A finite selection from outer measure.

Fix a bounded open graph-slope chart of directions, with slopes \(s\in\mathbb R^3\), writing spatial lines as \[x(t)=b+ts,\qquad x=(y,w).\] A unit segment in one of these directions has a nondegenerate time projection. Every such segment contains the graph over some interval with rational endpoints, and its intercept belongs to a bounded box with integer endpoints. These are countably many choices. Countable outer subadditivity therefore supplies one interval and one intercept box for which the set \(S_0\) of eligible slopes has positive outer Lebesgue measure, say \(a>0\). After a fixed affine coordinate change, the common time interval is \([0,1]\), and all these slopes and intercepts lie in fixed bounded boxes. This coordinate change preserves Hausdorff dimension. No direction-to-witness map has been chosen.

Choose \(0<\sigma<1\) with \(\dim_{\mathrm H}K<4-\sigma\). There are covers by balls of arbitrarily small maximum radius with \[\sum_i r_i^{\,4-\sigma}\le1.\] Only balls meeting the fixed bounded region containing the eligible graph segments are needed for the following argument; their centers are bounded as well. For \(k\ge1\), let \(U_k\) be the union of balls with \(2^{-k-1}<r_i\le2^{-k}\). There are at most \(C2^{k(4-\sigma)}\) such balls.

Choose a fixed \(c>0\) so that \(\sum_{k\ge1}ck^{-2}<1\). Every eligible witness has some \(k\) for which \[|\{t\in[0,1]:b+ts\in U_k\}|\ge ck^{-2},\] because the sum of these time measures is at least one. Define \(S_k\subset S_0\) by existence of a bounded witness with this property. The sets \(S_k\) need not be measurable, but \(S_0\subset\bigcup_kS_k\). Choose \(c'>0\) with \(c'\sum_{k\ge1}k^{-2}<1\). Outer subadditivity implies that for some \(k\) \[ |S_k|^*\ge c'a k^{-2}. \tag{24}\] Indeed the reverse strict inequality for every \(k\) would imply \(|S_0|^*<a\). The same scale thus has both the time-fraction condition defining \(S_k\) and [a01:productive-slopes]. As the maximum cover radius tends to zero, every possible such \(k\) tends to infinity.

Put \(\delta=2^{-k}\). A maximal \(\delta\)-separated subset of the bounded set \(S_k\) is finite and covers \(S_k\) by \(\delta\)-balls. Its cardinality \(M\) therefore satisfies \[ c k^{-2}\delta^{-3}\le M\le C\delta^{-3}. \tag{25}\] Choose one successful witness for each of these finitely many slopes. This is the only witness choice needed, and uses no measurable selection.

Smoothing and three density bounds.

Let \(N_0=\delta^{-1}\), initially \(L=1\), \(P_*=w\), and \(B=1\). Use the uniform mixture of the selected lines and perturb all three intercept and all three slope coordinates by independent uniform boxes of radius a fixed small multiple of \(\delta\). Select incidences in fixed enlargements of the balls in \(U_k\). Each original covered time stays in those enlargements under every allowed perturbation, so selected mass is at least \(ck^{-2}\).

This is a tempered version-\(2\) problem. The mixture may be represented by the internal indexed copies of 3; its count and density heights are polynomial, its supports are boxes, and the finite union of enlarged balls is a bounded-degree polynomial predicate of polynomial size. The separated slope centers and [a01:finite-slope-packing] give the full three-slope density ceiling \[ \frac{d\nu_s}{ds}\le Ck^2\le K_0. \tag{26}\] Conditional on the full slope, the \(y\)-intercept density is at most \(C\delta^{-2}\): within each mixture component this follows from independent intercept smoothing, and conditional mixture weights sum to one.

Homogenize jointly for \(0\le r\le1\), \(0\le u\le4\), where \(u\) is the depth of the two-dimensional \(V\)-bin. Write \[n(r,u)=N_0^{-f(r,u)},\qquad m(r,u)=N_0^r n(r,u),\qquad g(r,u)=r-f(r,u).\] The resulting limiting exponents obey \[ g(1,0)\ge\sigma,\qquad g(0,0)\le0,\qquad g(r,u)\le r+2-2u. \tag{27}\]

For the first inequality, at width \(\delta\) each enlarged covering ball uses \(O(1)\) hidden \(w\)-bins over a base \((t,y)\)-volume \(O(\delta^3)\). The total observation-space measure of these states is at most \(C\delta^{\sigma-1}\). The selected mass is inverse-subpower and the typical density is \(n(1,0)\), so \(n(1,0)\ge K_0^{-1}\delta^{1-\sigma}\). This gives \(g(1,0)\ge\sigma\).

For the third inequality, the \(V\)-marginal has bounded region and density at most \(K_0\), by integrating [a01:original-slope-cap] over the bounded remaining slope. A \(V\)-bin has probability at most \(K_0N_0^{-2u}\). At fixed \(t\), the conditional \(y(t)\)-density, given the full slope, is at most \(C\delta^{-2}\). Ignoring the hidden \(w\)-bin gives the pointwise joint ceiling \(K_0N_0^{2-2u}\), hence the claimed bound for \(g(r,u)\).

For the second inequality, the projected \(y\)-lines have a full-time plank cap at every fine mesh \(\rho=D_m^{-1}\): \[ \nu\{l:y_l([0,1])\subset\mathcal P\} \le K_0ab\rho^2 \tag{28}\] for spatial radii \(a\rho,b\rho\). Containment at \(t=0,1\) confines the velocity to the corresponding rectangle of radii \(2a\rho,2b\rho\), and its density is at most \(K_0\).

Suppose instead that \(n(0,0)\) grows by a fixed positive power. Choose \(\alpha>0\) smaller than that power, and flatten at a sufficiently fine fixed polynomial mesh \(\rho\sim N_0^{-D}\). Retain typical cells. Their projected base cell masses have the lower bound \(N_0^\alpha\rho^3\), after decreasing \(\alpha\) if needed; the bounded \(w\)-range costs only subpower many unit bins. There are therefore at most \(K_0N_0^{-\alpha}\rho^{-3}\) visited base cells. Prune short labeled durations using the product prior, leaving inverse-subpower total mass and duration at least \(K_0^{-1}\) per used trace. Write \(m_{\mathrm{ret}}\ge K_0^{-1}\) for the retained incidence mass, and mark the visited time bins of each trace. Each mark covers at most \(\rho\) units of time. In the finite weighted approximation used for 16, this gives \[\sum_i\omega_i\#J_i\ge\frac{m_{\mathrm{ret}}}{\rho}.\] Dividing by the visited-cell bound, and enlarging the subpower factor, yields the raw weighted average multiplicity \[\frac{\sum_i\omega_i\#J_i}{\#E_{\mathrm{vis}}} \ge\frac{m_{\mathrm{ret}}/\rho} {K_0N_0^{-\alpha}\rho^{-3}} \ge K_0^{-1}N_0^\alpha\rho^2.\] On the other hand, [lem:weighted-planks,a01:projected-full-plank-cap] bound it by \(C_\varepsilon K_0D_m^\varepsilon\rho^2\). Choose \(D\varepsilon<\alpha/2\). This is impossible for large \(N_0\), proving \(g(0,0)\le0\).

Selecting an endpoint and an angular scale.

The bounds on \(g\) force positive growth between hidden depths zero and one and give an upper bound at large velocity depth. We use both facts to select an endpoint with the one-sided comparisons required by the cap class. Choose small constants \(R,S>0\), for example with \(R+4S<\sigma/2\) and \(R,S<1/10\), and maximize \[ g(r,u)-Rr+Su \quad\hbox{on }[0,1]\times[0,4]. \tag{29}\] Continuity follows from homogenization. At a maximizer \((r_0,u_0)\), we have \[ r_0>0,\qquad u_0<4,\qquad g(r_0,u_0)\ge\sigma/2. \tag{30}\] Indeed the value at \((1,0)\) is at least \(\sigma-R\). At \(r=0\), monotonicity in \(u\) and \(g(0,0)\le0\) bound the value by \(4S\). At \(u=4\), the last inequality in [a01:initial-density-inequalities] bounds it by \(r-6-Rr+4S<0\). Finally maximality gives \(g(r_0,u_0)\ge\sigma-R-4S\).

Partition the \(y\)-intercept and \(V\) coordinates into full-time boxes \(\mathcal B\) of side \(a_{u_0}\), with dyadic rounding. Make them worlds with conditional trajectory priors and weights proportional to their original probabilities \(\nu(\mathcal B)\). Discard extremely light boxes and homogenize their weights. Subtract the central affine \(y\)-line \(y_{\mathcal B}(t)\) and divide \(y\) by \(a_{u_0}\), retaining \(w\). At fixed exact \((t,y)\), the old \(V\)-bin determines the intercept bin up to boundedly many possibilities, since \(y_0=y-tV\). Thus appending the crop label \(\mathcal B\) to \((p_s,[V]_{a_{u_0}})\) is a bounded-list refinement at every depth \(s\).

For clarity, let \(\rho_{\mathcal B}(s)\) be the density of this refined observation in the old, unnormalized law. Time is unchanged, the base change \(y=y_{\mathcal B}(t)+a_{u_0}y_{\mathrm{new}}\) has Jacobian \(a_{u_0}^2\), and conditioning the trajectory prior divides by \(\nu(\mathcal B)\). For transported bins the within-world density is therefore exactly \[\rho_{\mathrm{new},\mathcal B}(s) =\frac{a_{u_0}^2}{\nu(\mathcal B)}\rho_{\mathcal B}(s).\] This is the specialization of [a01:exact-density-transform] to a full-time crop. The actual rounded grids have bounded overlap. Combining that overlap and the bounded crop-label ambiguity with 4 gives the typical equivalence, for \(0\le s\le r_0\), \[ n_{\mathrm{new}}(s)\sim_{\log} n(s,u_0)\frac{a_{u_0}^2}{\nu(\mathcal B)}. \tag{31}\] A new angular bin of depth \(u\) corresponds, with the same bounded-overlap convention, to the old joint angular refinement at depth \(u_0+u\).

Maximality in [a01:endpoint-maximization] gives \[\begin{align*} f(r_0,u_0)-f(s,u_0) &\le(1-R)(r_0-s),&&0\le s\le r_0, \tag{32}\\ f(r_0,u_0+u)-f(r_0,u_0) &\ge Su,&&0\le u\le4-u_0. \end{align*}\] Consequently the new exact problem, with length \(L=r_0\), belongs to \(C(1-R)\) for the chosen \(S\) and, for example, \(\eta=(4-u_0)/(2r_0)>0\).

It remains to prove positive horizon. We return a true terminal chart’s mass floor to the original prior, then bound the prior mass of all trajectories that can satisfy its geometric tests. Comparing these bounds will force its time interval to be short.

The original-prior floor of a terminal chart.

Consider any true terminal chart in this new problem, including one obtained after a further permitted selection or on a subsequence. Let its time length be \(q\), its spatial Jacobian in the cropped coordinates be \(J_{\mathrm{new}}\), and set \[J_{\mathrm{phys}}=a_{u_0}^2J_{\mathrm{new}}.\] Multiplying its true floor by \(\nu(\mathcal B)\) and using [a01:crop-density], its selected mass in the original, unnormalized prior satisfies \[ \frac{m_{\mathrm{old},c}}q \ge K_0^{-1}n(r_0,u_0)J_{\mathrm{phys}} =K_0^{-1}m(r_0,u_0)a_{r_0}J_{\mathrm{phys}}. \tag{33}\] This calculation cancels \(\nu(\mathcal B)\) even when the crop has a very small polynomial mass. Neither the crop prior nor the original prior is renormalized to the analytically successful incidences.

Every trajectory contributing positive selected time to this floor satisfies the whole-block spatial, small-value, and upper-derivative tests in the original coordinates, and satisfies \(|P_w|\ge K_0^{-1}\) at some time. The original prior mass of these trajectories is at least \(m_{\mathrm{old},c}/q\). Unused trajectories in the chart assignment are not needed for this lower bound.

The slope-volume bound.

Fix this chart. Let \(\Sigma\subset\mathbb R^3\) consist of the bounded slopes \(s\) for which some bounded intercept \(b\) satisfies \[\sup_{t\in I}|D_{\mathrm{phys}}^{-1} (y_{b,s}(t)-y_c(t))|\le K_0,\qquad \sup_{t\in I}|P(t,b+ts)|\le K_0a_{r_0},\] and \[\sup_{t\in I}|P_w(t,b+ts)|\le K_0,\qquad \sup_{t\in I}|P_w(t,b+ts)|\ge K_0^{-1}.\] Here \(P(t,b+ts)\) means that the three spatial coordinates are split as \((y,w)\), and \(D_{\mathrm{phys}}\) is the chart’s original spatial \(y\)-matrix. Use fixed bounded slope and intercept boxes containing the original smoothed prior support.

The set \(\Sigma\) is defined by these geometric tests alone. We impose neither crop membership, finite witness identity, density selection, nor selected-time conditions. Thus \(\Sigma\) contains the slopes of all trajectories contributing to [a01:original-chart-floor].

We claim that \[ |\Sigma|\le K_0\frac{a_{r_0}J_{\mathrm{phys}}}{q^3}. \tag{34}\] The original full-slope density ceiling will then convert this geometric bound into an upper bound for the prior mass of an eligible chart. To prove the claim, we select one eligible intercept for each slope and compare the volume swept out by these lines with the volume allowed by the chart tests.

The definition of \(\Sigma\) uses a fixed number of real variables and fixed-degree polynomial conditions, with quantification over \(b\) and \(t\). All chart coefficients, bounds, and interval endpoints are parameters. Quantifier elimination, semialgebraic choice, and \(C^1\) cell decomposition therefore provide an eligible choice \(b=b(s)\) whose graph has uniformly bounded formula complexity; its \(C^1\) full-dimensional open pieces cover \(\Sigma\) up to a null set. We use here the parameter-uniform forms of these foundational results from real algebraic geometry; see (Basu et al. 2006). Their bounds depend on the formula format, not on the numerical magnitudes of the chart coefficients.

For each \(t\), put \(F_t(s)=b(s)+ts\). On a \(C^1\) piece, \[ \det DF_t(s)=\det(Db(s)+t\,\mathrm{Id}), \tag{35}\] a monic cubic in \(t\). There is also a uniform bound on the number of regular preimages of any \(F_t\). Indeed the fiber has fixed semialgebraic complexity, uniformly in \(t\), the target point, and the chart parameters, so has a uniformly bounded number of connected components. A preimage with nonzero Jacobian is isolated in its fiber by local inversion inside its open \(C^1\) piece, and hence is a separate component. This proves the bound without any bound on \(Db\), and excludes a multiplicity growing with \(N_0\).

Times with a large Jacobian and derivative.

For a fixed slope \(s\), remove from \(I\) intervals of radius \(cq\) around the real parts of the three complex roots of the monic cubic in [a01:monic-jacobian]. The total removed length is at most \(6cq\). Outside them each factor has modulus at least \(cq\), so \[|\det DF_t(s)|\ge c^3q^3.\] Also, \(g_s(t)=P_w(t,b(s)+ts)\) is real affine and has \(\sup_I|g_s|\ge K_0^{-1}\). For small fixed \(\epsilon>0\), \[ |\{t\in I:|g_s(t)|<\epsilon/K_0\}| \le4\epsilon q. \tag{36}\] To verify this, let \(M=\sup_I|g_s|\). If the variation of \(g_s\) on \(I\) is less than \(M/2\), then \(|g_s|\ge M/2\) throughout. Otherwise its slope has modulus at least \(M/(2q)\), and the inverse image of \((-\epsilon/K_0,\epsilon/K_0)\) has length at most \(4\epsilon q\).

Choose \(c,\epsilon\) so that each bad set has length at most \(q/4\). Let \(\Sigma_t\) be the slopes in the open \(C^1\) pieces passing both tests. Then \[ \int_I|\Sigma_t|\,dt\ge\frac q2|\Sigma|. \tag{37}\] The two good-time conditions therefore overlap uniformly for each slope; no independence of them is asserted.

The area formula and the uniform regular-fiber bound give \[\begin{align*} q^4|\Sigma| &\lesssim q^3\int_I|\Sigma_t|\,dt \lesssim \int_I|F_t(\Sigma_t)|\,dt\tag{38}\\ &\le K_0qJ_{\mathrm{phys}}a_{r_0}. \tag{39}\end{align*}\] For the last inequality, the allowed \(y\)-region has area at most \(K_0J_{\mathrm{phys}}\). At fixed \(t,y\), the conditions \[|P(t,y,w)|\le K_0a_{r_0},\qquad |P_w(t,y,w)|\ge\epsilon/K_0\] confine \(w\) to total length at most \(K_0a_{r_0}\), after enlarging the subpower factor. This is 10, applied at fixed \((t,y)\). Integrate this spatial-volume bound over \(I\). Dividing [a01:selector-volume] by \(q^4\) proves [a01:eligible-slope-volume].

Comparison with the mass floor.

Use the original full-slope density ceiling [a01:original-slope-cap]. The original prior mass of all geometrically eligible trajectories is at most \(K_0|\Sigma|\). Combining this with [a01:original-chart-floor,a01:eligible-slope-volume] and cancelling \(a_{r_0}J_{\mathrm{phys}}>0\) yields \[m(r_0,u_0)\le K_0q^{-3}.\] By [a01:chosen-endpoint], \(m(r_0,u_0)\ge N_0^{\sigma/2}\). Thus any true terminal system with \(q=N_0^{-h+o(1)}\) has \(h\ge\sigma/6\). The same estimate holds on every subsequence and further allowed selection, so \(H(E)\ge\sigma/6>0\). This proves the theorem. ◻

Jet scales and nested states.

Logarithmically pigeonhole the derivative gaps through order \(J\). After a subsequence their exponents are \(\beta_j\), meaning that the \(j\)-th gap has size \(N^{-\beta_j+o(1)}\) on the retained events. Exponents above a fixed large constant greater than one are cut off; at a cutoff we use only the corresponding upper bound. We have \[\beta_j\ge0,\qquad \beta_0\ge1,\qquad \beta_j=0\ \text{for some }1\le j\le D.\] At value scale \(N^{-a}\), choose a time radius \(N^{-r(a)}\) on which every Taylor term of the difference is at most \(N^{-a}\), up to subpower factors. The \(j\)-th term requires \(r(a)\ge(a-\beta_j)/j\). The corresponding accuracy for an individual \(j\)-th derivative is \(N^{-(a-jr(a))}\). This exponent need not increase with \(a\), so we retain its largest preceding value. Thus, for \(0\le a\le1\), define \[ r(a)=\max_{1\le j\le J}\frac{a-\beta_j}{j}, \qquad l_j(a)=\max_{0\le x\le a}\{x-jr(x)\}. \tag{45}\] Then \(r(0)=0\), and \(r\) is increasing, convex, and piecewise linear. Moreover, \[\begin{align*} l_0(a)&=a,& 0\le l_j(a)&\le\beta_j,& l_j(a)-l_{j+i}(a)&\le i r(a). \tag{46}\end{align*}\] The last inequality follows by evaluating \(l_{j+i}\) at a point where the maximum for \(l_j\) is attained. Since \(r(x)\ge x/D\), for \(a>0\) we also have \[ (J+1-j)r(a)>l_j(a). \tag{47}\] Indeed, if a maximizing point \(x\) for \(l_j(a)\) is positive, use \((J+1)r(x)>x\) and monotonicity of \(r\); if \(x=0\), then \(l_j(a)=0\) and \(r(a)>0\).

On an open piece where the unique active term in \(r\) is \(m\), convexity gives \[\begin{align*} r(a)&=(a-\beta_m)/m,\tag{48}\\ l_j(a)&=a-jr(a)\quad(0\le j\le m),\tag{49}\\ l_j(a)&>a-jr(a)\quad(j>m). \tag{50}\end{align*}\] For \(j\le m\), the function \(x-jr(x)\) is nondecreasing up to the active piece. For \(j>m\), moving slightly to the left within that piece strictly increases it. The active \(\beta_m\) has not been cut off, because the cutoff terms cannot maximize \(r\) on \([0,1]\).

We choose the finite grid before constructing its states or selecting their atom masses. Fix an integer \(C>5J\). Trim a small fixed amount from the ends of every active piece, also staying away from zero, and omit pieces shorter than the trim. On the remaining compact intervals, the gaps \[l_{m+i}(a)+i r(a)-\beta_m>0\quad(1\le i\le J-m)\] and all needed remainder gaps are uniformly positive. Choose a common small step \(b\), after these trims, and subdivide each remaining interval. Write \(A_0\) for its first subdivision point. We will use steps \([a,a+b]\) only when \(a+Cb\) still lies in that interval. Take a finite increasing grid containing \(1\), all these subdivision points, and every required \(a+b\) and \(a+Cb\). The construction that follows is performed anew for each such grid. All its selections and limits in \(N\) precede any decrease of \(b\) or of the trims.

For a level \(a\) in this fixed grid, the state \(F_a\) records the dyadic interval \(I_a\) containing the event time, of length comparable to \(N^{-r(a)}\), the bins of both arms’ derivatives of orders \(0,\ldots,J\) at the center of \(I_a\), with widths \(N^{-l_j(a)}\), and all earlier grid states. The dyadic partitions are nested. Taylor translation over \(I_a\) has uncertainty at derivative order \(j\) bounded by \[ K\sum_{i=0}^{J-j}N^{-l_{j+i}(a)-ir(a)} +K N^{-(J+1-j)r(a)} \le K N^{-l_j(a)}. \tag{51}\] Here [a02:translation-exponents,a02:remainder-gap] handle, respectively, the binned terms and the remainder. Applying the same calculation to the two arms, starting from the event gaps and \(l_j\le\beta_j\), shows that their center bins differ by only \(K\) neighbors, at every recorded level.

An index from one color and \(I_1\) determine that arm’s entire history. The other arm’s history then has only subpowerly many neighbor choices. Therefore \[ |\mathop{\mathrm{supp}}F_1|\le N^{d+r(1)+o(1)}. \tag{52}\] For this fixed grid, make a further subpower selection on which the masses of occurring atoms, measured before this selection, are \(N^{-u_a+o(1)}\), uniformly at each grid level. Extremely light atoms can first be omitted using [a02:state-support]. The exponents are nonnegative and nondecreasing, and \[ u_1\le d+r(1). \tag{53}\] If a retained coarse atom at \(a\) has retained descendants at \(c>a\), their number is at most \[ N^{u_c-u_a+o(1)}. \tag{54}\] To see this without any assertion about later conditional probabilities, denote the pre-selection law by \(p_{\rm pre}\). Each retained child has \(p_{\rm pre}\)-mass at least \(N^{-u_c-o(1)}\), and the disjoint children lie in a parent of mass at most \(N^{-u_a+o(1)}\). Summing their pre-selection masses proves [a02:descendant-count]. Write \(\mathbb P\) for the final normalized event law.

The upper bound [a02:state-upper] will contradict the growth forced by matching. More precisely, on almost all small steps in an active \(m\)-piece we will prove \[u_{a+b}-u_a\ge b\left(1+\frac1m-o_b(1)\right).\] Summing these increments, then decreasing \(b\) and removing the trims, will force the uniform upper bound \(d+r(1)\) to be at least \(1+r(1)\). The next two constructions establish this increment on most steps of the already fixed grid.

A palette with little entropy growth.

We use the entropy chain rule and conditional mutual information in their standard forms; see Cover and Thomas (2006, chap. 2). Fix \(D_1>5\). The palette \(Z\) is the pair of \(m\)-th derivative bins at the center of \(I_{A_0}\), now at width \(N^{-\beta_m-D_1b}\). Since \(l_m(A_0)=\beta_m\), \[H(Z\mid F_{A_0})\le2D_1b\log N+O(1).\] At a step retained in the grid construction, let \[L_a=\log\frac{\mathbb P(Z\mid F_{a+Cb})}{\mathbb P(Z\mid F_a)}\] at the sampled palette value. Conditional mutual information and the entropy chain rule give \[\sum_a \mathbb EL_a\le C H(Z\mid F_{A_0}).\] The negative part has bounded expectation. Indeed, for probability vectors \(p,q\), \[\sum_{p(z)<q(z)}p(z)\log\frac{q(z)}{p(z)} \le \frac1e\sum_zq(z)\le\frac1e.\] Apply this conditional on the fine state, with \(p\) its palette law and \(q\) the coarse palette law. There are order \(b^{-1}\) steps, so the average of \(\mathbb E(L_a)_+\) is \(O(b^2\log N+1)\). Markov’s inequality gives an average probability \(O(b^{3/4})+o_N(1)\) for \(L_a>b^{5/4}\log N\). A second application shows that, except for a proportion \(O(b^{1/2})+o_N(1)\) of the steps, this failure probability is at most \(b^{1/4}\). Call these steps good. Thus at a good step, \[ \mathbb P(Z=z\mid F_{a+Cb}) \le N^{b^{5/4}}\mathbb P(Z=z\mid F_a) \tag{55}\] at the sampled value, apart from probability tending to zero with \(b\).

The switching experiment.

Fix a good step. We will make \(m+1\) traversals, whose \(m\) internal arrival times supply the variables of a full-rank endpoint map. Its endpoint must lie in the small union of targets supplied by [a02:descendant-count]; comparing those volumes will force the required increment. Start with \(E^0\sim\mathbb P\), keep every time in the same \(I_a\), and proceed as follows. At the beginning of traversal \(p+1\), draw a fresh event \(\widetilde E^p\) from \(\mathbb P\) conditional only on \[F_{a+Cb}(\widetilde E^p)=F_{a+Cb}(E^p).\] Write \(t_p=t(E^p)\) for the running time and \(\widetilde t_p=t(\widetilde E^p)\) for the refreshed event’s witness time. Both lie in the same fine interval \(I_{a+Cb}\), but the refresh leaves the running time at \(t_p\). Choose a color \(\kappa_p\) by the rule below and set \[\varphi_p=f_{\kappa_p,i_{\kappa_p}(\widetilde E^p)}.\] Draw \(E^{p+1}\), including its time \(t_{p+1}\), freshly from \(\mathbb P\) conditional only on this arm’s color, index, and \(I_a\). Traverse \(\varphi_p\) from \(t_p\) to \(t_{p+1}\). The increments can have either sign.

Before any failures are removed, \(E^p\) and \(\widetilde E^p\) have marginals bounded by \(2^p\mathbb P\). Refreshing on a state preserves a marginal bound of this form. For a traversal, dominate the adaptively selected arm kernel by the sum of the two color kernels; each color kernel preserves \(\mathbb P\) after averaging over its index and interval. This proves the assertion inductively.

The possible coarse states stay in a subpower neighbor set of the start. A refresh preserves \(F_a\), since the finer state includes it. A traversal preserves the chosen arm’s bins at \(a\) and at all earlier levels: its index and \(I_a\) determine these data. At either endpoint the other arm is within \(K\)-neighbor bins. There are only finitely many coordinates at the fixed grid and only \(m+1\) traversals. Thus all possible coarse states lie in a start-determined set of size \(K\), after increasing \(K\).

Let \(t_a\) be the center of \(I_a\), and subtract the order \(J\) Taylor polynomial \(T\) of one starting arm there. On traversal \(p+1\), put \(\varphi=\varphi_p\) and set \[s=(t-t_a)N^{r(a)},\qquad Y_j=N^{l_j(a)}\partial_t^j(\varphi-T)(t), \quad 0\le j<m.\] Along a traversal, \[ \frac{dY_j}{ds}=Y_{j+1}\quad(j<m-1),\qquad \frac{dY_{m-1}}{ds}=U+O(N^{-2b}), \tag{56}\] where \(U\) is a constant of size at most \(K\). Indeed the factor in the last derivative is \(N^{\beta_m}\). Taylor expansion of this residual \(m\)-th derivative at \(t_a\) shows that its variation is bounded by \[K\sum_{i=1}^{J-m} N^{\beta_m-l_{m+i}(a)-ir(a)} +K N^{\beta_m-(J+1-m)r(a)}.\] The strict trimmed gaps make this smaller than \(N^{-2b}\), for \(b\) sufficiently small.

Choose \(U\) from the fresh palette, the higher jet bins at \(A_0\) recorded in the coarse history, and the fixed frame \(T\). Taylor translation from the center of \(I_{A_0}\) to \(t_a\) gives an error \(O(N^{-D_1b})\) in its \(m\)-th term, while the higher-bin errors have exponents \[l_{m+i}(A_0)+ir(A_0)-\beta_m>0.\] The remainder has the same strict gap as before. This discretizes \(U\) to the precision required in [a02:controlled-jets] using only the start, the coarse state, the palette, and the color. The two controls available in a fresh pair differ by at least \(K^{-1}\): the \(m\)-th derivative gap at its event has exponent \(\beta_m\), and all the approximation errors are smaller by a fixed power. Choose either color on the first leg. Thereafter at least one of the two available controls is separated from the preceding control by \(1/(2K)\); choose such a color and enlarge \(K\).

At a switch, the old and new arms have \(K\)-neighbor fine bins at \(a+Cb\). The running time \(t_p\) and the witness time \(\widetilde t_p\) lie in that finer interval, so the bins control the jump at the actual switching time \(t_p\). Applying [a02:translated-bin-width] there gives a jump in \(Y_j\) of size at most \[K N^{l_j(a)-l_j(a+Cb)} =K N^{-(1-j/m)Cb} \le N^{-2b},\qquad j<m.\] Here \(C>5J\) leaves room for subpower factors. This also accounts for the comparison to the starting arm at the first switch.

Kill a step if [a02:palette-likelihood] fails at its refresh. Also kill it if its chosen color-index/interval pair has \(\mathbb P\)-mass less than \(\pi_{\kappa,i}|I_a|/\log N\), with the weights from [a02:index-marginal]. For each color the sum of these latter thresholds over all indices and intervals is \(1/\log N\). The marginal domination just proved therefore makes the total failure probability small. On a retained pair, the new-time conditional density is bounded by \[\frac{K\pi_{\kappa,i}} {\pi_{\kappa,i}|I_a|/\log N} \le \frac K{|I_a|}.\] Consequently the \(m\) internal arrival times can also be required to be pairwise separated by \(|I_a|/K\), with negligible additional loss: test each new internal time against the preceding ones and choose the final reciprocal-subpower separation threshold smaller than the reciprocal of the preceding density bound. The experiment has successful probability bounded below when averaged over starts, first for a sufficiently small fixed \(b\) and then for large \(N\).

Domination conditional on the entire past.

Fix the exact start \(E^0\). For every coarse state in its start-determined neighbor set, push its conditional palette law to each of the two possible discretized controls, using that state’s earlier data and the fixed frame \(T\). Sum these pushforwards to obtain a measure \(\Lambda\) on controls, with \(\Lambda(\mathbb R)\le K\).

Let \(\mathcal H_p\) contain the entire sampling history before the next refresh, including all hidden event data. This history fixes the current finer and coarse states. The next refresh has exactly the finer-state conditional law, independently of the rest of \(\mathcal H_p\). On its good palette values, [a02:palette-likelihood] dominates that law by the corresponding coarse palette law. The chosen arm can depend on the full history and the full refreshed event, but its point mass is bounded by the sum of the two candidate-control point masses. Finally, conditional on that complete refreshed event and its actual chosen index, the retained new-time density is at most \(K\,ds\). Therefore the unnormalized successful step kernel satisfies \[ Q_{\mathcal H_p}(dU\,ds) \le K N^{b^{5/4}}\Lambda(dU)\,ds \quad\text{for every admissible }\mathcal H_p. \tag{57}\] The same \(\Lambda\) works for every such history with this start.

For a nonnegative necessary test depending only on the start and the successive controls and times, [a02:full-past-kernel] may be iterated backwards. Integrate the final step with its full previous history fixed. The upper integral then depends only on the previous displayed controls and times, so repeat with the previous step. No successful kernel is renormalized and no kernel is conditioned on future survival. This is why possible hidden dependence of future events on the refreshed line identity does not change the product upper bound.

The endpoint map and its target.

Write \(U_1,\ldots,U_{m+1}\) and \(s_1,\ldots,s_{m+1}\) for the controls and arrival times, put \(s=s_{m+1}\), and keep the starting time \(s_0\) fixed. Solve [a02:controlled-jets] with exact constant controls and no switching errors. Its endpoint differs from the actual low jets by \(O(N^{-2b})\). At fixed final time \(s\), the control contribution to its \(j\)-th coordinate is \[ \frac1{(m-j)!} \left[ U_1(s-s_0)^{m-j} +\sum_{p=1}^m(U_{p+1}-U_p)(s-s_p)^{m-j} \right]. \tag{58}\] The omitted part depends only on the initial jets and \(s\). The formula follows by telescoping the integrals of constant controls; it remains valid for signed increments.

For fixed controls the map from \((s_1,\ldots,s_m,s)\) to the final time and low jets is polynomial of fixed degree. Differentiating [a02:endpoint-polynomial] with respect to \(s_p\) gives \[-\frac{U_{p+1}-U_p}{(m-j-1)!}(s-s_p)^{m-j-1}.\] Thus the absolute Jacobian determinant is exactly \[ \frac{\displaystyle \prod_{p=1}^m|U_{p+1}-U_p| \prod_{1\le p<q\le m}|s_p-s_q|} {\displaystyle\prod_{k=0}^{m-1}k!}. \tag{59}\] It is at least \(K^{-1}\) on successful paths. The number of preimages on this regular locus is bounded in terms of the degree and dimension: each is an isolated solution of a fixed-degree polynomial system, and the fixed-format semialgebraic fiber bound applies uniformly in the coefficients.

Set \(c=a+b\). A possible final child state \(F_c\) records its own interval \(I_c\), and the final time lies in that interval. At each such time, translate each final-arm jet bin from the center of \(I_c\), subtract \(T\), and apply the coarse normalization. The uncertainty in derivative order \(j<m\) is bounded by \[\begin{align*} &K N^{l_j(a)} \left( \sum_{k=j}^J N^{-l_k(c)-(k-j)r(c)} +N^{-(J+1-j)r(c)} \right) \\ &\hspace{35mm}\le K N^{l_j(a)-l_j(c)} =K N^{-(1-j/m)b}. \tag{60}\end{align*}\] This uses the child’s interval, not the whole coarse interval. It includes every higher-jet uncertainty and the Taylor remainder. The simulation error \(N^{-2b}\) fits these widths. The normalized final-time interval has length \(O(N^{-b/m})\). Hence, for either choice of final color, the target associated with one child has volume at most \[ K N^{-b\left(\frac1m+\sum_{j=0}^{m-1}(1-j/m)\right)} = K N^{-b\left(\frac1m+\frac{m+1}{2}\right)}. \tag{61}\] Its center may move rapidly with the final time; Fubini’s theorem uses only the widths of the fibers in [a02:final-target-widths].

Take the union of these targets over both colors and all retained children of every possible coarse neighbor of the start. By [a02:descendant-count], it has at most \(N^{u_{a+b}-u_a+o(1)}\) constituent targets. Membership in this union, together with the internal-time and consecutive-control separations, is a necessary test depending only on the start, controls, and times. Other hidden success conditions can be discarded for the upper bound.

For each fixed separated control tuple, the change-of-variables formula, [a02:endpoint-jacobian], and the bounded regular-fiber multiplicity bound the time integral by \(K\) times the target volume. This estimate is uniform in the controls. We may therefore integrate it against \(\Lambda^{m+1}\), even when \(\Lambda\) is atomic, and use [a02:full-past-kernel]. The conditional successful probability is at most \[N^{u_{a+b}-u_a -b\left(\frac1m+\frac{m+1}{2}\right) +(m+1)b^{5/4}+o_N(1)}.\] The averaged lower bound for success gives, on every good step, \[ u_{a+b}-u_a \ge b\left(1+\frac1m-o_b(1)\right), \tag{62}\] since \((m+1)/2\ge1\).

For this fixed trimmed grid, first let \(N\to\infty\), passing to a subsequence that fixes the set of good steps. Sum [a02:entropy-increment]; all omitted increments are nonnegative. Only after this fixed-grid limit do we let \(b\to0\), so that the proportion of excluded steps tends to zero. Then let the trims tend to zero. The sum of the \(b/m\) terms tends to \(\int_0^1r'(a)\,da=r(1)\), and the sum of the \(b\) terms tends to one. Thus [a02:state-upper] implies \(d\ge1\), a contradiction. The homogeneous exponents can depend on the fixed grid; only their uniform upper bound is used when passing to finer grids.

Finally, if no uniform small fixed-power tolerance existed, choose counterexamples with loss exponents tending to zero and \(N\) tending to infinity. Their lists can be bounded using \(d_0\), and they would form the subpower counterexample sequence just excluded. This proves the last assertion. ◻

A common spatial frame.

In version two, consider a pair whose old narrow normals differ in sine angle by more than \(gN^{Re}\); there is something to prove only when \(gN^{Re}<1\). The velocity lies in two strips of widths \(K_0g\), because the old spatial chart bounds hold throughout their respective subblocks. Their intersection has diameter at most \(K_0N^{-Re}\).

We bound the exceptional pair mass using the angular cap before any conditioning on that exception. Fix \(A\). At the opposite time, the spatial point lies in its extrapolated rectangle of area at most \[q_0^2gN^{C_0e}.\] At each exact base, apply 10 with the value width from [a02:old-test-extrapolation], the opposite-time derivative floor \(N^{-e}\), and hidden-bin width \(\delta\). This gives at most \(N^{O(e)}\) terminal hidden bins. Enlarge \(C_0\), independently of \(R\), so that the count is at most \(N^{C_0e}\). Given the terminal observation, the old knowledge of \(A'\) leaves only \(K_0\) choices. Each disagreeing pair confines the velocity to \(K_0\) angular bins at depth \(Re\). For the opposite-time marginal of the unnormalized pair law, integrate the capped density over the spatial rectangle and these hidden and angular bins, then sum over \(A\). The bound is \[\frac{K_0N^e}{n_Q} \bigl(q_0^2gN^{C_0e}\bigr) \bigl(N^{C_0e}\bigr) \bigl(n(L)N^{(1-SR)e}\bigr).\] The four factors are, respectively, the old-label count, spatial area, hidden-bin count, and joint density cap; the partner-label and angular-bin multiplicities are absorbed in \(K_0\). The normalized time integral is \(\int_I dt/q_1=1\), and \(n_Q=n(L)q_0^2g\) in version two. Thus the exceptional mass is at most \[ K_0N^{(2+2C_0-SR)e}. \tag{68}\] Choose \(R\) large enough that this is power-small. In version one no normal comparison is needed.

Put \(g^\#=\min(1,gN^{Re})\), with \(g^\#=1\) in version one. We now choose a coarse moving box \(D\), deterministically from \(A\), of spatial radii \(q_1,q_1g^\#\) in version two, and \(q_1\) in version one. When \(g^\#<1\), grid the unoriented normal at spacing \(g^\#\). In that rounded frame grid the normal component of the old center velocity at spacing \(g^\#\), and grid its position at the new block midpoint at spacings \(q_1,q_1g^\#\). Use the resulting midpoint and normal velocity, and take zero tangential center velocity. When \(g^\#=1\), use fixed axes and a static midpoint grid of spacing \(q_1\). The entire spatial trace of the line on \(I\) lies in this coarse moving box up to subpower factors.

Moreover, \(A'\) determines a list of only \(K_0\) possible \(D\)’s on retained pairs. Enumerate the boundedly many possible rounded orientations of \(A\) neighboring the normal of \(A'\). For each, project the midpoint and center velocity of \(A'\) in that same rounded frame and enumerate their subpower neighborhoods in the stated grids. To justify these neighborhoods, compare both old centers to the common line. Their midpoint errors have norm at most \(K_0q_1\) and old normal components at most \(K_0q_1g\); their velocity errors are at most \(K_0\) and \(K_0g\), respectively. A rotation error of size \(g^\#\) acts on these differences, giving \(K_0q_1g^\#\) and \(K_0g^\#\). It does not act on an uncontrolled absolute midpoint. This proves the stated compatibility lists.

Use coordinates \(z=(\tau,\xi)\) for \(D\): rescale time in \(I\) to a unit interval, subtract its moving center, and divide the spatial coordinates by the displayed radii. A common subpower dilation makes the relevant traces lie in a fixed box, with bounded speeds in normalized time. The spatial Jacobian back to physical coordinates is at most \[ N^{C_1e}q_0^jg. \tag{69}\] Here \(C_1\) can depend on the already fixed \(R\). Write \(z_0\) for the base point at \(t_0\).

Let \(J_D\) be the spatial Jacobian of these coordinates, including the common dilation. For a fixed entry \((A,D)\), or \((A',D)\), push the restricted unnormalized pair law to \(z_0=(\tau_0,\xi_0)\), integrating out the other sampled time. Its density with respect to \(d\tau_0\,d\xi_0\) is bounded by \[ N^{C_1e}n_Q. \tag{70}\] Indeed, the source’s own test or the witness’s extrapolated test at \(t_0\), together with the respective derivative lower bound, allows at most \(N^{C_0e}\) hidden terminal bins by 10, with the same value width and derivative floor as in the preceding count. The capped one-time density is \(N^en(L)\), and the change of base variables satisfies \[\frac{dt_0\,dy}{q_1} =J_D\,d\tau_0\,d\xi_0.\] Consequently the density is at most \(J_DN^en(L)N^{C_0e}\le N^{C_1e}n_Q\), after increasing \(C_1\) in [a02:coarse-jacobian]. No conditioning on the label and no new lower mass bound for that entry is used.

We will repeatedly use the following counting consequence. Suppose a group construction gives at most \(N^{C_1e}\) options per old label. Omitting entries of mass less than \[n_QN^{-C'e}\] costs a power-small total amount if \(C'\) is a sufficiently large fixed constant. This follows by multiplying the threshold by the option count and [a02:old-label-count], in each world/block, and then summing with the original weights. Apply this first to the entries \((A,D)\) and \((A',D)\). We retain their pre-deletion masses when using the resulting lower bounds.

Stable simple graphs and comparison mass.

We now construct the graphs whose agreement will later be recorded at the anchor. Write \(P_{A,D}\) for the source equation expressed in \(D\)’s real affine base coordinates, after the initial normalization \(w\mapsto Bw\), and use \(P_{A',D}\) for the witness equation in the same coordinates. These equations have total degree at most two and constant second \(w\)-derivative. For either equation write \(P\) temporarily, and choose an early constant \(C_A\) larger than the value and derivative losses in [a02:old-test-extrapolation]. If \[|P_{ww}|\ge\delta^{-1}N^{-C_Ae},\] replace this test for comparison by \(\widetilde P=w-F_{\rm ctr}(z)\). [a02:critical-center] and the derivative upper bound give \[|w-F_{\rm ctr}(z)|\le\delta N^{O(e)}\] throughout the new block. Otherwise let \(\widetilde P=P\). Denote the resulting equations by \(\widetilde P_{A,D}\) and \(\widetilde P_{A',D}\). The conversion, including the curvature test and any affine replacement, depends only on the old label and \(D\), before a source time or a trajectory is fixed. For every unreplaced test, the choice of \(C_A\) makes \(|P_{ww}P|\) much smaller than \(|P_w|^2\) at the tested point \(z_0,w(t_0)\), where \(|P_w|\ge N^{-e}\). Thus \[ N^{-O(e)}\le \mathcal D(z_0):= \widetilde P_w^2-2\widetilde P_{ww}\widetilde P \le N^{O(e)}. \tag{71}\] For replaced tests the discriminant is one.

For each converted equation, the coefficients of its discriminant polynomial in \(z\) are bounded by \(N^{O(e)}\), uniformly over the retained entries. Indeed, the entry’s pre-deletion mass is at least \(n_QN^{-C'e}\), whereas its source-base density is at most \(n_QN^{C_1e}\). Its projection therefore has Lebesgue measure at least \(N^{-(C'+C_1)e}\). On that projection the upper bound in [a02:source-discriminant] holds. Polynomial Remez therefore bounds the coefficients and hence the discriminant and its derivatives on a fixed complex enlargement. This argument uses the original heavy entry, before later joint deletions.

Fix a net of real balls \(U\) of radius \(N^{-C_Ue}\) covering the \(z_0\)-box, with mesh at most half that radius. Choose \(C_U\) using the uniform discriminant derivative bounds and the lower bounds in [a02:source-discriminant]. For each pair, a net center within half a radius of \(z_0\) then gives a ball whose fixed complex enlargement \(8U\) has both of that pair’s discriminants nonzero. Assign the first such ball in a fixed ordering. Admissibility is a condition on the pair and the ball; no ball is required to avoid the zeros of every equation. The pair’s two converted equations have simple holomorphic branches on \(8U\), real on the real ball. Choose the nearest real branch to \(w(t_0)\). For an unreplaced test [a02:nearest-quadratic-root] applies; for a replaced test use its affine graph. In either case, \[ |w(t_0)-f(z_0)|\le\delta N^{O(e)}. \tag{72}\]

We claim that, apart from power-small pair mass, the two chosen sheets differ in norm on \(2U\) by at most \[ \delta N^{C_*e} \tag{73}\] for a fixed sufficiently large \(C_*\). To prove this, consider discrepancies with norm between \(M\) and \(2M\), for dyadic \(M\ge\delta N^{C_*e}\). Group by \(M,D,U\), world and new block, and by the two center jet cubes at spacing \(M\), in the scaled coordinates of \(U\), through a fixed sufficiently high order. Paired cubes are bounded neighbors by 19. The same lemma shows that all members of such a group are within \(O(M)\) of a common reference sheet in norm.

Each old label has at most \(N^{O(e)}\) group options, with an exponent independent of \(C_*\). The \(D\)-compatibility list is subpower, the fine ball grid has \(N^{O(e)}\) members, and there are at most two branches per ball and boundedly many neighboring jet cubes. The range of \(M\) is polynomial in \(N\), so it contributes only \(O(\log N)\) choices. For this last assertion, the chosen branch value near \(z_0\) is polynomially bounded by the trajectory coefficient bounds and [a02:near-source-graph]. Implicit differentiation, using the derivative lower bound from [a02:source-discriminant], bounds its fixed jets polynomially. Affine replacements use an inverse curvature only above their cutoff. Finite-jet norm control bounds the whole comparison norm. The polynomial exponent here may depend on the exact problem; its logarithmic contribution is absorbed in \(N^e\) for large \(N\). An unused far quadratic root is irrelevant to this argument.

Let \(m_G\) be the mass of one group in its world/block. At a given base point, all its values \(w(t_0)\) lie in a moving band of width \(O(M)\), using [a02:near-source-graph] and choosing \(C_*\) after its loss constants. If \(M\le1\), choose a capped grid depth with \(M\le a_r\le MN^e\). Only boundedly many hidden bins of that width meet the band. The cap-class inequality gives \[n(r)\le n(L)(a_r/\delta)^d.\] With [a02:coarse-jacobian], this proves \[ m_G\le N^{O(e)}n_Q(M/\delta)^d. \tag{74}\] When \(M>1\), use depth zero. The native value and incidence derivative tests in version two allow only a subpower number of unit hidden bins at each exact base, by 10; in version one \(w\) is subpower bounded. Since \(d\ge0\) for a nonempty cap class, the depth-zero cap again gives [a02:matching-group-mass].

In each color omit group indices, consisting of a label and branch, whose marginal mass before the joint deletion is less than \(n_QN^{-C'e}\). Choose \(C'\) so that the total cost, summed using the preceding group-option count and the old label count, is less than \(N^{-2e}\). The remaining lists have cardinality \[\le N^{O(e)}(M/\delta)^d\] by [a02:matching-group-mass]. If the total mass of these large-discrepancy groups were at least \(N^{-e}\), some group would retain mass \(m_G'\ge m_G/2\) after both color deletions. Normalize that group’s surviving law to probability. For a color, start with index weights \(n_Q/m_G'\). Their sum is at most \(2N^{C'e}\), because each retained index had its stated pre-deletion floor. Normalize these weights to probability. [a02:source-base-density], after rescaling the ball, then bounds each index-base density by \(N^{O(e)}\) times its normalized index weight.

The lists now agree in value to \(\delta N^{O(e)}\), have paired norm discrepancies comparable to \(M\), and satisfy all other hypotheses of 22. Every displayed loss \(N^{O(e)}\) is a sufficiently small fixed power of \(M/\delta\) if \(C_*\) is chosen large enough. The constants determining these losses and the option count were fixed before \(C_*\). The corollary gives a contradiction. This proves [a02:local-sheet-match] outside power-small mass.

2. Representative equations at a common anchor.

The graphs now agree near their sampled source point, but those comparison balls vary with the point. We next record their agreement at one common complex point for each frame \(D\). This will give a bin that the witness label can list and a real representative equation that fits the source trajectory on the whole new block. The source time remains averaged throughout this part.

Common complex anchors.

For each \(D\), including its world and new-block context, choose a random anchor \(\alpha_D\) uniformly in a fixed bounded complex box with interior. Also choose independently a random \(\sigma_D\) in a fixed bounded complex disk. For a source point \(z_0\), use the complex line \[z=z_0+s(\alpha_D-z_0)\] and the broken path \(0\longrightarrow\sigma_D \longrightarrow1\) in its \(s\)-plane.

For each retained pair, with probability \(1-N^{-\Omega(e)}\) over these choices, both converted discriminants have lower bounds \(N^{-O(e)}\) on a tube of radius \(N^{-O(e)}\) around this common path. Here are the quantitative details. The coefficient bounds and the lower bound at \(z_0\) imply a polynomial norm lower bound on the fixed anchor domain. Polynomial sublevel estimates therefore make both values at \(\alpha_D\) at least \(N^{-B e}\), except with power-small probability, for a fixed sufficiently large \(B\). Restrict a discriminant to the complex line; its coefficients in \(s\) are bounded by \(N^{O(e)}\). Its nonzero values at \(0,1\), together with these derivative bounds, place those endpoints at distance at least \(N^{-O(e)}\) from all zeros in the bounded path region.

Remove disks of a much smaller radius \(N^{-Be}\), with \(B\) increased, around the boundedly many such zeros. A segment from either endpoint to a uniformly chosen \(\sigma_D\) meets one disk only on a set of power-small probability: the disk subtends an angle bounded by its radius divided by its distance from that endpoint, and the \(\sigma_D\)-density is bounded. On a path avoiding these disks, factor the one-variable polynomial and compare each factor with its value at zero. Nearby zeros cost only fixed powers of \(N^{-e}\); far zeros cost fixed constants. This gives a lower bound \(N^{-O(e)}\) all along the path. The coefficient bounds in the full \(z\)-variables extend it to the stated smaller tube. These exceptional probabilities may be averaged against the original pair law, so their absolute cost is power-small.

Choose common endpoint comparison balls of radius \(N^{-O(e)}\), with fixed margins inside these tubes and, at the source, inside the original stable ball. Continue the two chosen branches. First use 19 to extend [a02:local-sheet-match] to the interior complex source domain, and then use 20. At the anchor their difference, and all required fixed jets in \(D\)’s \(z\)-coordinates, are at most \(\delta N^{O(e)}\).

For each \(D\), bin a sufficiently long jet of the source branch at \(\alpha_D\), separating real and imaginary parts, at one common spacing \[ \delta N^{C_b e}, \tag{75}\] where \(C_b\) is fixed after the propagation constants. On retained pairs these anchor bins are bounded neighbors. The reached jets have polynomial range: their source norms have polynomial bounds by the preceding near-branch argument, and 20 propagates the norm of the branch itself. Thus we can prescribe polynomial ranges for all bins used on successes, even when an unused far root has much larger coordinates.

Given \(A'\), the source \(D\) belongs to a subpower compatibility list. For each such \(D\), enumerate the simple roots at the anchor of the converted equation \(\widetilde P_{A',D}\), including its affine replacement when the early curvature test required one. There are at most two, and their jet tuples are determined by those roots. These are the same converted sheets used in the local comparison. Different continuation paths can interchange these two tuples but cannot create further ones. Taking the neighboring bins of both tuples gives a list of only subpowerly many possible source bins. This is genuine old-\(p_L\) list knowledge: its size is subpower, rather than merely \(N^{O(e)}\).

A representative test for each bin.

The source label \(A\) now supplies the branch to be fitted on the new block. The opposite-time label \(A'\) supplies only the list just constructed, which makes the eventual output label visible at \(p_L\). For each \(D\) and used bin choose a representative equation \(\widehat P\) among the converted real equations \(\widetilde P_{A,D}\) from all old source subblocks. Require the fixed discriminant coefficient bounds, its prescribed \(N^{-O(e)}\) lower absolute bound at \(\alpha_D\), and a branch whose jet lies in that bin. These are fixed equation/anchor eligibility tests, allowing affine replacements. Choose the first eligible finite option in a predetermined order. This table depends on the anchor, \(D\), and bin, and not on \(t_0\) or the sampled trajectory. Every source on a retained success is itself eligible.

On a common anchor ball with an inverse-power radius, the representative branch \(\hat f\) is simple and holomorphic. For the source branch \(f\) in the same bin, the finite-jet part of 19 gives \[\|f-\hat f\|\le\delta N^{O(e)}.\] The derivative of \(\widehat P\) on \(\hat f\) is bounded by the square root of its discriminant, hence by \(N^{O(e)}\); also \(|\widehat P_{ww}|\le\delta^{-1}\). Taylor expansion in \(w\) therefore yields on an interior anchor ball \[ |\widehat P(z,f(z))|\le\delta N^{O(e)},\qquad |\widehat P_w(z,f(z))|\le N^{O(e)}. \tag{76}\] For example, the quadratic error in the first estimate is bounded by \(\delta^{-1}(\delta N^{O(e)})^2 =\delta N^{O(e)}\).

Return these evaluations to the source along the source-safe path. Each is a fixed-degree polynomial evaluation of the continued branch of \(\widetilde P_{A,D}\), so 20 applies. Only the source discriminant is required to stay nonzero on this return path. In particular, the representative branch is not continued back, and no assertion is made that it has a real root near every source point.

On a real segment near \(t_0\) of relative new-block length at least \(N^{-O(e)}\), the actual line remains in the source endpoint ball and \[|w-f(z)|\le\delta N^{O(e)}.\] For a replaced source this is its affine-center estimate. For an unreplaced source, \(P_{A,D,w}\) is affine along the line, has the upper bound in [a02:old-test-extrapolation], and has its own-time lower bound \(|P_{A,D,w}(z_0,w(t_0))|\ge K_0^{-1}\). This is the original selected-incidence floor for \(A\); the opposite-time estimate for \(A'\) is not needed for this return. Shrink the segment by a fixed power of \(N^{-e}\) so that this lower bound persists within a constant factor; shrink it again to fit the source endpoint ball, using the bounded \(z\)-speed. The early cutoff \(C_A\) still makes \(|P_{ww}P|\ll|P_w|^2\) there. [a02:nearest-quadratic-root] thus applies continuously to the same source branch chosen at \(t_0\). If \(t_0\) is near the edge of the new block, use the one-sided portion of this segment; it has the same lower order of length.

Use \(|\widehat P_{ww}|\le\delta^{-1}\) once more to replace \(f(z)\) by the actual \(w\) in the returned estimates [a02:anchor-evaluations]. Polynomial extrapolation along the line then gives, on the entire new block, \[ \sup_I|\widehat P(z,w)|\le\delta N^{C_2e}, \qquad \sup_I|\widehat P_w(z,w)|\le N^{C_2e} \tag{77}\] for a fixed \(C_2\). In version two all trajectory coordinates are affine. In version one the special form \(w-F(t,y)\) still has degree at most two along a trajectory. Thus the extrapolation has fixed degree in both cases and costs only the indicated fixed powers.

We also need a lower bound for the representative discriminant at the real source: \[ |\widehat{\mathcal D}(z_0)| = |\widehat P_w^2-2\widehat P_{ww}\widehat P| \ge N^{-C_3e}. \tag{78}\] This fails only on power-small pair mass, with an exponent \(C_3\) chosen before the later curvature split. To prove this, fix the anchors and the resulting representative table. Each representative has the prescribed inverse-power discriminant lower bound at its anchor. Polynomial norm comparison transfers it to the fixed real \(z\)-box \(B_{\rm real}\): for an already fixed exponent \(b_{\rm anc}\), uniformly over the table, \[\|\widehat{\mathcal D}\|_{B_{\rm real}} \ge N^{-b_{\rm anc}e}.\] Let \(\theta>0\) be a fixed polynomial sublevel exponent for these degrees and this box. Then \[\bigl|\{z\in B_{\rm real}: |\widehat{\mathcal D}(z)|<N^{-C_3e}\}\bigr| \le C N^{-\theta(C_3-b_{\rm anc})e}.\]

For summation, fix a source label \(A\). Its \(D\) is fixed. At the fixed anchor its converted equation \(\widetilde P_{A,D}\) has at most two root/jet tuples, hence at most two source bins and a bounded number of associated representatives. The number of possible homotopy classes of a varying path does not enter this count. Charge the exceptional volumes against [a02:source-base-density] for this source, then sum using [a02:old-label-count]. In each world and block the bad pair mass is at most \[\begin{align*} &\frac{K_0N^e}{n_Q} \bigl(n_QN^{C_1e}\bigr) C N^{-\theta(C_3-b_{\rm anc})e}\\ &\hspace{12mm}\le K_0N^{[C_1+1-\theta(C_3-b_{\rm anc})]e}. \end{align*}\] The bounded representative count is absorbed in \(K_0\). Summing with \(\omega_\gamma q_1\) gives the same global bound. Choose \[C_3>b_{\rm anc}+\frac{C_1+2}{\theta}.\] This makes the exception power-small using only constants already fixed. In particular it is independent of the later curvature threshold. The charge is made while \(t_0\) is still averaged, using the representative table’s independence of \(t_0\).

3. Construction of the longer atlas.

The representative equations now satisfy the whole-block bounds [a02:representative-upper]. Outside power-small pair mass, their discriminants also satisfy [a02:representative-disc-lower] at the source. We use these facts to construct trajectory assignments on the longer blocks. The source label will determine an assignment, but will not be part of its output label.

Fixing the source and assigning anchor bins.

Let \(\mathbf a\) denote the anchor and detour table, and let \(\mathcal S_{\gamma,I}(\ell,t_0,t;\mathbf a)\) be the event that the pair passes all restrictions through [a02:representative-disc-lower]. For \(\vartheta=(\mathbf a,t_0)\), define \[s_{\gamma,I}(\vartheta) =\int\int_I \mathbf 1_{\mathcal S_{\gamma,I}}(\ell,t_0,t;\mathbf a) \frac{dt}{q_1}\,d\nu_\gamma(\ell).\] Sample \(\vartheta\) using the anchor and detour laws above and independent uniform \(t_0\in I\). For each world and block, choose \(\vartheta_{\gamma,I}=(\mathbf a_{\gamma,I},t^{\rm src}_{\gamma,I})\) with mass at least its average. The preceding estimates give \[M_{\rm fr}:= \sum_{\gamma,I}\omega_\gamma q_1 s_{\gamma,I}(\vartheta_{\gamma,I}) \ge \sum_{\gamma,I}\omega_\gamma q_1 \mathbb E_\vartheta s_{\gamma,I}(\vartheta) =N^{-o(1)}.\] The lower bound concerns this weighted sum; individual blocks may have much smaller mass. In a fixed world and block, write \(t_0=t^{\rm src}_{\gamma,I}\). The frozen successful measure is \[d\mu^{\rm fr}_{\gamma,I}(\ell,t) =\mathbf 1_{\mathcal S_{\gamma,I}} (\ell,t_0,t;\mathbf a_{\gamma,I}) \,d\nu_\gamma(\ell)\,\frac{dt}{q_1}.\] It remains an unnormalized restriction of the original product prior. We now define geometric assignments containing its successful trajectories.

At the fixed source time, require old selected source-chart membership. This determines the old label \(A\), its coarse frame \(D\), and the nearest simple branch of \(\widetilde P_{A,D}\). Require its discriminant to be nonzero on the fixed broken path, with the prescribed simple-branch conditions at its endpoints, and continue the source branch to the anchor. Require its value and jets to lie in the stored polynomial ranges. Let \(b\) be its anchor-jet bin, and require that this bin have a representative table entry. Retain only trajectories satisfying the whole-block spatial bound, both bounds in [a02:representative-upper], and [a02:representative-disc-lower] at \(t_0\), for the table entry \(\widehat P_{D,b}\). Every frozen success satisfies these conditions.

Assign each such trajectory the preliminary label \[c_0=(D,b),\] merging all source labels that give the same pair. The fixed source time and disjoint old source assignments give at most one label per trajectory in each new block. In particular this assignment is independent of the later time \(t\). Its representative equation is the fixed table entry \(\widehat P_{D,b}\); the old source identity is no longer recorded.

For these assignments retain the original later incidences whose old terminal label \(A'\) supplies \(D\) and \(b\) in its compatibility lists. This selection contains the frozen successes, has weighted mass at least \(M_{\rm fr}\), and is still a restriction of \(d\nu_\gamma\,dt/q_1\). It is specified by the old incidence predicates, the fixed tables, and geometric tests; the analytical density masks are not part of the selection. At \(p_L\), the old label \(A'\) belongs to a subpower list. Each such label gives a subpower list of frames, and for each frame at most two anchor-jet tuples with boundedly many neighboring bins; intersect this list with the stored polynomial bin range. Thus the preliminary label \(c_0\) is known by \(p_L\).

It remains to turn each representative equation into an admissible real test. The two curvature cases below preserve these assignments; only the low-curvature case refines their labels.

Normalizing the representative tests.

The bounds [a02:representative-upper] and the choice of \(C_3\) are already fixed. Choose \(C_4>C_3/2+2\), then choose \[C_s>C_2+C_4+1,\] and finally choose \[ T>\max(C_2+C_3+1,\ C_s+C_4+1). \tag{79}\] First consider a representative with \[|\widehat P_{ww}|\le\delta^{-1}N^{-Te}.\] On every assigned trajectory, the product of curvature and value in [a02:representative-upper] is at most \(N^{(C_2-T)e}\), smaller than the discriminant lower bound in [a02:representative-disc-lower]. At the real source, \[\widehat P_w^2 =\widehat{\mathcal D}+2\widehat P_{ww}\widehat P.\] It follows that the discriminant is positive and \(|\widehat P_w(z_0,w(t_0))|\ge N^{-C_4e}\). Consequently the trajectory’s block maximum satisfies \[N^{-C_4e}\le M_\partial:=\sup_I|\widehat P_w(z,w)| \le N^{C_2e}.\] Let \(v\) be the lower endpoint of the dyadic bin containing \(M_\partial\), refine the label to \(c=(D,b,v)\), and use the test \(\widehat P_{D,b}/v\). These refinements remain disjoint trajectory assignments throughout the block. The full range of \(v\) contains only \(O(\log N)\) values, so the refined label is still known by \(p_L\). In version one the derivative is identically one, and we take \(v=1\).

The normalized derivative has block maximum between fixed positive constants. The value bound is at most \(\delta N^{(C_2+C_4)e}\), and the curvature bound is \[\delta^{-1}N^{(-T+C_4)e}.\] By [a02:late-constant-order], these fit the thickness \[\delta_s=\delta N^{C_s e}.\] The derivative upper bound is of order one. For its incidence lower bound, omit later times where its absolute value is below an inverse-subpower number \(\kappa\). On each assigned trajectory this derivative is affine and has order-one block maximum, so the omitted set has relative length \(O(\kappa)\). Disjointness of the assignments gives total weighted loss at most \(O(\kappa)\) against the original trajectory and time priors. The same bound holds for the selected submeasure. Choose \(\kappa=N^{-o(1)}\) with \(\kappa=o(M_{\rm fr})\); the loss is then negligible compared with the available mass. This estimate uses no density bound for a law conditioned on pair success.

For the remaining representatives, \[|\widehat P_{ww}|>\delta^{-1}N^{-Te},\] use the real affine critical center and the test \(w-F_{\rm ctr}(z)\), keeping the label \(c=c_0\). By [a02:critical-center,a02:representative-upper], \[|w-F_{\rm ctr}(z)|\le\delta N^{(T+C_2)e}\] on the entire new block. This fits thickness \(\delta_h=\delta N^{(T+C_2+1)e}\), with derivative identically one and zero curvature. A real representative root near the source is not needed in this case.

Choose a curvature case carrying at least half the remaining mass, and pass to a subsequence if necessary to obtain a common thickness exponent. All constants were chosen in the order \(C_A\), the stable-ball and comparison constants, the continuation and binning constants giving \(C_2\), then \(C_3,C_4,C_s,T\). In particular \(C_3\) is independent of \(T\): its sublevel estimate was proved for the representative table while the source time was still averaged.

Let \(C\) exceed both final thickness exponents and all needed smallness constants. Restrict \(e<cL\), with fixed \(c>0\) small enough that \(L-Ce>0\) and \(Re\le\eta L\). The chosen thickness is \(a_{L'}\), with \(0<L'<L\) and \(L-L'\le Ce\). The block length \(q_1\) has exponent \(h-e\).

Finite encoding and return to the original coordinates.

The assignments just constructed use polynomially many old labels, coarse boxes, anchor bins, and representative table entries. The chosen source branches and their propagated jets have polynomial ranges, as established above; unused far roots need not be stored. The continuation predicates have bounded semialgebraic complexity by 20, and fixed-order implicit differentiation gives the same property for the endpoint jets. Nearest-root choices, geometric inequalities, and affine block maxima also have fixed-degree descriptions. A whole-block bound quantifies only over one time variable for each individual test. Combining these conditions with the old predicates and finite tables therefore preserves temperedness.

All equation and coordinate coefficients have polynomial bounds. The affine-center constructions invert curvatures only above their specified cutoffs, and the low-curvature normalization divides only by numbers at least a constant times \(N^{-C_4e}\). Thus the final tests and inverse coordinate changes remain polynomially controlled. The measurable choices made during the proof enter only through the fixed source times, anchors, equations, and tables, not through analytical masks or their searches.

Finally compose the tests with the real affine coordinate map \(z=z(t,y)\) and undo \(w\mapsto Bw\). In version one this preserves the form \(w-F(t,y)\); in version two it preserves total degree at most two and restores the derivative unit \(B\). The moving spatial frames contain the assigned full traces and have largest singular value at most \(q_1\) times a subpower factor. We have constructed tempered disjoint trajectory assignments with the required whole-block fits, inverse-subpower selected incidence mass, and labels known by the old \(p_L\). They form the asserted known atlas. ◻

Coordinates and masks at every potential parent.

For now retain all potential parent nodes in the finite grid. If \(A\) is a label of cumulative depth \(b\), let \(q_A\) be its original time length, \(J_A\) its spatial determinant, and \(\nu_{\mathrm{old}}(A)\) its full assigned trajectory mass in its old world. Saturation gives \[\nu_{\mathrm{old}}(A) \sim_{\log} n_{\mathrm{old}}(b)J_A.\] Normalize its base coordinates as in 14. The parent prior \(\nu_A\) is the old trajectory prior conditioned on this full assignment, and parent time is uniform on \([0,1]\). In version 2 rescale the hidden coordinate so that its parent derivative unit is \(B=1\). Its value need not be bounded in tangent units; the native signal \(X=P_*\) has the native bounds.

At width \(p=N^{-r}\), a parent hidden bin corresponds, at fixed base and label, to boundedly many old bins at depth \(b+\epsilon_n r\). Remove the original high-density exceptions at these depths, for all required parents and finite grids. The within-world density factor under parent normalization is \(J_A/\nu_{\mathrm{old}}(A)\). Consequently the remaining unnormalized density is at most \[K\,\frac{J_A}{\nu_{\mathrm{old}}(A)} n_{\mathrm{old}}(b+\epsilon_n r) \le K\,\frac{n_{\mathrm{old}}(b+\epsilon_n r)} {n_{\mathrm{old}}(b)} \le Kp^d.\] The last inequality uses the relative-profile errors \(o_n(\epsilon_n)\) fixed when choosing \(i\). The native derivative and curvature bounds give the same ceiling for \(X\)-bins. These are ceilings for restricted measures, so they remain valid under all subsequent deletions.

Prepare angular ceilings on the same finite collection of parents. Refinement of the original terminal observation by a known parent label has typical density at least \(n(L)/K_{\mathrm{path}}\) under the unnormalized final path measure. Remove the exceptional denominators and the old joint angular exceptions before projecting velocity. A normalized velocity cell of side \(p\) pulls back to an old velocity ball of radius \(Kp\). In the positive-narrowness case the omitted minor component is already much smaller than this radius, since the parent is at an interior original depth. For every tested cell the remaining angular numerator is therefore bounded by \(Kp^S\) times the original refined terminal denominator. Integrating gives the corresponding bound relative to the parent masses before these masks. No exceptional portion is included in this numerator sum.

We also prepare the spatial fullness that will be needed in original units. For \(0<\ell<\infty\), use the cumulative principal axes of each potential parent. Its minor spatial scale per unit time is smaller than the major scale by an actual power of \(N_0\). By 13, cumulative child largest widths per unit time are full up to actual subpower factors. Make these prunings simultaneously for the finitely many required ancestor–child pairs. For a descendant at a specified relative path depth, let \(D_s\) be its actual relative spatial matrix and \(q_s\) its actual relative time length. Then \[\rho_s=\|D_s^{\mathsf T}e_Z\| \ge q_s/K_{\mathrm{path}}.\] Indeed \(\|D_s\|/q_s\) is actually subpower bounded. If its major row were deficient by a fixed power, composition with the parent’s matrix, whose minor scale already has a fixed-power deficiency, would make the cumulative child largest scale deficient as well. This contradicts the fullness just prepared.

All density exceptions have power-small mass at fixed tolerances in each exact problem. The original order \(n,i,N_0\) allows the finite grids, tolerances, and fullness prunings to be chosen with total loss \(o(1)\) relative to the reference path. At this stage no entropy window has been selected.

The fixed probability law and the entropy window.

Normalize the surviving path once and call its probability law \(\mu_{\mathrm{prep}}\). For a potential parent \(A\), let \(s_A\) be its surviving success fraction in \(\nu_A\times dt\). For \(s_A>0\), set \[\lambda_A^0 =\frac{(\nu_A\times dt)|_{\mathrm{prepared\ path\ in}\ A}}{s_A}, \qquad p_A=\mu_{\mathrm{prep}}\{A\}.\] At any one parent node, \(\mu_{\mathrm{prep}}=\sum_Ap_A\lambda_A^0\). The weight \(p_A\) is proportional to its old world weight times \(q_A\nu_{\mathrm{old}}(A)s_A\).

Parents with too small a success fraction, or too small a surviving denominator relative to their pre-mask mass, have negligible total weight: sum the former costs with \(\sum_A\omega_{\mathrm{old}}q_A\nu_{\mathrm{old}}(A)\le1\), and the latter with the pre-mask parent masses. These parents may be omitted when using the chosen window. On the parents retained for estimates, \[ \lambda_A^0\le K\nu_A\times dt, \qquad \pi_A(B_p)\le Kp^S, \tag{97}\] where \(\pi_A\) is the velocity marginal of \(\lambda_A^0\). The division by \(s_A\) costs only \(K\), so the pure ceilings above also hold for these conditional probabilities.

Choose the window using the single law \(\mu_{\mathrm{prep}}\), before any further graph or likelihood deletion. Let \(V_*\) be the indicated original velocity bin, of depth \(\epsilon_n U_n\). For disjoint successive windows, the increments \[I(V_*;\hbox{history through the window end} \mid\hbox{history through its start})\] are nonnegative and sum to at most \(H(V_*)\), by the chain rule for nested histories. The original velocity range gives \(H(V_*)\le O(U_n\log N)+o(\log N)\). There are \(\gtrsim(D_n\epsilon_n)^{-1}\) windows, so one has increment \(o(\log N)\), since \(U_nD_n\epsilon_n\to0\).

Use the normalized coordinates of this window’s parent. In version 1 and for \(\ell=0\), put \(v=V\); for positive finite narrowness, put \(v=Z'\). Given the parent, the required \(v\)-bin has at most \(K\) possibilities from \(V_*\), and hence its conditional mutual information with the end history is larger by at most \(\log K=o(\log N)\). For \(\ell=0\), both parent scales per unit time are full to tangent factors \(K\), which suffices for this ambiguity estimate; positive narrowness uses the major-coordinate fullness prepared above. The requested velocity grid is no finer than \(N^{-U_n}\) in its own units.

Fix \(p_A\), \(\lambda_A^0\), and \(\pi_A\) from now on. Omitting the exceptional parents and imposing graph masks restricts this mixture without redefining its reference laws or its entropy estimate. Substantial coverage will mean positive probability relative to the original reference path.

Path scales and the two time conventions.

For a descendant \(Q=Q_s\), put \(a=N^{-s}\), \(h=N^{-ks}\), and, in version 2, \(g=N^{-\ell s}\). The relative-profile and path conclusions give \[ a_s=N^{-s},\qquad q_s\sim_{\log,N}h, \qquad J_s\sim_{\log,N}h^jg, \qquad \nu_Q\sim_{\log,N}a^dJ_s. \tag{98}\] In version 1 omit \(g\). Write \(P_s\) for the fitted true test. Its actual node thickness \(a_Q\) is comparable to \(a\) in logarithmic order. The true value, derivative, and curvature bounds continue to hold in the actual units of that node. The full trajectory assignments remain disjoint within each time block and nested in the fixed parent prior.

At an exact base, the true value and incidence-derivative bounds permit only \(K\) hidden bins of width \(a_Q\). The map to unit base coordinates \(u\) of \(Q_s\) has Jacobian \(q_sJ_s\); denote this map by \(\operatorname{base}_Q\). The pure cap therefore gives \[ d(\operatorname{base}_Q)_*(\lambda_A^0|_Q)(u) \le Ka^dq_sJ_s\,du \le Kq_s\nu_Q\,du. \tag{99}\] This is a cap in parent-time mass. Dividing by the block length \(q_s\) gives the cap \(K\nu_Q\) in normalized \(Q_s\)-time. If the trajectory prior is normalized on the full assignment of \(Q\), the pure ceiling is \(Kp^d/\nu_Q\) in parent base coordinates. After changing to unit \(Q\)-base coordinates and unit block time, it is \(Kp^dJ_s/\nu_Q\). Here \(p\) remains the hidden-bin width in parent units; the denominator is the full assignment mass, not the current selected incidence mass. The budgets are \[\sum_{Q\ \mathrm{in\ one\ block}}\nu_Q\le1, \qquad \sum_{Q\ \mathrm{at\ one\ node}}q_s\nu_Q\le1.\] They allow light labels to be removed whenever current incidence floors are needed, with thresholds chosen below the current selected mass. Spatial enlargements by subpower factors only change \(K\).

For positive finite narrowness, subtract a multiple of the \(Z\)-row of \(D_s\) from its \(Y\)-row to make the rows orthogonal. The subtraction coefficient is at most \(\|D_s^{\mathsf T}e_Y\|/\rho_s\), and the new transverse row has length \(|\det D_s|/\rho_s\). Thus the child is a tilted box of widths \(h,hg\), and its assigned velocities satisfy \[Y'=A_s\cdot(1,v)+O(Kg).\] The row \(A_s\) is determined by \(Q_s\) and has actual subpower coefficients. Its affine center velocity is controlled by the whole-block position bounds on occupied trajectories. In actual units the transverse width per time is comparable, up to actual subpower factors, to \(J_s/q_s^2\).

Stable graphs and point-state lists.

Later arguments need velocity bounds after recording finite-mesh point states. We first construct the finite list of possible states; its size will be the input to the appended-entry floor below. Use normalized base coordinates \(u\) in \(Q_s\), keeping the hidden variable \(w\) in parent units with \(B=1\). Let \(K_g=N^{o(1)}\) contain the present test bounds. In version 2 they give \[|P_s|\le K_ga_Q,\qquad K_g^{-1}\le |P_{s,w}|\le K_g,\qquad |P_{s,ww}|\le K_g/a_Q\] on the counted incidences. First omit labels whose current mass is too small relative to \(q_s\nu_Q\). The assignment budget controls the total loss, and [a03:packet-base-cap] then gives a base projection of inverse-subpower volume in every used label.

The polynomial discriminant \[\mathcal D(u)=P_{s,w}^2-2P_{s,ww}P_s\] is independent of \(w\). Its values on this projection are bounded by \(K\); polynomial Remez therefore bounds \(\mathcal D\) and its first derivatives by \(K\) on the needed complex enlargements of the normalized packet box. Choose a subpower \(H\) with \(H/K_g^3\to\infty\), and split the tests at \(\lvert P_{s,ww}\rvert=(a_QH)^{-1}\). Above this cutoff the affine quadratic center satisfies \[|w-F_{\mathrm{ctr}}(u)| =\frac{|P_{s,w}(u,w)|}{|P_{s,ww}|} \le K_gHa_Q\le Ka.\] Below the cutoff, \[ 2|P_{s,ww}P_s|\le \frac{2K_g}{H} =o(K_g^{-2})=o(|P_{s,w}|^2), \qquad \mathcal D\ge\tfrac12K_g^{-2} \tag{100}\] at those incidences. Partition the normalized box into core subboxes of inverse-subpower radius, each with a fixed-factor complex enlargement and a further margin. Choose the radius so that the variation of \(\mathcal D\) on each occupied enlargement is much smaller than this lower bound. For a quadratic test, each such enlargement has two stable simple holomorphic branches, real on its real core. The branch with the incidence derivative sign approximates \(w\) to \(Ka\), by [a02:nearest-quadratic-root]. For a linear equation the same argument keeps its coefficient nonzero.

There are \(K\) core/branch entries per label, including affine center entries. Delete light entries using the same \(q_s\nu_Q\) budgets and a smaller reciprocal-subpower threshold. The cap [a03:packet-base-cap] supplies inverse-subpower real base volume for each retained entry. Compose its graph with the native test \(P_*\). The native derivative and curvature bounds give an error \(Ka\) in \(X\) at the incidences, and \(|X|\le K\) there. The composition is an algebraic holomorphic sheet; 19 bounds it and its required derivatives by \(K\) on the used interiors. In version 1 use the explicit graph directly. These bounds now belong to the fixed graphs and persist under later incidence restrictions.

An end history refining \(Q_s\) localizes its normalized base coordinates to uncertainty at most \(Kq_{\rm end}/q_s\). Indeed, nested relative frames have largest width bounded by relative time length times path slacks, and occupied chart centers move at bounded speed up to those slacks. For a prescribed mesh in \((t,y,X)\), first choose \(a\) much finer than that mesh and then choose the end depth so that the graph derivative bound times \(Kq_{\rm end}/q_s\) is also much finer. Each graph then gives only \(K\) possible mesh states. The finite core/branch list proves the required ambiguity bound before any palette floor is used. Cores lie inside the derivative-controlled interiors; bounded overlapping covers handle their boundaries.

These preparations use arbitrary fixed mesh exponents first, with sufficient window depth, and then slowly increasing grids. They may be made after the entropy-window choice with negligible total loss, without redefining \(\lambda_A^0\), and before arbitrary later sparse selections.

Original observation lists and full-layer witnesses.

Finite-mesh point states in \((t,y,X)\), together with the path labels in the window, have actual-list accuracy given the original \(p_L\). Condition on the actual path charts in its lists. The base is exact, and the much finer original terminal hidden bin, with the native derivative and curvature bounds at the incidence, controls \(X\). This is actual knowledge in the original problem; the graph lists separately supply the \(K\)-ambiguity needed in tangent estimates.

For stored full-layer incidence witnesses, impose the same original pure-cap masks on the fixed density grids needed in the coordinate transfers. Their costs are power-small at every fixed tolerance in the exact problem. If a layer label loses at least half its incidence mass, saturation bounds its old block length times its full trajectory mass by an actual path factor times the lost incidence mass. Sum with the old world weights. The cost of dropping all such labels from the final path is therefore negligible when \(N_0\) is sufficiently late. This prepares capped witnesses in each needed layer. Their earlier floors refer to those witnesses; they are not floors for the final path, and they do not provide several-time palette estimates.

All these preparations use the already fixed diagonal. Their initial cap masks retain \(1-o(1)\) of the reference path, and their graph and witness prunings have negligible additional relative loss. The parent weights and conditional probabilities remain the reference laws fixed at the entropy-window step. ◻

Coverage by one fixed candidate scenario.

First fix a gain \(c>0\), a candidate type, and a bounded range of path nodes in tangent units. Use a fixed power slack \(N^\sigma\) in the tests and mass threshold, where \(\sigma>0\) will satisfy the bounds below. In each chart \(Q\), choose tests successively that carry the threshold mass on lines not yet assigned.

Assign to each chosen test all still available \(Q\)-trajectories satisfying its line inequalities. For time candidates the value and derivative upper bounds hold throughout the block, and the derivative lower bound holds somewhere in it. Only occurrences where that lower bound holds count toward the mass threshold. Eligibility may be measured under the unnormalized parent prior restricted to prepared path incidences.

Disjointness makes this priority assignment stop after at most \(N^{dc+\sigma}\) choices per \(Q\) for time candidates, or \(N^\sigma\) for strips, with a \(K\) loss if path normalization is used. Run it separately for each node and scenario. The retained geometric line predicates record the chosen tests; they need not encode the analytical mass calculation that certified their eligibility.

Choose a countable menu of scenarios. Each fixes a type, a bounded tangent-depth range, a positive gain \(c\), and a sufficiently small power slack \(\sigma\); include arbitrarily fine choices of gains and slacks. The smallness conditions below are uniform on compact ranges of positive gains. If a residual candidate has gain \(c_n\to c_*>0\), choose a menu gain sufficiently close to \(c_*\) and a slack \(\sigma\) for which the difference of gains, multiplied by every fixed exponent in the tests and mass floor, is less than \(\sigma/4\). The candidate’s subpower losses and the depth-grid rounding use the remaining slack for all sufficiently late stages. In particular the change in the factor \(N^{-dc}\) is included in this choice; it is not ignored. Every candidate with positive limiting gain is thus eligible for some fixed scenario eventually.

For each available scenario and node, make the finite greedy assignment maximal. The selections for different scenarios are made separately on the same prepared path. Suppose that the union of every fixed finite menu has reference coverage tending to zero. Choose \(J_n\to\infty\) sufficiently slowly that the union of the first \(J_n\) scenarios still has coverage tending to zero. Remove all path incidences lying on their assigned trajectory pieces throughout the respective blocks. The remaining path has substantial mass. The candidate hypothesis gives, on a further subsequence, a candidate of positive limiting gain in a bounded depth range. The preceding menu choice supplies a fixed compatible scenario, eventually among the first \(J_n\). Its witnessing trajectories are still free for that scenario in the relevant chart, since every incidence on every earlier assigned piece was removed. This contradicts maximality.

Hence a fixed finite menu, of size \(J\), covers a fixed reference fraction \(\eta_0>0\) on a subsequence. Pigeonholing in that finite menu fixes one scenario on a further subsequence. Choosing one available node per sample, among at most \(G_n\), leaves coverage \[ \beta_n\ge\frac{\eta_0}{J G_n}. \tag{112}\] The node can vary with the sample; its depth remains in the fixed scenario’s bounded range. By [a03:mass-diagonal], \(\beta_n\log N_0\to\infty\). Moreover \(\beta_n^{-1}=N^{o(1)}\) by the diagonal preparation. These are distinct facts: the latter permits the analytical comparison, while the former supplies the actual output mass.

The following comparisons lose only negligible fractions of this coverage or bounded factors. In the time case retain the derivative lower bound \(N^{-2\sigma}\) at the covered incidences. The derivative on each assigned line is constant or affine, with maximum absolute value at least \(N^{-\sigma}\), so the excluded times have relative length \(O(N^{-\sigma})\). Disjointness of assignments and [a03:prepared-parent-law] bound their total mass by a fixed negative power, hence by \(o(\beta_n)\).

Conversion of strip coverage.

Take a true label deeper within the same parent, at a fixed target depth large enough that its velocity-strip thickness is \(\ll gN^{-c}\); here \(\ell>0\). Let \(A_{\rm deep}\) be its row, and put \(D=R-A_{\rm deep}=(D_0,D_1)\). On a counted incidence, \[|D_0+D_1v|\le CgN^{-c+\sigma}.\] If \(\|D\|>gN^{-c/2}\), boundedness \(|v|\le K\) implies either that no such incidence exists or that \(|D_1|\ge gN^{-c/2}/K\), after enlarging \(K\). The possible velocities then lie in an interval of length \[ \frac{2CgN^{-c+\sigma}}{|D_1|} \le KN^{-c/2+\sigma}\le N^{-c/3}. \tag{113}\] For each deep history \(E\), its ancestor \(Q_s\) has at most \(N^\sigma\) assigned rows. The joint numerator bound [a03:good-joint-numerator] and the angular ceiling therefore charge all these misaligned rows by at most \[K N^{\sigma-Sc/3}\lambda_{A,\mathrm{pre}}(E).\] Use fine velocity bins and bounded enlargements to cover the intervals. Summing over histories, labels, and parents with weights \(p_A\) gives total mass at most \(KN^{\sigma-Sc/3}\). For example, choose \(\sigma<\min(c/12,Sc/12)\). This is a fixed-power saving and hence is \(o(\beta_n)\), since \(\beta_n=N^{-o(1)}\). Thus all but a negligible fraction of counted coverage satisfies \[ \|R-A_{\rm deep}\|\le gN^{-c/2}. \tag{114}\]

The unbinned rows stored in the priority assignment also have coefficients bounded by a fixed power of \(N\). A row with much larger norm and any eligible lines would confine bounded velocities to an interval of arbitrarily small fixed-power length. The same joint bound, summed over the preceding histories in \(Q\), charges that interval by its palette cost times \(Kq_s\nu_Q\). Here the preceding history mass is at most \(Kq_s\nu_Q\) by [a03:prepared-parent-law]. For a sufficiently large fixed power this contradicts the greedy eligibility threshold.

Set \(\tau=(J_s/q_s^2)N^{-c/4}\), using the actual relative \(J_s,q_s\) at \(Q\), and bin the two coefficients of \(R\) at width \(\tau\), with values \(R_0^b,R_1^b\). On aligned events the deeper path label gives an actual list for this bin at original \(p_L\), since \(gN^{-c/2}\ll\tau\). The row \(R\) itself then has coefficients bounded by \(K_{\rm path}^{O(1)}\), using the actual deep-row bounds from \(Z\)-projection fullness. Omit predictors violating these bounds from the output. On assigned lines, \[ |Y'-R_0^b-R_1^b Z'|\le K_{\rm path}^{O(1)}\tau \tag{115}\] by the actual bound on \(v=Z'\) and the positive power buffer. Also bin \(Y-R_1^bZ\) at the block midpoint at width \(q_s\tau\). Given the observed positions and velocity-row bin, this position bin has actually subpower ambiguity. The output label consists of the old label \(Q\), its binned row, and this position bin. Retain each trajectory’s earlier priority assignment and merge assignments with the same three entries. This assignment is fixed throughout the block. The row and position bins, rather than the predictor’s identity, are the data to be recovered from the original observation.

Keep the old \(Z\)-center. For \(Y-R_1^bZ\), use the selected partition value moving with derivative \(R_0^b\), and take spatial columns \(q_s(1,R_1^b)\) and \(q_s(0,\tau)\) in parent \((Z,Y)\) coordinates. This geometry has actual path-slack bounds throughout the block. Its determinant in parent coordinates is \(q_s^2\tau=J_sN^{-c/4}\), so composition through the parent multiplies the old cumulative determinant at \(Q\) by \(N^{-c/4}\). Keep the true test, depth, and time. The resulting atlas has the forbidden narrowness improvement and the required aggregate list success on the original path; its tempered implementation is given below.

Comparison of the time candidates.

We compare candidate graphs compatible with the same original observation. They agree at its base point to the candidate fitting accuracy. We will prove that, except on a negligible fraction of compatible pairs, they are also close on a whole short comparison box. Their finite jets can then replace the candidate indices as listable chart labels.

Work in each assigned \(Q_s\), using its actual normalized base coordinates \(u\) (unit-block time and spatial coordinates via its moving affine center and frame); \(w\) remains in parent units. Write \(a_Q\) for the actual node thickness in these parent units, \(a_Q/a\sim_{\log,N}1\). Any subpower factors in the following preparatory comparisons can be absorbed using \(N^\sigma\), for fixed \(\sigma\) and sufficiently late stages. The constants in exponents \(O(\sigma)\) below can be fixed independently of \(\sigma\) small. Write \(m_Q=p_A q_s\nu_Q\), so \(\sum_Q m_Q\le1\) summing over parents and blocks at the chosen node.

Each index (assigned test)’s base marginal on the counted occurrences costs at most \[ m_Q N^{-dc+O(\sigma)}\,du. \tag{116}\] Indeed the candidate value and lower derivative tests require only \(O(N^{O(\sigma)})\) hidden bins of width \(a'\) at each exact base (use monotone intervals of the quadratic); use the \(K(a')^d\) cap and Jacobian \(q_s J_s\), with \(\nu_Q\sim_{\log,N}a^d J_s\). This bound holds also on each test’s original matched incidence set witnessing the greedy mass threshold. That set’s projection in \(u\) has volume \(\ge N^{-O(\sigma)}\), by the threshold and [a03:candidate-marginal] (likewise if using the unnormalized parent prior for eligibility).

Convert each assigned test and the true \(Q_s\) test into local graphs in \(w\). Write the test as \(P\), with \(a_{\rm fit}=a'\) for a candidate and \(a_{\rm fit}=a\) for the true test. At late stages its whole-block value bound is \(a_{\rm fit}N^\sigma\), its whole-block derivative upper bound is \(N^\sigma\), and its derivative lower bound at counted points is \(N^{-2\sigma}\), up to harmless fixed constants. If \(|P_{ww}|\ge N^{-C\sigma}/a_{\rm fit}\), for a sufficiently large fixed \(C\), say \(20\), use the affine quadratic center. It stays within \(a_{\rm fit}N^{O(\sigma)}\) of the entire assigned line piece; if an equation is needed, use \(w\) minus this center.

For the remaining tests the discriminant \[ \mathcal D=P_w^2-2P_{ww}P \tag{117}\] is independent of \(w\) and at counted points positive and comparable to \(P_w^2\). On complex enlargements of the entire normalized chart box (\(K_{\rm path}\)-sized ranges in \(u\) allowed) this polynomial and its first derivatives are bounded by \(N^{O(\sigma)}\). Indeed its upper bound on the matched projection just discussed follows immediately from the tests there, for both the assigned candidate and true test; extend by polynomial Remez. This uses the initial matching before greedy enlargement for each assigned candidate, not a new floor on the path in each subbox.

Subdivide the time of \(Q_s\) dyadically at relative size \(\rho\) comparable to \(N^{-C'\sigma}\), with \(C'\) sufficiently large and fixed. In each short block, bin every line’s normalized spatial position at the midpoint into \(\rho\)-boxes. There are at most \(N^{O(\sigma)}\) resulting base subboxes, including time, per \(Q\). Given \(Q\) and the exact base position, these subboxes have lists at the original \(p_L\) of size bounded by actual path-slack powers: motion in normalized coordinates over a short block is at most \(K_{\rm path}^{O(1)}\rho\).

Around each subbox center, take comparison balls or polydiscs with radius a fixed multiple of \(K_{\rm path}^{O(1)}\rho\), large enough to contain the corresponding real line pieces with a fixed extra holomorphic margin. Keep only test–subbox choices certified by a counted occurrence, which supplies the derivative lower bound at one point. The chart derivative bound for \(\mathcal D\), together with the choice of \(C'\), makes its variation throughout this comparison domain much smaller than \(N^{-4\sigma}\). For each nonreplaced quadratic test, the discriminant therefore stays nonzero and gives simple holomorphic branches throughout the domain, real on the real box. For a linear equation in \(w\), the same check keeps its coefficient nonzero.

For such nonreplaced tests, on any assigned line for these choices the low-curvature cutoff guarantees \(2|P_{ww}P|\ll\mathcal D\) over the whole real short piece, so the sign of \(P_w\) is constant there, picking one branch \(f\). At every time \(w\) has distance at most \(O(a_{\rm fit}N^{O(\sigma)})\) from that branch: \(P_w\) has the same sign there and at \(w\), comparable in absolute value to \(\sqrt{\mathcal D}\), so the derivative lower bound works on the intervening interval. Replaced tests simply use their affine centers. Thus branch signs for true and candidate tests can be assigned invariantly over the short block. Denote the true comparison graph in one such choice by \(f_0\).

Let \(\mu_{\mathrm{cov}}\) be the unnormalized law of counted covered occurrences, including the parent weights. Set \(O=(p_L,Q_s,\text{subbox},\text{true branch})\), with history included in \(Q_s\), and let \(\kappa_O\) be the conditional probability of an occurrence under this law. Sample \(O\) with its covered-law pushforward, then sample two independent posteriors: \[ \mu_{\mathrm{pair}} =\int \kappa_O\otimes\kappa_O\,d(O_*\mu_{\mathrm{cov}})(O). \tag{118}\] Each marginal is \(\mu_{\mathrm{cov}}\), and the pair law retains its unnormalized total mass, at least \(N^{-o(1)}\). The candidate graphs agree at the sampled common base point to \(a'N^{O(\sigma)}\), because original \(p_L\), given the parent, determines \(w\) much more finely. In version 1 the subtracted parent graph is fixed by the chart and exact base, so the same conclusion holds.

Fix \(d_0=(1+d)/2<1\). Let \(\varepsilon_*\) be the tolerance in 22 for these fixed comparison domains and degrees. Decrease it to at most one, and set \[ \varepsilon=\varepsilon_*/2,\qquad \theta=\varepsilon_*/8. \tag{119}\] We will choose \(\sigma\) after these constants. We claim that, outside an \(o(1)\) relative fraction of the pair law, the candidate graphs agree in norm on the comparison real box, with its fixed margin around the used pieces, to \(aN^{-2\theta c}\).

First discard entries (index, subbox, candidate and true branch signs) with marginal mass below \(m_QN^{-dc-C''\sigma}\). The index and subbox counts make their total mass negligible on either marginal when \(C''\) is sufficiently large. For each remaining entry, the candidate graph differs from \(f_0\) by at most \(aN^{O(\sigma)}\) on its incidence base projection. By [a03:candidate-marginal], this projection has relative measure at least \(N^{-O(\sigma)}\) in the comparison box. Apply 19 on the interiors with fixed margins to obtain the same norm bound, with an adjusted \(O(\sigma)\) exponent.

Suppose a nonvanishing relative fraction of the pair law were bad. The light-entry omissions leave a group \((Q,\text{subbox},\text{true branch})\) of bad pairs of mass at least \(m_QN^{-O(\sigma)}\); the weighted count justifying this choice is given below. Normalize that group’s bad pairs to probability. We compare its two candidate lists by 22, relative to the common true graph \(f_0\), with \(M=a\) and \(\Delta=a'\). After rescaling the base, [a03:candidate-marginal] gives index-base ceilings with weights \(N^{-dc}\) and losses \(N^{O(\sigma)}\). To check that they remain valid for probability weights in each color, we also account for the bad-pair normalization. Indeed, if \(W_c\) is the sum of \(N^{-dc}\) over that color’s retained indices and \(B\) is the unnormalized bad-pair mass in the selected group, the index density bound becomes \[ \frac{f_i}{B} \le N^{O(\sigma)} \frac{m_Q W_c}{B}\, \frac{N^{-dc}}{W_c}. \tag{120}\] The list count gives \(W_c\le N^{O(\sigma)}\), and the group choice gives \(B\ge m_QN^{-O(\sigma)}\). Thus the multiplier of the probability weight \(N^{-dc}/W_c\) is \(N^{O(\sigma)}\), as required. The existence of such a group uses the sum \(\sum_Qm_Q\le1\) and the polynomial-in-\(N^\sigma\) subbox count. If a nonvanishing relative fraction of the covered pair law were bad, its absolute mass would be at least a fixed multiple of \(\beta_n=N^{-o(1)}\). Light-entry errors are power-small and hence \(o(\beta_n)\); the factor \(\beta_n^{-1}\) is absorbed into the fixed-power comparison allowance.

Here is the tolerance check. Put \(R=M/\Delta=N^c\), with \(M=a\) and \(\Delta=a'\). All functions are holomorphic sheets with bounded-degree relations. A bad pair has norm gap at least \[aN^{-2\theta c}=MR^{-2\theta}>MR^{-\varepsilon}.\] The list counts are at most \(R^{d+C\sigma/c+o(1)}\). The norm upper bounds relative to the true graph \(f_0\), the pointwise matching precision, and the normalized marginal multipliers just computed have losses at most \(R^{C\sigma/c+o(1)}\), for finitely many fixed constants \(C\). Choose \(\sigma\) so that every \(C\sigma/c<\varepsilon_*/8\). At late stages the remaining \(o(1)\) losses, including \(\beta_n^{-1}\), fit within another \(\varepsilon_*/8\). The normalized bad-pair law has mass one, so it also satisfies the required lower mass bound. Since \(d<d_0\), all hypotheses of 22 hold with \(\varepsilon=\varepsilon_*/2\), a contradiction. This proves the claim.

The improved time charts.

Bin sufficiently many candidate branch jets at the subbox center (in comparison-ball scaled coordinates) to steps \(aN^{-\theta c}\). Pairs with the proved closeness give bounded neighbor bins by the norm rule. For clarity, if \(\pi_O\) is the posterior law of the actual jet bin at a conditioning observation \(O\), and \(\mathcal N(b)\) is the bounded neighbor list of a bin \(b\), then \[ \sum_b\pi_O(b)\pi_O(\mathcal N(b)) = \mathbb P\{B_2\in\mathcal N(B_1)\mid O\}. \tag{121}\] Choose a maximizing \(b\) separately for each \(O\). Its neighbor list has success at least the displayed average. Integrating preserves \(1-o(1)\) of the covered mass once the bad-pair fraction is \(o(1)\); in particular it preserves a fixed positive fraction. This argument retains aggregate covered mass; it does not select an individual candidate that may carry only inverse-subpower mass.

Define the preliminary geometric label \[c_{\mathrm{geom}}=(Q_s,\text{short block},\text{spatial subbox}, \text{true branch sign},\text{candidate jet bin}).\] Keep the earlier priority assignment and its invariant branch signs, and merge assignments with the same \(c_{\mathrm{geom}}\). This gives a trajectory assignment fixed throughout the short block. The original candidate index is not part of \(c_{\mathrm{geom}}\). The conditioning entries preceding the jet bin have only actual path-slack ambiguity at the original \(p_L\), so the posterior-neighbor lists give lists for this full preliminary label with the same type of bound.

For each nonempty jet bin in its group, choose one representative equation \(\widehat P\) and its assigned branch \(\hat f\). The representative is a fixed function of \(c_{\mathrm{geom}}\). Every member branch in this bin differs from \(\hat f\) by \(O(aN^{-\theta c})\) on the real box: sufficiently many center jets control the norm of their holomorphic algebraic difference. Thus \[|w-\hat f|\lesssim aN^{-\theta c}\] on each member’s entire short line piece, after choosing \(\sigma\) small enough in the graph approximations. We now turn the representative into an admissible chart test.

• If \(|\widehat P_{ww}|\le (aN^{-\theta c/2})^{-1}\), normalize by the absolute \(w\)-derivative on the branch at box center (this equals 1 for explicit tests, notably in version 1). In the simple-root case this factor is \(\ge N^{-O(\sigma)}\) and comparable uniformly, to constants, to the absolute branch derivative throughout by discriminant stability. Passing from branch to member \(w\) changes the derivative by \(O(N^{-\theta c/2})\). Thus the normalized equation has actual order-one derivative bounds, value \(\lesssim aN^{-\theta c}\) and curvature \(\le a^{-1}N^{\theta c/2+O(\sigma)}\) on the required data. These fit at thickness \(a_Q N^{-2\theta c/3}\).

• For larger curvature, the representative branch is within \(aN^{-\theta c/2+O(\sigma)}\) of its affine center by the branch derivative upper bound. Use \(w\) minus that affine center as test with thickness \(a_Q N^{-\theta c/3}\).

Choose \(\sigma\) after \(\theta\) to meet the preceding matching tolerances and graph-approximation bounds. Also impose \(C'\sigma<k\theta c/12\). The strict buffers absorb the subpower discrepancy \(a_Q/a\). Select a representative type carrying a fixed fraction of the list success, so the output has a common depth.

Use the true spatial frame scaled by \(\rho\), with center adjusted to the subbox’s midpoint spatial bin in moving chart coordinates. Its bounds hold throughout with actual path slacks. Restore original coordinates by 14: thickness and tests acquire the parent-depth factor, with derivative units restored. In version 2, \(w\) uses the resulting parent \(B\)-scale; in version 1 the parent graph subtraction preserves the special test form. The path labels, subbox, true branch sign, and deterministic bin lists together cost only actual path-slack and logarithmic powers.

Keep the preliminary label \(c_{\mathrm{geom}}\); its chosen representative already determines the curvature type and the final test. Retain only assigned trajectories satisfying the resulting whole-block geometric, value, and derivative upper bounds, and impose the derivative lower bound on selected incidences. The comparisons above show that the retained list successes satisfy these tests. We have therefore constructed geometric charts and full-label lists on the covered analytical law.

The depth gain is at least \((\theta c/3)\epsilon_n\), whereas the additional time exponent is \(C'\sigma\epsilon_n+o(\epsilon_n)\). Our choice of \(\sigma\) makes this less than half \(k\) times the depth gain. Either representative type therefore gives the forbidden improved-depth atlas.

Finite encoding and return to the original law.

Both constructors have now specified their chart geometry, tests, full labels, and block-invariant trajectory assignments. We verify that these objects have bounded original-coordinate complexity and recover their lists on the original tempered path.

Store the ordered quadratic tests or row predictors at the single chosen node. The assignments use membership in the old chart’s trajectory set, whole-block line inequalities, and finite priority rules. Subbox certification flags, branch signs, merger tables, and representative tests are fixed data for each chart and test; a simple branch is selected by its midpoint derivative sign. Nonempty jet bins may be reindexed by discrete identifiers, so numerical jets need not enter a predicate. The analytical masks and the posterior search used to choose the lists are not part of these geometric assignments.

Coefficient bounds.

Here is an explicit coefficient bound for every time candidate. In normalized chart coordinates write \(P(u,w)=A w^2+B(u)w+C(u)\), with \(B\) affine and \(C\) quadratic. Curvature gives \(|A|\le N^{s+c+O(\sigma)}\). The original threshold witness, before greedy enlargement, has base projection of volume \(\delta\ge N^{-C_{\mathcal S}\sigma}\) by [a03:candidate-marginal]; its chart weight cancels when the floor is divided by the ceiling. On that witness \(|w|\le W\le N_0^{P_0(M_{\mathrm{path}})}\) by original problem and path bounds. The derivative and value estimates yield \[ |B(u)|\le N^\sigma+2|A|W,\qquad |C(u)|\le a'N^\sigma+|A|W^2+|B(u)|W . \tag{122}\] Fixed-degree polynomial Remez from this projection costs at most \(\delta^{-C}\); it bounds all coefficients of \(B,C\) by \(N_0^{P_{\mathcal S}(M_{\mathrm{path}})}\). This uses a bound for \(w\) in original exponent units and does not assume that its affine part is bounded in tangent units. Inverse curvatures for affine centers are needed only above cutoffs; rescaling by a branch derivative uses the stability lower bound. Raw strip coefficients were bounded above. Coordinate compositions back to the original world are all polynomially controlled. Counts of new choices per path chart cost only fixed tangent powers. The tests for an entire line/block here have fixed-degree semialgebraic complexity (if quantifying over block time do that only for each individual test separately).

Actual lists.

On the masked successes the full output labels have lists at the original \(p_L\) of size at most \((K_{\mathrm{path}}\log N_0)^{r_{\mathcal S}}\), with \(r_{\mathcal S}\) fixed by the scenario. Apply 35 to the original unnormalized reference path and its masked covered submeasure, with the geometric assignment just defined. This produces a tempered cell-list selection with the same list size and only \(O(N_0^{-1})\) original mass loss. In particular, one does not flatten the analytical masks or encode their posterior rules. The final selector observes only the original exact base and original discrete observation entries.

Complexity and mass.

The coefficient and predicate bounds above, followed by 35, bound every original-coordinate complexity exponent by a polynomial in \(M_{\mathrm{path}}\) depending only on the fixed scenario. This includes the flattening depth and list-table count. Only the chosen scenario is encoded. Its polynomial bound fits \(M^\#\) at all sufficiently large stages by [a03:complexity-guard]; its fixed power of actual path factors and logarithms likewise fits the allowance \(n\).

The comparison and representative choice retain some fixed fraction \(c_0>0\) of \(\beta_n\). After original-law list recovery, the original final success is therefore at least \[ \frac{c_0\eta_0}{J G_n}\,\mathfrak p_{\mathrm{path}} -O(N_0^{-1}) > \frac{\mathfrak p_{\mathrm{path}}}{\log N_0} \tag{123}\] for large stages, by [a03:mass-diagonal]. The original mass loss is negligible since \(N_0^{-1}\log N_0/\mathfrak p_{\mathrm{path}} \le N_0^{-1}K_{\mathrm{path}}\log N_0\to0\). The fixed positive tangent gain exceeds \(\gamma_n\), and its time cost is less than half \(k\) times its gain. All requirements of 29 are now satisfied, giving the contradiction. ◻

A finite family of heavy strips.

Use unit normal and offset pairs \((n_v,b_v')\) for those predicted lines, with Euclidean parameter distance modulo simultaneous sign. Parent velocities are bounded by \(T\ge1\) of order actual path-slack powers; offsets needed then cost \(O(T)\). On almost all the narrow events the predicted strip, say enlarged to \(2N^{-u}\) using finer velocity bins, has parent palette \(\pi\)-weight at least \(\kappa_N=1/K\) for sufficiently small reciprocal-subpower threshold. Indeed those of smaller palette mass cost negligibly by summing the unnormalized joint bound of velocity with the end history using the subpower list there (choose \(\kappa_N\) small relative to all those losses). State predictions need only be specified on occurring states; their choices are finite here.

Per parent take maximal parameter-separated representatives at spacing \(r_1=N^{-u/2}\) for such heavy strips. Their number \(M_1\) is \(\le K\). For two separated ones with common narrow-band points in \(|v|\le T\), the angle sine is at least a constant times \(r_1/(1+T)\): take the sign with closer normals; offset difference there is at most \(4N^{-u}+T|n_{v,1}-n_{v,2}|\). Thus intersection costs a ball of diameter \(O((1+T)N^{-u}/r_1)\le N^{-u/3}\), hence a fixed angular power in \(\pi\). By Cauchy–Schwarz on the sum of strip indicators, \[ (M_1\kappa_N)^2\le M_1+K N^{-Su/3} M_1^2, \tag{124}\] giving the claim. Moreover for representatives separated by \(>r_2=N^{-u/4}\), their enlarged strips of widths \(10(1+T)r_1\) still have intersections of negligible palette mass even summed over all pairs: the same reasoning gives diameter \(\lesssim (1+T)^2r_1/r_2\) or no common point in the used range.

Trajectory assignments and the narrower atlas.

On narrow events the velocity therefore lies in an enlarged representative strip, and any choice of such containing strip agrees to \(O(r_2)\) in parameter (modulo sign) with the predicted pair outside negligible mass. Indeed there is one within \(r_1\) of the prediction by maximality, and the excluded overlaps depend just on parent and velocity and cost little by the palette bound. Assign the first containing representative by velocity alone on parent trajectories, invariantly over the block. Bin its parameters (with one orientation each) to rectangular \(r_2\)-bins; this costs only boundedly many options given the state prediction on successful events. Fix one unit normal and offset pair \((\hat n,\hat b)\) per occurring bin consistent with it. Then \[ |\hat n\cdot y'-\hat b|\lesssim (1+T)r_2 \tag{125}\] on assigned pieces. Bin also the corresponding \(\hat n\)-position at parent-block midpoint to width \(r_2\) in parent units, known with actual slack up to lists via observed base and this velocity prediction. Merge assignments using those two bin labels. Use a frame with unit tangential column and width \(r_2\) along \(\hat n\) in parent spatial coordinates (move the transverse bin with derivative \(\hat b\)); compose with the parent frame, keeping parent test, depth and horizon. All slacks thus cost only actual path bounds to fixed powers, and the determinant has gained the new narrowness factor.

On the successful incidences, the assumed state knowledge supplies actual lists for the full output labels at the original \(p_L\). Store the finite representative and bin assignments together with the original path predicates. Apply 35 to this geometric label map on the original path and its successful submeasure. The analytical list success is then realized by tempered tables with the same list size and only \(O(N_0^{-1})\) original mass loss; the state predictions themselves need not be encoded. As in the preceding construction, the original mass and fixed-complexity bounds fit 29. This contradicts the width improvement excluded at the parent.

In version 1 the corresponding avoidance of narrow velocity intervals on substantial families already follows from the angular cap and the deep-history joint bound (summing the subpower state ambiguities there). These almost-all breadth/separation conclusions can use reciprocal-subpower accuracies by taking fixed-power thresholds to zero sufficiently slowly. In applications requiring general position, one can prepare them before sparse selections and keep them through selections of whole conditioning states (or with small fixed-fraction losses controlled there). No breadth in version 2 on every \(K^{-1}\) velocity-dependent selection is claimed. ◻

Preparing the current labels.

We perform the following preparation before fitting each row component at each hierarchy length \(H\). Denote the current labels by \(E_l^{\rm cur}\), and write \[m(t)=\nu\{l:t\in E_l^{\rm cur}\},\qquad \delta=\int_0^1m(t)\,dt\ge K^{-1}.\] Choose a reciprocal-subpower threshold \(\tau=o(\delta)\). First remove times with \(m(t)<\tau\). Next remove each dyadic \(H\)-block whose remaining incidence mass is less than \(\tau H\). The two removals cost at most \(\tau\) each, so the resulting labels \(E_l^{\rm pre}\) have mass at least \(\delta/2\). Every surviving block has mass at least \(\tau H\ge H/K\), and at every surviving active time the instantaneous mass is still \(m(t)\ge\tau\): both removals kept or discarded all current labels at a time. Since \(\nu\) is a probability, each block has mass at most \(H\). Thus at least \(\delta/(2H)\) blocks survive. We now run the local construction separately in each of these blocks.

Local normalization.

Fix a surviving block \(I=[t_I,t_I+H]\), put \(u=(t-t_I)/H\), and define \[U_l=\{u\in[0,1]:t_I+Hu\in E_l^{\rm pre}\cap I\}.\] Averaging and Cauchy–Schwarz provide \(u_*\) such that \[\int \mathbf 1_{\{u_*\in U_l\}}|U_l|\,d\nu(l)\ge K^{-1}.\] During the local construction retain trajectories labeled at \(t_*=t_I+Hu_*\), and put \[F_*=F(t_*),\qquad B_l=x_l(t_I)-F_*y_l(t_I).\] The difference \(B_l-S_l(t_*)\) is \(O(KH)\). Thus the occupied \(H\)-bins \(C\) of \(B_l\) number at most \(KH^{-d}\), and each has prior mass at most \(KH^d\). For the latter statement, sum [a04:absolute-cap] at time \(t_*\) over the \(KH^{-j}\) tangential \(H\)-bins compatible with \(|B_l-S_l(t_*)|\le KH\). In each \(C\), divide the restricted prior by \(H^d\) and call the result \(\nu_C\). Deleting groups with \(\int |U_l|\,d\nu_C<K_1^{-1}\) loses at most \(K/K_1\) of the original normalized incidence mass. Choose a sufficiently large subpower \(K_1\), so that a dense family of good groups remains.

For a lower endpoint \(b_C\), define \[\begin{align*} Y_l&=y_l(t_I),& z_l&=(B_l-b_C)/H,& q_l&=x_l'(t_I)-F_*y_l',\\ \phi(u)&=(F(t_I+Hu)-F_*)/H,& L_l(u)&=z_l+u q_l-\phi(u)Y_l. \end{align*}\] The normalized row \(\phi\) measures the variation that we must fit on this block. The scalar model \(L_l\) separates that row from the frozen position \(Y_l\) and residual slope \(q_l\). These quantities are bounded by \(K\) on the labels, and \(\phi\) obeys a Lipschitz comparison with slack \(\zeta/H\). Writing \(x_{2,l}\) for the quadratic coefficient, direct expansion gives the exact identity \[ \frac{S_l(t_I+Hu)-b_C}{H} =L_l(u)+H\bigl(u^2x_{2,l}-u\phi(u)y_l'\bigr). \tag{143}\] The final term is \(O(KH)\). Consequently at \(p=H^{1/4}\), and also at \(p_0=H^{1/8}\), the \(L_l(u)\)-supports retain the upper \(d\)-count. Moreover, \[ \nu_C\{l:u\in U_l,\ (Y_l,L_l(u))\text{ in given }p\text{-bins}\} \le Kp^{j+d}. \tag{144}\] Indeed, in the original variables the corresponding region has tangential widths \(Kp\) and normal width \(KHp\). It is covered by \(KH^{-j}\) cubes of side \(Hp\), and division by \(H^d\) converts their total cap to \(Kp^{j+d}\). The same proof gives [a04:local-joint-cap] with \(p_0\). Finally, the residual comparison at \(t_*\) gives \[ |q_l-q_{\tilde l}| \le K(|Y_l-Y_{\tilde l}|+H). \tag{145}\] Here \(y_l(t_*)=Y_l+O(KH)\), the relevant scalar values differ by \(O(KH)\), and replacing \(x_l'(t_*)\) by \(x_l'(t_I)\) costs \(O(KH)\). Thus the quadratic part of \(x_l\) contributes only the recorded \(O(KH)\) errors. At the coarser scale \(p=H^{1/4}\), freezing the other coordinates of \(Y_l\) makes \(q_l\) an approximately Lipschitz function of the remaining coordinate. We now use the planar theorem to fit the corresponding component of \(\phi\).

Fitting one row component.

Slice the other \(j-1\) coordinates of \(Y_l\) into \(p\)-bins; no slice is needed when \(j=1\). A group consisting of \(C\) and one such slice has prior mass at most \(KH^dp^{j-1}\), by the spatial cap at \(t_*\). After division by \(H^dp^{j-1}\) and deletion of light groups, choose a sliced measure \(\nu_s\) with \[\nu_s(\text{all trajectories})\le K,\qquad \int|U_l|\,d\nu_s(l)\ge K^{-1}.\] Let \(v_l\) be the remaining coordinate of \(Y_l\). Choose representatives of \(q_l\) on occupied \(v\)-bins and interpolate between their centers. [a04:frozen-q-comparison] gives a function \(G\) satisfying \[|G(v)-G(v')|\le K(|v-v'|+p).\] Translate \(L_l(u)\) by the other components of \(\phi(u)\) times the slice centers. The resulting labeled values satisfy \[ L_l^a(u)=z_l+uG(v_l)-\phi_1(u)v_l+O(Kp). \tag{146}\] They have the local upper \(d\)-count, and \[ \nu_s\{l:u\in U_l,\ (v_l,L_l^a(u)) \text{ in given }p\text{-bins}\}\le Kp^{1+d}. \tag{147}\]

Choose \(u_0\) with \(\int\mathbf 1_{\{u_0\in U_l\}}|U_l|\,d\nu_s\ge K^{-1}\). For pairs of trajectories labeled at both \(u_0\) and \(u\), count collisions of their \(L^a(u)\)-bins using the unnormalized measure \(d\nu_s(l)\,d\nu_s(l')\,du\). Cauchy–Schwarz and the \(Kp^{-d}\) output-bin count give collision mass at least \(p^d/K\). Fixing the first trajectory’s starting scalar bin at \(u_0\) leaves at least \(p^{2d}/K\) of this mass. On the retained pairs, [a04:sliced-model] implies \[\begin{align*} &(u-u_0)(G(v_l)-G(v_{l'})) -(\phi_1(u)-\phi_1(u_0))(v_l-v_{l'})\\ &\hspace{35mm}=L_{l'}^a(u_0)-L_l^a(u_0)+O(Kp). \tag{148}\end{align*}\] At fixed \(u\) and fixed \(p\)-bins of \(v_l,v_{l'}\), the first trajectory has weight at most \(Kp^{1+d}\) by its fixed starting scalar bin. The equation confines the second starting scalar value to \(K\) bins, so its weight has the same bound. Each \((u,v_l,v_{l'})\)-cell therefore carries at most \(Kp^{3+2d}\). The total pair weight at a fixed \(u\) is at most \(Kp^{2d}\): sum [a04:sliced-cap] over the tangential bins for the first starting value and the second matching value.

For each time \(p\)-bin \(J\), let \(T_J\subset J\) be its active pair times for the retained collisions, before any quadrant restriction. Discard bins with \(|T_J|<p/K_1\), and keep these pre-quadrant sets \(T_J\) attached to all surviving bins. There are \(O(p^{-1})\) time bins, so the discarded pair mass is at most \(Kp^{2d}/K_1\). Take \(K_1\) large enough to preserve the lower bound. The represented triples \[a=(u-u_0,-\phi_1(u)+\phi_1(u_0)),\quad b=(G(v_l),v_l),\quad b'=(G(v_{l'}),v_{l'})\] then have \(p\)-covering number at least \(p^{-3}/K\).

Retain a quadrant for \(a\) containing a fixed fraction of the pair mass. Let \(\mathcal H\) be the image of these retained collision triples, let \(\mathcal A\) be its first-coordinate projection, and let \(\mathcal B\) be the union of its two trajectory-coordinate projections. Thus \(\mathcal H\subset\mathcal A\times\mathcal B\times\mathcal B\), with the same lower covering count up to a fixed factor. Reflect both planar sets by the same coordinate reflection, dilate each by a positive scalar between \(K^{-1}\) and \(1\), and translate only the second set to place the sets in \([0,1]^2\). Translation of the second set cancels in \(b-b'\); no translation of the first set is made. Their graph comparisons give the upper covering condition [a04:planar-regularity] with a subpower constant: covering a horizontal interval for the first graph, or a vertical interval for the second, by \(s\)-intervals gives at most \(Kw/s\) graph cells. Normalization changes this constant and the triple lower count only by subpowers.

Fix \(0<\varepsilon<1-d\). For sufficiently large \(N\), the \(p^{\pm\eta}\) allowances in 41 absorb these subpower losses. Its expansion alternative is impossible: by [a04:collision-dot], all dot products are within \(Kp\) of a translate of the starting scalar support, so they use at most \(K(|J|/s)^d\) \(s\)-bins in \(J\). This would imply \[(|J|/s)^{1-\varepsilon-d}\le K, \qquad |J|/s\ge p^{-\eta},\] a contradiction. The additive error and inverse dilations cost only subpowers at \(s\ge p\); intervals larger than the unit cutoff can be split into subpower many intervals in the bounded range.

We therefore obtain at least \(p^{\varepsilon-1}/K\) time bins near one line in the graph of \(\phi_1\). The line is a graph over time with slope at most \(K\): two represented points have horizontal separation at least \(p^\varepsilon/K\), larger than their \(Kp\) proximity errors, and the graph comparison bounds their slope. Let \(\mathcal J_I\) be the resulting family of represented time bins. In each \(J\in\mathcal J_I\), the graph comparison extends the affine fit from a represented near-line point to every time in the stored set \(T_J\), with error \(Kp\). Indeed, their time separation is at most \(p\), and both the graph comparison and the fitted line have slope bounded by \(K\). Use precisely the time set \[T_I=t_I+H\bigcup_{J\in\mathcal J_I}T_J.\] Every \(T_J\) has length at least \(p/K_1\); the disjoint bins and their number therefore give \[|T_I|\ge Hp^\varepsilon/K.\] In particular, the fitting set consists of prepared active times, not the whole selected bins. Let \(\varepsilon\) tend to zero sufficiently slowly, after the corresponding \(\eta,\delta_0\) and large-\(N\) thresholds have been fixed. Then \(|T_I|\ge H/K\), while the normalized fit error remains \(Kp\).

In the original variables this is an affine component fit to \(F\) with error \(KH^{5/4}\) on \(T_I\). At these times retain all prepared labels \(E_l^{\rm pre}\), including those outside the temporary sliced group. Their total incidence mass satisfies \[ \sum_{I\ \mathrm{surviving}}\int_{T_I}m(t)\,dt \ge\tau\sum_I|T_I| \ge\tau\frac{\delta}{2H}\frac{H}{K} =\frac{\delta\tau}{2K}. \tag{149}\] This is inverse-subpower. It uses both the instantaneous floor and the number of surviving blocks, each established before the local search. For \(j=2\), repeat with the second component on these new labels, recomputing \(m,\delta,\tau\) first. Do the same at every length in the fixed finite hierarchy. Subsequent label restrictions preserve the earlier fits and all upper inputs. We obtain affine rows \(P_I(t)\), with slopes \(A_I\) of size at most \(K\), satisfying \[ |F(t)-P_I(t)|\le KH^{5/4} \tag{150}\] on every retained active time in the corresponding interval. Let \(I_*(t)\) be the finest dyadic block containing \(t\), using a fixed half-open endpoint convention. Define \[A(t)=A_{I_*(t)}\] on the finest blocks with retained labels, and set \(A(t)=0\) elsewhere. This is a fixed time-only row bounded by \(K\). Subsequent pruning restricts labels without changing this row.

Comparison across lengths.

We now compare the affine fits through the intermediate lengths fixed before any selections. Let \(\mu_{\rm fit}\) be the current labeled measure. At every hierarchy length \(H\), delete the labels in blocks \(I\) with \(\mu_{\rm fit}(I)<H/K_1\), testing all blocks against this same measure. Each length costs at most \(1/K_1\), so choose \(K_1\) large enough to make the total loss negligible. Every child block that still contains a retained incidence had \(\mu_{\rm fit}\)-mass at least \(H'/K_1\). Since \(\nu\) is a probability, its fitting times have length at least \(H'/K_1\). Both its own affine fit and its parent’s hold on those times, whether or not all of them survive the deletion. Choose two separated by \(H'/K\). Comparing the fits there gives \[|A_{I'}-A_I| \le K\frac{H^{5/4}}{H'}\le KH^{7/32}.\] Summing through the fixed finite chain compares the finest slope \(A(u)\) with \(A_I\) by \(KH^{7/32}\). Together with [a04:vector-affine-fit], this gives \[|F(v)-F(u)-(v-u)A(u)| \le K\bigl(H^{5/4}+H^{1+7/32}\bigr)\le KH^{11/10}.\] This proves [a04:first-taylor], which persists under the label restrictions below.

Concentrating the residual slopes.

The affine fits now hold on a dense set of incidences. We next restrict the trajectory labels so that their residual slopes admit short lists once the time and scalar value are known. At the finest length \(H=H_*\), repeat the unsliced \(C\)-group normalization on the retained labels. With \(p_0=H^{1/8}\), the affine approximation to \(\phi\) gives an affine approximation to \(L_l\) with slope \[k_l=q_l-A_IY_l\] and error at most \(Kp_0\). The slopes vary by at most \(Kp_0\) in each \(Y\)-cell of side \(p_0\), by [a04:frozen-q-comparison]. Put \(\gamma=(1-d)/4\). We claim that some interval of width \(p_0^\gamma\) contains slopes of labeled incidence mass at least \(K^{-1}\) in every good group, with uniform subpower constants.

If this failed, there would be a sequence of groups and a fixed \(c>0\) on which every such interval has incidence mass at most \(N^{-c}\). Let \(\tau=N^{-c/4}\). Delete trajectories with labeled length below \(\tau\), and then \(Y\)-cells whose remaining trajectory mass is below \(p_0^j\tau\). The two losses are at most \(K\tau\). The remaining output collisions, integrated in time, still have mass at least \(p_0^d/K\), by Cauchy–Schwarz.

For a fixed first trajectory, second trajectories with slope within \(p_0^\gamma\) occupy at most \[K N^{-c}\tau^{-2}p_0^{-j}\] retained \(Y\)-cells. Indeed, their slopes, including the \(Kp_0\) variation within each cell, lie in a bounded number of intervals of the assumed small incidence mass. Their labeled lengths are at least \(\tau\), so their prior mass is at most \(KN^{-c}/\tau\); each retained cell has prior mass at least \(p_0^j\tau\). For each such cell the matching output weight at a time is at most \(Kp_0^{j+d}\), by [a04:local-joint-cap]. Thus the close-slope collision contribution is at most \[Kp_0^d N^{-c}\tau^{-2}=Kp_0^dN^{-c/2}.\] For the other pairs, the two affine approximations can agree to \(Kp_0\) only on times of length at most \(Kp_0^{1-\gamma}\). Their total contribution is at most that quantity times a subpower bound for the product prior mass. Both contributions are \(o(p_0^d/K)\), because \(1-\gamma>d\). This contradicts the collision lower bound and proves the claim. If the claim’s subpower constants were not uniform across groups, an offending sequence of groups would give exactly the contradiction just proved.

Keep one such slope interval in each good group. Since \[ J_1(l,t)-k_l =H\bigl(2u x_{2,l}-\phi(u)y_l'-uA_Iy_l'\bigr)=O(KH), \tag{151}\] this gives a deterministic interval cover for \(J_1\). For fixed \(t,S_l(t)_H\), only \(K\) base bins \(C\) are possible: \(|B_l-S_l(t)|\le KH\). Hence \(J_1\) lies in \(K\) intervals of width \[Kp_0^\gamma=KH^{(1-d)/32}.\] At any reading scale \(r\ge H^{(1-d)/64}\), these are \(K\) \(r\)-bins. The cover survives appending all tuple times to \(T\) and imposing any later event. The formula for \(J_1\), whose rows are now fixed and bounded, also gives \(K\) \(r\)-bins when \(T,l_r\) is fixed. Apply [a04:common-information] and [a04:entropy-independence] anew on each dense tested event, with fine scalar \(S_H\) and coarse scalar \(S_r\). This proves [a04:first-jet-entropy]. The argument rederives the entropy estimate on the specified event, using covers that persist under restriction. Together with the Taylor estimate already proved, this completes the proposition. ◻

An independent time tree.

Choose \(u\) uniformly in \([0,1]\). Add a second time uniformly in its \(H_1\)-interval. At the next level, split each of these times into itself and a new independent uniform draw in its \(H_2\)-interval. Continue for \(M-1\) pair levels. Separately draw \(t'\) uniformly in the \(h\)-block of \(u\), independently of all the other draws conditional on that block. This entire experiment is independent of the trajectory before any labels are imposed. Let \(T\) contain the world, the whole tree, and \(t'\).

All tested times are labeled with dense probability. Here is a direct verification. For a fixed trajectory and a fixed containing interval, write \(a\) for the fraction of labeled times. At a split, conditional on the smaller partition interval, the two child times are independent uniform points of that interval. The mean of the product of their subtree success probabilities is the square of the mean subtree probability. Jensen’s inequality, repeated up the finite tree, gives a lower bound by \(a^{2^{M-1}}\) in each \(h\)-block. The additional independent \(t'\) multiplies this by \(a\). Averaging over blocks, trajectories and worlds, and using Jensen once more, bounds total success below by the \((2^{M-1}+1)\)-st power of the single-time label mass. This is inverse-subpower.

We may also require that every pair gap at scale \(H_i\) be at least \(H_i/K\). Before label restriction, each excluded gap has probability at most \(2/K\). Since the number of pairs is fixed, take this \(K\) large enough compared with the reciprocal all-label probability. The loss is then negligible relative to that probability. These restrictions are performed on the presampled tuple law. We shall reapply 39 on each further dense event; we do not assume independence after restriction.

Inductive information statement.

Put \(F_0=F\) and \(F_1=A\). Starting at the leaves, a subtree rooted at time \(t\) will carry rows \(F_1(t),\ldots,F_m(t)\) and jets \[ J_i(t)=x_l^{(i)}(t)-F_i(t)y_l(t)-iF_{i-1}(t)y_l', \qquad 1\le i\le m. \tag{154}\] Every row is a function of its subtree times and the world only. In particular it is independent of the trajectory and does not use the separate time \(t'\). The rows are bounded by \(K\) on the retained tuples. At the readings required below, we maintain \[ \mathsf H(J_i(t)_r\mid T,S_l(t)_r)=o_* \tag{155}\] on each further dense event in the original trajectory–tuple experiment. At the leaves this is [a04:first-jet-entropy] for the fixed row \(A\). Its proof remains valid with the whole tree and \(t'\) appended to \(T\): the deterministic fine-scalar and coefficient-bin covers survive, and 39 is proved anew on the chosen event.

Consider a pair \(t,w\) at scale \(H\), whose two subtrees have jets through order \(m\). Set \(q=H^{1.01}\), with dyadic rounding. Their internal scales will have been chosen fine enough for the child assertions at \(q\). Taylor expansion of \(x_l,y_l\), and [a04:first-taylor], give \[ S_l(w)=S_l(t)+(w-t)J_1(t)+O(KH^{1.1}). \tag{156}\] The error is smaller than \(q\). Thus \(S_l(w)_q\), and then all the \(q\)-jets at both endpoints, have negligible entropy given \(T,S_l(t)_q\). This uses the child information statements on the current event and chain rule. Write \[\nabla Z=\frac{Z(w)-Z(t)}{w-t},\] where endpoint rows use their own subtree data. Since \(|w-t|\ge H/K\), the \(\nabla J_i\) are determined to accuracy \(KH^{.01}\), with entropy cost \(o_*\), by \(T,S_l(t)_q\).

For \(i\ge1\), quadraticity of \(x_l\) and affinity of \(y_l\) give the exact identity \[ \nabla J_i =x_l^{(i+1)}(t)-(\nabla F_i)y_l(t) -\bigl(F_i(w)+i\nabla F_{i-1}\bigr)y_l'. \tag{157}\] For \(i=1\), the difference quotient of \(x_l'\) equals \(x_l''\); for \(i\ge2\), both sides of the corresponding derivative identity are zero. The remaining terms in [a04:jet-difference] are simply the exact product difference for an affine \(y_l\). There is therefore no unrecorded higher-order scalar remainder.

Proceed through \(i=1,\ldots,m\), using the exponents fixed above and maintaining \[ \nabla F_{i-1}=F_i(t)+O(KH^{c_{i-1}}), \qquad c_{i-1}>0. \tag{158}\] Initially [a04:first-taylor], divided by the pair gap, supplies the chosen exponent \(c_0=.1\).

First, \(\nabla F_i\) is bounded by a subpower after negligible loss. If a fixed-power tail \(|\nabla F_i|\ge N^\alpha\), \(\alpha>0\), had dense mass on a subsequence, divide [a04:jet-difference] by that norm. The other displayed coefficients are bounded by \(K\), using [a04:row-compatibility], and \(x_l^{(i+1)},y_l'\) have size \(K\). The equation would therefore encode the projection of \(y_l(t)\) in the \(T\)-known direction \((\nabla F_i)/|\nabla F_i|\), with negligible entropy, at some fixed positive power accuracy coarser than \(q\). For example, if \(H=N^{-b+o(1)}\), take a reading \(N^{-\beta}\) with \(0<\beta<\min(\alpha,b)\). All nuisance terms after division are below this reading, as is the divided \(q\)-bin error. This contradicts [a04:projection-entropy] on that very tail event. It follows that every fixed-power tail is negligible relative to the current dense tuple mass, and a slow choice of cutoffs yields \(|\nabla F_i|\le K\).

We now also have \(F_i(w)-F_i(t)=O(KH)\). Recall that \[r_i=\min(.009,c_{i-1}/2)>0.\] [a04:jet-difference,a04:row-compatibility] encode \[ U_{i+1} =x_l^{(i+1)}(t)-(\nabla F_i)y_l(t) -(i+1)F_i(t)y_l' \tag{159}\] to width \(H^{r_i}\), at entropy cost \(o_*\), given \(T,S_l(t)_q\). Indeed, the error \(KH^{.01}\), the \(O(KH)\) row difference, and the \(O(KH^{c_{i-1}})\) compatibility error are all smaller than this width.

If \(i<m\), subtract the already known jet \(J_{i+1}(t)\). This encodes \((\nabla F_i-F_{i+1}(t))y_l(t)\) to the same width. On an event where the row norm is at least \(H^{r_i/2}\), division would encode its unit projection at width \(H^{r_i/3}\), again contradicting [a04:projection-entropy] on any dense such event. After a negligible deletion, \[\nabla F_i-F_{i+1}(t)=O(KH^{r_i/2}).\] This proves the compatibility assertion with the chosen \(c_i=r_i/2\), allowing us to proceed to the next value of \(i\).

For \(i=m\), define \(F_{m+1}(t)=\nabla F_m\) and keep the older rows at \(t\). This row is computed only from the two child rows and their times; no trajectory or \(t'\)-dependent choice is made. The new jet equals \(U_{m+1}\). Its bin at \(r=H^{r_m}\), and at coarser required powers, has negligible entropy given \(T,S_l(t)_q\). Given \(T,l_r\), the bounded rows and coefficient-bin diameter allow only \(K\) bins for that jet. Applying [a04:common-information] with \[Z=J_{m+1}(t)_r,\quad A=S_l(t)_q,\quad B=l_r,\quad C=S_l(t)_r\] and invoking [a04:entropy-independence] on the current event gives [a04:hereditary-jets] for the new jet. The older jets retain their hereditary assertions.

There are finitely many rows and discrepancy tests for fixed \(M\). For each fixed positive threshold exponent, a nonnegligible relative bad mass would itself yield a dense event on a subsequence, contradicting the projection estimate just used. Choose threshold exponents decreasing to zero so slowly that the union of the finitely many bad sets has relative mass tending to zero. The resulting row bounds are deterministic on the retained tuples. Every later dense event inherits those deterministic bounds; its entropy assertions are then derived anew from the child assertions, [a04:common-information], and 39. We never infer hereditary entropy by conditioning an earlier average. These cutoff tests concern rows already defined from the time tree. They do not introduce a dependence on the trajectory or on \(t'\).

We can now verify the schedule. A jet formed at a pair scale \(H_i\) is available at width \(H_i^{r_m}\) for its relevant order \(m\), and at all coarser required readings. Since \(r_m\ge r_{\min}\), the root widths satisfy \[KH_1^{r_m}\le KH_1^{r_{\min}}\le Kh^{M+11} \ll p:=h^{M+10}.\] At a child scale \(H_i\), \(i\ge2\), the corresponding bound is \[KH_i^{r_m}\le KH_{i-1}^{1.02}\ll H_{i-1}^{1.01},\] which supplies the reading \(q\) requested by its parent. Older jets inherited from finer subtrees obey the same requirements. For the leaf jets, [a04:first-jet-entropy] applies at every reading at least \(H_*^{\alpha_0}\); by [a04:jet-scale-schedule], this is finer than \(H_{M-1}^{1.01}\), with the same fixed margin. Finally, \(\zeta\le H_*^3/K\) covers the first-jet construction, its fine scalar scale \(H_*\), and all the intermediate readings just used. The projection readings in the tail and compatibility tests are coarser than their scalar conditioning scale \(q\). Thus the entire induction uses available scales.

Testing the polynomial at a separate time.

At the root \(u\), set \[P_F(v)=\sum_{i=0}^M\frac{F_i(u)}{i!}(v-u)^i,\qquad E=F(t')-P_F(t'),\qquad J_0(u)=S_l(u).\] The coefficients of \(P_F\) are bounded by \(K\), including when expanded in ordinary powers of \(v\), because \(M\) is fixed and \(u\in[0,1]\). Crucially, the coefficients are functions of the world and the tree excluding \(t'\). Exact evolution of the quadratic \(x_l\) and affine \(y_l\) gives \[ S_l(t') =\sum_{i=0}^M\frac{J_i(u)}{i!}(t'-u)^i -E\,y_l(t')+O(Kh^{M+1}). \tag{160}\] The last term is the single extra degree produced when the degree \(M\) polynomial \(P_F\) multiplies the affine \(y_l\).

Suppose \(|E|\ge h^{M+1/4}\) on a dense event. Work on that event and condition on \(T,S_l(u)_p\). The root jet bins at \(p\) have total additional entropy \(o_*\); after specifying them, [a04:final-jet-expansion] places \(S_l(t')\) in an interval of length \(K|E|\). Put \(r=h^{1/4}\). The local support count [a04:scalar-support] allows at most \(Kr^{-d}\) values at width \(|E|r\) in that interval. Both the width and interval length are \(T\)-known. The variable width satisfies \[|E|r\ge h^{M+1/4}h^{1/4}=h^{M+1/2}\ge K\zeta,\] by [a04:jet-floor]; its containing interval is larger still. Thus both are within the available scale range; dyadic enlargement or the unit cutoff costs only \(K\).

After division by \(|E|\), the Taylor remainder is \(O(Kh^{3/4})\); the root-jet bin error is smaller, and the scalar-bin error is \(O(r)\). Also \(y_l(t')=y_l(u)+O(Kh)\). Write \(\mathcal J\) for the tuple of root-jet bins at width \(p\). Given \(T,S_l(u)_p,\mathcal J\), the expansion therefore confines the width-\(r\) bin of \((E/|E|)y_l(u)\) to at most \(Kr^{-d}\) possibilities. Since \(\mathsf H(\mathcal J\mid T,S_l(u)_p)=o_*\), chain rule gives \[\mathsf H\bigl(((E/|E|)y_l(u))_r\mid T,S_l(u)_p\bigr) \le d\log(1/r)+o_*.\] [a04:projection-entropy], applied on this same event, gives the lower bound \(\log(1/r)-o_*\). Since \(d<1\) and \(h\) is a fixed positive power, this is impossible. We retain dense tuple mass with \(|F(t')-P_F(t')|\le Kh^{M+1/4}\).

Recovering labeled time fibers.

Let \(z\) consist of the world, the trajectory, and the rest of the time tree, excluding \(t'\). The polynomial \(P_F\) is fixed at \(z\), and before restrictions the conditional law of \(t'\) is uniform in its \(h\)-block. If the good tuple event has mass \(p_g\ge K^{-1}\), write \(g(z)\) for the fraction of successful \(t'\)’s in that block. Then \(\int g(z)\,dP(z)=p_g\). For any positive \(\delta\), \[\int_{\{g<p_g\delta\}}g(z)\,dP(z)\le p_g\delta.\] Choose \(\delta\to0\) sufficiently slowly and retain only the other fibers. They still carry dense tuple mass, and every retained fiber has successful labeled length at least \(hp_g\delta\ge h/K\).

The pre-restriction marginal law of \((l,t')\) is the original prior times uniform time: uniform \(u\) selects \(h\)-blocks with their length weights, and \(t'\) is uniform inside the chosen block. Projecting the good event therefore gives a dense set of incidences \((l,t')\). Each has a witnessing polynomial fitting on a successful fiber of length \(h/K\), including that incidence. No independence is asserted after conditioning on good success or on a discrepancy test. If desired, round the bounded polynomial coefficients on a sufficiently fine finite grid, adding error much smaller than \(h^{M+1/4}\), and test its successful length on the original labels. This makes the existential qualification test measurable and preserves the asserted bounds.

Every construction began with upper inputs and dense mass, so it may be performed anew on a dense residual. This proves the proposition. ◻

The parameter zoom.

The physical cubes \(C\) were used to count scalar-normal strips. To obtain the conditional spatial mass bounds, we now condition on a full coefficient bin \(B_\rho\) and rescale its trajectories. Run the following construction in the family of worlds \(\mathfrak q=(\gamma,B_\rho)\), where \(\gamma\) is a parent and \(B_\rho\) is a full parameter block of positive \(\nu^\circ_\gamma\)-mass. Coordinates, cells and rows in this construction always include \(\mathfrak q\) in their labels. Within one such world subtract its central coefficient curves \(X_c^{\mathfrak q}(t),y_c^{\mathfrak q}(t)\) and divide positions by \(\rho\): \[x=\frac{X-X_c^{\mathfrak q}}{\rho},\qquad \widetilde y=\frac{y-y_c^{\mathfrak q}}{\rho}.\] Time is unchanged. All new trajectory coefficients are bounded; aligned width-\(r'\) parameter bins are precisely the subbins \(B_{\rho r'}\) of \(B_\rho\). Choose a time-only row \(F_{\mathfrak q}(t)\) representing \(G_y\) at active points in each sufficiently fine time bin of length at most \(\rho\), and extend it boundedly at inactive times. For the within-world calculations we suppress the superscript on the central curves and write \(F=F_{\mathfrak q}\). From [a05:field-lipschitz], \[ |F(t)-G_y(W_b)|\leq K\rho,\qquad |F(t)-F(u)|\leq K(|t-u|+\rho) \tag{176}\] at active occurrences. The physical positions lie within \(K\rho\) of a central curve of speed \(K\). At fixed time, \(\rho(x-F(t)\widetilde y)\), up to a common translation in each of the at most \(K\) nearby cubes \(C\), differs from the strip coordinate by at most \(K(\rho^2+b)\). This is negligible compared with \(\rho\zeta\). Thus [a05:strip-count] supplies the scalar-normal local \(d\)-support bound down to \(\zeta\), also on enlarged upper intervals by at most \(K\) subdivisions.

Take the fixed prior \(\nu^\circ(\,\cdot\mid B_\rho)\), retaining all original trajectory tags. At the required resolutions \(\zeta\leq q'\leq r'\leq1\), for a participating subbin \(B_{\rho r'}\), the fixed-time numerator in a zoomed spatial \(q'\)-cube is at most \[K(\rho q')^{j+d}\pi([V]_{\rho r'}).\] Its prior denominator is at least \((\rho r')^{j+d}\pi([V]_{\rho r'})/K\), by [a05:parameter-floor]. Consequently the conditional spatial cap is \[ K(q'/r')^{j+d}. \tag{177}\] The case \(r'=1\) gives the unconditional spatial cap. These are caps for the original parameter-conditioned prior. They are not renormalized when labels are subsequently restricted.

The residual velocity.

Put \(R=x'-F(t)\widetilde y'\). Within a fixed \(W_b,B_\rho\), its range is much smaller than \(\zeta\). Indeed, writing \(v_c=y_c'\), one has \[\rho R =G_0+G_yv_c-X_c'(t) +(G_y-F(t))(V-v_c)+O(b^\kappa).\] Here \(|V-v_c|\leq K\rho\), \(|X_c''|\leq K\), and the time variation within \(W_b\) is at most \(b\). The choices of \(b,\rho,\zeta\) give the asserted range.

Fix \(s'\geq\zeta\) and put \(\beta=\rho s'\). A \(\beta\)-parameter block uses at most \(K/s'\) zoomed \(s'\)-cells in its unique \(\rho\)-zoom. Delete block/cell entries of incidence mass below \(s'\nu^\circ(B_\beta)/K_1\). The summed loss is at most \(K/K_1\), before any normalization inside \(B_\rho\). In one zoomed cell the time interval has length \(s'\), whereas the physical spatial cross-section at each time has side \(K\beta\) and moves with the central curve. For a fixed velocity bin \(J=[V]_\beta\), integrating [a05:fixed-time-cap] along that moving cross-section gives \[\lambda(\text{zoomed cell},J) \leq Ks'\beta^{j+d}\pi(J).\] Every surviving block pays at least \[\frac{s'\nu^\circ(B_\beta)}{K_1} \geq K^{-1}s'\beta^{j+d}\pi(J).\] Both estimates contain \(s'\), so there are at most \(K\) blocks. No union over physical cells of time width \(\beta\) is used. Within one block the zoom coefficients have diameter \(O(s')\); their absolute sizes are \(K\). Across the cell’s time interval, \(x'\) varies by \(Ks'\), and [a05:zoom-row] varies by at most \(K(s'+\rho)\leq Ks'\). Thus one block contributes at most \(K\) width-\(s'\) residual bins for \(R\).

We next remove the velocity condition with a summed denominator argument. Write \(K_{\rm pre}=N^{o(1)}\) for the accumulated bounds, and let \(m_0(W)\) be the earlier unnormalized fine-state weights, including mixture weights, and retain \[\lambda(W,J)\leq K_{\rm pre}m_0(W)\pi(J),\qquad \sum_Wm_0(W)\leq K_{\rm pre}.\] For a coarse velocity bin \(I=[V]_\rho\), the parameter list allows \(K_{\rm pre}\) blocks \(B_\rho\) given \(W,I\), and each has \(K_{\rm pre}\) zoom cells meeting \(W\), since \(K b/\rho\ll\zeta\). There are at most \(K_{\rm pre}^2\) refined entries \(E=(W,B_\rho,s'\text{-cell})\) over \(W,I\). Deleting those of mass below \(m_0(W)\pi(I)/K_{\rm cut}\) loses at most \[\frac{K_{\rm pre}^2}{K_{\rm cut}}\sum_Wm_0(W)\sum_I\pi(I) \leq \frac{K_{\rm pre}^3}{K_{\rm cut}}.\] Choose the subpower \(K_{\rm cut}\) larger than all accumulated inverse-subpower density losses. On each survivor, \[ \lambda(J\mid E)\leq K\frac{\pi(J)}{\pi(I)}. \tag{178}\] This uses neither a new prior nor a current incidence floor from [a05:parameter-floor].

Every occurring residual bin has a retained entry witnessing it. Near constancy within that entry and [a05:zoom-posterior] make its supporting velocity bins have total palette mass at least \(\pi(I)/K\), within boundedly many neighboring residual bins. A fixed fine velocity bin supports at most \(K\) residual bins by the preceding block/cell count. Summing against the common palette in \(I\) gives at most \(K\) residual bins in the zoomed cell.

Residual comparison and the scalar-normal input.

It remains to make a common residual choice in each zoomed cell. Carry out the preceding deletions on a fixed finite grid \(\{s_1,\ldots,s_L\}\subset[\zeta,1]\), including its endpoints up to dyadic rounding. Choose the thresholds successively so that the summed loss is at most half the entering dense mass. The two losses at a tested width have the forms \(K/K_1\) and \(K_{\rm pre}^3/K_{\rm cut}\) obtained above, so on a fixed grid the required thresholds remain subpower. Denote the remaining unnormalized measure, with the original parent weights, by \(\mu_0\). Its mass \(\delta_0\) is at least \(K^{-1}\).

In world \(\mathfrak q=(\gamma,B_\rho)\) use \[Z_{\mathfrak q}(e)=(t,\widetilde y_l(t),x_l(t)),\qquad R_{\mathfrak q}(e)=x_l'(t) -F_{\mathfrak q}(t)\widetilde y_l'.\] The dimension of \(Z_{\mathfrak q}\) is \(D=j+2\), and time retains its parent units. Freeze the width-\(s_i\) residual lists in every \(s_i\)-cell of these coordinates. The preceding argument bounds their sizes by \(K_i=N^{o(1)}\). The palette paying for each list is common to the cell because \(B_\rho\) fixes \(I=[V]_\rho\). The finer entry \((W,B_\rho,C)\) supplied a denominator and a witness for this list; it is not an additional key for the residual choice.

Apply 46 with \(Y=R_{\mathfrak q}\). Choose the slowly enlarging grid together with the preceding deletions and density losses as specified after that lemma. We obtain a restriction \(\mu_L\) of mass \(\delta\geq K^{-1}\) such that \[ |R_{\mathfrak q}(e)-R_{\mathfrak q}(\widehat e)| \leq K\bigl(|Z_{\mathfrak q}(e)-Z_{\mathfrak q}(\widehat e)| +\zeta\bigr) \tag{179}\] for every pair of retained incidences in the same world. Coordinates and rows need not agree across worlds. This time the selection may remove individual edges: the last use of breadth in this construction was strip filling, and its remaining steps use only the inherited upper bounds.

Put \(a_{\mathfrak q}=p_\gamma\nu^\circ_\gamma(B_\rho)\). These weights sum to at most one. Hence some positive-weight world satisfies \(\mu_L(\mathfrak q)/a_{\mathfrak q}\geq\delta\). Divide by this parent weight and parameter-block mass. Since \(\lambda\leq K\nu^\circ\,dt\), the retained labels have product-prior mass at least \(K^{-1}\) for the fixed probability \(\nu^\circ_\gamma(\,\cdot\mid B_\rho)\,dt\).

We can now apply the scalar-normal results of 5 in this world. The trajectories are \((x,\widetilde y)\), the row is \(F_{\mathfrak q}\), and the prior remains \(\nu^\circ_\gamma(\,\cdot\mid B_\rho)\). The strip count gives the scalar-normal support bound; [a05:zoom-conditional-cap] gives both spatial caps, including conditioning on a parameter bin; [a05:zoom-row] gives row comparison; and [a05:zoom-residual-comparison] gives residual comparison. These are exactly the hypotheses of 38, with the floor already chosen for 42. Apply 42. It produces a degree-\(M\) polynomial with coefficients at most \(K\) fitting \(F\), hence \(G_y\), to \(Kh^{M+1/4}\) along labeled portions of an indexed line of duration at least \(h/K\), including the counted occurrences. Use [a05:velocity-graph-estimate] and \(G_0=X'_l-G_yV_l+O(b^\kappa)\) to fit \(G_0\) too. All witness times on such a line in the block have the same true \(Q_s\), by its block-invariant trajectory assignment. Write \(K'\) for the combined subpower in this discovery. For the fixed \(\epsilon>0\) and all sufficiently large \(N\), the coefficient bound is smaller than \(N^\epsilon\), the duration is at least \(hN^{-\epsilon}\), and \(K'h^{M+1/4}\leq h^M/4\). Coefficient rounding still leaves error below \(h^M\) and coefficients below \(N^\epsilon\). These reference-labeled witnesses qualify occurrences lying in \(U_{N,\epsilon}\), a contradiction. The discovered mass can be inverse-subpower: any positive mass contradicts the definition of this failure set. Therefore, for every fixed \(\epsilon>0\), \[\frac{\Lambda_{\rm ref}(U_{N,\epsilon})}{L_{\rm ref}} \longrightarrow0.\]

For \(\epsilon_r=1/r\), choose cutoffs beyond which this fraction is at most \(1/r\), and let \(r(N)\to\infty\) slowly enough to respect those cutoffs and the preceding finite-grid preparations. Set \(K_{\rm qual}=N^{1/r(N)}\) and \(U_N=U_{N,1/r(N)}\). Then \(K_{\rm qual}\) is subpower and [a05:qualification-exception] follows. For any required fixed finite collection of applications, divide the failure budget among its members before enlarging the collection. This proves a relative \(o(1)\) exception in the fixed substantial mixture; it gives neither finite-\(N\) nullness nor a rate that can be passed to an arbitrary later sparse restriction. ◻

Version 1.

Choose a heavy intercept group. In normalized block time, [a05:potential-fit] compares \(P\) with the quadratic \(X-C\) along each of its lines. Polynomial Remez gives \[|D_{\bf v}^3P|\leq Ka_2\] there. Conditional product sampling and palette averaging fix four separated very fine velocity bins with witnesses on common states carrying an inverse-subpower fraction of the group. Use their centers as directions. The locations in a state differ by at most \(Kb/h\) in \(u\), and fixed derivatives of \(P\) cost \(Kh\). Finer velocity bins and \(b/h\ll a_2/h\) make the transferred errors negligible. By [a05:heavy-group,a05:intercept-base-cap], these common tests hold on inverse-subpower base volume. Polynomial Remez extends them to the enclosing box. A binary cubic is determined by its values in four separated directions \((1,V_i)\); inversion of this Vandermonde system costs \(K\). Hence \(\|\nabla^3P\|\leq Ka_2\). The quadratic Taylor polynomial of \(P+C\) fits to \(Ka_2\) on the block box. The test \(w\) minus this polynomial has hidden derivative one and zero hidden curvature, and its heavy-group mass is the required one.

Branches in version 2.

Here \(w\) may be uncontrolled in tangent units, so we use the native test rather than assume a bound for \(w\). In each heavy group consider \[ P_*(z,w)=P(u)+C,\qquad z=z_c+hu. \tag{190}\] At path observations the residual is \(O(Ka_2)\), the absolute hidden derivative lies between \(K^{-1}\) and \(K\), and the hidden curvature is at most \(K\). The discriminant expression \[\Delta(u) =(P_{*,w})^2 -2P_{*,ww}\bigl(P_*(z,w)-P(u)-C\bigr)\] is independent of \(w\), because the equation is quadratic in \(w\) with constant quadratic coefficient. It lies between \(K^{-1}\) and \(K\) on the observations. The heavy-group floor divided by [a05:intercept-base-cap] gives inverse-subpower base volume. Polynomial Remez bounds the coefficients and derivatives of \(\Delta\) on an enlarged complex box by \(K\). Thus, about every used point, \(\Delta\) is zero-free on a complex ball of inverse-subpower radius with a fixed margin.

Each observed \(w_l\) is within \(Ka_2\) of a real simple root of [a05:branch-equation]. For a quadratic, choose the root with the same derivative sign as at \(w_l\); the derivative on the intervening segment has size at least \(K^{-1}\), after enlarging \(K\), by the positive discriminant and small residual. For a linear equation, divide by its nonzero coefficient. The roots extend holomorphically to the stable balls, by a holomorphic square root of \(\Delta\), or by linear division.

Choose a lattice of real comparison cores, each inside a fixed several-fold complex enlargement, with radius common up to \(K\) among the groups. Assign a core by the \((\mathcal D,C)\)-center. Since \(Kb/h\) is negligible, it contains the state’s used points with an interior margin. Boundary points can be assigned in a bounded overlapping cover. Add the sign of the nearby root. Discard sign fractions smaller than a sufficiently small fixed fraction; the core choice itself removes no fraction at a state. Delete line/intercept/core/sign portions with weighted duration below \(h/K_1\). Each line has one intercept and at most \(K\) core/sign choices, so the total loss is at most \(Kh\nu_Q/K_1\). Remove states with excessive fractional loss. The fixed-fraction comparisons to the earlier good state laws, with sufficiently small initial pencil probabilities, retain general position and palette domination.

There are at most \(K(a/a_2)^d\) group/core/sign choices. One choice therefore has mass at least \[ K^{-1}h\nu_Q(a_2/a)^d. \tag{191}\] Let \(U\) be its real inner ball and \(f\) its branch. Every contributing line has an immediately preceding witness set, in this same choice, of ordinary length at least \(h/K\), including its currently counted incidences. All witnesses lie in \(U\), with \(f\) holomorphic on a sufficient complex enlargement. The convex cores and that margin ensure that the line chords used below stay in the controlled domain.

Analytic estimates before tensor inversion.

Set \[ R(u)=P_s(z_c+hu,f(u))/a. \tag{192}\] We will apply the two-copy cubic test to \(\nabla^3f\) and the three-copy tests to the higher derivatives of \(R\). First we need analytic bounds on \(R\) that hold before moving any directions to an exact conic. At the earlier witnesses, the true packet test obeys \[|P_s(z,w_l)|\leq Ka,\qquad |P_{s,w}(z,w_l)|\leq K,\qquad |P_{s,ww}|\leq K/a, \qquad |w_l-f(u)|\leq Ka_2.\] Its exact quadratic Taylor formula in \(w\) gives \[|R(u)|\leq K\bigl(1+a_2/a+(a_2/a)^2\bigr)\leq K.\] The base density of this choice is bounded by the original group ceiling \(Kh\nu_Q(a_2/a)^d\). Together with [a05:branch-mass], this gives real projection volume at least \(K^{-1}\). The function \(R\) is holomorphic algebraic of bounded relation degree. 19 bounds its norm and every required fixed derivative by \(K\) on the used interiors, before any tensor inversion.

Along a contributing line, compare \(f\) with the affine hidden trajectory \(w_l\), and compare \(R\) with the quadratic \(P_s(z,w_l)/a\), on its earlier long witness set. Normalized time on its chord has length at least \(K^{-1}\); the witness set has inverse-subpower relative length and the chord has the stated complex margin. The one-variable algebraic norm bound yields, at the incidences, \[ |D_{\bf v}^2f|\leq Ka_2,\qquad |D_{\bf v}^3R|\leq Ka_2/a. \tag{193}\]

Let \[S_0=\inf_{L\ {\rm affine}}\|f-L\|_{C^0(U)}.\] For \(S_0>0\), apply 19 to \(f-L\) with \(\|f-L\|_{C^0(U)}\leq2S_0\). All fixed derivatives of order at least two are bounded by \(KS_0\) on a slightly larger interior. If \(S_0=0\), \(f\) is affine and these derivatives vanish.

Sample seven velocities in each retained state, sequentially avoiding all preceding points and joining lines. The prepared breadth gives a positive probability that all triple determinants of \({\bf v}_i=(1,v_i)\) have size at least \(K^{-1}\). Palette domination turns this into a product-palette mass at least \(K^{-1}\) of eligible bin tuples. Averaging fixes one tuple with common directional witnesses on states carrying an inverse-subpower fraction of the current mass. Using fine bin centers transfers their tests to actual base points in those states. For \(R\), its preceding derivative bounds make all mesh errors negligible relative to \(a_2/a\). For \(f\), the Hessian and higher derivative errors are bounded by \(KS_0\) times the normalized position and velocity errors. Choose those errors at most \(h^2/K\), independently of \(S_0\). Common-state mass and the base ceiling again give base volume at least \(K^{-1}\). Apply 19 to the directional derivative functions themselves; differentiation of the simple root preserves bounded algebraic relation degree on the stable domain. We obtain throughout \(U\), also after the fixed further differentiations needed below, \[ |D_{{\bf v}_i}^2f|\leq K(a_2+S_0h),\qquad |D_{{\bf v}_i}^3R|\leq Ka_2/a. \tag{194}\] The weaker \(S_0h\) allowance absorbs the \(S_0h^2\) transfer error and subpower losses. All ball radii and margins here cost only \(K\).

Differentiate the first tests once and apply the injective cubic two-copy test in 48. Its polynomial conditioning is subpower, so \[ \|\nabla^3f\|\leq K(a_2+S_0h). \tag{195}\] If \(S_0\leq a_2/h\), a quadratic Taylor polynomial \(f_2\) approximates \(f\) on \(U\) within \(Ka_2\). The test \(w-f_2((z-z_c)/h)\) is quadratic in parent variables, has derivative one and curvature zero, and fits the long witnesses. This includes \(S_0=0\).

The conic case.

Suppose \(S_0>a_2/h\). Let \(u_c\) be the ball center and \(L'\) the affine Taylor polynomial of \(f\) there. Define \[q(x)=\frac{1}{2S_0}\nabla^2f(u_c)[x,x].\] Taylor’s theorem and [a05:third-f] give \[ \frac{f(u)-L'(u)}{S_0} =q(u-u_c)+O(Kh). \tag{196}\] The relative derivative bound gives \(\|q\|\leq K\). Conversely the definition of \(S_0\) yields \[1\leq \|q\|_{C^0(U-u_c)}+Kh\leq K\|q\|+Kh,\] so \(\|q\|\geq K^{-1}\). The Hessian tests imply \(|q({\bf v}_i)|\leq Kh\). Choose one parameter \(A=N^{o(1)}\) controlling all coefficient norms, direction sizes, and inverse triple determinants. Since \(h\) is a fixed negative power, 49 applies. It gives a determinant lower bound \(K^{-1}\) for \(q\), and exact real zeros \({\bf v}'_i=(1,v_i')\) with \[|{\bf v}'_i-{\bf v}_i|\leq Kh, \qquad |\det({\bf v}'_i,{\bf v}'_j,{\bf v}'_k)|\geq K^{-1}.\] Thus even a nearly singular or nearly definite proposed quadratic is quantitatively controlled before inversion.

The independent derivative bounds for \(R\) from [a05:normalized-test] show that these shifts alter its rank-three, rank-four and rank-five contractions by at most \(Kh\). This is negligible compared with \(a_2/a=h^{e_2}\), because \(e_2<1\). The differentiated tests in [a05:seven-tests] therefore remain \(Ka_2/a\) in the shifted directions. Apply the exact-conic kernels of 48. The fifth derivatives of \(R\) are bounded by \(Ka_2/a\); at \(u_c\), the cubic and quartic Taylor tensors are within \(Ka_2/a\) of \(q\) times a linear form and a multiple of \(q^2\), respectively. Taylor expansion through degree four gives \[ R(u)=R_2(u)+q(u-u_c)L_1(u) +\lambda_0q(u-u_c)^2+O(Ka_2/a), \tag{197}\] where \(R_2\) is quadratic, \(L_1\) is affine, and all displayed coefficients are bounded by \(K\). To justify division in these factors, the coefficient norm of a product of polynomials of fixed degrees is bounded below by a positive constant times the product of their coefficient norms. Normalize the factors and use compactness and the absence of zero divisors. Since \(\|q\|\geq K^{-1}\), division by \(q\) or \(q^2\) costs only \(K\). A constant term of \(L_1\) can be absorbed into \(R_2\).

On the earlier long witnesses, put \[H'(u,w)=\frac{w-L'(u)}{S_0}.\] Because \(a_2/S_0<h\), [a05:f-conic] and \(|w-f(u)|\leq Ka_2\) imply \(H'=q(u-u_c)+O(Kh)\) and \(|H'|\leq K\). Define \[ P_{\rm new}(z,w) =P_s(z,w) -a\left(R_2(u)+H'(u,w)L_1(u) +\lambda_0H'(u,w)^2\right), \qquad u=(z-z_c)/h. \tag{198}\] Every term has total degree at most two in \((z,w)\): \(u,H',L'\), and \(L_1\) are affine in the indicated variables. Substituting \(H'\) for \(q\) costs \(Kah\), which is below \(a_2\) by \(h^{1-e_2}\). Replacing \(f\) by the actual \(w\) in \(P_s\) costs at most \[K|w-f|+(K/a)|w-f|^2 \leq Ka_2+Ka_2^2/a\leq Ka_2.\] Together with [a05:R-restoration], this proves the \(Ka_2\) value bound on the long witnesses.

The exact derivative perturbations are \[\partial_w\bigl(P_s-P_{\rm new}\bigr) =\frac a{S_0}(L_1+2\lambda_0H'),\qquad \partial_{ww}\bigl(P_s-P_{\rm new}\bigr) =\frac{2a\lambda_0}{S_0^2}.\] Consequently \[\begin{align*} \bigl|\partial_w(P_s-P_{\rm new})\bigr| &\leq K a/S_0 <K h^{1-e_2}\ll K^{-1},\tag{199}\\ |\partial_{ww}P_{\rm new}| &\leq K/a+K a/S_0^2 \leq K/a_2+(K/a_2)h^{2-e_2} \leq K/a_2. \tag{200}\end{align*}\] The old derivative lower bound at the counted observations survives [a05:derivative-margin]; the upper bound holds on the earlier witnesses. No magnitude bound for \(w\), \(L'\), or \(S_0\) was used, only the normalized value \(H'\) and the displayed lower bound on \(S_0\).

The same time block.

In both cases the test is an allowed quadratic. On each contributing version 2 trajectory, \(z(t)\) and \(w(t)\) are affine, so its value is a polynomial of degree at most two in time and its hidden derivative has degree at most one. The earlier witness set has physical length at least \(h/K\) inside the same \(h\)-length \(Q_s\)-block. After normalizing that block to unit length, polynomial Remez costs only \(K\). Thus the value and derivative upper bounds hold throughout this block. The lower derivative is needed only on the counted incidences and was proved there. Curvature is constant in \(w\) for these quadratic tests. The version 1 comparison has the same fixed-degree time property. Extrapolation is never made from an \(h/K\)-set to a unit parent time interval.

The retained choice has mass \(h\nu_Q(a_2/a)^d/K\), by [a05:branch-mass] or the heavy intercept floor in version 1. Degree restoration makes no further discard. Dividing by its time length \(h\) gives \(\nu_Q(a_2/a)^d/K\), exactly the candidate mass requirement. ◻

Projection covers and point-mass witnesses.

Freeze the present incidence law as \(\mu_{\rm cov}\). Delete lines with too little labeled time or incidence weight. Prior-times-time domination leaves each used trajectory with a \(\mu_{\rm cov}\)-witness time set of length at least \(1/K\). It visits at least \(1/(K\xi)\) witness point cells, throughout which its projection varies by at most \(K\xi\). Replacing its true velocity by the bin representative costs only \(K\xi\) in both projection coordinates.

For each direction bin, projected values separated by a sufficiently large multiple of \(K\xi\) consequently use disjoint witness sets. Since the total number of point cells is at most \(K\xi^{-2}\), its currently occurring projection can be covered by at most \(K\xi^{-1}\) mesh cells. The law \(\mu_{\rm cov}\) will continue to supply these covers, even after the incidence law is restricted further.

Discard slope bins of palette weight below \(\xi^2\). The joint cap, summed over at most \(K/\xi\) such bins and the position cells, makes their total mass negligible. Pigeonhole comparable palette weights on dense incidence mass, leaving \(M\) bins. Their total palette weight is at least \(1/K\). The uniform law on them therefore has an interval bound \(K\rho^{S'}\), for some fixed \(S'>0\) and \(\xi\le\rho\le1\).

Normalize this current incidence law to probability and freeze it as \(\mu_{\rm pt}\). Every joint point-direction cell has mass at most \(K\xi^2/M\). Delete point cells of marginal mass below \(\xi^2/K_1\), and point-direction edges of mass below \(\xi^2/(K_1M)\), with a sufficiently large subpower \(K_1\). The numbers of cells and edges pay for these deletions. At least \(\xi^{-2}M/K\) unweighted edges remain. Deleting low-degree points leaves at least \(\xi^{-2}/K\) point cells of degree at least \(M/K\).

Every surviving point still has its floor in the frozen \(\mu_{\rm pt}\)-marginal. Thus a later selection of \(\xi^{-2}/K\) such point cells will lift to dense mass by restoring all their \(\mu_{\rm pt}\)-incidences, regardless of which graph directions were selected. This second frozen law supplies point-mass floors; the first supplies the long-line projection covers.

Two projections and the third-direction sumset.

At each surviving point there are at least \(M^3/K\) ordered triples of incident slopes separated pairwise by at least \(1/K\). Indeed the interval bound makes the fraction within a sufficiently small reciprocal-subpower neighborhood negligible compared with the degree fraction. Pigeonhole the first two directions \(v_1,v_2\). They have at least \(\xi^{-2}M/K\) common-neighbor incidences with third directions also separated from them. Retain third directions with at least \(\xi^{-2}/K\) such neighbors. There are at least \(M/K\) retained directions; use their uniform law.

Let \(\mathcal A\) and \(\mathcal B\) be the projections of the common-neighbor point cells under \(\mathcal P_{{\bf u}_1}\) and \(\mathcal P_{{\bf u}_2}\), rounded to the \(\xi\)-lattice. The earlier projection covers give \(|\mathcal A|,|\mathcal B|\le K\xi^{-1}\). Each endpoint pair has at most \(K\) point preimages, since the basis determinant is at least \(1/K\) and its entries are bounded by \(K\). For each retained third direction, its common-neighbor points therefore define a graph \(G_\kappa\subset\mathcal A\times\mathcal B\) with at least \(\xi^{-2}/K\) distinct edges.

For that direction write \[{\bf u}_d=c_1{\bf u}_1+c_2{\bf u}_2,\qquad c_1=\frac{v_2-v_d}{v_2-v_1},\qquad c_2=\frac{v_d-v_1}{v_2-v_1},\qquad \kappa=\frac{c_2}{c_1}=\frac{v_d-v_1}{v_2-v_d}.\] The map \(\mathcal P_{\bf u}\) is linear in \({\bf u}\). Thus, for the unrounded projections \(a,b\) of a common point, \[a+\kappa b=c_1^{-1}\mathcal P_{{\bf u}_d}(z,V_0).\] Both \(c_i\) and their reciprocals are subpower bounded. Rounding and this dilation cost only \(K\), so the third-direction projection cover gives \[N_\xi\{a+\kappa b:(a,b)\in G_\kappa\} \le K\xi^{-1}.\]

We verify the remaining planar hypotheses. The graph sizes imply \(|\mathcal A|,|\mathcal B|\ge\xi^{-1}/K\), and both endpoint sets are bounded by \(K\). Separation from the basis slopes gives \(K^{-1}\le|\kappa|\le K\). The ratio law has the same positive-exponent interval bound up to \(K\), since \[|v_d-v_{d'}|\le K|\kappa_d-\kappa_{d'}|\] on the retained choices. Finally, prescribing length-\(\xi\) intervals for the first endpoint coordinates \((a_1,b_1)\) restricts \(z\) to at most \(K\) cells by the invertible basis map. There are only \(K\) point options above those cells. Each graph therefore has only \(K\) possible edges at such a first-coordinate rectangle. Its uniform edge law obeys the required cap, as does the uniform law on any subset of at least \(\xi^{-2}/K\) edges. Every such subset uses at least \(\xi^{-2}/K\) distinct surviving point cells.

Recovering the row and restoring incidence mass.

Apply 57 with \(h=\xi\). For a represented \(\kappa_0\), it retains at least \(\xi^{-2}/K_*\) original edges of \(G_{\kappa_0}\) in parallel strips of width \(K_*\xi\), with common slope and intercepts bounded by a new subpower \(K_*\). No fixed-power dependence on the earlier \(K\) is needed. Absorb \(K_*\) into the subsequent subpower factor.

On the corresponding point cells, \[V_0\cdot{\bf u}_i =b_i+\lambda_0z\cdot J{\bf u}_i+O(K\xi),\qquad i=1,2.\] The estimate depends only on \((z,V_0)\) and remains valid throughout each selected cell. Inverting the basis and undoing \(V_0=(\xi/E)T\) gives \(T=B+\lambda Jz+O(KE)\) there.

Bounded endpoint multiplicity leaves at least \(\xi^{-2}/K\) distinct selected point cells. Each has frozen \(\mu_{\rm pt}\)-mass at least \(\xi^2/K_1\), so their whole-cell mass is dense. Restore those incidences. The row estimate applies even when their directions were not among the selected graph edges, and no incidence removed before freezing \(\mu_{\rm pt}\) is restored.

It remains to bound the coefficients after undoing the scaling. If \(|\lambda|\) were power-large, the bound \(|T|\le K\) would confine this dense selected \(z\)-mass to a ball of radius \(K/|\lambda|\). Summing the joint cell cap over velocities bounds its mass by \[K(K/|\lambda|+\xi)^2,\] a contradiction. Hence \(|\lambda|\le K\), and an occupied point then bounds \(|B|\) by \(K\) as well. ◻

Smoothing the mesh law.

Let \(\nu\) be the given trajectory prior and \(\mu_{\rm act}\le K\nu\otimes dt\) the original active incidence measure. Round the five coefficients of \(y,x\) on a mesh \(\sigma'\ll\sigma a^2\) by a fixed power. Index the positive-weight rounded vectors by tags \(j\), and let \(\omega_j\) be the \(\nu\)-weight of tag \(j\). Mark a dyadic time \(\sigma\)-bin \(I\) for this tag when \[\mu_{\rm act}(j\times I)\ge \omega_j\sigma/K_1.\] Write \(M_j\) for the union of its marked bins. The unmarked aggregate mass is negligible when the subpower \(K_1\) is large enough.

Let \(\rho_\sigma\) be the product of five independent uniform noise laws of width \(\sigma\). The new trajectory index is \((j,\mathbf u)\), and its coefficients are the rounded vector of tag \(j\) plus \(\mathbf u\). Define the noisy prior and its marked incidence measure by \[\nu_{\rm jit}=\sum_j\omega_j\delta_j\otimes\rho_\sigma, \qquad d\mu_{\rm mark}(j,\mathbf u,t) =\mathbf1_{M_j}(t)\,d\nu_{\rm jit}(j,\mathbf u)\,dt.\] Thus the noise distribution is the same for every tag; the independence is between its five coordinates. The domination \[\mu_{\rm act}\{(j,t):t\in M_j\} \le K\sum_j\omega_j|M_j|=K\mu_{\rm mark}(\mathrm{all})\] shows that \(\mu_{\rm mark}\) has dense mass. This is a tempered single-world exact problem in base \(N\), version 1 with hidden \(w=x\); the mark and tag tables have polynomial sizes.

For the marked noisy law, the exact-base density in an \(x_p\)-bin is at most \(Kcp^d\). Fix \((t,y)\). The marked tags that can contribute have total prior weight at most \(K\sigma cp^d\): their original incidences satisfy the prescribed position constraints enlarged by \(K\sigma\), because motion over a time bin and the coefficient perturbation both cost at most \(K\sigma\). Summing the original cell caps and using the mark threshold gives this weight bound. The noisy intercept of \(y\) contributes density at most \(K/\sigma\), proving the exact-base ceiling.

The independent slope noise gives a further factor \(K\min(1,\rho/\sigma)\) when \(y'\) is confined to an interval of width \(\rho\). If the optional angular estimate is assumed, the same mark calculation at terminal \(x\)-width gives \[Kca^dN^{-S_1u}\] for the joint exact-base density with a velocity bin of width \(N^{-u}\), for every tested \(u<M\) in its range. The slope enlargement is much smaller than that bin. All these unnormalized ceilings survive further restrictions.

The noisy terminal support, measured by base Lebesgue measure and counting \(x_a\)-bins, is at most \(Kc^{-1}a^{-d}\). Every marked noisy cell is among \(K\) neighbors of a cell carrying original active incidences. Homogenize the profiles by the counting rules of 2, and pass to a subsequence on which \(C_0=\lim\log_Nc\). Dense mass, the terminal ceiling, and the support bound give \[n(s)\sim_{\log,N}ca^d,\qquad f(r)\ge dr-C_0, \qquad f(s)-f(r)\le d(s-r).\] If the angular estimate was assumed, the resulting problem belongs to \(C(d)\); choose its positive angular depth range below \(M\). Then 51 gives true packets with arbitrarily small horizon exponents on subsequences. Without that estimate, we first obtain it by cropping the slope and intercept.

Obtaining an angular gain.

When no angular gain is assumed, also homogenize the joint profiles \(f(r,u)\) with velocity bins through \(U=4(M+s+1)\). Put \(R=(1-d)/2>0\), fix a sufficiently small \(\epsilon>0\), and maximize \[(1-R)r-f(r,u)+\epsilon u, \qquad (r,u)\in[0,s]\times[0,U].\] At \((s,0)\) its value is \(C_0+Rs\). The slope-noise ceiling bounds the objective above by \(C_0+Rr+\epsilon u-(u-M)_+\). Continuity of the profiles therefore gives a maximizer \((r_0,u_0)\) with \[s-r_0\le\epsilon U/R,\qquad u_0\le M/(1-\epsilon),\qquad f(r_0,u_0)\le f(s,0)+\epsilon U.\] In particular \(r_0>0\) and \(u_0+r_0\le U\). Optimality with \(u=u_0\) fixed gives a terminal slope cap \(1-R\) ending at \(r_0\). Optimality with \(r=r_0\) fixed gives \[f(r_0,u_0+b)-f(r_0,u_0)\ge\epsilon b, \qquad 0\le b\le r_0.\]

Crop the noisy trajectories by the intercept and velocity of \(y\) at dyadic width \(G\) of order \(N^{-u_0}\). Make each crop \(\mathcal C\) a world with its conditional prior and world weight \(p_{\mathcal C}\), its prior probability. Subtract the central affine motion and divide \(y\) by \(G\). There are polynomially many crops. Discard extremely light crops, renormalize their world weights if necessary, and homogenize \(p_{\mathcal C}=N^{-\alpha+o(1)}\) on the used worlds. The resulting data remain tempered and have dense mass: the reciprocal crop probabilities and the affine coordinate changes have polynomial bounds.

The new pure density exponent is \[f'(r)=f(r,u_0)+u_0-\alpha.\] Indeed, given exact \((t,y)\) and the old velocity bin, the crop label has only boundedly many possibilities, including for its intercept since \(y(0)=y(t)-ty'\). Refinement by that label preserves the typical old exponent, while the within-world density is multiplied by \(G/p_{\mathcal C}\). Summing exceptional masses with the new world weights costs only a subpower factor. The old and new velocity grids at corresponding joint precisions boundedly overlap once the crop is fixed, as in 14. Thus the joint formula at \(r_0\) and relative velocity depth \(b\) replaces \(f(r,u_0)\) by \(f(r_0,u_0+b)\). The cropped problem belongs to \(C(1-R)\) with angular exponent \(\epsilon\). Apply 51 again to obtain arbitrarily small horizon exponents.

Lifting such packets to original noisy coordinates, write \(\beta\) for their spatial width scale including the \(G\) factor. Their mass per block time length within the single-world noisy law is at least \[K^{-1}p_{\cal C}\,N^{-f'(r_0)}(\beta/G) \ \ge\ \beta c a^d N^{-\epsilon U-o(1)}\] by the floor in 9 and the profile formulas. They have an explicit quadratic fit over that time block at thickness \(N^{-r_0+o(1)}\) (the spatial frame change was affine in \(t,y\)). Disjointness over worlds/labels lifts to disjointness within each block using the crop assignments. Total selected success lifts to dense total noisy mass by the choice of world weights (any overall renormalization for discarded worlds here costs at most a subpower).

Full-time fits and return to the original law.

In the cropped construction, let \(\epsilon\) and the horizon exponents tend sufficiently slowly to zero. In the optional angular case, use the terminal systems obtained without cropping and let only the horizon exponents decrease. At each fixed tolerance, first take samples sufficiently late along the corresponding successful subsequence. The resulting diagonal has dense noisy incidence mass in true assignments with block length \(q\ge1/K\), interval scale \(\beta\le K\), per-label mass at least \(qca^d\beta/K\), and quadratic error \(Ka\) throughout the assigned block. These are tangent subpower conclusions.

The interval scale satisfies \(\beta\ge\sigma/K\). The block bound confines \(y'\) to width \(K\beta/q\le K\beta\) about the moving-center slope. At exact base the graph test uses only \(K\) bins of \(x_a\). Integrate the exact-base ceiling, including its additional slope-noise factor, over base area \(Kq\beta\), and compare with the mass floor. This forces the asserted lower bound. In the optional angular case the same comparison with the augmented ceiling, at a sufficiently small positive \(u\), excludes every fixed-power deficiency of \(\beta\) from one.

The quadratic predictor also has norm at most \(K\) in moving interval coordinates. Normalize time to the block and use \((y-y_c(t))/\beta\) for the spatial variable. Since \(|x|\le K\), the mass floor and base ceiling show that \(|F|\le K\) on a set of Lebesgue measure at least \(1/K\) in a \(K\)-bounded box. Polynomial Remez bounds its norm. As \(q\ge1/K\), extrapolation extends this bound to unit time at cost \(K\). Extrapolate the affine position bounds and the quadratic fit errors along each assigned trajectory in the same way. The moving center is bounded by \(K\) as well, because an occupied chart contains a bounded assigned trajectory.

Keep one block’s assignments carrying dense selected \(\mu_{\rm mark}\)-mass; there are only \(K\) blocks since \(q\ge1/K\). Their floors give \(\sum_{\rm labels}ca^d\beta\le K\). Averaging over the noise coordinate \(\mathbf u\) in the displayed product prior fixes one common coefficient jitter vector for which the assigned marked-time mass, measured by \(\sum_j\omega_j\delta_j\otimes dt\), is still dense. Assign every original trajectory of a selected tag to that tag’s label in this block.

Undo this common jitter in every label. If its affine and quadratic components are \(\Delta y(t),\Delta x(t)\), use center \(y_c-\Delta y\) and predictor \(F(t,y+\Delta y(t))-\Delta x(t)\). The bounded norms and full-time fits persist for the original trajectories: their remaining rounding errors are \(O(\sigma')\), negligible in moving coordinates compared with \(a\) because \(\beta\ge\sigma/K\).

It remains to return from marked-time mass to \(\mu_{\rm act}\). Every marked tag-bin that contributes at the fixed jitter has original active mass at least \(\omega_j\sigma/K_1\). Restore its entire original active part. Full-time extrapolation permits this also for bins meeting block edges. The mark threshold therefore retains at least \(K_1^{-1}\) times the successful marked-time mass, so the restored original mass is dense. Finally discard labels whose restored mass is too small compared with \(ca^d\beta\), using an enlarged subpower denominator and the summed numerator bound above. This costs negligible mass. Further dyadic pigeonholing makes \(\beta\) common up to subpower factors and proves the assertion.

Subpower conclusions over an ensemble of eligible models with common exponent bounds can be taken uniformly: a fixed-power failure along a sequence of individually chosen models would contradict existence with subpower losses on a further subsequence. Equivalently use the worst infimum log-loss over models at each stage then diagonalize. 60 can thus be used anew on dense restrictions meeting its hypotheses. ◻

Comparing the projected fits at two domain sizes.

Compare \(\Phi_i\) and \(\Phi\) on long domains of time length \(\rho\sim_{\log,N}\beta_{\min}/\beta_{\max}\), slope-bin width \(\beta_{\max}\), and moving spatial width \(\beta_{\min}\). Let \(A\) be the common time-times-width Jacobian. An entry of color \(i\) has reference weight \[T_i=A w_\alpha a_i^d.\] As in 61, slicing gives a summable budget of these weights and retained marginal floors \(T_i/K\). The individual index-base ceilings are \(KT_i\).

The polynomial discrepancies on these long domains are at most \(KaN^{C_du}\) after negligible deletion. To prove this, use neighboring-jet groups at discrepancy scale \(M_0\) and match width \(a\), as in 61. The list in color \(i\) has at most \(K(M_0/a_i)^d\) members. Since \(a_1=aN^{-u}\) is the smallest thickness, whenever \(M_0/a\ge N^{C_du}\), \[(M_0/a_i)^d\le(M_0/a)^dN^{du} \le(M_0/a)^{d+d/C_d} \le(M_0/a)^{(1+d)/2}.\] The strict inequality \(d+d/C_d<(1+d)/2<1\) leaves room for subpower losses. Normalize the current pair law with the separate color priors \(T_i/W_i\), using its bad mass \(B\) and the reference totals \(W_i\) exactly as in the earlier proof. Thus 22 excludes any fixed-power excess over \(aN^{C_du}\). This argument keeps each color’s factor \(a_i^d\) in its own prior.

On the whole short domain of a retained pairing this difference \(F_{\Phi_i}-F_\Phi\) now varies by \(\ll a_i\): the normalized diameter needed in the long coordinates measured from the pairing is at most \(K(h_s/\rho + b_s/\beta_{\min})\), allowing slope error \(K g_s\) over the short time. Polynomial bounds on fixed enlargements suffice. Likewise on the whole fiber box \(Z=Z_\alpha(t)+O(K\chi),\ Y=Y_Q(t,Z)+O(K b_s)\) in this time interval the same estimate applies. Thus at retained events \[X=F_\Phi(t,Y)+D_i+O(K a_i)\] where \(D_i\) can be a constant determined by \(Q,\alpha,\Phi,\Phi_i\) (evaluate the difference at the block center on the specified central fiber); \(|D_i|\le K a\).

There are at most \(K(a/a_i)^d\) possible \(\Phi_i\)’s and hence constants given \(Q,\alpha,\Phi\): on each relevant short domain the participating \(\Phi_i\) have their tests within \(K a\) of \(F_\Phi\), so their marginal floors sum by the cap. Conversely given a short domain, \(\alpha,\Phi_i\), there are only \(K\) possible paired \(\Phi\).

The reverse bound and the true-label comparison list give only \(K\) refinements of each sliced \(\Phi_1\) entry. After pruning light source entries, typical source entries consisting of \((\alpha,D,\Phi_1,\Phi,Q)\) have active weight at least \(K^{-1} q_s w_\alpha a_1^d b_s\). Their normalized \((t,Y)\) density on the short domain is \(\le K\) times uniform by the finest label test and cap (at sufficiently fine meshes).

Use horizontal coordinates \(z=(t-t_*,Z-Z_*)/q_s\) (constant centering per \(Q\)) and \(U=(Y-Y_Q(t,Z))/b_s\). Expand \[F_\Phi(t,Y)=f_0^\Phi(z)+f_1^\Phi(z)U+f_2^\Phi U^2 .\] Here \(f_1^\Phi\) is affine and \(f_2^\Phi\) constant. We seek one polynomial \[P_{\rm nc}(z,U)=C_1(z)U+C_2U^2, \qquad e=aN^{-(1-d)u/2},\] with \(C_1\) affine and \(C_2\) constant, such that on dense mass \((f_1^\Phi(z),f_2^\Phi)=(C_1(z),C_2)+O(Ke)\). Subtracting it will leave, to error \(Ke\ll a_2\), only the \(z\)-dependent coefficient and the already counted offsets \(D_2\). We can then count scalar bins without retaining \(U\) and project in \((t,Z)\).

The coefficients \(f_1^\Phi,f_2^\Phi\) are bounded by \(K\) near their active fibers by the moving-interval norm, since \(Kb_s\) fits inside the interval scale. Evaluation throughout bounded \(Q\)-coordinates, or subpower enlargements, requires only fixed-power accuracies: the original coefficients cost at most \(K\beta_{\min}^{-O(1)}\), and occupied interval centers cost \(K\). Taking \(\chi\) sufficiently small therefore makes replacement of \(Z\) by \(Z_\alpha(t)\) negligible.

Coefficient agreement at a fine state.

Condition on a fine state consisting of \(Q\) and \((z,U,X)\) in tiny cells. All comparison precisions, including the direction grid used below, may be prepared at arbitrarily fine fixed powers. After light-state deletion, conditional \(v\)-bin weights are bounded by \(K\pi\). This follows from the \(K\) fine-state options per end label and does not assert independence from the projected labels.

Normalize only the whole dense current law to sample a source event, then sample a target independently conditional on the same state. Source entries \((\alpha,D,\Phi_1,\Phi,Q)\) retain their unnormalized \(\Phi_1\) ceilings after state deletion; no floor for their individual surviving masses is needed. Their reference weights \(q_sw_\alpha a_1^db_s\) sum to at most \(K\). For the target velocity we instead use the conditional kernel bound \(K\pi(d_v)\). We claim that the two nonconstant coefficient pairs agree to \(Ke\) except with negligible sampling probability.

Fix a source entry and a target velocity bin \(d_v\) of width \(\chi\). Their joint law is bounded by the source law times \(K\pi(d_v)\). We now choose a containing list of target polynomials before considering their offsets.

First restrict source time to a sufficiently short fixed-power bin. The fine state ties the source and target base positions much more closely than \(\chi\), so the target intercept lies within \(K\chi\) of \(Z_\alpha(t)-tv_{d_v}\). Throughout this time bin there are only \(K\) possible target \(\alpha\)’s, uniformly in \(U,X\). For each, the whole tilted \(Q\)-box meets only \(K\) slope/midpoint short domains, and the comparison list gives \(K\) coarse predictors \(\Phi\) on their union. A finer target label chooses one of at most \(KN^{du}\) offsets \(D_1\) for such a predictor. It does not enlarge the coarse polynomial list, and its offset does not change the derivative polynomial. The source entry already fixes its own polynomial and offset.

Use source variables \(t,U_\alpha\), \(U_\alpha=(Y-Y_Q(t,Z_\alpha(t)))/b_s\); here \(|U_\alpha|\le K\). Expand both candidate polynomials for these variables at source \(Z_\alpha(t)\). Their values with offsets (using \(i=1\) in the offset formula) have to agree within \(K a_1\): fine-state agreement controls \(X,t,Z,Y\) between the events, and in the new expression only \((t,Y)\) are used as arguments of either polynomial. A power-negligible error also covers transport of the two pairs of nonconstant coefficients to this source evaluation from their respective \(z\)’s. All these errors, including those for then freezing the polynomial coefficients on the indicated tiny source time bin, can fit inside \(K a_1\) by the fixed power bounds, \(\chi\) accuracy and still finer bins.

Put \(\tau=(1-d)u/8\). For each frozen polynomial pair with nonconstant coefficient gap at least \(e\), the derivative gap exceeds \(eN^{-\tau}\) outside a set of \(U_\alpha\)-length \(KN^{-\tau}\). If the quadratic coefficient difference is much smaller than \(e/K\), the linear term gives this uniformly; otherwise apply the sublevel estimate for the affine derivative. Sum this exceptional length over the \(K\) coarse polynomial choices. Offsets introduce no further exceptional sets.

On the complement, monotonicity on a bounded number of intervals for each polynomial and offset bounds the total length where values can match to \(Ka_1\) by \(KN^{du}a_1N^\tau/e\). The two bounds are therefore \[KN^{-\tau},\qquad KN^{du}\frac{a_1N^\tau}{e} =KN^{-3(1-d)u/8}.\] Both save a fixed power. The errors in freezing the coefficients are already within \(Ka_1\), so a coefficient gap of \(Ke\) at the actual events is covered by these estimates. Thickening the at most \(KN^{du}\) intervals by sufficiently fine fixed-power mesh steps preserves the saving.

Integrate the fine time bins with their lengths. No factor equal to their number is incurred. State deletion only decreases the source entry’s unnormalized ceiling, and normalization of the whole dense sampling law costs \(K\); there is no division by the surviving mass of an individual entry.

At such fine source meshes in normalized time and \(U_\alpha\), the source upper weight costs \(\le K q_s w_\alpha a_1^d b_s\) times mesh area per entry, by the joint cap and the \(\Phi_1\) test. This holds with normalization to our overall sampling probability as well. Thus the length tests integrate against this upper source measure and the palette probability with power-small cost times the reference weights; sum those weights. This proves the claimed coefficient agreement.

A common coefficient field. Conditional averaging now assigns a deterministic coefficient pair to each fine state, agreeing to \(Ke\) with the source pairs on dense mass. Prune light states to retain velocity domination, then choose \(Q\) with dense mass relative to \(q_s\nu_Q\), using \(\sum_Qq_s\nu_Q\le1\). Its normalized \(z\)-marginal is bounded by \(K\) times area. We next show that these statewise pairs agree on dense mass with \((C_1(z),C_2)\), for one affine \(C_1\) and constant \(C_2\), both bounded by \(K\).

Choose first a fixed-power grid step \(\omega\ll\chi\), small enough that all coefficient evaluation errors below are \(\ll e\). This choice is independent of the eventual basis, since the admissible bases and their inverses cost only \(K\). Prepare a direction grid and the common conditioning-state mesh much finer than \(\omega\). These direction bins refine the width-\(\chi\) bins stored in \(\alpha\); each meets only boundedly many of those older bins.

Now sample from \(\pi\). Each typical state’s compatible direction bins have total palette weight at least \(1/K\); the ball cap gives separated pairs weight at least \(1/K\). Averaging selects two fine bins with representatives \((1,v^1),(1,v^2)\), separated to reciprocal-subpower order, such that dense state mass has a witness from each. Use these vectors as the axes of the step-\(\omega\) product grid in \(z\), and assign each state to a nearby vertex.

Along a grid fiber parallel to either axis, its witnesses have only \(K\) possible \(\alpha\)’s over the entire fiber. Indeed the fiber prescribes an affine motion of that slope in \((t,Z)\), with contact errors at most \(Kq_s\omega\). The witness slope bin is much finer than \(\omega\), so the full-time intercept has only \(K\) bins at width \(\chi\). The earlier comparison then gives only \(K\) possible \(\Phi\)’s for this fiber and \(Q\). Their coefficient pairs restrict to an affine function and a constant on the fiber. At each compatible state one option on each fiber agrees with the assigned pair to \(Ke\). Select one polynomial pair per fiber by averaging; dense state mass satisfies both selected fits. Keep all matching occurrences, including those with directions outside the two witness bins.

There are now at least \(K^{-1}\omega^{-2}\) vertices each carrying weight at least \(\omega^2/K\) with that agreement (by the \(z\) upper bound and discarding light vertices), out of a product of side counts \(\le K/\omega\). A dense fraction up to powers of \(K\) of \(3\)-by-\(3\) product grids have all vertices present: the expected proportion of common neighbors of three independently chosen first coordinates is at least the cube of the overall density, by convexity; cube again for three independent opposite coordinates. Each side count is also \(\ge 1/(K\omega)\), so deleting near coincidences (on step-\(\omega\) grids) permits demanding reciprocal-subpower coordinate gaps by taking those cutoffs sufficiently small compared with this density, still with many grids. Fix the first two anchor coordinates on each side so there are \(\ge\omega^{-2}/K\) target pairs completing such grids.

Bilinear interpolation from the four anchors fits the first entry at the target vertices to \(Ke\); a constant from one anchor fits the second. Choose the four first-entry values from the selected affine polynomials on the two anchor rows. The interpolant agrees identically with those row polynomials. On a target column it is therefore within \(Ke\) of the selected column polynomial at both anchor rows. Affine interpolation along that column, using the inverse-subpower gaps, gives the target fit. The fiber constants compare through their crossings with one anchor row. Occupied anchor bounds and the gaps bound both coefficient sets by \(K\). Grid accuracy transfers the fits to events, and the vertex floors and target count give dense fitted mass.

It remains to remove the mixed bilinear term. Let \(\mu_Q\) be the unnormalized law in \(Q\) before the fiber and anchor selections. At that stage \(Q\) belongs to the fine state with conditional velocity domination, and \(\mu_Q(\mathrm{all})\le Km_Q\), where \(m_Q=q_s\nu_Q\). For either selected axis \(v^j\), this earlier law satisfies \[\mu_Q\{|v-v^1|<r\ \text{or}\ |v-v^2|<r\} \leq K m_Q\sum_{j=1}^2\pi(B(v^j,2r)) \leq K m_Q r^{S'}.\] This upper bound survives every subsequent restriction. If the final fitted law has mass \(m_Q/K_{\rm ret}\), choose the reciprocal-subpower radius \(r\) slowly enough that \(K K_{\rm ret}r^{S'}=o(1)\), first proving the assertion at fixed power radii. A dense non-axis residual remains. The law is dominated by \(K\) times the original trajectory prior restricted to the full assignment of \(Q\), times Lebesgue measure in time. That restricted prior has total mass \(\nu_Q\), so some indexed line has retained duration at least \(q_s/K\). Its fixed \(\alpha\) and \(Q\) permit only \(K\) coarse predictors, so one \(\Phi\) holds for duration \(q_s/K\). On those retained times the bilinear interpolant agrees to \(Ke\) with the affine function \(f_1^\Phi\). In the chosen basis the components of \((1,v)\) are \[\frac{v^2-v}{v^2-v^1},\qquad \frac{v-v^1}{v^2-v^1}.\] Both are bounded below in absolute value by a reciprocal subpower. The quadratic coefficient of the bilinear interpolant on this line is its mixed coefficient times their product. Remez on the retained time set therefore bounds the mixed coefficient by \(Ke\). Removing that term gives the claimed affine \(C_1\), while the fitted second entry gives the constant \(C_2\). The argument used the earlier unnormalized direction bound throughout.

Scalar support after subtraction. The common field just proved supplies the proposed \(P_{\rm nc}=C_1(z)U+C_2U^2\). Subtract it from \(X\) in \(Q\). The resulting quadratic signal is subpower bounded on assigned trajectories. On a sufficiently fine \(z\)-mesh, with \(U\) no longer recorded, each cell has at most \(K(a/a_2)^d\) residual bins of width \(a_2\) after pruning. Indeed, at fixed \([v]_\chi\), the values are \(f_0^\Phi(z)+D_2+O(Ka_2)\): there are \(K\) choices for \(\alpha,\Phi\) and \(K(a/a_2)^d\) for \(D_2\). The error \(Ke\) fits within this width because \(c_2<(1-d)u/4\).

To sum this list count over velocities, use the output state consisting of the \(z\)-cell and residual scalar bin in the chosen \(Q\). It has only \(K\) options per sufficiently deep end label, since \(P_{\rm nc}\) has fixed-power coefficient bounds. Delete light appended entries against the earlier end-history weights, whose sum inside \(Q\) is at most \(Kq_s\nu_Q\). The retained output states then have conditional velocity domination, hence compatible velocity bins of palette mass at least \(1/K\). Integrating the preceding count against \(\pi\) proves the claimed residual-bin count. The fine \(z\)-mesh determines the intercept to width \(\chi\) up to \(K\) choices and freezes \(f_0^\Phi(z)\) to \(Ka_2\); all these meshes are prepared simultaneously.

Normalize the current mass by \(q_s\nu_Q\). In unit \(Q\)-time and scaled \(Z\), the mass per unit projected base volume in a scalar bin of width \(a_2\le p\le1\) is at most \(K(p/a)^d\), also with the angular factor from \(\pi\). For a \(z\)-cell of side \(\sigma_*\), integrating over \(|U|\le K\) uses old base volume at most \(Kq_s^2b_s\sigma_*^2\). On still finer meshes the \(X\)-bin can follow the offset \(P_{\rm nc}\). Now use \(\nu_Q\sim_{\log,N}a^dh_sb_s\) and the palette ball bound at sufficiently small positive exponents. The scaled velocity remains \(v\).

The prior for the next projection is the true assignment of \(Q\). Restrict it, if necessary, to trajectories with bounded coefficients on the normalized full block. Every trajectory in the final dense events has these bounds by construction. Since \(Q\) was selected heavy relative to its full trajectory weight, this conditioning costs only \(K\). Apply 60 to \((X-P_{\rm nc},Z)\), with \(c=a^{-d}\), terminal width \(a_2\), and its positive angular cap. The terminal density is then \(ca_2^d=(a_2/a)^d\). The support count just proved is its second hypothesis. It gives labels of full horizontal width up to \(K\), hence a quadratic prediction in \((t,Z)\) throughout the block, with normalized mass at least \((a_2/a)^d/K\). Restoring \(P_{\rm nc}\) gives the asserted quadratic in the full base and the required mass.

These last comparison meshes may have arbitrarily fine fixed-power widths. The equations with errors \(O(a_i)\) are evaluated on true base variables; the prepared pure and joint caps and palette domination apply to the finer native point states. No improvement of those equation errors is needed. If this construction is used inside an earlier true parent after affine changes, select heavy descendants using their full assignments nested in that parent’s prior, whose weights sum to at most that prior in each block. ◻

From projected intervals to a row list.

Choose \(r>s\) with \((k+\ell)r>B_0>\ell r\). Given \(Q_r\), its \(Q_s\)-history, and sufficiently fine horizontal position and velocity bins, there are only \(K\) possible \(\beta_s\)-bins for \(Y'\). Indeed \(\alpha\) is known up to neighbors. The \(Q_s,\Phi_s\) comparison domains then have \(K\) possibilities: \(Q_r\) determines \(Y\) to \(Kb_r\), while \(Q_s\) determines the \(g_s\)-slope bin. Translating from the occurrence time to a comparison-subblock midpoint costs \(Kg_sq\), its prescribed width, and \(b_r\lesssim_{\log,N}\min(b_s,\beta_s)\). Thus only \(K\) midpoint bins, and hence \(K\) projected labels, are possible. Each label’s full-time moving interval confines \(Y'\) to width \(K\beta_s\). Take the horizontal and velocity meshes sufficiently fine also relative to \(\chi\).

Normalize the true parent \(Q=Q_r\). Translate time and \(Z\) by constants and divide by \(q_r\), keeping horizontal velocity \(V_1=(1,v)\). Subtract \(Y_Q\) and divide the transverse coordinate by \(q_rg_r\). The normalized transverse velocity is \[\frac{Y'-A_QV_1}{g_r},\] bounded by \(K\), and the preceding list has accuracy \(E=N^{-(B_0-\ell r)}\). A sufficiently deep true label has row \[T=(A_{\rm deep}-A_Q)/g_r\] whose evaluation \(TV_1\) predicts this velocity to \(\ll E\). Fix base and velocity meshes \(\xi\ll E\), fine enough for these lists.

We verify the remaining row-fit hypotheses. At a state consisting of the deep row label and fine horizontal location, use sufficiently deep histories, append this state, and delete light states to obtain conditional velocity domination by \(K\pi\). Its supporting velocity bins have palette mass at least \(1/K\). The angular cap therefore supplies pairs separated by a reciprocal subpower, with palette product weight at least \(1/K\). The two evaluations of the fixed row at actual witnesses bound \(T\) by \(K\).

For any fixed such pair at a horizontal cell, the two lists for the normalized transverse velocity constrain compatible \(E\)-row bins to \(K\) possibilities by interpolation. Bounded rows and velocities make the binned-velocity error negligible. Choose one state per row-bin option and sum with the palette product law. Since every option has witness-pair weight at least \(1/K\), there are at most \(K\) row options per horizontal cell.

The horizontal joint cell cap, before dividing mass by \(q_r\nu_Q\), is \[K\xi^2q_r\nu_Q\,\pi(v_\xi).\] It follows by summing the joint state bounds with their earlier weights on which the \(Q\)-base cap holds. The palette interval bounds through \(\xi\) follow from the power-scale caps and interpolation. Choose a \(Q_r\) heavy relative to \(q_r\nu_Q\), and use its full conditioned trajectory prior. All hypotheses of 58 now hold in the normalized chart.

The horizontal equation and its scales.

The row fit gives normalized transverse velocity \((B+\lambda Jz)\cdot V_1+O(KE)\) on dense mass. If \(|\lambda|\) is power-small on a subsequence, this is a strip candidate: multiply by \(g_r\) and restore \(A_QV_1\). Otherwise \(|\lambda|\sim_{\log,N}1\). Subtract \(B\cdot z\) from the transverse coordinate and divide by \(\lambda\). Reusing \((t,Z,Y)\) for these coordinates, we obtain \[ |Y'-Jz\cdot V_1|\le N^{-e_0} \tag{217}\] for some fixed \(e_0>0\), decreased with room if needed. The expression on the left before absolute values is constant along each trajectory.

At a small relative depth \(s'>0\), a child \(Q_{r+s'}\) has block length \(H=q_{r+s'}/q_r\sim_{\log,N}h_{s'}\), horizontal diameter at most \(KH\), and a fixed row predicting transverse velocity to \(KN^{-\ell s'}\). Compare this row with \(Jz\) using separated velocities in a typical fine-location state. Interpolation gives row error at most \(K(N^{-\ell s'}+N^{-e_0})\). These comparisons use unnormalized domination and earlier weights summing inside \(Q_r\) to at most \(Kq_r\nu_Q\); delete light refinements in that normalization.

Choose \(s'\) small enough that the \(N^{-e_0}\) error is negligible at both relative scales. After summable deletion, each used child carries normalized mass at least \(H\nu_{Q_{r+s'}}/(K\nu_Q)\). Prior-times-time domination then forces its active time extent to be at least \(H/K\). But its fixed row predicts the second component of \(Jz\), namely \(t\), to \(KN^{-\ell s'}\). Therefore \[N^{-ks'}\le KN^{-\ell s'},\] so \(k\ge\ell\). If \(k>\ell\), replace \(Jz\) in [a06:horizontal-equation] by its value at an active child center. The error from the horizontal diameter is smaller by a power than the nominal velocity width, giving a strip candidate at \(Q_{r+s'}\). Restoring earlier coordinates multiplies velocity error by \(|\lambda|g_r\); dividing the displayed child mass floor by \(H\) and restoring the \(\nu_Q\) denominator gives the required candidate mass. Thus it remains only to consider \(k=\ell\).

Whole horizontal boxes.

Choose an occupied center \((z_c,Y_c)\) of a used child and set \[ \widehat z=(z-z_c)/H,\qquad \widehat Y=(Y-Y_c-Jz_c\cdot(z-z_c))/H^2. \tag{218}\] The true strip row is within \(KH\) of \(Jz_c\), by the preceding two-velocity comparison at a point within horizontal distance \(KH\) of the center. With \(k=\ell\), its true transverse spatial width is \(KH^2\) up to subpower factors. The coordinate changes used to obtain the horizontal equation alter these comparisons only by \(K\). A slope discrepancy of \(KH\) over horizontal distance \(KH\) costs \(KH^2\), so these centered coordinates describe the entire true tilted box, up to \(K\), rather than only its retained incidences.

The two-velocity comparison can use common-location meshes much finer than the box widths: the row is fixed, and the variation of \(Jz\) between the nearby true-velocity witnesses is harmless. Prepare finitely many requested depths simultaneously, then let the grids grow slowly, discarding labels and states too light relative to their earlier weights. Time is unit time in \(Q_r\), and the prior is its full true line assignment conditioned to probability. Descendant mass floors and deletions continue to use the full nested assignment weights even if only some assigned lines participate in the later restrictions. ◻

Counting support by whole true boxes.

Before the group restrictions, discard true \(Q_s\)’s of current mass less than \(m_{Q_s}/K_1\), and freeze the retained witnesses. The budget [a06:horizontal-budgets] makes the loss at most \(1/K_1\). These are witnesses from the bounded-residual law after choosing \(L\), so its pure base cap applies to them. The full assignment supplies the reference budget, not the witness measure. Later contact with a group will locate the entire box containing these witnesses.

Fix a time bin of length \(a_s\). Each contributing true box meets the group’s reference motion to \(KD\). In a true time block of length \(H_s\), all these boxes lie in one enlarged horizontal box. To verify this, note that the reference motion satisfies \(Y_c'=Jz_c\cdot(1,v_{d_0})\) exactly and that \(D\le h_s^2\). At contact the error therefore fits both horizontal and transverse widths. Use the plane with row \(Jz_c(t_*)\) at the reference block midpoint. Each meeting box has horizontal extent \(Kh_s\); changing its centered plane to this reference plane costs at most \(Kh_s^2\). Thus its whole domain lies in the common enlargement, whose base volume is at most \(Kh_s^4\).