entirely above root cutoff \(k'\), proving \[H_{k',2k'}\ge c_1r \quad\text{for every integer }k'\in[A+0.61r,A+0.64r].\] There are at least \(c_2r\) such cutoffs for large \(r\). The left side of (75) is therefore at least \(c_1c_2r^2\ge c_1c_2\delta^2a^2\), contradicting its linear upper bound. This proves the assertion for the chosen gap. The finite number of gaps allows one common lower threshold for \(a\). ◻

Proof of Theorem 33. First let \(a\) be large with \(2a\) regular. By Lemma 35, transient classes can occur only within distance \(c\) of one of the finitely many degrees \(s_i a\). Lemma 34 excludes degrees above the last such band. Thus only a bounded number of integer degrees can contribute. We now bound the dimension in each of them uniformly.

For high degrees, the reflection local ranks of Theorem 10 give \[ \dim B_a^j \le C\sum_i\#\{m\ge1:m\ell_i<a,\ |j-m\Delta_i|\le C_{\rm loc}+n\}. \tag{77}\] This follows by summing the crossing exact-sequence bounds, starting with the zero relative group below the first event. If \(\Delta_i>0\), the number in braces is bounded by \(1+2(C_{\rm loc}+n)/\Delta_i\), up to an integer rounding constant. If \(\Delta_i=0\), it is zero once \(j>C_{\rm loc}+n\). The finite prime list therefore gives a uniform bound on \(\dim B_a^j\) in every sufficiently high degree.

The fixed low-degree interval needs a different argument, which also accounts for all zero-mean-index iterates. Choose a regular \(a_+\in(a+D,a+D+1)\). By ordinary death, \[ \ker\bigl(H_j(X_a,M)\to H_j(\Lambda,M)\bigr) =\ker\bigl(H_j(X_a,M)\to H_j(X_{a_+},M)\bigr). \tag{78}\] The kernel on the right is bounded in dimension by the sum of local ordinary ranks in this interval of cut values of width at most \(D+1\). The number of its events is at most \[\sum_i\left(2+\frac{D+1}{\ell_i}\right),\] independently of \(a\). Uniform local ranks bound the kernel, and the global-image rank is bounded by the fixed global degree rank. Consequently \(\dim H^j(X_a,M)\) is bounded uniformly in \(a\) for each fixed low degree \(j\).

The sign-line Gysin sequence gives \[\dim B_a^j\le\dim B_a^{j-1}+\dim H^j(X_a,M), \qquad B_a^{-1}=0.\] Induction, including degree zero, bounds \(B_a^j\) uniformly throughout any fixed low-degree interval. Combining this with (77), every degree in the finitely many remaining bands has uniformly bounded dimension. Since the number of such degrees is also bounded, their sum bounds \(T(a)\) as asserted.

Cuts \(a\) in a bounded interval have only finitely many crossing stages, so they satisfy a common bound as well. Finally, if \(a\) is regular but \(2a\) is not, choose a nearby \(a'\) in the same no-critical interval such that \(2a'\) is regular. The critical values and their halves are locally finite. The no-critical restriction isomorphism between the root sublevels is natural with respect to global restriction, and hence preserves both \(B_a\) and its global image. In particular \(T(a)=T(a')\). This proves (71) for every positive regular cut. ◻