Proof. The good-cutoff construction and the integrated Bochner estimate give (i) and (7); see (Gigli and Tamanini 2021, Appendix A, Lemma A.2 and Equation (A.13)). For the local version of the cutoff construction one uses finitely many small good cutoffs inside \(U\), and a smooth truncation of their sum. This keeps the Laplacian bounded and Sobolev by the test algebra and chain rules.

Let \(P_t\) denote the heat semigroup. For \(h\in L^2\cap L^\infty\), positive-time regularization gives \(P_th\in\operatorname{Test}^\infty\) (Gigli and Tamanini 2021, Appendix A, Equation (A.7)). Spectral calculus and density of bounded \(L^2\) data show that these regularizations form a core. For each such regularization \(h\), \[\Delta(\chi_Rh) =\chi_R\Delta h+h\Delta\chi_R +2\langle\nabla\chi_R,\nabla h\rangle.\] The uniform cutoff bounds and the \(L^2\) tails imply convergence to \(\Delta h\) in \(L^2\), as well as convergence of the functions and gradients. This proves the compact-core assertion.

For (iii), choose nested cutoffs \(\chi,\zeta\) such that \(\chi=1\) near the region in question and \(\zeta=1\) near \(\mathop{\mathrm{supp}}\chi\). Put \[g_t=\chi P_t(\zeta g).\] The function \(\zeta g\) lies in \(\operatorname{Test}\). The product rule, spectral continuity and the fixed cutoff bounds give convergence of \(g_t\) to \(\chi\zeta g\) in the Laplacian graph norm. The Bakry gradient estimate gives \[\|\nabla g_t\|_\infty \le \|\chi\|_\infty e^{-Kt}\|\nabla(\zeta g)\|_\infty +\|\nabla\chi\|_\infty\|\zeta g\|_\infty.\] Apply (7) to \(g_t-\chi\zeta g\) to obtain strong Hessian convergence. Since \(\chi\zeta g=g\) near the region, (iii) follows.

For (iv), use \(\xi_t=\chi P_t\xi\), with an outer cutoff \(\chi=1\) near \(\mathop{\mathrm{supp}}\xi\). The heat semigroup converges strongly in \(W^{1,2}\); the uniform Lipschitz estimate and the small-time heat-kernel moment bound give uniform convergence on the common compact support. The cutoff product gives the same conclusions for \(\xi_t\), and positivity is preserved. ◻

Proof. Expand \(\nabla_X(fW)=X(f)W+f\nabla_XW\). Every coefficient of \(f\) is a product of two \(L^2\) factors and bounded factors. The coefficients of \(\nabla f\) lie in \(L^2\). Uniform convergence of \(f_j\) pairs with the fixed \(L^1\) coefficient, and weak \(L^2\) convergence of their gradients pairs with the fixed \(L^2\) coefficient. This proves the weight-continuity assertions. Approximation then extends the identities.

For the field assertion, \[\nabla_{X_j}Y_j-\nabla_XY =(\nabla Y_j-\nabla Y)(X_j)+\nabla Y(X_j-X).\] The first term tends to zero in \(L^2\) by uniform boundedness of \(X_j\); the second does so by dominated convergence. The brackets therefore converge in \(L^2\) as well. Products of the resulting strongly convergent \(L^2\) quantities converge in \(L^1\). Any remaining uniformly bounded field factor is handled by almost-everywhere convergence and domination. This includes the last term of [eq:curvature-definition]. ◻

Proof. The fields in (14) are bounded, have \(L^2\) covariant derivative and bounded divergence, and have support in a fixed compact set. The Sobolev flow theorem applies (Ambrosio and Trevisan 2014, Theorems 4.3, 5.4, 6.7 and 8.3; Section 9.6). Applying it also to the time-reversed field, and using uniqueness, gives the inverse flow. The divergence bound is \(O(h)\), so both compression constants are \(1+O(h)\). The speed is bounded by \(h\|A\|_\infty+h^2\|S_A\|_\infty\), which gives the displacement and support assertions. Properness is used here to replace bounded closed neighborhoods by compact sets.

To check (15), the pushforward of \(m\) under the flow has density between \(1-O(h)\) and \(1+O(h)\), by the forward and inverse compression bounds. Replacing \(f(z)\) by \(f(F^h_{r,A}(z))\) in the pushforward changes the density by at most \(O(h)\operatorname{Lip}(f)\), again by compression and the displacement bound. Multiplication of the pushed-forward reference density by \(f\) changes it from \(f\) by at most \(O(h)\|f\|_\infty\). This proves (15) without requiring a spatially Lipschitz velocity field.

The continuity equation, tested with \(q\in W^{1,2}\), reads \[ \int q(\rho^h_{r,A}-f) =\int_0^r\int \langle\nabla q,hA+h^2tS_A\rangle\rho^h_{t,A}\,\mathrm dt. \tag{19}\] All pairings are finite because the densities are bounded and have bounded support. Divide by \(h\) and use (15). The right-hand side converges, uniformly in \(r\), to \(r\int f\langle\nabla q,A\rangle=r\int qL_A\). The uniform \(L^\infty\) bound on the left quotient, followed by density of compact Lipschitz functions in \(L^1\) on bounded sets, proves (16), also in space-time. Use (19) once more, with \(r=1\), to obtain \[\begin{align*} \int q(\rho^h_A-f) ={}&h\int f\langle\nabla q,A\rangle +h^2\int_0^1\int \left\langle\nabla q, \frac{\rho^h_{r,A}-f}{h}A+r\rho^h_{r,A}S_A\right\rangle\,\mathrm dr. \end{align*}\] The functions paired with the normalized density difference are fixed and integrable on its support. Equation (16) therefore gives (17).

For completeness, if \(v\in L^1\) on a common bounded neighborhood, then \[v\circ F^h_{r,A}\longrightarrow v \quad\text{in }L^1(\,\mathrm dr\,fm).\] Indeed this holds for continuous functions by the displacement bound, and follows for \(L^1\) functions by approximation and the uniform compression bound. Differentiation of a Lipschitz function along the flow gives \[\frac{g(F^h_{1,A}(z))-g(z)}h =\int_0^1 dg(A)(F^h_{r,A}(z))\,\mathrm dr +h\int_0^1r\,dg(S_A)(F^h_{r,A}(z))\,\mathrm dr.\] The preceding composition convergence proves (18). A cutoff may be inserted when \(g\) is only locally Lipschitz. ◻

Proof. At \(m\)-almost every base point \(z\), the Alexandrov tangent cone is Euclidean. Choose a geodesic from \(z\) to each endpoint \(F^h_{1,A}(z)\). For the finite weights \(p_A\), the radial vectors \(v_A\in T_zM\) satisfy \[\sum_{A,B}p_Ap_B|v_A||v_B|\cos\angle(v_A,v_B) =\left|\sum_Ap_Av_A\right|^2\geq0.\] Actual angles are at least the corresponding \(\kappa\)-comparison angles. Consequently, \[ \mathbb E\!\left[ a^h_Aa^h_B\cos\widetilde\angle_\kappa (a^h_A,a^h_B,e^h_{AB})\right]\geq0. \tag{22}\] Zero radial lengths contribute zero. All sides are \(O(h)\), so the model triangles are defined for sufficiently small \(h\), including when \(\kappa>0\).

For radial side lengths \(a,b\) and opposite side length \(e\), put \(z_0=(a^2+b^2-e^2)/2\). The model cosine law gives, uniformly for such small triangles, \[ ab\cos\widetilde\angle_\kappa =z_0-\frac\kappa6(a^2b^2-z_0^2)+O(h^6). \tag{23}\] For \(\kappa=0\) this is exact without the correction. For \(\kappa\ne0\), substitute the Taylor series of the model sine and cosine into their cosine law. After multiplication by \(ab\), the denominators are the nonvanishing analytic factors \(\mathfrak s_\kappa(a)/a\) and \(\mathfrak s_\kappa(b)/b\), where \(\mathfrak s_\kappa''+\kappa\mathfrak s_\kappa=0\), \(\mathfrak s_\kappa(0)=0\), and \(\mathfrak s_\kappa'(0)=1\). This also proves uniformity at degenerate triangles and zero sides.

We first identify the leading distances. By integration of the speed and the composition convergence used in Lemma 6, \[\left(\frac{a^h_A}{h}-|A|\right)_+ \longrightarrow0\quad\text{in }L^1(fm).\] There is a countable collection of scalar \(1\)-Lipschitz functions whose differentials norm every tangent vector almost everywhere; one may use the norming family in (Ambrosio and Trevisan 2014, Lemma 9.2), with local cutoffs when needed. For each member \(g\), (18) and its difference for two flows give the lower bounds \[a^h_A/h\geq |g(F^h_{1,A})-g|/h,\qquad e^h_{AB}/h\geq |g(F^h_{1,A})-g(F^h_{1,B})|/h.\] First taking a finite part of the norming family, and then increasing it, shows that the negative parts of \(a^h_A/h-|A|\) and \(e^h_{AB}/h-|A-B|\) tend to zero in measure. The radial convergence follows. All scaled distances are uniformly bounded, so this convergence also holds in every finite \(L^p(fm)\).

Equations (22)–(23), initially retaining only their quadratic terms, imply \[\mathbb E\int f(e^h_{AB}/h)^2 \leq 2\mathbb E\int f(a^h_A/h)^2+O(h^2) \longrightarrow 2\mathbb E\int f|A|^2.\] By (13), the last quantity is \(\mathbb E\int f|A-B|^2\). The pair-distance lower bounds already proved imply convergence of the negative errors in \(L^1\). The displayed upper bound therefore forces the positive errors to converge in \(L^1\) as well. Since the family is finite, this proves the pair convergence in (20) for every pair of positive probability.

We may now retain the quartic term in (23). The convergence just proved gives \[h^{-4}\left((a^h_Aa^h_B)^2 -\left(\frac{(a^h_A)^2+(a^h_B)^2-(e^h_{AB})^2}{2}\right)^2\right) \longrightarrow |A\wedge B|^2\] in \(L^1(fm)\). Thus \[ \liminf_{h\downarrow0}\frac1{h^4} \left(\mathbb E\int f(a^h_A)^2 -\frac12\mathbb E\int f(e^h_{AB})^2\right) \geq\frac\kappa6\int f\,\mathbb E|A\wedge B|^2. \tag{24}\] The radial energy dominates half the squared radial displacement, and \((F^h_{1,A},F^h_{1,B})_\#(fm)\) is a coupling of the endpoint measures. Hence \[2J^h_{0A}\geq\int f(a^h_A)^2, \qquad J^h_{AB}\leq\frac12\int f(e^h_{AB})^2.\] Their combination with (24) proves (21). ◻

Proof. Restriction of the plan. The endpoints have uniformly bounded densities and support in a common ball. All connecting geodesics lie in a larger fixed ball, which is compact. The standard interpolation bound gives the uniform density estimate. We record a sharper consequence needed below. Weaken the Ricci parameter to a fixed \(K'<0\). The zero-mass case is immediate; otherwise let \(\pi^h\) be the optimal dynamical plan and normalize it to a probability \(\pi\), with marginals \(r_t m\). By (Gigli et al. 2016, Theorem 1.1), \(\pi\) has a Borel conull set on which every evaluation \(e_t\), \(t<1\), is injective. Both endpoints are absolutely continuous. Applying the same theorem to the reversed plan, pulling its conull set back under time reversal, and intersecting with the first set gives a Borel conull \(\Gamma\) on which \(e_t\) is injective for every \(t\in[0,1]\).

Fix \(s\in(0,1)\), and let \(H\geq0\) be bounded Borel on \(\Gamma\), with \(Z=\int H\,\mathrm d\pi>0\). The probability \(\pi_H=(H/Z)\pi\) is optimal for its endpoint marginals: after scaling \(H\leq1\), a cheaper transport for \(H\pi\) could be spliced with \(\pi-H\pi\), contradicting optimality. Its endpoint densities are bounded and absolutely continuous. The integrated \(\mathrm{CD}(K',n)\) inequality supplies an optimal dynamical plan for these marginals, and uniqueness forces that plan to be \(\pi_H\). The measurable inverses of \(e_t|_\Gamma\) give, by Lusin–Souslin, \[h_t=H\circ(e_t|_\Gamma)^{-1},\qquad r_t^H=(h_t/Z)r_t,\qquad t\in\{0,s,1\},\] where \(h_t\) is extended by zero off \(e_t\Gamma\). Write \(\beta_t(\ell)=\tau_{K',n}^{(t)}(\ell)\) for the unreduced distortion coefficient and set \[F_s(\gamma)=r_s(\gamma_s)^{-1/n} -\beta_{1-s}(\ell)r_0(\gamma_0)^{-1/n} -\beta_s(\ell)r_1(\gamma_1)^{-1/n}.\] The integrated inequality for \(\pi_H\), after cancellation of \(Z^{1/n-1}\), is exactly \[\int H^{1-1/n}F_s\,\mathrm d\pi\geq0.\] There are no undefined inverse powers in this calculation: zero or infinite density values lie over marginal-null sets at the three fixed times, and on the common finite-measure compact support \(Q\), \[\int r_t(\gamma_t)^{-1/n}\,\mathrm d\pi =\int_Q r_t^{1-1/n}\,\mathrm dm<\infty.\] Taking \(H\) to be indicators of Borel sets proves \(F_s\geq0\) \(\pi\)-almost everywhere. Restoring the common mass preserves this homogeneous inequality. Its exceptional set may depend on \(s\); every fixed-time estimate below has the same constants, which suffices for the supremum over \(s\). Jointly Borel density versions and Fubini give the corresponding \(\,\mathrm ds\otimes\pi\)-almost-everywhere statements for time integrals.

The density norm bound. On the bounded range of possible geodesic lengths \(\ell\), the pointwise CD distortion coefficients, divided by their zero-curvature coefficients, have a positive lower bound and are at least \(1-C\ell^2\), uniformly in \(s\). The negative power mean is at most the arithmetic mean. Since the endpoint densities are bounded, the pointwise CD inequality therefore implies, along the optimal dynamical plan \(\pi^h\), \[ \sigma^h_s(\gamma_s) \leq (1-s)\rho^h_A(\gamma_0) +s\rho^h_B(\gamma_1)+C\ell(\gamma)^2. \tag{27}\] To see that the error remains valid when \(\ell\) is not small, let \(a(\ell)\) be a common positive lower bound for the two distortion ratios. The CD inequality first gives the negative power mean multiplied by \(a(\ell)^{-n}\). On the fixed length interval, \(a(\ell)^{-n}-1\leq C\ell^2\). This proves (27), including such lengths.

Integration against \(\pi^h\) gives \[\int(\sigma^h_s)^2 \leq(1-s)\int(\rho^h_A)^2+s\int(\rho^h_B)^2 +C\int\ell^2\,\mathrm d\pi^h.\] Semiconcavity of \(f\) gives \[-2\int f\sigma^h_s \leq-2(1-s)\int f\rho^h_A-2s\int f\rho^h_B +C\int\ell^2\,\mathrm d\pi^h.\] Here \(\int\ell^2\,\mathrm d\pi^h=2J^h_{AB}=O(h^2)\), by the configuration coupling. Adding the last two inequalities and \(\int f^2\) proves the norm bound in (25), using (15) for the endpoints.

First variations of velocity and density. The same coupling and (20) imply \[\limsup_{h\downarrow0}\int_0^1\int \sigma^h_s|v^h_s/h|^2\,\mathrm ds\leq\int f|D|^2.\] The continuity equation tested with \(u_B-u_A\), and then (17), gives \[\int_0^1\int\sigma^h_s\langle v^h_s/h,D\rangle\,\mathrm ds =\frac1h\int(u_B-u_A)(\rho^h_B-\rho^h_A) \longrightarrow\int f|D|^2.\] The norm estimate for the densities also gives \(\int_0^1\int\sigma^h_s|D|^2\,\mathrm ds\to\int f|D|^2\). Expanding the square proves (26).

Finally, for any fixed \(q\in W^{1,2}(M)\), \[\int q\frac{\sigma^h_s-f}{h} =\int q\frac{\rho^h_A-f}{h} +\int_0^s\int\sigma^h_t\langle\nabla q,v^h_t/h\rangle\,\mathrm dt.\] The first term converges to \(\int qL_A\). In the second, use (26) and Cauchy–Schwarz to replace \(v^h_t/h\) by \(D\); then use the density norm bound to replace \(\sigma^h_t\) by \(f\). The error tends to zero uniformly in \(s\). The limit is \(s\int f\langle\nabla q,D\rangle =s\int qL_D\). Boundedness of the normalized density differences in \(L^2(\,\mathrm ds\,\mathrm dm)\), and density of finite sums of separated space-time test functions, prove the weak convergence in (25). ◻

Proof. The energy identity. For a bounded-compression solution \(\sigma_s\) of the continuity equation with velocity \(v_s\), use \[p_s=h(u_B-u_A)+h^2q_s.\] Its strong \(W^{1,2}\) dependence on time and the continuity equation give the exact energy identity \[\begin{align*} \frac12\int_0^1\int\sigma_s|v_s|^2\,\mathrm ds ={}&\left[\int p_s\sigma_s\right]_{s=0}^{s=1} -\int_0^1\int\sigma_s \left(\partial_sp_s+\frac12|\nabla p_s|^2\right)\,\mathrm ds \\ &\quad+\frac12\int_0^1\int\sigma_s|v_s-\nabla p_s|^2\,\mathrm ds. \tag{31}\end{align*}\] One may verify the time differentiation directly by writing \(p_s\) as a sum of fixed Sobolev functions with affine time coefficients. The endpoint terms are finite by the bounded densities and bounded support; the remaining terms are finite by the kinetic-energy and \(W^{1,2}\) bounds. Thus no boundedness assumption on \(q_{AB}\) is needed.

The dual expansion. The term \(-s|D|^2/2\) in \(q_s\) cancels the quadratic part of the Hamilton–Jacobi residual. Thus the residual begins at order \(h^3\), and the first density variation supplies its contribution at order \(h^4\). Explicitly, \[ \partial_sp_s+\frac12|\nabla p_s|^2 =h^3\langle D,Q_s\rangle+\frac{h^4}{2}|Q_s|^2. \tag{32}\] For the optimal interpolation, Lemma 8 gives \[\begin{align*} \int_0^1\int\sigma^h_s\langle D,Q_s\rangle\,\mathrm ds ={}&\int_0^1\int f\langle D,Q_s\rangle\,\mathrm ds +h\int_0^1\int L_{C_s}\langle D,Q_s\rangle\,\mathrm ds+o(h),\\ \int_0^1\int\sigma^h_s|Q_s|^2\,\mathrm ds ={}&\int_0^1\int f|Q_s|^2\,\mathrm ds+o(1). \end{align*}\] The first limit uses weak \(L^2\) convergence against the fixed \(L^2\) function \(\langle D,Q_s\rangle\). The second uses convergence in measure and the uniform density bound, by truncation of the fixed integrable function \(|Q_s|^2\). Outside the common density support both statements are vacuous.

Apply (17) to the two fixed endpoint functions \(q_0,q_1\). Leave the endpoint contribution involving \(u_B-u_A\) unexpanded. The terms of order four are then precisely (29). For the cubic term, integration by parts gives \[\int(q_1L_B-q_0L_A) =\int f(\langle Q_1,B\rangle-\langle Q_0,A\rangle).\] Using \(Q_1=Q_0-T_D\) and \(\int_0^1Q_s\,\mathrm ds=Q_0-T_D/2\), its difference from the cubic residual is \[-\int f\left\langle B-\frac D2,T_D\right\rangle =-\int_0^1\int f\langle C_s,T_D\rangle\,\mathrm ds.\] Also \(q_1-q_0=-|D|^2/2\). Dropping the nonnegative last term in (31) proves (28).

The radial defect. For the radial flow use its actual velocity from (14), and use Lemma 6 in place of Lemma 8. Here \(D=A\), \(C_s=sA\), and \(Q_s=-s\nabla_AA\). The last term of (31), divided by \(h^4\), is \[\frac12\int_0^1s^2\int \rho^h_{s,A}|S_A+\nabla_AA|^2\,\mathrm ds \longrightarrow\frac16\int f|S_A+\nabla_AA|^2.\] The integrand in the spatial limit is fixed and integrable, and (15) justifies the limit. This proves the radial assertion. ◻

Proof. Integration by parts. We first justify the identity \[ \int\langle L_DC-L_CD,\nabla q\rangle =\int f\langle\nabla_DC-\nabla_CD,\nabla q\rangle, \qquad q\in W^{1,2}(M). \tag{37}\] For test \(q\), integration by parts in its two terms gives \[\begin{align*} \int L_D\langle C,\nabla q\rangle &=\int f\bigl(\langle\nabla_DC,\nabla q\rangle +\operatorname{Hess}q(D,C)\bigr),\\ \int L_C\langle D,\nabla q\rangle &=\int f\bigl(\langle\nabla_CD,\nabla q\rangle +\operatorname{Hess}q(C,D)\bigr). \end{align*}\] The Hessian terms cancel. The vector fields paired with \(\nabla q\) on both sides of (37) are in \(L^2\) and have bounded support. Density of test functions in \(W^{1,2}\) therefore proves the identity for the stated class, without assigning a Hessian to \(q\).

Square completion. The maps \(s\mapsto F_{C_s}\) and \(s\mapsto Q_s\) are polynomials with values in \(L^2\). They have derivatives \[F_{C_s}'=L_DC_s+L_{C_s}D-f\nabla_DC_s-f\nabla_{C_s}D, \qquad Q_s'=-\nabla_DD.\] Express the endpoint difference in \(K_{AB}\) as the integral of the derivative of \(\langle F_{C_s},Q_s\rangle/2\), and subtract the other terms in (29). Applying (37) gives \[K_{AB}=\frac12\int_0^1\int \left[-\langle F_{C_s},\nabla_DD\rangle -2f\langle\nabla_{C_s}D,Q_s\rangle-f|Q_s|^2\right]\,\mathrm ds.\] Completion of the square proves (34).

Averaging. For the averaging, set \[T=\mathbb E P(A,A;A,A),\quad S=\mathbb E P(A,A;B,B),\quad H=\mathbb E P(A,B;A,B).\] Independence and sign symmetry kill every term with an odd number of occurrences of either random field. Exchangeability identifies the two pure and the two unmixed terms. Expansion in the four slots therefore gives \[\mathbb E P(C_s,C_s;D,D) =((1-s)^2+s^2)(T+S)-4s(1-s)H.\] Both \(\int_0^1((1-s)^2+s^2)\,\mathrm ds\) and \(\int_0^14s(1-s)\,\mathrm ds\) equal \(2/3\). The contribution of the \(P\) term to \(\mathbb E K_{AB}\) is consequently \((T+S-H)/3\). In the radial case \(C_s=sA\), \(D=A\), \(Q_s=-s\nabla_AA\), so the square vanishes and \(K_{0A}=P(A,A;A,A)/6\). Finally, \[Q_s+\nabla_{C_s}D=\nabla q_{AB}+\nabla_AD.\] These facts prove (35).

The curvature contraction. To identify the curvature convention explicitly, write \[a=\nabla_AA,\qquad b=\nabla_BB,\qquad U=\nabla_BA,\qquad V=\nabla_AB.\] Separate symmetric polarization of (33) gives \[\begin{align*} P(A,B;A,B)=\int\biggl[ &-\frac14\langle L_AB+L_BA,U+V\rangle \\ &+f\left(\frac14|U|^2+\frac14|V|^2 +\langle U,V\rangle+\frac12\langle a,b\rangle\right) \biggr], \tag{38}\end{align*}\] whereas \[P(A,A;B,B)=\int [-L_A\langle A,b\rangle+f\langle a,b\rangle+f|V|^2].\] Exchangeability gives \(\mathbb E|U|^2=\mathbb E|V|^2\) and \[\frac12\mathbb E\int\langle L_AB+L_BA,U+V\rangle =\mathbb E\int L_B\langle A,U+V\rangle.\] Hessian symmetry for the gradient \(A\) gives \(\langle A,U\rangle=\langle B,a\rangle\); swapping the random choices then yields \(\mathbb E\int L_B\langle A,U\rangle =\mathbb E\int L_A\langle A,b\rangle\). Thus \[\begin{align*} 2\mathbb E[P(A,B;A,B)-P(A,A;B,B)] =\mathbb E\int\bigl[ &L_A\langle A,b\rangle-L_B\langle A,V\rangle \\ &+f(-\langle a,b\rangle-|V|^2+2\langle U,V\rangle) \bigr]. \tag{39}\end{align*}\] The first two terms of the specified integrated curvature formula, with \((X,Y,Z,W)=(A,B,B,A)\), combine to \[L_A\langle A,b\rangle-f\langle a,b\rangle.\] Its next two terms combine to \[-L_B\langle A,V\rangle+f\langle U,V\rangle.\] Finally \([A,B]=V-U\), and Hessian symmetry for \(B\) gives \[-f\langle\nabla_{[A,B]}B,A\rangle =-f\langle V-U,V\rangle.\] The sum is exactly the integrand in (39). This proves (36), including the sign of the bracket term. ◻

Proof. First keep all endpoint corrections \(S_A\) fixed. In twice the expected radial expansion from Lemma 9, minus the expected pair lower bound, every term displayed before order four cancels. Indeed independence and \(\mathbb E u_A=0\) give the exact identity \[\mathbb E\int(u_B-u_A)(\rho^h_B-\rho^h_A) =2\mathbb E\int u_A(\rho^h_A-f).\] This identity uses no further expansion of the endpoint densities. The quadratic terms cancel by (13). The radial cubic term is a constant multiple of \(\int f\langle A,\nabla_AA\rangle\), which is odd under sign reversal. The pair cubic term is a constant multiple of \[\int f\left\langle\frac{A+B}{2},\nabla_{B-A}(B-A)\right\rangle,\] which is odd under simultaneous sign reversal of both choices. Their expectations therefore vanish. Together with Lemma 7, this proves \[ \frac\kappa6\int f\,\mathbb E|A\wedge B|^2 \leq 2\mathbb E K_{0A}(S)-\mathbb E K_{AB}(S) +\frac13\mathbb E\int f|S_A+\nabla_AA|^2. \tag{41}\]

Now approximate \(-\nabla_AA\) in \(L^2\) by admissible fields \(S_A^{(j)}\). Such an approximation can be chosen explicitly: \(|A|^2/2\) is bounded, compactly supported, and Sobolev; heat regularize it, multiply by a fixed test cutoff equal to one on a neighborhood of its support, and take the negative gradient. Strong continuity of the heat flow in \(W^{1,2}\) gives \[S_A^{(j)}\longrightarrow-\nabla_AA\quad\text{in }L^2.\] Each approximant is a bounded compact test gradient with bounded divergence. Its bounds may depend on \(j\). For every fixed \(j\), the limit \(h\downarrow0\) leading to (41) has already been taken, so no uniform flow bounds in \(j\) are required.

The dependence of \(K_{AB}(S)\) on an endpoint correction is a pairing with one of the fixed \(L^2\) fields \(fQ_0,fQ_1\). It is therefore continuous under this approximation. The radial defects tend to zero. Taking \(j\to\infty\) in (41) gives \[\frac\kappa6\int f\,\mathbb E|A\wedge B|^2 \leq2\mathbb E K_{0A}-\mathbb E K_{AB}.\] Lemma 10 is exactly (40). ◻

Proof. Use the inf-convolutions \[f_\varepsilon(x) :=\inf_y\left(f(y)+\frac{d(x,y)^2}{2\varepsilon}\right), \qquad\varepsilon\downarrow0.\] Minimizers exist by properness. If \(L=\operatorname{Lip}(f)\), then \[0\leq f_\varepsilon\leq f, \qquad \|f_\varepsilon-f\|_\infty\leq\frac{\varepsilon L^2}{2}.\] In particular the supports lie in the fixed compact support of \(f\). Every minimizer \(y\) satisfies \(d(x,y)\leq\varepsilon L\): compare it with a point a short distance along a minimizing segment from \(y\) to \(x\), divide the resulting optimality inequality by that distance, and let the distance tend to zero. The same bound in the difference-quotient estimate for the inf-convolution gives local slope at most \(L\); since \(M\) is a length space, \(\operatorname{Lip}(f_\varepsilon)\leq L\).

For sufficiently small fixed \(\varepsilon\), all these minimizing distances are below a fixed model-comparison scale. Triangle comparison makes the functions \(x\mapsto d(x,y)^2\) uniformly semiconcave when these distances remain below that scale, including at \(x=y\); compare (Petrunin 2007, sec. 1.2). For an explicit normalization, choose \(a>0\) with \(-a^2\le\kappa\), weaken the curvature bound to \(-a^2\), and put \(h(r)=(\cosh(ar)-1)/a^2\). Model comparison gives \((h\circ d_y)''\le1+a^2h\circ d_y\) along unit-speed geodesics. Write \(r^2=\Psi(h(r))\). The function \(\Psi\) is increasing and concave, is smooth at zero, and satisfies \(0\le\Psi'\le2\). Thus on \(d_y\le R\) the squared distance has semiconcavity constant \(2\cosh(aR)\), uniformly in the center. Locally near each \(x\), the infimum defining \(f_\varepsilon\) may be restricted to a common small neighborhood of \(x\), by the minimizer bound. It is thus an infimum of functions with a common semiconcavity constant there. Infima preserve that semiconcavity inequality. The constant depends on \(\varepsilon,\kappa,L\), but not on the base point. Local semiconcavity with this constant yields the same inequality along each geodesic by subdivision. This proves the required semiconcavity of each \(f_\varepsilon\).

The common support and Lipschitz bound give boundedness in \(W^{1,2}\). Uniform convergence identifies every weak Sobolev subsequential limit with \(f\); hence any sequence \(\varepsilon_j\downarrow0\) has the asserted weak convergence. In (40), the two terms without curvature are integrals against fixed locally integrable functions. The curvature term converges by Lemma 4. Passing to the limit proves the last assertion. ◻

Proof. The tangent module is generated by gradients of Sobolev functions. Thus finite sums of such gradients with bounded measurable scalar coefficients are dense in \(L^2(TM)\). Approximating the coefficients by finite-valued functions and refining the resulting partitions shows that fields of the form \[\sum_{\ell=1}^L\mathbf1_{E_\ell}\nabla q_\ell, \qquad q_\ell\in W^{1,2}(M),\] are dense as well: on each piece a constant linear combination of gradients is the gradient of the corresponding linear combination of potentials. This construction may be made simultaneously for the finite list \(V_r\). Since \(f\) is bounded, it provides a finite measurable partition and choices \(q_{r,\ell}\) for which \[\sum_{r,\ell}\int_{E_\ell} f|\nabla q_{r,\ell}-V_r|^2<\eta/2.\]

Put \(K_f=\operatorname{supp}f\). On this compact set the measure with density \[f\left(1+\sum_{r,\ell}|\nabla q_{r,\ell}-V_r|^2\right)\] is finite and regular. Choose disjoint compact subsets \(K_\ell\subset E_\ell\cap K_f\) whose complement has arbitrarily small measure for this density. There are nonnegative Lipschitz functions \(\alpha_\ell\) summing to one, with \(\alpha_\ell=1\) on \(K_\ell\). For example, take Lipschitz bumps equal to one on the nonempty \(K_\ell\), supported in disjoint neighborhoods, and assign their remaining complement to the first function. On \(\bigcup_\ell K_\ell\), these functions agree with the corresponding partition indicators. On the complement, the change in the error integral is bounded by the sum appearing in the displayed density. Taking the omitted measure sufficiently small proves (42). ◻

Proof. First retain an arbitrary finite sign-symmetric law as above. Apply Lemma 13 to the finite list \[V_{AB}:=-\nabla_A(B-A).\] For each \(\ell\), apply (40), as extended in Lemma 12, with weight \(f\alpha_\ell\) and potentials \(q_{AB,\ell}\). Sum over \(\ell\). The curvature functional and its model term are linear in the weight, and \(\sum_\ell f\alpha_\ell=f\). Thus their sums are exactly their evaluations at \(f\), including the terms containing derivatives of the weight. The error is at most \(\eta/2\), after decreasing the tolerance in (42) if needed. Letting \(\eta\downarrow0\) gives \[ \mathbb ER(A,B,B,A)(f) \geq\kappa\int f\,\mathbb E|A\wedge B|^2. \tag{44}\] In particular there are no partition-derivative errors and no uniform derivative bound on the partitions is needed.

Choose the uniform law on the four potentials yielding the fields \(U,-U,V,-V\). In (44), pairs from the same signed field have zero sectional and wedge expressions. The mixed pairs have total probability \(1/2\). Multilinearity and the curvature symmetries identify all of them with the corresponding expression for \((U,V)\). Both sides of (44) are therefore one half of the respective sides of (43). This proves the proposition. ◻

Proof. For constant coefficient vectors \(a,b\in\mathbb R^N\), put \(E(a)=\sum_i a_iE_i\). This is again in \(\mathcal G\). Proposition 14 says that \[T_{a,b}(f):= R(E(a),E(b),E(b),E(a))(f) -\kappa\int f|E(a)\wedge E(b)|^2\] is nonnegative on nonnegative compact Lipschitz functions. It is therefore a positive Radon measure. Indeed, for weights supported in a fixed compact set, choose a nonnegative Lipschitz cutoff dominating one there. Positivity bounds the functional by its value on that cutoff times the uniform norm of the weight. The functional extends to continuous compactly supported functions and the Radon representation theorem applies.

Define functionals on compact Lipschitz weights by \[\begin{gathered} D_{ijkl}=R(E_i,E_j,E_k,E_l)-\kappa Q_{ijkl}m,\\ Q_{ijkl}=\langle E_i,E_l\rangle\langle E_j,E_k\rangle -\langle E_i,E_k\rangle\langle E_j,E_l\rangle. \end{gathered}\] For \(\alpha,\beta\in\mathbb N_0^N\) with \(|\alpha|=|\beta|=2\), let \[C_{\alpha\beta} :=\sum_{\substack{e_i+e_l=\alpha\\e_j+e_k=\beta}}D_{ijkl}, \qquad T_{a,b}=\sum_{\alpha,\beta}a^\alpha b^\beta C_{\alpha\beta}.\] Here \(e_i\) are the coordinate vectors, and the first sum is over ordered quadruples, retaining all monomial multiplicities. Polynomial interpolation at finitely many constant vectors expresses every grouped coefficient \(C_{\alpha\beta}\) as a finite real linear combination of the positive Radon measures \(T_{a,b}\). Thus each \(C_{\alpha\beta}\) is a signed Radon measure; no measure representation of an individual \(D_{ijkl}\) is needed.

On a fixed relatively compact region containing the support of the weight, put \[\lambda=\sum_{\alpha,\beta}|C_{\alpha\beta}|,\qquad c_{\alpha\beta}=\frac{dC_{\alpha\beta}}{d\lambda},\qquad P_x(a,b)=\sum_{\alpha,\beta}a^\alpha b^\beta c_{\alpha\beta}(x).\] For every rational pair \(a,b\), positivity of \(T_{a,b}\) gives \(P_x(a,b)\geq0\) for \(\lambda\)-almost every \(x\). Countability and polynomial continuity give a common conull set on which this holds for every real pair.

For variable test coefficients, first apply tensoriality of \(R\) over the test algebra. All scalar weights in the following identity are compact test functions: \[\begin{split} &R(X,Y,Y,X)(f)-\kappa\int f|X\wedge Y|^2\\ &\qquad=\sum_{i,j,k,l}D_{ijkl}(fa_i a_l b_j b_k)\\ &\qquad=\sum_{\alpha,\beta}C_{\alpha\beta}(fa^\alpha b^\beta)\\ &\qquad=\int f(x)P_x(a(x),b(x))\,\mathrm d\lambda(x)\geq0. \end{split}\] The last line uses only the continuous representatives of scalar test functions on the signed measures. No tangent field is evaluated on a singular measure support. This proves the claim, including when some \(C_{\alpha\beta}\) are singular with respect to \(m\). ◻

Proof. Choose a fixed compact neighborhood \(E'\) of \(E\) and apply Lemma 2(iii) on \(E'\). It gives the uniform gradient bound and all the asserted scalar convergences on this fixed neighborhood. Strong \(L^2\) convergence of the gradients gives almost-everywhere convergence after passage to a subsequence.

For \(X=\sum_i a_i\nabla b_i\), keep the original coefficients and approximate its finitely many potentials as just described. The product rules give \[\nabla X_j=\sum_i da_i\otimes db_{i,j}+a_i\operatorname{Hess}b_{i,j}, \qquad \operatorname{div}X_j =\sum_i\langle\nabla a_i,\nabla b_{i,j}\rangle+a_i\Delta b_{i,j}.\] The coefficients and their gradients are bounded, so both expressions converge strongly in \(L^2\) on the neighborhood. The vector bounds and convergence follow from the corresponding properties of the gradients. Use one common subsequence for the finite collection. ◻

Proof of Theorem 5. It remains to extend Proposition 14; the RCD and support conclusions were noted at the beginning of the section. Fix a nonnegative compactly supported test weight \(f\) and arbitrary \(X,Y\in\operatorname{TestV}(M)\). Lemma 16 gives approximating fields \(X_j,Y_j\) near \(\operatorname{supp}f\). Lemma 15 proves (12) for every pair \(X_j,Y_j\).

The field-continuity assertion of Lemma 4 passes the curvature evaluation to the limit on the fixed compact support of \(f\). The wedge term converges by dominated convergence on that support. This proves (12) for arbitrary test fields and compact test weights.

Finally let \(f\in\operatorname{Test}(M)\) be nonnegative without a support restriction. Choose \(0\leq\chi_R\leq1\) in the stronger test cutoff class, equal to one on \(B_R(o)\), with compact support and uniformly bounded gradient. Such an exhaustion exists for every fixed \(K,n\) (Gigli and Tamanini 2021, Appendix A, Lemma A.2). The product \(f\chi_R\) is a compact test function, so the desired inequality holds for this weight.

All the original tensor integrands are globally integrable. Test fields are bounded and in \(L^2\), and their derivatives and divergences are in \(L^2\). Terms with two derivatives are thus \(L^1\); a typical term with \(df\) is bounded by a constant times \(|W||\nabla Z|\), again in \(L^1\). The new terms containing \(d\chi_R\) obey the same bound, with an additional indicator of \(M\setminus B_R(o)\), and hence their integrals tend to zero. The model term is integrable, since \[f|X\wedge Y|^2 \leq\|f\|_\infty\|X\|_\infty^2|Y|^2.\] Dominated convergence proves (12) for the original global weight. This completes the proof. ◻

Proof. The first inequality is the \(L^1\) gradient estimate (Savaré 2014, Corollary 3.5); the second is the reverse Poincaré inequality. The Bakry–Ledoux estimate (Erbar et al. 2015, Proposition 4.9) gives the third: its coefficient of \(|\Delta P_sh|^2\), before rearrangement, is \((1-e^{-2Ks})/(nK)\), with value \(2s/n\) at \(K=0\). Apply (50) at time \(s/2\) to \(P_{s/2}h\), then apply (49) inside the remaining heat operator, to obtain (51).

These inequalities give the \(L^\infty\)-to-Lipschitz and \(L^\infty\)-to-\(L^\infty\) Laplacian bounds at positive time. Semigroup splitting, together with commutation in \(L^2\), therefore gives, for each integer \(j\ge0\), \[\|\Delta^jP_sh\|_\infty +\sqrt{s}\,\mathop{\mathrm{Lip}}(\Delta^jP_sh) \le C_{j,S}s^{-j}\|h\|_\infty.\] The time dependence is differentiable in the corresponding \(L^2\) generator domains on intervals away from zero.

Here is a local extension argument that does not assume \(m(M)<\infty\). Choose \(0\le\chi_R\le1\), equal to one on \(B_R(o)\), compactly supported, and set \(h_R=\chi_Rh\). For \(R'\ge R\), the difference \(d_R=h_R-h_{R'}\) vanishes on \(B_R(o)\) and satisfies \(|d_R|^2\le\|h\|_\infty^2\mathbf1_{M\setminus B_R(o)}\). The Gaussian kernel bound shows that \(P_s(d_R^2)\) tends to zero uniformly on compact spatial sets and compact subintervals of \(s>0\). Equations (49) and (51) then make the gradients and Laplacians Cauchy locally in \(L^\infty\), as well as the functions themselves. Distributional closure identifies their limits with those of \(P_sh\). Repeated splitting gives the same local conclusions for generator powers. The kernel bounds used here are valid for all centers, with constants depending only on \(K,n,S\); see (Jiang et al. 2016, Theorem 1.2).

For integrable nonnegative data, the general-measure Li–Yau inequality is (Gigli and Tamanini 2021, Theorem 3.4): \[|\nabla\log P_sh|^2 \le e^{-2Ks/3}\frac{\Delta P_sh}{P_sh} +\frac{nK}{3}\frac{e^{-4Ks/3}}{1-e^{-2Ks/3}}.\] Its value at \(K=0\) is understood by continuity. We may weaken the Ricci bound to a strictly negative constant. Rearranging the inequality, using \(|e^{2Ks/3}-1|\le Cs\) and the \(C/s\) bound for the remaining coefficient on \(s\le1\), gives (52). Apply this to \(h_R\) and pass to the local limit just established. Positive-time positivity and continuity supply a lower bound on each compact region, so division and the logarithmic chain rule pass to the limit. This proves the assertion also for bounded data that are not in any finite \(L^p\) space. ◻

Proof. The first heat datum is a constant plus a compactly supported \(W^{1,2}\cap L^\infty\) function. Constants are preserved by the heat flow, so Lemma 19 applies. Since \(u\) is bounded, \(H_t\) has positive global lower and upper bounds for fixed \(c\). Thus \(g=f/H_1\) belongs to \(L^1\cap W^{1,2}\cap L^\infty\), and the same lemma applies to \(J\). Positivity of the kernel gives \(J>0\); its continuous representative has a positive minimum on each compact space-time region under consideration.

The logarithmic Laplacian formula \[\Delta\log H=\frac{\Delta H}{H} -\frac{|\nabla H|^2}{H^2}\] and its counterpart for \(J\) give local boundedness and \(W^{1,2}\) regularity. The squared-gradient term is Sobolev by metric compatibility and the \(L^2\) Hessian estimate. Product and chain rules similarly show that \(\rho\) is a local test function. Generator differentiation before composition, followed by these same rules, gives strong local differentiation in the Laplacian graph norm; the Hessian estimate (7) then gives the Hessian statement.

If \(L_u=\mathop{\mathrm{Lip}}(u)\), then \(|\nabla e^{-u/(2c)}|\le (L_u/(2c))e^{-u/(2c)}\). Use (48), applied to the compact perturbation of the constant, to get \(|\nabla\phi_t|\le e^{-Kct}L_u\). Also \[\mathop{\mathrm{Lip}}(\log g)\le\mathop{\mathrm{Lip}}(\log f)+C/c.\] Applying (48) to \(g\) gives the corresponding bound for \(\nabla\psi\). This proves (58) and also bounds the gradients of the terminal expressions. Since \(\rho=HJ\), \(\nabla\rho=\rho w_c/c\); differentiating the two factors proves (59)–(60) with the displayed signs.

Symmetry and the semigroup property, with nonnegative integrands, give \[\int H_tP_{c(1-t)}(f/H_1)\,\mathrm dm =\int H_1(f/H_1)\,\mathrm dm=1.\] The maximum principle bounds \(\phi_t\) between \(\inf u\) and \(\sup u\). Finally \(J_t\to g\) strongly in \(W^{1,2}\). Heat continuity for bounded Lipschitz data, using the uniform small-time first moment of the kernel, gives uniform convergence of \(J_t\) to \(g\). Strong local convergence for the products and logarithms follows from their local positive lower bounds. This proves (61). ◻

Proof. The two-sided Gaussian estimate and Bishop–Gromov give constants \(a,C,c_0>0\), uniform in the base point, for which \[p_s(x,y)\le\frac{C}{m(B_{\sqrt{s}}(x))} e^{-d^2(x,y)/(as)},\qquad p_s(x,y)\ge\frac{c_0}{m(B_{\sqrt{s}}(x))} \quad(y\in B_{\sqrt{s}}(x)).\] The harmless bounded factors involving \(s\le1\) are absorbed in the constants. In addition, for \(j\ge0\), \[\frac{m(B_{(j+1)\sqrt{s}}(x))}{m(B_{\sqrt{s}}(x))} \le C(j+1)^n e^{C(j+1)\sqrt{s}}.\] The lower bound and the logarithmic Lipschitz condition give \(P_sg(x)\ge c_0e^{-A}g(x)\). On the annulus \(j\sqrt{s}\le d(x,y)<(j+1)\sqrt{s}\), \(g(y)^2\le g(x)^2e^{2A(j+1)}\). Therefore \[P_sg^2(x)\le Cg(x)^2 \sum_{j=0}^\infty(j+1)^n \exp\bigl(-j^2/a+(C+2A)(j+1)\bigr).\] This convergent series is uniform in \(x,s\). Dividing the two estimates proves the claim. ◻

Proof. Bounded compression. Apply (52) at heat time \(ct\), using (58). It gives \[\Delta\phi_t\le C(1+t^{-1}),\qquad -\Delta\psi_t\le C(1+(1-t)^{-1}).\] The constants are independent of \(c\). Since \(v_c=\nabla\phi+w_c\), the density equation can also be written \[ \partial_t\rho=-\mathop{\mathrm{div}}(\rho\nabla\phi)-c\Delta\rho. \tag{64}\] For \(p\ge2\), multiplication by \(p\rho^{p-1}\) gives \[\frac{d}{dt}\int\rho^p\,\mathrm dm =-(p-1)\int\rho^p\Delta\phi\,\mathrm dm +cp(p-1)\int\rho^{p-2}|\nabla\rho|^2\,\mathrm dm \ge-(p-1)C(1+t^{-1})\int\rho^p\,\mathrm dm.\] For fixed \(c\) the integrations are justified first with exhausting cutoffs: \(\rho\) is bounded and integrable, \(|\nabla\rho|\le C_c\rho\), and \(\Delta\phi\) is globally bounded away from the initial time. The cutoff errors vanish by integrability. One may equivalently use the strong interior \(L^2\) heat differentiation before removing the cutoffs. Integrate backward from \(\rho_1=f\), then let \(p\to\infty\). This proves the density bound, with the direction of integration fixed by (64).

The terminal lower bound. For the endpoint bound let \(r=1-t\), \(g=f/H_1\), and \(s=cr\). When \(r\le c\), the logarithmic Lipschitz bound for \(g\) is at most \(C/c\), and \(s\le c^2\). Lemma 21 and (50) consequently give \[\frac{|\Delta P_sg|}{P_sg} \le C s^{-1/2}c^{-1},\qquad c|\Delta\psi_t| \le C\left(1+\sqrt{c/r}\right).\] For \(r\ge c\), use the preceding Li–Yau bounds instead. Since \(b=(\psi-\phi)/2\), one can thus take, after increasing constants depending on \(\delta\), \[e_c(t)\le \begin{cases} C_\delta c(1+r^{-1}),&r\ge c,\\ C_\delta(c+1+\sqrt{c/r}),&0<r<c. \end{cases}\] For sufficiently small \(c\), its integral is at most \(C_\delta(c|\log c|+c)\). The Sobolev function \(\Delta b\) has the same lower bound outside a polar set in its quasi-continuous representative, by the capacitary facts in Section 2. ◻

Proof. The integrated Bochner identity. Choose a nonnegative space-time cutoff \(\chi\), zero for \(t\le\delta/2\), equal to one on \([\delta,1]\times E\), and compactly supported in space, such that \[|\partial_t\chi|+|\nabla\chi|+|\Delta\chi| \le C\sqrt\chi.\] For example take fourth powers of a smooth time cutoff and a good spatial test cutoff. We first work on an interval with terminal time \(T<1\). The equations and the definition of \(\Gamma_2\) yield \[\begin{align*} \frac d{dt}\int\rho\chi\Delta\theta\,\mathrm dm ={}&\int\rho(\partial_t\chi+ \langle\nabla\chi,v_c\rangle)\Delta\theta\,\mathrm dm\\ &-\bigl(\Gamma_2(\theta)+\Gamma_2(b)\bigr)(\rho\chi) -\int\rho\Delta b\bigl(c\Delta\chi+ \langle\nabla\chi,w_c\rangle\bigr)\,\mathrm dm. \tag{67}\end{align*}\] To check the last term, put \(q=\Delta b\). Twice integrating by parts, using \(\nabla\rho=\rho w_c/c\), gives \[\int\rho\chi(\langle w_c,\nabla q\rangle+c\Delta q) =\int\rho q(\langle w_c,\nabla\chi\rangle+c\Delta\chi).\] This identity is distributional; \(q\) is locally Sobolev and bounded, and the required multipliers are admitted by the Bochner measure calculus. Thus [heat:weighted-bochner-identity] does not assume a pointwise fourth derivative.

Weighted second-order bounds. Average (9) and (10). For either \(z=b,\theta\), \[\Gamma_2(z)\ge \left(K|\nabla z|^2+\tfrac12|H_z|^2+ \tfrac1{2n}|\Delta z|^2\right)m.\] The two cutoff terms in [heat:weighted-bochner-identity] are bounded by \(\varepsilon\int\rho\chi(|\Delta b|^2+|\Delta\theta|^2)\) plus a constant, using (58), (63), and Young’s inequality. The possibly negative \(K\)-terms have the same harmless bound. At the initial endpoint the primitive is zero. At the terminal endpoint its limit is \[-\int\nabla(f\chi_1)\cdot \bigl(\nabla\phi_1+c\nabla\log f\bigr)\,\mathrm dm,\] which is bounded uniformly in \(c\). This follows from (61); no Hessian of \(\log f\) is needed. After absorption, let \(T\uparrow1\). Monotone convergence for the squared terms gives the first four bounds in (65). In particular these bounds also hold uniformly for all shorter terminal intervals.

The fourth-order bound. Choose a spatial cutoff \(\eta\) equal to one on \(E\), with \(|\nabla\eta|\le C\sqrt\eta\). The divergence of \(\rho\eta|w_c|^2w_c\) gives \[\begin{align*} \frac1c\int\rho\eta|w_c|^4 ={}&-\int\rho\eta\bigl(|w_c|^2\Delta b +2H_b(w_c,w_c)\bigr)\\ &-\int\rho|w_c|^2\langle w_c,\nabla\eta\rangle. \end{align*}\] All terms are integrable on interior intervals by local test calculus. Integrate in time and then exhaust the terminal interval. If \(A_c=\int_\delta^1\int\rho\eta|w_c|^4\), Cauchy–Schwarz and the bounds just proved show \(A_c/c\le C\sqrt{A_c}\). For the cutoff term use also the uniform bound on \(w_c\) and the mass bound. Hence \(A_c\le C^2c^2\), proving the last estimate.

Vanishing weighted products. It follows that \(\int_\delta^1\int_E\rho^c|w_c|^2\to0\). For (66), truncate the fixed function \(T\): on \(\{|T|\le A\}\) use this convergence, and on its complement use the uniform bounds on \(\rho^c,w_c\). The latter contribution is bounded by \(C\int_{\{|T|>A\}}|T|^2\), which tends to zero as \(A\to\infty\). ◻

Proof. All assertions are local, so first insert the cutoffs just described. Since \(\nabla_a a=H_za=\nabla(|a|^2/2)\), the first two terms of the defining expression for \(R(Y,a,a,Y)(h)\) sum to \(H_{|a|^2/2}(Y,Y)(h)\). Its other terms sum to \[\int\mathop{\mathrm{div}}(ha)H_z(Y,Y) +2hH_z(\nabla_aY,Y)-h|H_zY|^2\,\mathrm dm,\] because \([Y,a]=H_zY-\nabla_aY\). This is exactly (69). The calculation uses Hessian symmetry and first-order products only; it does not require a third covariant derivative of \(z\).

For fixed \(c\), on a compact interior time interval the pairing on the left of [heat:riccati-evolution] is absolutely continuous. This follows from local graph-norm differentiation of \(\theta\), the Hessian estimate, and local boundedness of \(\partial_t\rho\). Differentiate that pairing, use \(\partial_t\rho=-\mathop{\mathrm{div}}(\rho v_c)\), and interpret its contraction against \(H_\theta\) through (68). Thus no continuity equation is being tested against an unproved Sobolev representative of \(H_\theta(Y,Y)\). Substitute (60) and apply (69) to \(z=\theta\). The \(\nabla_{v_c}H_\theta\) terms cancel, leaving [heat:riccati-evolution]. ◻

Proof. Exact cancellations. We work first at fixed \(c\) and at an interior time, suppressing these parameters. By (69) with \(z=b\), the error is the sum of \[\int\rho|H_bY|^2\,\mathrm dm,\qquad R(Y,w,w,Y)(\rho),\qquad I=(\nabla_wH_b)(Y,Y)(\rho)+cH_{\Delta b}(Y,Y)(\rho).\] The first term is nonnegative. The curvature bound gives \(R(Y,w,w,Y)(\rho)\ge \kappa\int\rho|Y\wedge w|^2\,\mathrm dm\), whose possible negative contribution tends to zero after time integration by (66).

We examine \(I\). Formula (68) and \(\nabla\rho=\rho w/c\) give \[ (\nabla_wH_b)(Y,Y)(\rho) =-\int\rho\left[ 2H_b(\nabla_wY,Y)+H_b(Y,Y) \left(\Delta b+\frac{|w|^2}{c}\right)\right]\,\mathrm dm. \tag{72}\] Put \(D=\mathop{\mathrm{div}}Y\), \(A=\nabla_YY\), and \(\alpha=Yb=\langle Y,w\rangle\). Twice using the definition of the distributional Hessian gives \[\begin{align*} cH_{\Delta b}(Y,Y)(\rho) ={}&\int c\Delta b\, \left[\mathop{\mathrm{div}}\bigl(Y\mathop{\mathrm{div}}(\rho Y)\bigr) +\mathop{\mathrm{div}}(\rho A)\right]\\ ={}&\int\rho\Delta b \left[H_b(Y,Y)+\frac{\alpha^2}{c} +2\langle w,A\rangle+2\alpha D +c\bigl(D^2+YD+\mathop{\mathrm{div}}A\bigr)\right]. \tag{73}\end{align*}\] The last divergence is a measure, not necessarily a function. Its product with \(\rho c\Delta b\) is understood using quasi-continuous representatives.

To justify [heat:error-double-divergence] for the actual test class, observe that the function \(\mathop{\mathrm{div}}(\rho Y)=\rho D+\rho\alpha/c\) belongs locally to \(W^{1,2}\). The product rule for its product with the bounded field \(Y\), proved first after truncation, gives an \(L^1\) divergence. The multiplier \(\Delta b\) is locally bounded Sobolev, so the truncations pass by \(L^2\) pairing for the gradient term and dominated convergence for the divergence term. Also \(A=\nabla(|Y|^2/2)\), whose divergence is the Bochner Laplacian measure; multiplication by the Lipschitz function \(\rho\) obeys its usual weak product rule. Expanding \(Y\alpha=H_b(Y,Y)+\langle w,A\rangle\) gives the second line of [heat:error-double-divergence].

The terms \(\rho\Delta b H_b(Y,Y)\) in (72) and [heat:error-double-divergence] cancel. There is a second exact cancellation: \[\begin{align*} \int\rho\bigl(\alpha^2\Delta b-|w|^2H_b(Y,Y)\bigr)\,\mathrm dm =\int\rho\bigl[ -2\alpha\langle w,\nabla_wY\rangle +|w|^2\alpha D+|w|^2\langle w,A\rangle\bigr]\,\mathrm dm. \tag{74}\end{align*}\] Indeed integration by parts in the first term on the left produces \(-\rho\alpha^2|w|^2/c-2\rho\alpha H_b(w,Y) -2\rho\alpha\langle w,\nabla_wY\rangle\). Writing \(H_b(Y,Y)=Y\alpha-\langle w,A\rangle\) and integrating the negative Hessian term gives the opposites of the first two expressions, which cancel them, with the remaining terms displayed on the right of (74).

The remaining Sobolev terms. After division by \(c\), each term on that right side is bounded in absolute value, on the whole time interval, by a constant times \[\left(\int_\delta^1\!\int_E \rho\frac{|w|^4}{c^2}\right)^{1/2} \left(\int_\delta^1\!\int_E \rho|w|^2|T_0|^2\right)^{1/2},\] where \(E\) contains the common support and \(T_0\) is a fixed \(L^2\) coefficient formed from \(\nabla Y,D\) and bounded trial fields. This tends to zero by Lemma 23. The remaining terms involving \(H_b\) or \(\Delta b\), one factor \(w\), and derivatives of \(Y\) are bounded the same way, using their weighted \(L^2\) bound in place of the first factor.

The Bochner measure term. It remains to control the final parenthesis in [heat:error-double-divergence]. Here \(D\) is only \(L^2\), so \(D^2\) need not be an \(L^2\) coefficient. We retain it in a nonnegative measure, to which the lower bound for \(c\Delta b\) applies, and leave an \(L^2\) remainder. Since \(Y=\nabla g_t\), the required decomposition is \[ D^2+YD+\mathop{\mathrm{div}}A=\mathcal M_t+\ell_t m, \quad \begin{cases} \mathcal M_t=(\Delta g_t)^2m+\Gamma_2(g_t)-K|Y_t|^2m\ge0,\\ \ell_t=2\langle Y_t,\nabla\Delta g_t\rangle+K|Y_t|^2. \end{cases} \tag{75}\] Here \(\ell_t\) has a uniformly bounded local \(L^2\) norm. The measures \(\mathcal M_t\) have uniformly bounded mass on \(E\): polarize \(\Gamma_2\) on the fixed finite list of test potentials and use boundedness of the smooth temporal coefficients. Equivalently, test their defining formula with a fixed outer cutoff. Thus no \(L^4\) bound on \(D\) is used.

Time measurability and terminal uniformity. To justify time integration of the singular part, fix a compact neighborhood \(E_*\) of the trial support and define the finite signed coefficient measures and their common dominating measure by \[\mathcal M_{ij}=(\Delta g_i)(\Delta g_j)m +\Gamma_2(g_i,g_j)-K\langle\nabla g_i,\nabla g_j\rangle m, \qquad \Lambda=\sum_{i,j}|\mathcal M_{ij}|\big|_{E_*}.\] Every coefficient measure neglects polar sets. The identity \(\mathcal M_t=\sum_{i,j}a_i(t)a_j(t)\mathcal M_{ij}\) shows that \(\mathcal M_t|_{E_*}\) has a jointly measurable, uniformly bounded \(\Lambda\)-density.

Choose a compact test cutoff \(\chi=1\) near \(E_*\). At fixed \(c\), on each compact interior time interval, \(\chi\rho_t c\Delta b_t\) is strongly continuous in \(W^{1,2}\) and has a common \(L^\infty\) bound. Indeed, for either positive heat factor \(U\), the spatial formula \[\nabla\Delta\log U =\frac{\nabla\Delta U}{U} -\frac{\Delta U\nabla U}{U^2} -\frac{2H_U\nabla U}{U^2} +\frac{2|\nabla U|^2\nabla U}{U^3}\] expresses the derivative using bounded Lipschitz heat powers, local positive lower bounds, and the strongly locally \(L^2\)-continuous Hessian \(H_U\). The latter continuity follows from the Laplacian graph estimate; continuity of the other factors follows by integrating the next generator powers. The fixed cutoff and \(\rho_t\) preserve these Sobolev continuity and boundedness properties.

Strong Sobolev convergence with a common bound has quasi-everywhere convergent subsequences. Since \(\Lambda\) is finite and neglects polar sets, dominated convergence and the subsequence criterion give continuity of these quasi-continuous multipliers in \(L^1(\Lambda)\). Choose a jointly measurable representative of this curve. At almost every time it agrees \(\Lambda\)-almost everywhere with the quasi-continuous multiplier. The lower bound for \(c\Delta b_t\) therefore holds \(\Lambda\)-almost everywhere at each admissible time, and Fubini applies without an uncountable union of polar exceptional sets. All the bounds in this paragraph are only fixed-\(c\), interior-time bounds.

The quasi-continuous lower bound in (63), and the density bound there, give \[\int\rho_t c\Delta b_t\,\mathrm d\mathcal M_t \ge-C_\delta e_c(t)\mathcal M_t(E).\] Bochner measures neglect polar sets, so an almost-everywhere bound on the Sobolev multiplier has been used here only after passing to its quasi-continuous representative. In particular, for every \(T<1\), \[\int_\delta^T\!\int\rho_t c\Delta b_t\,\mathrm d\mathcal M_t\,\mathrm dt \ge-C_\delta\sup_t\mathcal M_t(E) \|e_c\|_{L^1(\delta,1)}.\] This lower error bound tends to zero uniformly in \(T\). The absolutely continuous remainder is bounded by \[c\left(\int_\delta^1\!\int_E\rho|\Delta b|^2\right)^{1/2} \left(\int_\delta^1\!\int_E\rho|\ell_t|^2\right)^{1/2} \longrightarrow0.\] The Sobolev remainders were estimated in absolute value on \((\delta,1)\), and the remaining measure contribution was bounded below using \(\|e_c\|_{L^1(\delta,1)}\). After discarding the nonnegative terms, these bounds hold with the same error for every \(T<1\), proving (71). ◻

Proof. Integrate [heat:riccati-evolution] from \(\delta\) to \(T<1\). The initial pairing vanishes. Apply (B) to its curvature term and use \[2H_\theta(D_cY,Y)-|H_\theta Y|^2\le|D_cY|^2, \qquad D_cY=\partial_tY+\nabla_{v_c}Y.\] Lemma 25 bounds the last term uniformly in \(T\). The resulting right-hand side is integrable up to one, by boundedness of \(\rho,v_c\) and the fixed \(L^2\) derivatives of \(Y\).

For the terminal left-hand side, use the first-order expression (8). The convergences (61) imply strong local \(L^2\) convergence of \(\mathop{\mathrm{div}}(\rho_tY_t)\) to \(\mathop{\mathrm{div}}(fY_1)\), and of the gradients of \(\theta_t\) to that of \(\phi_1^c+c\log f\). The bounded density and field factors converge uniformly on the compact support. Both products in (8) therefore converge in \(L^1\). Let \(T\uparrow1\), with \(c\) still fixed. This proves (76). The remaining limits will be taken successively: first \(c\downarrow0\), and then removal of the positive terminal perturbation \(\lambda\bar f\) by \(\lambda\downarrow0\). ◻

Proof. Use the two-sided Gaussian bounds with exponents \(4-\eta\) and \(4+\eta\), for any fixed \(0<\eta<1\), and heat time \(ct\). The kernel is symmetric, so the volume factor can be taken to be \(m(B_{\sqrt{ct}}(x))\) at the evaluation point. These are the bounds of (Jiang et al. 2016, Theorem 1.2), after weakening the Ricci bound to a negative one if needed.

Here are the two Laplace estimates with their uniformity specified. Fix \(t\in[\alpha,1]\) and \(x\) in a compact set. In the upper kernel bound reserve a small fixed fraction \(\tau>0\) of its Gaussian exponent. Taking the infimum of the other part of the exponent gives \[H_t^c(x)\le C_{\eta,\tau} \exp\left[-\frac1{2c} Q_{(4+\eta)t/(4(1-\tau))}u(x)\right].\] Indeed the integral of the reserved Gaussian, divided by \(m(B_{\sqrt{ct}}(x))\), is uniformly bounded by the annular volume-comparison estimate used in Lemma 21. For the lower kernel bound, choose a minimizer for the corresponding modified quadratic cost and integrate over a small ball of fixed radius \(r>0\) around it. Boundedness of \(u\) keeps all these minimizers in a common compact set. The Lipschitz bound for \(u\), and the bounded distance of the minimizers from \(x\), change the exponent by at most \(C_\alpha r\). The logarithm of the volume prefactor, multiplied by \(c\), tends to zero uniformly: on the compact region, Bishop–Gromov and full support bound small-ball volumes below by a fixed multiple of the \(n\)-th power of the radius, and above by a fixed finite constant. Thus \[\limsup_{c\downarrow0}\phi_t^c(x) \le Q_{(4-\eta)t/4}u(x)+C_\alpha r,\] uniformly on the compact parameter set. Send first \(c\downarrow0\), then \(r,\eta,\tau\downarrow0\). The quadratic infima vary uniformly with their time parameter on compact positive-time intervals, since their minimizers have uniformly bounded distance. The two estimates give (77). Finally the uniform spatial Lipschitz bound gives weak local Sobolev compactness; closure of the differential identifies its limit with \(\nabla\Phi_t\). ◻

Proof. Let \(L\) be the velocity bound in (58) and \(r(x)=d(o,x)\). For a bounded nondecreasing Lipschitz function \(\chi\) that is constant at large arguments, the continuity equation, first with compact spatial tests and then after subtracting that constant, gives for \(s\le t\le1\) \[\frac d{dt}\int\chi(r+L(t-s))\,\mathrm d\mu_t^c =\int\chi'(r+L(t-s)) \bigl(L+\langle\nabla r,v_c\rangle\bigr)\,\mathrm d\mu_t^c\ge0.\] Compact Lipschitz tests are admitted by their strong Sobolev test approximations. At time one, local convergence of the density and preservation of its total mass justify the limit for these tests. Hence \[ \int\chi(r)\,\mathrm d\mu_s^c \le\int\chi(r+L(1-s))f\,\mathrm dm. \tag{78}\] Approximate \(r^2\) and then \(r^2\mathbf1_{\{r>R\}}\) monotonically by such functions. The first choice proves finiteness of the moments, and the second gives \[ \sup_{c,\,s\in[\delta,1]} \int_{\{r>R\}}r^2\,\mathrm d\mu_s^c \le\int_{\{r>R-L\}}(r+L)^2f\,\mathrm dm\longrightarrow0. \tag{79}\] The same bound also gives tightness.

The explicit heat factors give continuity against compact tests on the interior and at the terminal endpoint. Tightness gives narrow continuity, and (79) upgrades it to \(W_2\) continuity. The weak continuity equation extends to every \(h\in W^{1,2}\), with functional \[\mathcal L_t^c(h)=\int\langle\nabla h,v_c\rangle\rho_t^c\,\mathrm dm, \qquad |\mathcal L_t^c(h)| \le\left(\int\rho_t^c|v_c|^2\,\mathrm dm\right)^{1/2} \left(\int\rho_t^c|\nabla h|^2\,\mathrm dm\right)^{1/2}.\] To see this extension, approximate \(h\) in \(W^{1,2}\) by compact tests; the density bound makes the errors uniform in time on \([\delta,1]\).

For the exact domain in the cited continuity-equation criterion, write \(S^2(M)\) for the Sobolev class with square-integrable weak gradient, without requiring the function itself to belong to \(L^2\). The same formula defines \(\mathcal L_t^c\) on all of \(S^2(M)\), with the displayed dual-seminorm bound. The integrated continuity equation holds for every \(h\in L^1(m)\cap S^2(M)\): set \(h_M=\max\{-M,\min\{h,M\}\}\). Then \(|h_M|^2\le M|h|\), so \(h_M\in W^{1,2}\), and \[\nabla h_M=\mathbf1_{\{|h|<M\}}\nabla h, \qquad h_M\to h\text{ in }L^1, \qquad \nabla h_M\to\nabla h\text{ in }L^2.\] Locality makes the choice on the threshold level sets irrelevant. Bounded compression gives \[\sup_{t\in[\delta,1]} \left|\int(h_M-h)\,\mathrm d\mu_t^c\right| \le C_\delta\|h_M-h\|_1\longrightarrow0,\] while, on any fixed positive-time interval \(I\), \[\int_I|\mathcal L_t^c(h_M-h)|\,\mathrm dt \le L\sqrt{C_\delta}|I| \|\nabla h_M-\nabla h\|_2\longrightarrow0.\] Thus the integrated identity passes for every pair of times, with finite scalar integrals and an absolutely continuous representative. The identical argument applies to the limiting gradient functional with \(\nu,V\) below, since it has the same density and velocity bounds. The continuity-equation criterion now gives \(|\dot\mu_t^c|\le L\) almost everywhere. Finally properness, the uniform moment tails, and Arzelà–Ascoli in \(\mathcal P_2(M)\) give the claimed subsequence. The bound on the limiting densities follows by testing their narrow limit against compact continuous nonnegative functions. A diagonal choice handles intervals with arbitrarily small positive left endpoint. ◻

Proof. On \([\alpha,\beta]\) the fluxes \(j_c=\rho^cv_c\) are bounded in the fixed Hilbert space \(L^2(\,\mathrm dt\,\mathrm dm;TM)\), since \(\int|j_c|^2\le C_\alpha\int\rho^c|v_c|^2\). Extract a weak flux limit \(j\), and a weak-* density limit \(\rho^c\rightharpoonup^*\nu\) in \(L^\infty(\,\mathrm dt\,\mathrm dm)\). The latter agrees with the curve limit. Testing the convex inequality \(|j_c|\le L\rho^c\) against bounded fields of bounded support gives \(|j|\le L\nu\). Define \(V=j/\nu\) on \(\{\nu>0\}\) and zero elsewhere. The continuity equation passes to the limit. Its kinetic lower semicontinuity follows, for example, from \[\int\nu|V|^2 =\sup_Z\int\bigl(2\langle j,Z\rangle-\nu|Z|^2\bigr),\] where the supremum is over bounded \(L^2\) fields with bounded support.

For the first equality in [heat:exact-action-identity], write \(a=\nabla\phi\) and use \(v_c=a+w_c\). The equations give \[\frac d{dt}\int\phi\rho\,\mathrm dm =\int\rho\left[-\tfrac12|a|^2+c\Delta\phi +\langle a,a+w_c\rangle\right]\,\mathrm dm =\tfrac12\int\rho|a|^2\,\mathrm dm,\] because \(c\int\rho\Delta\phi=-\int\rho\langle w_c,a\rangle\). One first uses spatial cutoffs. For fixed \(c\), their errors vanish by mass integrability, boundedness of \(\phi,a\), and \(|\nabla\rho|\le C_c\rho\). The identity extends to \(\beta=1\) using the terminal traces and bounded gradients; no Laplacian bound at exactly time one is needed. The difference between the actions of \(a\) and \(v_c\) tends to zero on bounded spatial regions by Lemma 23. On their complement it is bounded by a constant times the density mass, uniformly small by (79). This proves the second equality.

Put \(h=\beta-\alpha\) and \[A=\int\Phi_\beta\,\mathrm d\mu_\beta- \int\Phi_\alpha\,\mathrm d\mu_\alpha.\] The bounded potentials, local uniform convergence, and tightness identify this with the limit of the left side of [heat:exact-action-identity]. The Hopf–Lax semigroup property says \(\Phi_\beta(y)-\Phi_\alpha(x)\le d^2(x,y)/(2h)\). The continuity-equation criterion and action lower semicontinuity therefore give the sandwich \[\begin{align*} \frac{W_2^2(\mu_\alpha,\mu_\beta)}{h} &\le\int_\alpha^\beta|\dot\mu_t|^2\,\mathrm dt \le\int_\alpha^\beta\!\int\nu|V|^2\,\mathrm dm\,\mathrm dt\\ &\le\liminf_{c\downarrow0} \int_\alpha^\beta\!\int\rho^c|v_c|^2\,\mathrm dm\,\mathrm dt =2A\le\frac{W_2^2(\mu_\alpha,\mu_\beta)}{h}. \tag{83}\end{align*}\] Thus all inequalities are equalities. Equality in the metric action bound proves the geodesic assertion and equality of the velocity norms.

To turn norm convergence into (81), expand the square. For its cross term, test the weak flux convergence against \(V\mathbf1_{B_R(o)}\); for its last term, test the density convergence against \(|V|^2\mathbf1_{B_R(o)}\). These are fixed admissible tests in the appropriate spaces. Boundedness of the velocities and tightness let \(R\to\infty\). The resulting limit is zero by equality of actions in [heat:action-sandwich].

Finally, for any \(A_0>0\), \[\begin{align*} \int\rho^c|T(v_c-V)|^2 \le A_0^2\int\rho^c|v_c-V|^2 +4L^2C_\alpha\int_{\{|T|>A_0\}}|T|^2. \end{align*}\] First let \(c\downarrow0\), then \(A_0\to\infty\). This proves (82). ◻

Proof. Initially normalize \(f\in\mathcal F\) to mass one. Equality [heat:action-sandwich] on every positive-time interval makes \(\mu_t\) a consistent constant-speed Wasserstein geodesic. Its speed is bounded, so completeness of \(\mathcal P_2(M)\) extends it to time zero. A dynamical optimal plan represents this geodesic; one may use metric superposition (Gigli and Han 2015, Proposition 2.1) and equality of endpoint cost and path action. Since \(\|Q_tu-u\|_\infty\le t\mathop{\mathrm{Lip}}(u)^2/2\), the dual equalities in [heat:action-sandwich] pass to time zero and give \[\int\Phi_1 f\,\mathrm dm-\int u\,\mathrm d\mu_0 =\tfrac12W_2^2(\mu_0,fm).\] The nonnegative pointwise duality gap therefore vanishes on almost every path, proving (46) for this plan. Its individual path lengths are bounded by \(\sqrt{2\operatorname{osc}(u)}\). The positive-time compression is the limiting bound \(\nu_t\le C_\delta\).

We specify the identification with \(V\). On a positive-time interval the plan is a test plan. Its pullback velocity, defined by differentiation of Sobolev functions, has norm equal to the metric path speed (Gigli 2018a, Theorem/Definition 2.33). Let \(W\) be its conditional average given \((t,\gamma_t)\). It solves the same continuity equation and its action is at most the path action, by conditional Cauchy–Schwarz. The kinetic lower bound makes this an equality. Both \(V\) and \(W\) solve the same equation; apply the same lower bound to \((V+W)/2\) and use the parallelogram identity. Equality of their actions with the minimum forces \(V=W\). Equality in conditional Cauchy–Schwarz then identifies the pullback velocity itself with \(V_t(\gamma_t)\) almost everywhere. Separability of the tangent module, available on this RCD space, permits the conditional-average construction by a countable family of module generators.

The prescription is independent of the positive datum. Take two normalized calibrated plans, reverse their time orientation, and form their equally weighted mixture. The mixture is still calibrated, and hence is optimal from its absolutely continuous initial measure, which is the mixture of the two terminal densities, to its other endpoint measure. The latter has finite second moment and may be singular. Uniqueness of dynamical optimal transport from an absolutely continuous measure (Gigli et al. 2016, Theorem 1.1) forces this reversed mixture to be induced by a map into whole geodesics. Disintegration consequently forces both component plans to choose the same path wherever their terminal densities overlap. Positive members of \(\mathcal F\) are equivalent to \(m\), so this agreement holds \(m\)-almost everywhere.

Fix the reference datum proportional to \(\bar f\) and choose its resulting Borel map \(S\). The argument just given shows that every heat limit for every positive datum is \(S_\#(fm)\), after restoring mass. The prescribed-potential uniqueness theorem (Gigli et al. 2016, Theorem 1.3) applied to the \(c\)-concave potential \(Q_1u\), \(c=d^2/2\), also gives uniqueness of the calibrated path for almost every terminal point.

For a compact datum \(f_0\ge0\), the ratio \(f_0/\bar f\) is bounded. Thus \(S_\#(f_0m)\) is boundedly dominated by the reference plan. This proves its compression assertion and transfers its path-velocity representative. In particular one can choose this representative using the reference plan once and for all. Restriction to a bounded set of terminal points preserves all these statements and, by the speed bound, gives bounded spatial support on positive-time intervals. ◻

Proof of Theorem 18. First take \(f\in\mathcal F\). On the common compact support of the trial, Lemma 27 and the uniform gradient bounds imply \[H_{\phi_1^c+c\log f}(Z,Z)(f) \longrightarrow H_{\Phi_1}(Z,Z)(f).\] Indeed (8) pairs the gradient of its potential with fixed \(L^2\) coefficients; weak local Sobolev convergence suffices. The additional gradient \(c\nabla\log f\) converges uniformly to zero.

For the right side of (76), use (82) with the fixed coefficient \(\nabla Y\). It replaces \(\nabla_{v_c}Y\) by \(\nabla_VY\) strongly in the weighted \(L^2\) norm. The remaining fixed squared expression is in \(L^1(\,\mathrm dt\,\mathrm dm)\) and has compact support, so weak-* convergence of the densities passes its integral to the limit. The curvature term passes in the same way using boundedness of \(Y\) and the velocities. Lemma 30 identifies the limiting expression with the path integral in (47). Let \(c\downarrow0\) in Proposition 26.

Finally, for the desired compact datum \(f_0\), use \(f_\lambda=f_0+\lambda\bar f\). The common prescription gives \(\pi_{f_\lambda}=\pi_{f_0}+\lambda\pi_{\bar f}\). Both sides of the already established inequality are linear in the terminal data. All terms involving \(\bar f\) are finite, by compact support of the trial, bounded speed, and positive-time compression. Let \(\lambda\downarrow0\). This proves (47), including the zero datum. ◻

Proof. Bounds along dyadic radii imply bounds at every radius by enlarging to a dyadic radius at most twice as large. Fix \(\varepsilon>0\), to be chosen sufficiently small in terms of \(s\), and put \[E_j=\{x\in E:\mathcal H^s_\infty(E\cap B(x,r)) \leq\varepsilon r^s\text{ for every }0<r<1/j\}.\] The sets \(E_j\) cover \(E\). Fix \(x\in E_j\) and \(R<1/j\). A content cover of \(E_j\cap B(x,R)\) can be recentered in \(E_j\): discard covering sets not meeting this set and replace a covering set of positive diameter \(d_i\) by a ball of radius \(2d_i\) centered at a point of its intersection with \(E_j\). Replace singleton covering sets by balls with arbitrarily small positive total \(s\)-cost. Choosing the original cover within an arbitrarily small error of the content gives child balls \(B(x_i,r_i)\) with \[x_i\in E_j,\qquad \sum_i r_i^s\leq\theta R^s, \qquad 0<\theta<1,\] provided \(\varepsilon\) was chosen small enough. In particular \(r_i\leq\theta^{1/s}R<1/j\).

Repeat this procedure on \(E_j\cap B(x_i,r_i)\), which may enlarge the part of the parent set being covered but still covers it. At generation \(k\) the total diameter cost is at most \(2^s\theta^kR^s\), and every diameter is at most \(2\theta^{k/s}R\). Both bounds tend to zero. Thus \(\mathcal H^s(E_j\cap B(x,R))=0\). Separability supplies countably many such root balls covering \(E_j\). Taking the countable union over \(j\) proves the claim. No finiteness assumption on \(\mathcal H^s(E)\) is used. ◻

Proof. The volume lower bound makes the average of \(P\) on \(B(y,r)\) within \(1/4\) of \(I\), simultaneously for all \(y\in B(x,r)\), if \(\varepsilon_0\) is small enough. Let \(\lambda\) be the enlargement factor in the Poincaré inequality, and choose \(L>2+5\lambda\). For \(t_j=2^{-j}r\), the difference between the averages on \(B(y,t_j)\) and \(B(y,t_{j+1})\) is at most \[C t_j^{1-n/2} \left(\int_{B(y,\lambda t_j)}|\nabla P|^2\,\mathrm dm\right)^{1/2}.\] If, for every \(j\geq0\), \[\int_{B(y,\lambda t_j)}|\nabla P|^2\,\mathrm dm <c_0t_j^s r^{n-2-s},\] the sum of these differences is bounded by \[C\sqrt{c_0}\sum_{j\geq0}2^{-j(s-n+2)/2}.\] Because \(s>n-2\), this is less than \(1/2\) for an appropriate \(c_0>0\). The averages converge to the precise value at \(y\), so this contradicts \(|P(y)-I|\geq1\). Consequently, each \(y\in E\) has a radius \(t_y\leq r\) for which \[ \int_{B(y,\lambda t_y)}|\nabla P|^2\,\mathrm dm \geq c_0t_y^s r^{n-2-s}. \tag{86}\] The \(5r\) covering lemma selects countably many disjoint balls \(B(y_i,\lambda t_i)\) whose fivefold enlargements cover \(E\). They lie in \(B(x,Lr)\). Summing (86) over these disjoint balls bounds \(\sum_i t_i^s\) by the right-hand side of (85), up to a fixed factor. Their fivefold enlargements give the required content cover. ◻

Proof of Proposition 31. Selection at all dyadic scales. Put \(r_k=2^{-k}\), and choose a countable \(r_k\)-net \(N_k\) in \(M\). For each integer \(A\geq4\), each \(y\in N_k\), and each integer \(q\geq1\), select one harmonic \(n\)-tuple on \(B(y,2Ar_k)\) with splitting error at most \(1/(n^2q)\), whenever such a tuple exists. The splitting-map estimates give a dimensional gradient bound and \[\begin{align*} \frac{1}{m(B(y,2Ar_k))}\int_{B(y,2Ar_k)}|P-I|\,\mathrm dm&\leq q^{-1},\\ \frac{(2Ar_k)^2}{m(B(y,2Ar_k))} \int_{B(y,2Ar_k)}|\operatorname{Hess}u^a|^2\,\mathrm dm&\leq q^{-1}, \end{align*}\] for its components \(u^a\) and their Gram matrix \(P\). The existence input is the harmonic splitting-map theorem near a Euclidean rescaling; see (Bruè et al. 2021, Proposition 1.4). To match that theorem’s product formulation, use the auxiliary synthetic dimension \(N=n+1\), Euclidean factor dimension \(n\), and a point as the \(\mathrm{RCD}(0,1)\) residual factor. Its radius is \(2Ar_k/5\), so its output ball is \(B(y,2Ar_k)\). The rescaled curvature bound tends to zero. This auxiliary parameter does not change the actual dimension \(n\), the measure \(\mathcal H^n\), or the volume estimates used here.

Subtract the values \(u^a(y)\) and multiply by a good test cutoff supported in \(B(y,2Ar_k)\) and equal to one on \(B(y,Ar_k)\). These are compact test functions. Indeed, on the support of the cutoff \(\eta\), the functions and their gradients are bounded and their Hessians are locally in \(L^2\), and \[\Delta(\eta u^a)=u^a\Delta\eta +2\langle\nabla\eta,\nabla u^a\rangle.\] The right-hand side is in \(W^{1,2}\) by the product rule and the local Hessian bounds. The construction therefore changes none of the Gram entries on the inner ball. The collection of all selected components is countable. Adjoin the prescribed countable family.

Fix a regular point \(x\), a bounded region containing it, and an integer \(A>L+1\), with \(L\) as in Lemma 33. Choose \(y_k\in N_k\) with \(d(x,y_k)\leq r_k\). The rescalings at \(y_k\) converge to Euclidean space at each fixed enlargement of \(r_k\). To check this, take any subsequence of rescalings at \(x\). Their limit is Euclidean by regularity. The moving basepoints are at bounded rescaled distance and hence have a further convergent subsequence; the resulting Euclidean pointed space is isometric to the one pointed at the origin. This proves the assertion for the original sequence as well. The same reasoning applies to normalized measures.

It follows that, for each fixed \(q\), the selected harmonic tuple with threshold \(q\) is available at \((y_k,2Ar_k)\) for every sufficiently large \(k\). Choose \(q_k\leq k\) maximal among the available thresholds, ignoring the finitely many initial scales where there is no choice. Then \(q_k\to\infty\). On \(B(x,Lr_k)\), the resulting Gram matrix \(P_k\) agrees with that of the harmonic tuple. Metric compatibility, bounded gradients, and (11) imply \[ \int_{B(x,Lr_k)} \bigl(|P_k-I|+r_k^2|\nabla P_k|^2\bigr)\,\mathrm dm \leq C_Aq_k^{-1}r_k^n=o(r_k^n). \tag{87}\] Thus the conclusion holds at every sufficiently small dyadic scale, rather than only along a sparse subsequence.

Common precise representatives. Every pairwise inner product of the countably many gradients is a bounded Sobolev function. On the present complete, locally doubling Poincaré space, such functions have \(L^2\)-Lebesgue representatives, given by limits of ball averages outside a set of \(2\)-capacity zero, and these are their quasi-continuous representatives; see (Björn and Björn 2019, Propositions 4.8–4.9). The equal-norm identification of Newtonian and RCD weak-gradient Sobolev spaces follows from (Ambrosio et al. 2013, Theorem 7.4 and Section 8.2); see also (Gigli 2018b). Completeness makes \(M\) its own completion, so the path-openness hypothesis of Proposition 4.9 and the measurable-inclusion qualification in the appended correction are automatic here. To identify the polar sets, let \(C_W\) be the outer Sobolev capacity and \(C_N\) the Newtonian capacity, using the same norm normalization. On an open set \(U\), a Newtonian competitor is a Sobolev competitor of the same norm. Conversely, truncate a Sobolev competitor to \([0,1]\) and take its equal-norm Newtonian representative. This representative equals one at every one of its \(L^2\)-Lebesgue points in \(U\). The exception is \(C_N\)-polar, so resetting it to one there preserves its Newtonian class and norm. Thus the two capacities agree on open sets. Their outer regularity identifies their polar sets, and uniqueness identifies the resulting quasi-continuous representatives. Remove one common polar set for this countable collection. The rational quadratic forms of the Gram matrices are nonnegative almost everywhere, and all their \((n+1)\)-minors vanish almost everywhere. On the bounded range of any finite list of Gram entries, each such polynomial is Lipschitz. Their \(L^2\)-mean convergence at a common Lebesgue point therefore implies that the polynomial averages converge to the polynomial of the precise values. This passes the almost-everywhere positivity and minor identities to these values. The polynomials are themselves bounded Sobolev functions. Removing a further countable union of polar sets makes these identities hold simultaneously in the precise representatives. This proves the positivity and rank assertions outside a single polar set \(N\), which may be taken Borel.

Size of the rank defect. Let \(S\) be the set of regular points outside \(N\) where no \(n\)-tuple is positive definite. The representatives and \(\mathcal R_n\) may be taken Borel, so \(S\) is Borel. Each chosen \(n\)-tuple is singular at every point of \(S\), and hence \(|P_k-I|\geq1\) there. Apply Lemma 33 and (87). For every regular \(x\) and every \(s>n-2\) this gives \[\mathcal H^s_\infty(S\cap B(x,r_k))=o(r_k^s).\] The constants are uniform on each bounded region. Lemma 32, followed by a bounded exhaustion, gives \(\mathcal H^s(S)=0\). In particular we take \(s=n-1\). When \(n=2\) this is \(s=1\); the telescoping series in Lemma 33 has ratio \(2^{-1/2}\), so no borderline exponent is used. ◻

Proof. Put \(T=b-a\) and \(M_\pi=\pi(C([a,b],M))\). Sobolev differentiation along bounded-compression plans and the elementary one-dimensional bound on the supremum of an absolutely continuous function give \[ \int\sup_{t\in[a,b]}|u(\gamma_t)|\,\mathrm d\pi \leq\sqrt{CM_\pi} \bigl(\|u\|_{L^2(m)}+LT\|\nabla u\|_{L^2(m)}\bigr), \tag{88}\] where on the left we initially use the continuous path representative. The chain rule and its representative formulation are part of the RCD Sobolev calculus; see (Gigli 2018b; Savaré 2014).

If \(N\) is polar, choose open neighborhoods \(U_j\supset N\) and \(u_j\in W^{1,2}(M)\) with \(u_j\geq1\) almost everywhere on \(U_j\) and \(\|u_j\|_{W^{1,2}}\to0\). For almost every path, membership in the exceptional null subset of \(U_j\) occupies zero time, by bounded compression and Fubini. A path hitting \(U_j\) has a nontrivial one-sided time interval in \(U_j\). Its continuous representative of \(u_j\) therefore has supremum at least one. Estimate (88) shows that the measure of paths hitting \(N\) is zero.

For the Hausdorff assertion set \(L_*=\max\{1,L\}\). A path hitting \(B(z,r)\) spends at least \(r/L_*\) units of time in \(B(z,2r)\) whenever \(r<L_*T\). This remains true for a hit at an endpoint, using a one-sided interval. Consequently \[\pi\{\gamma:\gamma([a,b])\cap B(z,r)\ne\varnothing\} \leq\frac{CL_*T}{r}m(B(z,2r))\leq C' r^{n-1}.\] The last constant is uniform on the bounded region under consideration. Covering an \(\mathcal H^{n-1}\)-null set by balls with arbitrarily small total \((n-1)\)-cost proves its avoidance.

Finally, approximate each of the specified Sobolev functions sufficiently rapidly in \(W^{1,2}\) by Lipschitz functions. A subsequence converges to its quasi-continuous representative outside a polar set. Estimate (88), applied to successive differences, makes the approximations converge uniformly along almost every path. Those paths avoid the polar set just obtained; their uniform limit is therefore the quasi-continuous representative at every time. It also equals the Sobolev path representative, since they agree for almost every time. Take the countable intersection of the resulting conull sets. ◻

Proof. Almost every path is regular at the endpoints and at every time in a fixed countable dense subset of \((a,b)\). Consider one such nonconstant path and fix an interior time \(t\). Take any sequence of rescalings converging to a tangent \(Y\) at \(\gamma_t\). At a nearby regular time \(t_j\), the same rescalings converge to Euclidean \(n\)-space. Deng’s estimate, valid along every geodesic, implies for each fixed radius \(R\) that \[d_{\mathrm{pGH}}\bigl((B_R(Y),o_Y), (B_R(\mathbb R^n),0)\bigr) \leq C R|t-t_j|^\alpha.\] The constants can be kept fixed while \(t_j\) stays in a compact interior interval containing \(t\); see (Deng 2025, Theorems 1.1–1.2). Letting regular \(t_j\) tend to \(t\) proves that every pointed ball in \(Y\) is Euclidean. The tangent sequence was arbitrary, so \(\gamma_t\in\mathcal R_n\). In the noncollapsed setting the normalized tangent measures are the corresponding Euclidean measures. Constant paths are already covered by regularity at one fixed time. Endpoint regularity came directly from bounded compression, not from the interior estimate. ◻

Proof. Fibers and connection. Lemmas 34 and 35 ensure that, along almost every entire path, the common Gram representatives have rank \(n\) at every time. Sobolev differentiation of their entries gives \[ \frac{d}{dt}\langle e_i,e_j\rangle =\langle\nabla_v e_i,e_j\rangle +\langle e_i,\nabla_v e_j\rangle. \tag{90}\] All terms are in \(L^2\) in time along almost every path, since the fields are bounded, their covariant derivatives are in \(L^2(m)\), and speed and compression are bounded. Countability permits these assertions simultaneously for every pair of specified fields.

One concrete definition of a fiber is the linear span of the symbols \(e_j\) modulo the kernel of their Gram form. All finite Gram matrices have rank at most \(n\), and some \(n\)-tuple is positive definite, so this fiber has dimension \(n\). Continuity of Gram entries makes a positive-definite \(n\)-tuple a frame on an open time interval. Compactness of \([a,b]\) gives a finite subdivision such that each closed piece lies in a frame interval and its inverse Gram matrix is bounded. Frame changes have \(W^{1,2}\) coefficients, as follows by multiplying cross-Gram entries by the inverse Gram matrix.

At almost every time these fibers are the original tangent fibers and the frame spans the tangent space. Thus \(\nabla_v e_i\) has \(L^2\) coefficients in a frame, and (89) is meaningful. To verify that it is independent of a representation, suppose \(W=\sum_i a_ie_i\) is zero. Its pairing with each frame vector is identically zero. Differentiate that pairing using (90); the term pairing \(W\) with the derivative of the frame vector is zero almost everywhere. It follows that the right-hand side of (89) pairs to zero against a basis almost everywhere and hence vanishes. The same calculation proves metric compatibility. In each frame the parallel equation is a linear ODE with \(L^2\), hence \(L^1\), coefficients. Its absolutely continuous solutions exist uniquely. Metric compatibility preserves their pairings, and gluing at the subdivision endpoints constructs an orthonormal parallel frame on \([a,b]\).

Parallel velocity. Put \(T=b-a\). Apply (Gigli and Tamanini 2021, Theorem 5.13(i)) to the bounded-support optimal geodesic test plan after the affine reparameterization \(\widetilde\gamma_\tau=\gamma_{a+T\tau}\). The functions \(g_i\) are in \(H^{2,2}\), as required there. If \(\nabla\widetilde\varphi_\tau\) is its Kantorovich velocity, the first differentiation identity identifies the actual velocity in the original parameter as \(v_t=T^{-1}\nabla\widetilde\varphi_{(t-a)/T}\) along the plan almost everywhere. The second differentiation identity has a factor \(T^2\) on both sides, giving, for \(e_i=\nabla g_i\), \[ \frac{d}{dt}\langle e_i,v\rangle =\operatorname{Hess}g_i(v,v)=\langle\nabla_v e_i,v\rangle. \tag{91}\] Its identity in \(L^2(\pi)\) yields the pathwise distributional identity by Fubini; its right-hand side is in \(L^2(dt\,d\pi)\) by bounded speed and compression. Equivalently one may apply the theorem to boundedly weighted restrictions of the plan, which remain optimal. Choose jointly measurable representatives \(u_i(t,\gamma)=\langle e_i,v\rangle(\gamma_t)\) and \(f_i(t,\gamma)=\operatorname{Hess}g_i(v,v)(\gamma_t)\) in \(L^2(dt\,d\pi)\). Their continuous scalar representatives can be specified by \[ u_i^*(t,\gamma)=\frac1T\int_a^b u_i(s,\gamma)\,\mathrm ds +\int_a^t f_i(r,\gamma)\,\mathrm dr -\frac1T\int_a^b(b-r)f_i(r,\gamma)\,\mathrm dr. \tag{92}\] This formula is jointly measurable. Along almost every path it belongs to \(W^{1,2}\), has derivative \(f_i\), and has the same average as \(u_i\). It therefore represents the original scalar Sobolev class. Countability gives these assertions on one conull path set.

In a positive-Gram frame put \(c=P^{-1}u^*\), where \(u^*\) is the vector of the corresponding scalar representatives. The coefficients \(c\) are \(W^{1,2}\). Write \(\nabla_v e_i=\sum_j C_{ji}e_j\). Metric compatibility and (91) give \[P'=C^{\mathsf T}P+PC,\qquad u^*=Pc,\qquad (u^*)'=C^{\mathsf T}Pc.\] Differentiation yields \(c'+Cc=0\), proving \(\mathcal Dv=0\) from the scalar Sobolev identities. Its almost-everywhere norm is the constant metric speed, so continuity gives the same norm at every time.

Measurable endpoint traces. The Gram identities defining the endpoint fibers agree with the actual tangent-module identities outside an \(m\)-null set. Endpoint compression excludes this set. Countability of the family makes the identification simultaneous for all specified fields. The endpoint vector can also be chosen measurably: enumerate the countable \(n\)-tuples and, at \(b\), select the first tuple \(I=(i_1,\ldots,i_n)\) with positive Gram determinant. This defines a measurable countable partition of the good paths. In a measurable realization of the tangent module, set \[ v_b=\sum_{\alpha=1}^n \bigl(P_I(b,\gamma)^{-1}u_I^*(b,\gamma)\bigr)_\alpha e_{i_\alpha}(\gamma_b). \tag{93}\] This is measurable. The chosen minor remains positive on a neighborhood of \(b\) along that path, so the construction agrees with the continuous pathwise velocity just obtained. All its other specified gradient pairings equal \(u_j^*(b)\) by continuity, and its norm equals the path speed. The same construction applies at \(a\). Thus the later endpoint \(L^2(\pi)\) pairings use a measurable bounded vector defined by scalar traces, independently of any choice of values for the original time-equivalence class. ◻

Proof. Choose a smooth scalar \(\eta\) with \(\eta(\delta)=0\) and \(\eta(1)=1\). For every interval with rational endpoints compactly contained in \((\delta,1)\), choose a countable subset of \(C_c^\infty\) dense in its \(W^{1,2}_0\) space. Let \(\mathcal A\) be their union and its rational linear span. Define \(\mathcal Y_{\delta,Z}\) to consist of \[\eta(t)Z+\sum_{\ell=1}^k a_\ell(t)e_{j_\ell}, \qquad a_\ell\in\mathcal A,\] with arbitrary finite lists of indices. This is countable and has the required global endpoint identities.

Work on a path satisfying Proposition 36. Subtract \(\eta Z\) from \(W\). In a global parallel frame the resulting field has \(W^{1,2}_0\) coefficients. Cutting it off near \(\delta\) and \(1\) converges in \(W^{1,2}\); this is the usual zero-trace approximation on an interval. A smooth temporal partition of unity then expresses the remaining compactly supported field in finitely many of the local test-gradient frames. Refine to intervals with rational endpoints whose closures remain in these frame intervals. The coefficient functions belong to \(W^{1,2}_0\) on the corresponding intervals. Approximate them by elements of the chosen countable dense subsets. Such approximation converges uniformly as well as in \(W^{1,2}\) on an interval. Uniform convergence controls multiplication by the \(L^2\) connection coefficients in (89). The fields therefore converge in the stated norm. ◻

Proof. Choose a compactly supported Lipschitz function \(\widetilde q\) equal to \(q\) on a neighborhood of \(\overline{B(p,R_\kappa)}\), and compact test functions \(h_j\to\widetilde q\) in \(W^{1,2}\). Include this deterministic countable sequence in the family of Proposition 31 before applying the pathwise statements. Test-function density here follows from heat regularization and good cutoffs; see (Gigli 2018b).

Let \(\pi\) denote the radial restriction above, and fix one \(\delta<1\). If \((e_t)_\#\pi\leq C m\) on \([\delta,1]\), then for any Lipschitz \(e\in W^{1,2}\) and \(0<h<1-\delta\), Sobolev differentiation and Jensen’s inequality give \[ \left\|\frac{e(\gamma_1)-e(\gamma_{1-h})}{h} \right\|_{L^2(\pi)}^2 \leq\frac{R_\kappa^2}{h}\int_{1-h}^1 \int|\nabla e|^2(\gamma_t)\,\mathrm d\pi\,\mathrm dt \leq CR_\kappa^2\|\nabla e\|_{L^2(m)}^2. \tag{96}\] For each fixed \(j\), the continuous representative of \(\langle\nabla h_j,v\rangle\) gives \[\frac{h_j(\gamma_1)-h_j(\gamma_{1-h})}{h} \longrightarrow\langle\nabla h_j(\gamma_1),v(1)\rangle \quad\text{in }L^2(\pi).\] Dominated convergence applies with bound \(R_\kappa\operatorname{Lip}(h_j)\); no uniform bound in \(j\) is required. Terminal compression and \(|v(1)|\leq R_\kappa\) also imply \[\|\langle\nabla(\widetilde q-h_j)(\gamma_1),v(1)\rangle \|_{L^2(\pi)} \leq R_\kappa\sqrt C\, \|\nabla(\widetilde q-h_j)\|_{L^2(m)}.\] On the radial paths the quotient for \(\widetilde q\) is exactly \((1-h/2)r(\gamma_1)^2\). Apply (96) to \(e=\widetilde q-h_j\), let \(h\downarrow0\) first, and then let \(j\to\infty\). This proves \[\langle\nabla q(\gamma_1),v(1)\rangle=r(\gamma_1)^2.\] Locality justifies replacing \(\widetilde q\) by \(q\) at the terminal points. Since \(|\nabla q|\leq r\) and \(|v(1)|=r(\gamma_1)\), equality in Cauchy–Schwarz proves (95), including the zero-speed case. The terminal measure is \(m\) on the ball, giving the last assertion. ◻

Proof. Use the radial data (94). For the fixed \(Z\), adjoin its potential and the approximations in Lemma 39 to the countable family. Fix \(0<\delta<1\). For every trial (45) satisfying the global endpoint identities \(Y_\delta=0\) and \(Y_1=Z\), Theorem 18 gives \[ \operatorname{Hess}q(Z,Z)(f)\leq\int_{B(p,R_\kappa)} f(x)E_Y(x)\,\mathrm dm(x), \tag{98}\] where \(f\geq0\) is compactly supported and Lipschitz in the ball, the common calibrated path ending at \(x\) is denoted by \(\gamma^x\), and \[ E_Y(x)=\int_\delta^1 \bigl(|\partial_tY+\nabla_vY|^2 -\kappa(|Y|^2|v|^2-\langle Y,v\rangle^2) \bigr)(\gamma^x_t)\,\mathrm dt. \tag{99}\] The common terminal prescription is essential here: the right-hand side is integration against a function of \(x\), without averaging over several paths ending at \(x\). The velocities agree under reweighting of these paths, since their pairings with test gradients are the derivatives of the same test functions along the paths. Bounded compression of \(\pi_\chi\), bounded speed, and the \(L^2\) covariant derivatives of each fixed trial show that \(E_Y\in L^1_{\mathrm{loc}}(B(p,R_\kappa),m)\).

We first extract a measure using one fixed trial, for example \(Y_t=(t-\delta)Z/(1-\delta)\), and denote its energy by \(E_0\). The functional \[f\longmapsto\int E_0 f\,\mathrm dm-\operatorname{Hess}q(Z,Z)(f)\] is nonnegative on nonnegative compact Lipschitz functions. On a fixed compact subset, comparison with a Lipschitz cutoff equal to one there bounds its absolute value by a constant times \(\|f\|_\infty\). Density in \(C_c\) and the Riesz representation theorem therefore give a positive Radon measure \(\nu\) representing it. Thus \[\mu:=\operatorname{Hess}q(Z,Z)=E_0m-\nu\] is a signed Radon measure, with nonpositive singular part. Its representation of the fixed functional is unique, so the measure does not depend on the initially chosen trial or on \(\delta\). Write \(\mu=h_Zm+\mu^{\mathrm s}\) for its Lebesgue decomposition.

Apply (98) to each member of the countable collection \(\mathcal Y_{\delta,Z}\) from Lemma 37. Each inequality says that \(E_Ym-\mu\) is a positive Radon measure. Consequently \[ h_Z(x)\leq\inf_{Y\in\mathcal Y_{\delta,Z}}E_Y(x) \quad\text{for }m\text{-a.e. }x. \tag{100}\] The null set can be chosen simultaneously because the list is countable. The same is true for the pathwise calculus under \(\pi_\chi\) and its radial restriction.

On such a path, \(v\) is parallel and has norm \(r=r(x)\). Lemma 39 identifies its terminal direction with \(\nabla q(x)\). Parallel translate the two components \(Z_\parallel(x)\) and \(Z_\perp(x)\) to obtain parallel fields \(P_\parallel\) and \(P_\perp\) on \([\delta,1]\). For \(r>0\) take the pathwise field \[W_t=\frac{t-\delta}{1-\delta}P_\parallel(t) +\frac{\mathfrak s_\kappa((t-\delta)r)} {\mathfrak s_\kappa((1-\delta)r)}P_\perp(t).\] Its initial value is zero and its terminal value is \(Z(x)\). The denominator is positive by the choice of \(R_\kappa\). At \(r=0\) use the linear coefficient on the whole field. Lemma 37 approximates \(W\) in the pathwise norm by members of \(\mathcal Y_{\delta,Z}\). The energy (99) is continuous in this norm because \(v\) is bounded. Hence the infimum in (100) is bounded above by the energy of \(W\). Only this pathwise upper bound on a countable infimum is needed; no measurable selection of the fields \(W\) or their approximating sequences is required.

The parallel component contributes \(|Z_\parallel|^2/(1-\delta)\). If \(j(t)=\mathfrak s_\kappa((t-\delta)r)/ \mathfrak s_\kappa((1-\delta)r)\), then \(j''+\kappa r^2j=0\), \(j(\delta)=0\), and \(j(1)=1\). Thus the transverse contribution is \[|Z_\perp|^2\int_\delta^1(j'^2-\kappa r^2j^2)\,\mathrm dt =|Z_\perp|^2[jj']_\delta^1 =r\frac{\mathfrak s_\kappa'((1-\delta)r)} {\mathfrak s_\kappa((1-\delta)r)}|Z_\perp|^2.\] We have proved \[h_Z\leq\frac{|Z_\parallel|^2}{1-\delta} +r\frac{\mathfrak s_\kappa'((1-\delta)r)} {\mathfrak s_\kappa((1-\delta)r)}|Z_\perp|^2\] almost everywhere, with the continuous zero-speed value \(|Z|^2/(1-\delta)\). Choose a countable sequence \(\delta\downarrow0\). The same density \(h_Z\) occurs for every \(\delta\), so outside their common null set we may pass to the limit pointwise. Together with \(\mu^{\mathrm s}\leq0\), this gives (97) as a measure inequality and therefore against every indicated Lipschitz weight. ◻

Proof. Write \(r=d(p,\cdot)\) and \(q=r^2/2\), and fix \(0<a<b<R_*\). Choose a smooth nonincreasing function \(\theta\colon[0,\infty)\to[0,1]\), equal to one on \([0,a^2/2]\) and zero on \([b^2/2,\infty)\), and set \[\chi(0)=0,\qquad \chi'(q)=\theta(q)\frac{\mathfrak s_\kappa(\sqrt{2q})}{\sqrt{2q}},\qquad F_p=\chi(q).\] The quotient has a smooth extension at zero by its power series. Consequently \(F_p\) is bounded and globally Lipschitz, is constant outside \(B_b(p)\), and has the first value in (103). It need not lie in \(L^2(m)\); boundedness and a bounded gradient are the hypotheses needed for the weight.

For a compact test gradient \(Z=\nabla g\), the definition (8) and the chain rule give \[ H_{F_p}(Z,Z)(h) =H_q(Z,Z)(h\chi'(q)) +\int h\chi''(q)\langle\nabla q,Z\rangle^2\,\mathrm dm. \tag{104}\] This identity involves only first derivatives of \(q\), so it holds for the distributional Hessian. Local cutoffs keep its first term inside \(B_{R_*}(p)\). For \(r>0\), put \(Z_r=\langle\nabla q,Z\rangle/r\); then \(|Z_r|\le|Z|\), since the Lipschitz bound for \(r\) gives \(|\nabla q|\le r\) almost everywhere. The Hessian comparison of Theorem 40 and \(\chi'\ge0\) bound (104) above by \[\int h\left[A(r)|Z_r|^2 +B(r)(|Z|^2-|Z_r|^2)\right]\,\mathrm dm,\] where, after cancellation of the model-distance factors, \[\begin{align*} A(r)&=\chi'(r^2/2)+r^2\chi''(r^2/2) =\theta(r^2/2)\mathfrak s'_\kappa(r)+r\theta'(r^2/2)\mathfrak s_\kappa(r),\\ B(r)&=\theta(r^2/2)\mathfrak s'_\kappa(r). \end{align*}\] The nonpositive singular part of \(H_q(Z,Z)\) remains nonpositive on multiplication by \(\chi'\). Both coefficients extend continuously with \(A(0)=B(0)=1\), and vanish for \(r\ge b\). Since \(\theta'\le0\) and \(\mathfrak s_\kappa\ge0\) on \([0,b]\), we have \(A\le B\): flattening the distance adds a nonpositive radial term. Hence the bounded continuous function \[G_p(x)=\theta\bigl(r(x)^2/2\bigr)\mathfrak s'_\kappa\bigl(r(x)\bigr)\] gives (101) globally. On \(r<a\), it equals \(\mathfrak s'_\kappa(r)=1-\kappa F_p\), as required. ◻

Proof. Theorems 18 and 40 supply the hypothesis of Lemma 42. Apply Theorem 41 to \(F_p,G_p\). For every geodesic \(\sigma\colon[0,1]\to B_a(p)\) of length \(\ell\), (103) yields \[ y''+\kappa\ell^2y\le\ell^2, \qquad y(t)=\mathfrak m_\kappa(d(p,\sigma_t)). \tag{105}\]

Consider a geodesic triangle \([pxy]\) of perimeter less than \(a\), and let \(\sigma\) be its chosen side from \(x\) to \(y\). The triangle inequality along the two routes through the vertices gives \(d(p,\sigma_t)<a\), so (105) holds on this entire side. The comparison triangle \([\bar p\bar x\bar y]\) in the two-dimensional model of curvature \(\kappa\) exists at this scale. For its corresponding side \(\bar\sigma\), the model cosine law gives \[\bar y(t)=\mathfrak m_\kappa(d(\bar p,\bar\sigma_t)),\qquad \bar y''+\kappa\ell^2\bar y=\ell^2, \quad \bar y(0)=y(0),\quad\bar y(1)=y(1).\] This also holds for degenerate triangles by continuity. Let \(h=y-\bar y\). It belongs to \(H^1_0(0,1)\), and \(-h''-\kappa\ell^2h\ge0\) in distributions. Testing with the negative part \(h_-\), by nonnegative \(H^1_0\) approximation, gives \[\|(h_-)'\|_2^2\le\kappa\ell^2\|h_-\|_2^2.\] If \(\kappa\le0\), this forces \(h_-=0\). If \(\kappa>0\), our scale choice gives \(\kappa\ell^2<\pi^2\), and the Dirichlet Poincaré inequality gives the same conclusion. Since \(\mathfrak m_\kappa\) is strictly increasing on the distances under consideration, \[ d(p,\sigma_t)\ge d(\bar p,\bar\sigma_t) \qquad(0\le t\le1). \tag{106}\] Thus every sufficiently short triangle, with any chosen sides, satisfies point-on-side comparison.

We spell out the localization for the final geometric step. Fix \(o\in M\) and choose \(r>0\) with \(20r<a\). The local converse to point-on-side comparison gives the four-point \(\mathrm{CBB}(\kappa)\) inequality on \(B_r(o)\), as follows. Point-on-side comparison makes the comparison angle of a short hinge nonincreasing in either radial parameter. Its limit defines the angle, which is at least the comparison angle. The adjacent-angle inequality follows from the model gluing lemma. For a quadruple \(p,x,y,z\in B_r(o)\), take \(w\in(p,z)\) with \(d(p,w)<r\). The radial legs from \(w\) to these points have length at most \(3r\), so every triangle used to compare points on these legs has perimeter at most \(12r<a\). Write \(\widetilde\angle_\kappa(w;x,y)\) for the comparison angle at \(w\) in the triangle \(wxy\). For the actual angles at \(w\), insert the direction \(wp\) between \(wx\) and \(wy\) in the angle triangle inequality. The resulting terms pair with \(wz\) into two adjacent-angle sums, each at most \(\pi\). Each comparison angle is at most its actual angle, so \[\widetilde\angle_\kappa(w;x,y) +\widetilde\angle_\kappa(w;y,z) +\widetilde\angle_\kappa(w;z,x)\le2\pi.\] Letting \(w\to p\), continuity of the comparison angles gives the same inequality with \(w\) replaced by \(p\). This is the local four-point inequality, obtained by localizing the converse proof in (Alexander et al. 2024, Theorem 8.14). All segments here are ambient minimizing segments; no convexity of a metric ball is needed. Hence \(M\) is locally \(\mathrm{CBB}(\kappa)\) by the local four-point criterion in (Alexander et al. 2024, Theorem 8.30, condition (1)). Completeness and the length-space property allow the globalization theorem (Alexander et al. 2024, Theorem 8.31), so \(M\) is \(\mathrm{CBB}(\kappa)\).

Finally, \(m=\mathcal H^n\) is finite on bounded sets and has full support. Every nonempty bounded ball therefore has positive finite \(n\)-dimensional Hausdorff measure, and hence Hausdorff dimension \(n\). A countable exhaustion by such balls shows \(\dim_H M=n\). This is the dimension of the resulting Alexandrov space. ◻