The problem and its development

The problem belongs to the theory of summability of Fourier expansions. Riesz’s work on one-dimensional summation methods (Riesz 1923) preceded Bochner’s spherical means for multiple Fourier series (Bochner 1935, 1936). The multiplier \((1-|\xi|^2)_+^\delta\) softens the boundary of the frequency ball; the question is how much smoothing is needed to preserve an \(L^p\) norm. Radial inversion and radial summability were studied by Herz and Welland (Herz 1954; Welland 1975). For general functions the geometry of frequency directions is essential. Using a Kakeya construction, Fefferman proved that in dimension at least two the order-zero ball multiplier is unbounded on \(L^p\) for \(p\ne2\) (Fefferman 1971), while Carleson and Sjölin established the conjectured positive-order range in the plane (Carleson and Sjölin 1972).

Geometric decompositions connect this problem with the arrangement of long thin tubes and with Fourier restriction. Córdoba’s work on radial multipliers and Kakeya maximal functions developed this connection through rectangle decompositions and almost orthogonality (Córdoba 1975, 1977). Bourgain combined geometric maximal-function estimates with improved restriction and multiplier bounds in three dimensions (Bourgain 1991). The bilinear restriction theorems of Wolff for the cone and Tao for the paraboloid and elliptic surfaces (Wolff 2001; Tao 2003) led, through Lee’s multiplier reduction, to sharp-order Bochner–Riesz estimates for \(\max\{p,p'\}\ge10/3\) (Lee 2004); here \(p'\) is the Hölder conjugate of \(p\). The threshold exponent follows from Lee’s printed strict range by interpolation with \(L^2\), keeping a positive margin in the order.

Polynomial partitioning and subsequent multiscale refinements brought further progress on restriction (Guth 2016; Wang 2022). For the multiplier itself, Wu obtained the conjectured strict-order range when \(\max\{p,p'\}\ge13/4\) (Wu 2023). Guo, Oh, Wang, Wu, and Zhang developed a pseudoconformal approach and an induction on two scales for transferring restriction methods to Bochner–Riesz estimates (Guo et al. 2025). More recently, Gao, Wu, and Xi established the strict-order multiplier range \(\max\{p,p'\}\ge22/7\) (Gao et al. 2026, Corollary 1.3). Wang and Wu’s related diagonal extension estimate holds for \(p>22/7\) (Wang and Wu 2024, Theorem 0.2). The connection also runs in the other direction: Tao showed that the Bochner–Riesz conjecture implies the spherical restriction conjecture (Tao 1999). Our argument directly establishes the scalar multiplier bound on \(L^3(\mathbb R^3)\) for every fixed positive order. Zipeng Wang has also proposed a proof of the Bochner–Riesz conjecture in all dimensions (Wang 2026, version 6, Theorem One).

Let \(S^2=\{\omega\in\mathbb R^3:|\omega|=1\}\) and let \(\sigma\) be surface area measure on \(S^2\). For bounded measurable \(g:S^2\to\mathbb C\), define \[Eg(x)=\int_{S^2}g(\omega)e^{2\pi i x\cdot\omega}\,d\sigma(\omega).\] Tao’s implication applied to (3) gives, for every real \(p>3\), \[\norm{Eg}_{L^p(\mathbb R^3)} \le C_p\norm{g}_{L^\infty(S^2,\sigma)},\] where \(C_p<\infty\) depends only on \(p\). This is the spherical extension estimate also proved in OpenAI (2026c, Theorem 1.1). For the separate diagonal extension theorem on compact smooth positively curved surfaces, possibly with smooth boundary, see OpenAI (2026b, Theorem 1.1); that general-surface statement is not deduced here from the spherical multiplier theorem.

An independent route through critical local smoothing gives the same full strict fixed-radius range in (3), uniformly in the radius (OpenAI 2026a, Corollary 11.2). It also gives strong maximal Bochner–Riesz bounds and almost-everywhere summation for \(3\le p<\infty\) and \(\delta>1-3/p\) (OpenAI 2026a, Theorem 11.1). These results are not inputs to the argument here.

The geometric input here is the sharp planar Furstenberg estimate of Ren and Wang (Ren and Wang 2025). It belongs to a line of projection theory that includes the classical work of Marstrand and Kaufman (Marstrand 1954; Kaufman 1968) and the discretized sum-product and projection methods of Bourgain (Bourgain 2010). Oberlin formulated a sharp prediction for exceptional projection directions (Oberlin 2012). Important advances toward the Furstenberg problem include the packing-dimension results of Orponen and Shmerkin (Orponen 2020; Shmerkin 2022), their joint Hausdorff-dimension improvement (Orponen and Shmerkin 2023), and their proof of the conjectured bound when the associated family of affine lines has equal Hausdorff and packing dimensions (Orponen and Shmerkin 2026). Building on that regular case, Ren and Wang removed the additional equality assumption. We use their finite incidence theorem. Section 3 derives the form needed for probability weights, auxiliary tags that may share a slope, and incidence graphs of positive product mass; Section 2 supplies the finite conditioning used in later applications.

From geometric growth to oscillatory cancellation

The proof has two stages. We first control the information gained by observing positive masses moving along lines. We then transfer that control to wave packets, whose squared amplitudes are positive but whose superposition is oscillatory. The main obstruction is at the uncertainty boundary: an earlier packet cannot be assigned a position as precisely as a later observation requires. The second stage records this discrepancy as fine velocity information and shows that too much growth would make that information exceed its finite capacity.

Finite laws and order of limits

Entropy permits a common language for these geometric and analytic steps. We use finite entropy and relative entropy (Shannon 1948; Kullback and Leibler 1951), conditional copies and submodularity (Tao 2010), and homogeneous profiles across scales. Multiscale entropy and comparable-partition methods have important precedents in Hochman and Shmerkin (2012; Hochman 2014). Section 2 proves the precise finite-law statements used here.

Small normalized mutual information yields a comparison with a product law after trimming; it does not give exact independence. Theorem 8 specifies which labels are fixed and which remain free, so later conditioning uses the correct comparison population. For the characteristic-plane argument, first fix a homogeneous component and its limiting profile, then choose the approximation scale, and only afterwards take the finite precision small relative to that scale. Theorem 26 and Corollary 27 state these successive steps.

Packet composition also keeps two finite populations distinct. Its upper estimate concerns a retained superposition, while its earlier lower test uses the whole assigned cap class. Proposition 41 records the loss from retention. Section 11 carries the remaining fine velocity information through a fixed finite number of cuts, using the one-step statement of Proposition 48.

Throughout, \(\mathsf H,\mathsf I\) denote entropy and mutual information divided by \(\log(1/\epsilon)\), whereas \(\mathsf H_{\mathrm{nat}},\mathsf I_{\mathrm{nat}}\) use natural logarithms. The notation \(\log_{1/\epsilon}\!\left(r\right)\) uses the normalized units. The Planck depth is always \(H_{\mathrm P}\), distinct from entropy. A positive depth increment is fixed before the small-parameter limit; when increments subsequently shrink, that additional limit is stated separately.

Entropy estimates with unrestricted parent alphabets

Our conventions are the usual finite entropy and relative entropy (Shannon 1948; Kullback and Leibler 1951). Conditional copies and entropy submodularity also feature in Tao’s entropy calculus (Tao 2010); the comparison and retention bounds required here are proved below.

We use the chain rule, nonnegativity of conditional mutual information, and \(\mathsf H_{\mathrm{nat}}(X)\le\log|\mathcal X|\) for an \(\mathcal X\)-valued observation. Here are the particular consequences, including quantitative restriction estimates, that we will need.

We call a determination of zero cost when the right side of (6), divided by \(\ell\), tends to zero. This includes bounded ambiguity between comparable translated grids, provided the translation is recorded. When \(|\mathcal B|\le\epsilon^{-C}\), vanishing exceptional probability is sufficient for the last term.

Joint bins give a sharper version for increments. Suppose \(G\) is any parent, possibly containing \(T\), and on retained triples \((g,b,d)\), \[\epsilon^{b_0+a}\le\mathsf P(B=b\mid G=g)\le\epsilon^{b_0-a}, \qquad \epsilon^{d_0+a}\le\mathsf P(B=b,D=d\mid G=g)\le\epsilon^{d_0-a}.\] Dividing these inequalities gives conditional surprises for \(D\mid(G,B)\) in \([d_0-b_0-2a,d_0-b_0+2a]\) and at most \(\epsilon^{-(d_0-b_0+2a)}\) retained children per \((g,b)\). Indeed, the sum of the lower joint masses cannot exceed the upper parent mass. Lemma 2 therefore bounds entropy errors under a density loss \(K\) by \(2a+\log K/\ell\), independently of the size or entropy of the parent. These are estimates for retained support and for sampled conditional values; no assertion about every untrimmed conditional fiber is needed.

From small information to explicit product comparisons

The next theorem specifies the laws compared when labels are fixed. A point may be an exact finite particle or a tuple of spatial observations. Additional finite tags may be included in a label.

We spell out two consequences of the theorem to make its uses precise. First, take \(Z\) trivial and \(i\in J\). For typical fixed values \(b_J\), \[ R_P(\,\cdot\mid B_J=b_J) \le\frac{c^{n-1}}{m\eta}q_P(\,\cdot\mid B_i=b_i). \tag{32}\] Thus if at label \(b_i\) a coordinate tuple \(\Phi_i(P)\) has a bound \(q_{\Phi_i\mid b_i}\le A_i\prod_l\mu_{il}\), its law after fixing all of \(B_J\) has bound \(c^{n-1}A_i/(m\eta)\) against exactly those factors. This is true for every \(i\in J\) simultaneously. Applying the core lemma to the combined list supplies actual marginals for all of them in one retained law. Counts on the original pair support are preserved. If \(Z\) is nontrivial, the comparison is instead the conditional law \(q_P(\,\cdot\mid B_i=b_i,z_{b_J}(P)=z)\) in (29); additional coordinate product bounds must be conditioned by (10), with their factors explicitly identified.

Second, suppose \(\rho(a,b)\le K\mu(a)\lambda(b)\) and \(C=f(A)\). Outside \(\rho_C\)-mass at most \(\eta\), \[ \rho(a,b\mid C=c)\le\frac K\eta \mu(a\mid C=c)\lambda(b). \tag{33}\] The measure on children is the original \(\mu\) conditioned on its parent; there is no factor \(\mu_C(c)^{-1}\) in addition to the displayed likelihood loss. Any number of parent cells, in particular polynomially many, is allowed. Subsequent simultaneous marginal replacement uses the actual child and parameter populations retained in that cell. An observation depending on a free label does not have the factorization used in (31); it must first be included among fixed labels or handled by a separately verified comparison law.

For example, if a final retained point law has \[\rho_{X,U_0}\le A\rho_X\rho_{U_0},\qquad \rho_{X,N,L,Z}\le B\rho_X\rho_{N,L}\rho_Z,\] then conditioning the second inequality on \((X,N)=(x,n)\) gives \[ \rho(l,z\mid x,n) \le B\frac{\rho_X(x)\rho_N(n)}{\rho_{X,N}(x,n)} \rho(l\mid n)\rho_Z(z). \tag{34}\] The factor in the fraction exceeds \(1/\eta\) on a set of \(\rho_{X,N}\)-mass at most \(\eta\). This follows by summing \(\rho_{X,N}(x,n)\) over the exceptional values, so it is valid also for labels recorded at the finest precision. The first displayed comparison remains available on the same law; it is not replaced by a marginal from an earlier trim.

All these conclusions concern a fixed finite star. They supply no lower bound for the density of an arbitrary graph of cyclic incidences. If \(K\), \(M_0^{-1}\), \(m^{-1}\), \(u^{-1}\), \(\eta^{-1}\) and \(\alpha^{-1}\) have logarithms \(o(\ell)\) and \(n,N_{\mathrm{fac}}\) are fixed, every displayed loss is \(\epsilon^{-o(1)}\). For a normalized retained incidence law \(\rho\le A\mu\otimes\lambda\), its support has \((\mu\otimes\lambda)\)-mass at least \(A^{-1}\), simply by integrating the domination inequality. This is the dense-graph conclusion used in the projection theorem.

Countable profiles and the order of the limits

The use of entropy at successive resolutions is related to the local entropy averages and comparable-partition arguments of Hochman and Shmerkin (2012, sec. 3.3 and 4.2) and Hochman (2014, sec. 3). Here we need a finite-law diagonal with explicit retention properties, which we establish directly.

A conditional observation is homogeneous with exponent \(d\) if its realized conditional surprise, divided by \(\ell\), converges to \(d\) in probability and its normalized conditional entropy also converges to \(d\). When further restrictions will be made, we first retain the uniform bins of Lemma 5. Convergence in probability by itself is insufficient for this purpose. For example, let \(\ell\to\infty\), give one atom mass \(1-e^{-\sqrt\ell}\), and distribute the remaining mass uniformly among \(\lceil e^\ell\rceil\) atoms. Normalized surprise converges to zero in probability, but restriction to the latter atoms has density \(e^{\sqrt\ell}\) and entropy tending to one in normalized units. The support and bin requirements in Lemma 5 exclude precisely this issue.

In applications with a block length \(h\), “subpower loss” means \(\log K=o(h\ell)\), not merely \(o(\ell)\). For example the logarithm of a conditional \(k\)-factor core constant in Theorem 8 is bounded by a fixed linear combination of \[n\log(K/\alpha),\quad\log(1/M_0),\quad\log(1/m),\quad \log(1/\eta),\quad k\log(2N_{\mathrm{fac}}/u).\] These quantities must be chosen \(o(h\ell)\) with \(h\) fixed first. The finite numbers of observations and replicas are also fixed at that stage. Lemma 10 then permits increasing them slowly and shrinking \(h\) afterwards. When an increment has a much larger parent entropy, (9) and the uniform child counts give the same \(o(h)\) error without multiplying by that parent entropy.

Finally, the use of universal bounds after conditioning is a selection principle for a specified array of conditional laws. At each scale fix a class \(\mathcal A_\epsilon\) of laws and parameters such that every sequence selected from these classes is admissible for the asserted universal bound. In particular, all vanishing-error hypotheses must have common deterministic envelopes within the array: a hypothesis \(\log K=o(\ell)\) is represented there by \(\log K\le r_\epsilon\ell\) with one \(r_\epsilon\to0\). If \(F_\epsilon(\mathsf P,\vartheta)\) has \(\limsup F_\epsilon\le0\) along every such selected sequence, then \[\sup_{(\mathsf P,\vartheta)\in\mathcal A_\epsilon} F_\epsilon(\mathsf P,\vartheta)\le o(1).\] Otherwise choose a violating law and parameter at each of a sequence of scales. The same upper bound holds for any average over conditional laws in this array, regardless of the number of conditioning cells. The envelopes may depend on the construction; this argument asserts no uniform convergence over all possible sequences satisfying an unspecified subpower condition. When the universal statement is initially given for fixed parameters, this argument requires stability under convergent parameters; compactness alone does not supply that stability. The geometric applications below verify it by bounded box enlargements and \(O(\|\vartheta-\vartheta'\|)\) changes in depth weights.

Apertures and the universal growth rate

Use nested grids for time and write \(C_t\) for the cell of \(c\) of length comparable to \(\epsilon^t\), together with its chosen representative. At \(t=0\) one cell contains the time domain and its representative is zero. The finitely many boundary choices in this convention have entropy \(O(1/\log(1/\epsilon))\). For an unoriented normal \(R\), let \(Q_R\) be an orthogonal frame whose first vector is tangent and whose second vector is normal. Orientations of aspect \(e\) are recorded in cells of angular size comparable to \(\epsilon^e\). Nested angular grids, or a finite collection of nested slope grids, may be used. Choose a representative normal for each recorded angular cell; both rectangular grids use its frame.

Depths can be rounded on a mesh \(\xi=\xi_\epsilon\downarrow0\) with \(\log(1/\xi)=o(\log(1/\epsilon))\). The number of shape choices in a bounded triangle is then subpower. In this terminology a factor is subpower if its normalized logarithm tends to zero. Dyadic spatial widths give a finite menu of apertures: choose orientation spacing comparable to the minor-to-major axis ratio and translates spaced by the corresponding widths. Every rectangle is contained in a fixed enlargement of one in this menu. Indeed an angular error at most a constant times the axis ratio moves a tangent uncertainty into at most a constant multiple of the normal uncertainty. This verifies the assertion for both columns. Discarding extra labels never increases particle information. Define the root supremum by \[\tau_L(T)=\sup_{D\text{ at duration }0} \{\mathbb Ew(D)-\mathsf I(T;D)\}.\] The supremum is taken for the current particle law and the current precision, before any limit. For a fixed finite menu it is a supremum over all stochastic assignments to compatible caps. Rounding depths down on the mesh and adjoining the actual target cells costs \(O(\xi)+O(1/\log(1/\epsilon))\) in root score, uniformly on fixed domain, horizon, and enlargement bounds. Thus these menus recover the unrestricted root supremum in the limit.

The score balances three quantities. The aperture weight rewards finer localization, particle information charges what the observation reveals about \(T\), and conditional time entropy measures the time information left after the particle is known. The root supremum provides the zero-duration benchmark for this balance.

Write \(S_t=\mathsf H(C_t\mid T)\) and \[ \mathcal F_t(G)=\mathbb Ew(G)-\mathsf I(T;G)+2S_t, \qquad \beta=\sup\limsup_{\epsilon\to0} \frac{\mathcal F_t(G)-\tau_L(T)}{t}. \tag{42}\] The supremum ranges over all fixed \(0<t\le L<\infty\) and all sequences of the systems just defined. Domain bounds are fixed within each sequence. Fixed affine normalizations put them in one bounded domain without changing a limit. The label \(G\) determines, up to bounded ambiguity, a root aperture of the same shape: recover \(A=X(C_t)-C_tV\) and coarsen its position rectangle. Consequently \(\mathcal F_t(G)-\tau_L(T)\le2S_t+o(1)\le2t+o(1)\). For a single particle and independent uniform time, the choices \(b=a=L-t\) give \(\tau_L=2L+o(1)\) and \(\mathcal F_t=2L+o(1)\). Thus \[ 0\le\beta\le2. \tag{43}\]

We prove this bound by examining a sequence whose growth approaches \(\beta\). Localization and the least-aspect construction produce a saturated path. Its tangent reduction has two alternatives: an isotropic system, excluded in Section 5, or a path whose aspect grows in proportion to time. Sections 6 and 7 constrain the local information rates of the latter path; Section 8 combines those constraints with saturation to finish the proof. Until that point we use only (43) and the definition of \(\beta\).

The rate in (42) is unchanged if the underlying probability spaces are required to be finite. Fix the complete finite root menu and project \(G\) to its geometric coordinates, which can only decrease particle information. For every possible output/time pair \((g,k)\), use the two compatibility tests for \(V\) and \(A+\bar c_kV\), where \(\bar c_k\) is the recorded time representative. Partition particles by all these tests and by membership in every root cap. The finite signature \(\kappa(T)\) depends on \(T\) alone, not on the sampled time. Choose an actual \((V,A)\) in each positive signature, with finite particle index \(T^{\mathrm f}\). Give \((T^{\mathrm f},G,C_t)\) the old law of \((\kappa(T),G,C_t)\), and realize each time label by a point in its actual cell. The resulting finite law is compatible and has the same expected weight, with \[\mathsf I(T^{\mathrm f};G)\le\mathsf I(T;G),\qquad \mathsf H(C_t\mid T^{\mathrm f})\ge\mathsf H(C_t\mid T).\] For roots, averaging a compatible kernel within signatures preserves its output marginal and weight and decreases information; conversely, every signature kernel lifts through \(\kappa(T)\) with equal information. The frozen-menu root suprema therefore agree exactly. The uniform menu error above is \(o(1)\), so every general-law score excess is at most that of a finite law plus \(o(1)\). We take these finite laws before forming the conditional systems and independent copies below.

We make explicit the uniformity implicit in that definition. For any fixed domain bound \(B\), \(K<\infty\), and \(t_{\min}>0\), there is a function \(\omega_\epsilon(B,K,t_{\min})\to0\) such that \[ \mathcal F_t(G)\le\tau_L(T)+\beta t+ \omega_\epsilon(B,K,t_{\min}),\qquad t_{\min}\le t\le L\le K. \tag{44}\] Here the menu, its depth mesh, and its fixed enlargement convention are chosen as deterministic functions of \(\epsilon\), common to all systems in the supremum. It also applies to every conditional system with these bounds. To see this, a failure selects systems and parameters with a fixed positive excess and a convergent subsequence of \((t,L)\). Changing depths by \(\nu\) costs at most \(C_{K,\gamma}\nu+o(1)\): cover the larger rectangle by the finer cells, including its changed frame in the count. Coarsening time by \(\nu\) loses at most \(\nu+o(1)\) of conditional time entropy. Transport to the coarser time uses an uncertainty \(\epsilon^{t-\nu}\) times the measured velocity rectangle, which fits the earlier position rectangle. Finally, coarsening both depths of a root test by at most \(\nu\) loses at most \(2\nu+o(1)\) in weight; this proves the same continuity for the root supremum when its horizon changes by \(\nu\). Fixed parameter slack therefore converts the selected systems to a sequence contradicting (42). The same argument at \(t=0\) uses the root supremum directly, with no division by time.

Whenever we use conditional arrays below, we first fix the finite list of observations, the depth mesh, all trimming thresholds, and the positive time increments. Their exceptional probabilities and normalized errors are then bounded by one deterministic function \(q_\epsilon\to0\) on every retained cell. Cell coordinate bounds and covering multiplicities are common constants. Thus every selection of one retained system per precision remains an admissible sequence in (42). The conditional-array selection principle at the end of Section 2 applies to precisely this array; it does not require a uniform rate over unspecified subpower envelopes.

Splitting and cap bounds

Call an observation saturated if, along the sequence under discussion, \(\mathcal F_t(G)-\tau_L(T)\to\beta t\). An extremizer below always means such a sequence, not a maximizing measure at finite precision.

Anisotropic localization

The case \(r=d=0\) removes the common depth \(b\). Replacing the local horizon by its smallest permitted value \(e+t\) cannot lower a score minus its root supremum; the universal upper bound then forces saturation. Replacing \(\epsilon\) by \(\epsilon^t\) makes the duration one and divides all depths, weights, and normalized entropies by \(t\).

The least aspect and saturated predecessors

We first isolate the rectangle comparison used twice below. Two equal-shape apertures \((b,b+e)\) with angular separation of depth \(v\in[0,e]\) have a common coarsening of shape \((b,b+v)\) and an intersection aperture of shape \((b+e-v,b+e)\). The common label is determined by either original label, once an angle-depth bin of width \(\xi\) is supplied, at information cost \(O(\xi)+o(1)\). For the intersection, one normal constraint has width \(\epsilon^{b+e}\); solving the other normal constraint along its tangent gives width \(C\epsilon^{b+e}/|\sin\theta|\). The safe endpoint of the angle bin gives the asserted tangent depth with an \(O(\xi)\) loss. Coarsening to normal depth \(b+v\) proves the common-rectangle assertion. These arguments apply separately to both columns, with the common time factor when the duration is positive. The weight gain is \[ w(b+e-v,b+e)+w(b,b+v)-2w(b,b+e)=2\gamma(e-v). \tag{48}\]

Figure 1 shows the two rectangles responsible for this weight gain.

Isotropic tangents or a matched aspect

Finite incidence counts and regular increments

In the next two elementary estimates \(\rho\) is a mesh size, and all coordinates belong to a fixed bounded set. Constants may depend on that set, but not on \(\rho\). A point cell in dimension \(d+1\) has coordinates \((c,x)\in\mathbb R\times \mathbb R^d\), and a slope bin has representative \(v\in\mathbb R^d\) and diameter \(O(\rho)\). The intercept map is \[\pi_v(c,x)=x-cv.\]

We will use the following explicit form of the star calculation in Theorem 8. Let \(q\) be a normalized point–slope law with actual marginals \(\mu,\nu\), and suppose \[ q(p,v)\leq K\mu(p)\nu(v),\qquad \mu(p)\leq M^{-1}. \tag{51}\] Sample \(r\) slopes independently conditional on \(p\). The resulting law \(q_r\) satisfies both comparisons \[\begin{align*} q_r(p,v_1,\ldots,v_r)&\leq K^r\mu(p)\prod_{i=1}^r\nu(v_i), \tag{52}\\ q_r(p,v_1,\ldots,v_r)&\leq K^{r-1}q(p,v_j)\prod_{i\ne j}\nu(v_i)\quad(1\leq j\leq r). \tag{53}\end{align*}\] These follow by multiplying \(q(v_i\mid p)\leq K\nu(v_i)\), keeping one factor unestimated for the second inequality. If \(\lambda\) is the slope-tuple marginal, then the tuples on which \(\lambda/\prod_i\nu(v_i)<L^{-1}\) have \(\lambda\)-mass at most \(L^{-1}\), by summing against the product probability. Outside that set, \[ q_r(p\mid v_1,\ldots,v_r)\leq K^r L\mu(p). \tag{54}\] Thus fixing a typical tuple leaves at least \(M/(K^rL)\) point cells. An event of independent slope probability \(\eta\) has star probability at most \(K^r\eta\). All applications below first trim pairs, discard point fibers of inadequate retention, and replace comparison marginals by the actual retained marginals using Theorem 8. Consequently (51) holds for the normalized retained law. The fixed labels will always be precisely these two or three slope replicas; no cyclic incidence configuration is used.

Here is the scale-selection fact needed twice below. Let \(F_u\), \(0\leq u\leq1\), be a filtration and let \(X_h\) have normalized entropy at most \(Dh\). For an optional fixed conditioning variable \(Z\), put \(g_h(u)=\mathsf I(X_h;F_u\mid Z)\). Nestedness and the chain rule give \[ \int_0^{1-h}\mathsf I(X_h;F_{u+h}\mid Z,F_u)\,du =\int_0^{1-h}(g_h(u+h)-g_h(u))\,du \leq h\mathsf H(X_h\mid Z)\leq Dh^2. \tag{57}\] The middle inequality follows by canceling the two integrals on their common interval. For a sequence \(h_j\) with \(\sum_j\sqrt{h_j}<\infty\), Markov’s inequality and the elementary Borel–Cantelli argument show that, for almost every \(u\), the integrand divided by \(h_j\) is at most \(\sqrt{h_j}\) eventually. The same points may be required to be differentiability points of any fixed finite collection of Lipschitz entropy profiles.

Our anchored observations are nested only to bounded ambiguity. Formula (57) is applied to their limiting profiles. Its exact finite counterpart can instead use the nested dyadic cells of the physical point \((c,A+cV)\): at every fixed depth \(u\) these and \((C_u,[A+C_uV]_u)\) determine each other with a bounded number of possibilities. All entropy differences and conditional mutual informations consequently differ by \(O(1/\log(1/\epsilon))\). This error is taken to zero before \(h\).

The isotropic entropy bounds

For a vector \(Y\), write \([Y]_d\) or \(Y_d\) for its isotropic grid observation at scale \(\epsilon^d\). Assume the system in Proposition 23 exists. Write \[O_u=(C_u,[A+C_uV]_u),\qquad O=O_1,\qquad \alpha=2-2s+\beta.\] The exact particle \(T\) determines \(A,V\). In an anchored observation it leaves only the time label random, so \(\mathsf H(O_u\mid T)=su\). Write \(\tau\) for the root supremum, equivalently its uniformized cap level in the limiting system. The terminal weight is zero. Saturation therefore gives \[ \beta=-\mathsf I(T;O)+2s-\tau=3s-\mathsf H(O)-\tau. \tag{58}\] Since \(\tau\geq0\) and \(\mathsf I(T;O)\geq0\), we have \(\beta\leq2s\) and \[ \mathsf H(O)\leq3s-\beta,\qquad \mathsf H(V_d\mid O)\geq\alpha d\quad(0\leq d\leq1). \tag{59}\] To prove the second inequality, test at duration \(1-d\) with velocity depths \((d,d)\) and position depths \((1,1)\). Denote this observation by \(\widetilde O\). Its weight is \(2d\), and its conditional time entropy is \(s(1-d)\). Universality and (58) imply \[\mathsf I(T;\widetilde O)-\mathsf I(T;O)\geq(2-2s+\beta)d.\] The pair \((O,V_d)\) determines \(\widetilde O\) to bounded ambiguity: transport over a time displacement \(O(\epsilon^{1-d})\) with velocity uncertainty \(O(\epsilon^d)\) has position error \(O(\epsilon)\). The information difference is therefore at most \(\mathsf H(V_d\mid O)\) in the limit. At \(d=1\) the identical inequality follows from the definition of the root supremum at duration zero. Notice that \(\alpha\geq\beta>0\).

Homogenize jointly \(O\), \(V_d\), and their conditioning variables, for the finite lists of depths used below. Formula (59) then means, on the trimmed joint population, \[ \mathbb P(V_d=q\mid O=o)\leq\epsilon^{\alpha d-\eta} \tag{60}\] with arbitrarily small \(\eta>0\) after taking the inner limit. Indeed the conditional probability exponent is the difference of the homogeneous joint and marginal exponents, and that difference is at least \(\alpha d\). The same construction records the time probabilities conditional on the exact particle. It is legitimate on saturated homogeneous parts by the splitting and cap conclusions of Proposition 21.

Concentration of slopes in a thin strip

The profile \(u\mapsto\mathsf H(O_u)\) is nondecreasing and \(3\)-Lipschitz, starts at zero, and has endpoint at most \(3s-\beta\). There is consequently a set of positive measure of regular points \(k\in(0,1)\) with \[ p:=\partial_k\mathsf H(O_k)<3s-\beta/2. \tag{61}\] Choose such a point satisfying (57) with \(X_h=V_h\), whose entropy is at most \(2h\). Along a sequence \(h\downarrow0\), \[ \mathsf H(O_{k+h}\mid O_k)=(p+o(1))h,\qquad \mathsf I(V_h;O_{k+h}\mid O_k)=o(h). \tag{62}\]

Inside a typical \(O_k\)-cell translate time and position to the cell center and dilate both by \(\epsilon^{-k}\). Put \(\rho=\epsilon^h\), and call the child point observation \(P\). Bounded changes from anchored to physical point cells do not affect any exponent. Homogeneity, the conditional fiber rule, and Lemma 6, followed by the actual-marginal conclusion of Theorem 8, give a normalized retained pair law \(q(P,V_h)\) satisfying the following bounds, with \(e=e(h)\to0\): \[\begin{align*} \#\mathop{\mathrm{supp}}P&\leq\rho^{-(p+e)},& \mu(P=z)&\leq\rho^{p-e},& q&\leq\rho^{-e}\mu\otimes\nu. \tag{63}\end{align*}\] These conclusions hold outside base cells of probability tending to zero. One may enlarge \(e(h)\) to absorb all fixed numerical constants and all \(o(1)\) errors in (62). To be explicit about the conditioning: the mean information in block units tends to zero; discard base cells where it exceeds its square root, apply the likelihood-ratio truncation in each remaining cell, and discard point fibers losing more than half their mass. Homogeneous joint and marginal bounds then supply the two point estimates in (63). This procedure introduces only another quantity tending to zero into \(e\).

We also trim on the exact particle fibers before sampling replicas. A retained time child, conditional on \(T,C_k\) in the original base law, has probability at most \(\rho^{s-e}\). Lift all pair restrictions to that law, and discard particle fibers whose conditional retention there is below \(\rho^e\). If the pair restrictions have total retention \(m_0\), these discarded fibers have normalized retained mass at most \(\rho^e/m_0\), by (11). Here \(m_0\) is bounded below before this step; more generally a subpower lower bound is absorbed by increasing \(e\). Each remaining particle has at least \(\rho^{-s+2e}\) distinct retained time children, by dividing its retained mass by the original child mass bound. Given \(T,C_k\), the anchored base observation is fixed. Thus this is a statement in the very same base cell used for \(P\); no particle entropy bound is involved. For a slope bin of size \(\rho\), all these points have intercept within \(O(\rho)\) of the particle’s intercept. Lemma 24 gives \[ \#\pi_v(\text{retained neighbors of }v) \leq C\rho^{-(p-s+3e)}. \tag{64}\] Apply one final simultaneous marginal core to this pruned law. Its retention is at least one half, by Lemma 7, and the pair comparison again uses its actual point and slope marginals. Increasing \(e\) absorbs the finite factors, so (63) and (64) hold together for this final pair law. The core need not preserve the rich particle fibers: the intercept support bound was established before it, and survives by support inclusion. Use this same order of operations in the two-replica application below.

Fix \(\vartheta>0\) so small that \(12\vartheta<\beta/8\). We claim that, in every retained base cell, some strip of thickness \(C\rho^\vartheta\) has slope mass at least \(\rho^{5e}\), after enlarging \(e(h)\to0\) once more. If all such strips had mass less than \(\rho^{5e}\), two independent slopes would have distance at least \(\rho^\vartheta\) except with probability \(C\rho^{5e}\); a ball of that radius is contained in a strip of comparable thickness. Conditional on the first two, the third would lie at distance at least \(\rho^\vartheta\) from their line except with the same probability. Hence, outside an independent event of probability \(C\rho^{5e}\), \[\abs{\det((1,v_1),(1,v_2),(1,v_3))}\geq\rho^{2\vartheta}.\] For the three-replica star, (52) bounds the exceptional probability by \(C\rho^{2e}\). Choose \(L=\rho^{-e}\) in (54). A good tuple then leaves a point population of at least \(\rho^{-p+5e}\) cells, and each projection has the support bound (64). Lemma 25 implies \[2p-10e\leq3(p-s+3e)+12\vartheta+o_{\rho\to0}(1), \qquad\text{hence}\qquad p\geq3s-12\vartheta-19e-o_{\rho\to0}(1).\] This contradicts (61) for all sufficiently small \(h\), with \(\rho\) taken sufficiently small afterwards. We take \(e>0\) even when an error vanishes, so that the exceptional probabilities above tend to zero in this inner limit.

Choose one such strip measurably in each base cell, using a finite grid menu with bounded enlargement, and restrict incidences to it. The total retention is at least \(\rho^{5e}\) times the mass of the retained base cells. No strip index is fixed globally: its index is a function of \(O_k\) and will be included in the earlier test through that base label. With \[ l=\vartheta h, \qquad \abs{V\cdot n(O_k)-w(O_k)}\leq C\epsilon^l, \tag{65}\] round the unit normal and offset at precision \(\epsilon^l\). This changes \(C\) only. The Radon–Nikodym bound for the restriction is \(\epsilon^{-O(eh)}\). Since \(\vartheta\) is fixed and \(e(h)\to0\), its normalized logarithmic cost is \(o(l)\).

The bins for \(O\), the required \(V_d\), and the particle-conditional time probabilities are trimmed before this restriction. Lemma 5 and its conditional retention rule therefore preserve their exponents with errors \(o(l)\), including the particle-conditional time exponents and (60). The absolute root cap bound gives \(\tau_{\rm new}\leq\tau+o(l)\). The terminal numerator before subtraction of \(\tau\) changes by \(o(l)\), because its observation and conditional time exponents were recorded. Consequently the new terminal score is at least \(\beta-o(l)\), and universality bounds it above by \(\beta+o(l)\). We henceforth use this retained law and its own root supremum; its terminal score is \(\beta+o(l)\).

An earlier anisotropic observation

Take \(h\) small enough that \(k<1-l\), and put \(a=1-l\). In the frame with normal \(n=n(O_k)\) and tangent \(n^\perp\), form \[ B=\bigl(O_k,O_a,[(A+C_aV)\cdot n]_1\bigr). \tag{66}\] The coarse observation \(O_a\) locates tangent position to depth \(a\); its normal position is additionally located to depth \(1\). The strip supplies normal velocity to depth \(l\), and tangent velocity is bounded. The rounded frame is known to depth \(l\). Thus \(B\), with boundedly many additional data cells if necessary, is an admissible aperture with velocity depths \((0,l)\), duration \(a\), horizon \(1\), and weight \((1-\gamma)l\).

It is determined by \(O\) to bounded ambiguity. Indeed \(O\) coarsens to \(O_k,O_a\) to bounded ambiguity, hence determines the chosen strip up to the same fixed number of choices. For its fine normal position use \[ (A+C_aV)\cdot n=(A+C_1V)\cdot n+(C_a-C_1)w+O(\epsilon), \tag{67}\] where the error is bounded by \(C\epsilon^a\epsilon^l=C\epsilon\). Conversely every component of the anchored \(B\) is fixed by \((T,C_a)\), since \(C_a\) fixes \(C_k\). In particular, \[ \mathsf H(O\mid T,B)=sl+o(l). \tag{68}\] The conditional time profile on the retained law follows from the ratios of the recorded probabilities at depths \(a\) and \(1\); the strip restriction costs \(o(l)\) on typical particle fibers. Adjoining standard data cells to \(B\) uses deterministic functions of \((T,C_a)\) and leaves this argument unchanged.

We prove the information saving \[ \mathsf I(T;O)-\mathsf I(T;B)\geq sl-o(l). \tag{69}\] The main point is a lower bound for the last block’s joint time and tangent position entropy.

Slope bounds inside the last block

For \(0\leq u\leq1\) let \(F_u\) record, in addition to \(B\), time to depth \(a+ul\) and tangent position at that time to the same depth. Within a \(B\)-cell translate time and tangent position to their depth-\(a\) centers and dilate both by \(\epsilon^{-a}\). The resulting coordinates are bounded, the new mesh parameter is \(\rho_*=\epsilon^l\), and the tangent slope \(X=V\cdot n^\perp\) remains bounded. The filtration \(F_u\) has capacity \(2\) per unit depth. Its base \(F_0\) adds at most bounded information to \(B\), and \(F_1\) is determined by \((O,B)\) to bounded ambiguity. Given \(T,B\), its conditional entropy profile is \(su+o(1)\) in block units, because its new data are fixed by the particle and the time cell.

For each fixed \(d\in(0,1]\) and each fixed filtration depth \(u\), the retained conditional tangent slope law satisfies \[ \mathbb P\bigl(X\in J\mid B,F_u\bigr) \leq \rho_*^{\alpha d-o(1)} \quad\text{for intervals }J\text{ of length }\rho_*^d. \tag{70}\] Here, as throughout, the inequality is uniform on a further trimmed subprobability, followed by normalization on fibers of subpower retention. We verify this transfer, since a bare conditional entropy inequality would not suffice.

First apply (60) at velocity depth \(dl\) on the jointly trimmed law. Adjoin the finite list of labels \((B,F_u)\) to \(O\). Each such list has at most \(K\) possible values given \(O\), where \(K\) is fixed when the list is fixed. For a positive tolerance \(\zeta\), discard realized values whose conditional likelihood given \(O\) is below \(\rho_*^\zeta\). Their total probability is at most \(K\rho_*^\zeta\). On the remaining values, conditioning on the adjoined label multiplies a conditional velocity-bin probability by at most \(\rho_*^{-\zeta}\). More explicitly, write \(\kappa_h=O(e(h)/\vartheta)\) for an upper bound on all prior restriction costs in last-block units. Whenever a retained subprobability has mass at least \(\rho_*^{\kappa_h}\), discard conditioning fibers whose retention is below \(\rho_*^{\kappa_h+\zeta}\). Formula (11) bounds their normalized retained mass by \(\rho_*^\zeta\); normalization costs at most \(\kappa_h+\zeta\) in the conditional exponent. For the fixed finite number of such operations, the total extra loss is \(C\kappa_h+C\zeta\). It tends to zero by first taking the inner limit, then \(h\downarrow0\) and \(\zeta\downarrow0\). It incurs no factor involving the number of \(O\)-cells or \(B\)-cells.

For a fixed \(B\)-cell, its normal strip has thickness \(O(\epsilon^l)\). Its intersection with a tangent interval of length \(\epsilon^{dl}\) meets only \(C\) velocity grid cells of depth \(dl\), since \(d\leq1\). Thus the same uniform upper bound holds for this tangent interval conditional on \(O,B,F_u\). Averaging over compatible \(O\)-labels gives (70), with the explicit extra exponent loss \(\zeta\) and the retention losses. Taking those losses to zero proves the asserted form. Every bound is imposed on the retained joint population before averaging; no exponentially small bound for untrimmed exceptional probabilities is needed.

We now make the order of limits precise. Fix \(\vartheta\) first. Choose the regular sequence \(h\downarrow0\) above. For each fixed \(h\) record the finite lists of \(O,V_d\), and particle-conditional time bins before the strip choice. Construct \(B,F_u\) and perform the transfer just proved after choosing the strip. The later increment binnings are performed after this transfer. Take \(\epsilon\) small enough that all inner mesh and bin errors, divided by \(l\), tend to zero. A diagonal, with \(h\downarrow0\), gives last-block systems with parameter \(\rho_*\), exact limiting time profile \(su\), and (70) at a countable dense list of \((u,d)\). Bounded coordinate capacities extend the entropy profiles to other depths. For the conditional mass bound at a subsequently chosen fixed filtration depth \(u\), repeat the direct \(O\to(B,F_u)\) determination and likelihood truncation above; it has the same bounded multiplicity. A requested slope depth \(d\) is obtained by using recorded depths \(d'<d\) and then letting \(d'\uparrow d\). Thus the mass bound is inherited from that comparison, independently of continuity of entropy. A later increment length \(r>0\) is held fixed before each inner limit; only then is \(r\downarrow0\). Thus errors \(o(l)\) from the first construction are never divided by an unchosen infinitesimal \(r\). No bound on \(\mathsf H(B)/l\) is asserted or needed.

Two-slope expansion and the score contradiction

We show, in the last-block limits just constructed, that \[ \mathsf H(F_1\mid B)\geq2s \quad\text{in units }\log(1/\rho_*). \tag{71}\] Write \(f(u)=\mathsf H(F_u\mid B)\) in these units. It is nondecreasing and \(2\)-Lipschitz, with \(f(0)=0\). Apply (57) to \(X_r\), the tangent slope prefix of depth \(r\), conditional on \(B\). Its entropy is at most \(r\), so the integrated mutual information is at most \(r^2\). At almost every regular interior \(u\) there is consequently a sequence \(r\downarrow0\) for which the increment has negligible information about \(X_r\), in units of its own length.

In a typical \((B,F_u)\)-cell let \(P_1\) be the time–tangent-position increment, at mesh \(\rho=\rho_*^r\). Homogenize its probabilities and the pair with \(X_r\). Let \(p_1\) denote its point exponent. The likelihood-ratio and actual-marginal steps used in (63) give, with \(e\to0\) along the regular sequence, \[\#\mathop{\mathrm{supp}}P_1\leq\rho^{-(p_1+e)},\qquad \mu_1(z)\leq\rho^{p_1-e},\qquad q_1\leq\rho^{-e}\mu_1\otimes\nu_1.\] The particle-conditional time exponent is \(s\). Trimming exact particle fibers and applying Lemma 24 therefore bounds every retained intercept support by \(C\rho^{-(p_1-s+3e)}\).

Fix \(0<a_0<1\). Formula (70), at depth \(a_0r\), gives the actual retained slope marginal the interval bound \(\nu_1(J)\leq\rho^{\alpha a_0-o(1)}\) for intervals of length \(\rho^{a_0}\). The two-replica star consequently has \(\abs{v_1-v_2}\geq\rho^{a_0}\) outside a set of probability \(O(\rho^{\alpha a_0-2e-o(1)})\). Since \(\alpha>0\), this probability tends to zero by taking \(r\) sufficiently far along the regular sequence and then the inner small-parameter limit. Pin a good tuple using (54) with \(L=\rho^{-e}\). At least \(\rho^{-p_1+4e}\) point cells remain. Their original intercept support bounds are still valid. Formula (56) now gives \[p_1-4e\leq2(p_1-s+3e)+2a_0+o(1), \qquad p_1\geq2s-2a_0-10e-o(1).\] First let the regular-increment errors vanish, and then let \(a_0\downarrow0\). This proves the lower entropy rate \(p_1\geq2s\).

Here is why the cellwise argument yields \(f'(u)\geq2s\) almost everywhere, even if new homogeneous parts have different point exponents. Their conditional increment entropies average to \(f(u+r)-f(u)\), and all have capacity at most \(2r+o(r)\). If \(f'(u)<2s\) by a fixed amount, a positive fraction of the conditional population must have point exponents below \(2s\) by a smaller fixed amount. The mean conditional information is \(o(r)\), so Markov’s inequality removes a vanishing fraction while imposing the pair comparison above. The uniform time and slope bounds survive on the remaining typical fibers. Homogenizing a finite list costs a subpower, and thus selects a part with all these properties and the same fixed deficit. The preceding statement about inherited bounds can also be checked directly: in a fixed conditional cell, restriction to a part \(E\) of mass \(m\) gives \(\nu_E(J)\leq m^{-1}\nu(J)\). Here \(m=\rho^{o(1)}\), so the slope exponent is preserved. Discard exact particle fibers retaining less than \(m\rho^\zeta\) of their original conditional time law; their normalized retained mass is at most \(\rho^\zeta\). The time exponent therefore loses at most \(\log_{1/\rho}(1/m)+\zeta=o(1)+\zeta\). These estimates justify the simultaneous selection without any count of conditioning cells. The two-slope inequality contradicts the fixed deficit. Since \(f\) is Lipschitz, integrating \(f'(u)\geq2s\) proves (71).

Returning to the original units, (71) and the determination of \(F_1\) by \((O,B)\) give \[\mathsf H(O\mid B)\geq2sl-o(l).\] Together with (68), this gives \(\mathsf I(T;O\mid B)\geq sl-o(l)\). Since \(\mathsf H(B\mid O)=o(l)\), the information chain rule proves (69).

Finally let \(Q_G\) denote the undivided score numerator, including subtraction of the root supremum of the retained law. The same root supremum occurs for \(B\) and \(O\). Their time entropies differ by \(sl+o(l)\), so \[\begin{align*} Q_B-Q_O &=(1-\gamma)l+\mathsf I(T;O)-\mathsf I(T;B)-2sl+o(l)\\ &\geq(1-\gamma-s)l-o(l). \end{align*}\] On the other hand terminal saturation and the universal duration bound give \(Q_O=\beta+o(l)\) and \(Q_B\leq\beta(1-l)+o(l)\). Divide by \(l\) and pass through the specified limits to obtain \[1-\gamma-s\leq-\beta, \qquad s\geq1+\beta-\gamma>1.\] This contradicts the scalar time capacity \(s\leq1\) and proves Proposition 23.

The information profile and the contact relation

Work in one bounded slope chart, using the row frame from the end of Proposition 21. The particle \(T\) carries \((V,A)\). At label time \(v\) put \(E_v=(C_v,R_{mv})\) and set \[ X=V_1,\qquad Y=A_1+C_vX,\qquad W=V_2-R_{mv}X,\qquad Z=A_2+C_vV_2-R_{mv}Y. \tag{72}\] For data depths \(\mathbf L=(L_X,L_Y,L_W,L_Z)\) let \(O(\mathbf L,v)\) record \(E_v\) and these four variables at their respective grid precisions. We allow the depths to vary on compact subsets of \[ \begin{aligned} L_Y-L_X&\le v,& L_W-L_X&\le mv,\\ L_Z-L_W&\le v,& L_Z-L_Y&\le mv, \qquad 0<v<1. \end{aligned} \tag{73}\] They may range over any fixed bounded extension of the depths of the apertures; a negative depth gives only boundedly many bins.

When \(m=1\), impose \(L_Y=L_W=L_U\) at this point and regard \(U=(Y,W)\) as one datum of capacity two, with \(\mathbf L=(L_X,L_U,L_Z)\). The profile and all the trace statements below are then defined on this grouped domain, and the middle datum has one joint trace rate.

The finite matched apertures occupy a particular plane in the depth domain. At time \(u\), an aperture with common depth \(i\) has velocity depths \((i,i+mu)\) and position depths \((i+u,i+(1+m)u)\). Define \[ \begin{aligned} \Pi_m(i,u)&=(i,i+u,i+mu,i+(1+m)u,u), &&m\ne1,\\ \Pi_1(i,u)&=(i,i+u,i+2u,u), &&m=1. \end{aligned} \tag{74}\] The coordinate orders are \((L_X,L_Y,L_W,L_Z,v)\) and \((L_X,L_U,L_Z,v)\), respectively. The finite matched observation is \(O(\mathbf L,u)\) with \((\mathbf L,u)=\Pi_m(i,u)\). Every inequality in (73) is an equality on this plane. The part corresponding to admissible apertures is \[ \mathcal A_m=\{(i,u):0<u<1,\quad i\ge0,\quad i+(1+m)u\le L\}. \tag{75}\] The path observations are the points \(i=b(u)\), up to the zero-cost grid choices already established for the common row frame. The bounded extension of the data depths gives a neighborhood within this plane across the boundary of \(\mathcal A_m\). The full depth domain still has the one-sided faces in (73).

We now make one homogeneous choice before defining the limiting functions. Start with a countable dense family of admissible depth configurations and the countable family of all finite tuples formed from their observations, including the versions conditional on \(T\). Also include the score observations at dyadic path times. At each stage use only a finite initial list and finitely many path times. Apply the cap regularization of Lemma 17, then the uniform bins of Lemma 5. By Lemma 16, select typical components whose finitely many score deficits lie under one deterministic vanishing envelope. Let the lists grow and the bin meshes shrink slowly with the precision. The diagonal of Lemma 10, as in Remark 22, gives one sequence of finite laws \(p_\epsilon\) carrying all the listed homogeneous profiles. Every listed path observation is saturated along this sequence in the limit. Only after this sequence is chosen do we define \[f_\epsilon(\mathbf L,v)=\mathsf I^{p_\epsilon}(T;O(\mathbf L,v)),\qquad S_\epsilon(v)=\mathsf H^{p_\epsilon}(C_v\mid T).\] Lemma 10 and the box comparisons below give a limiting profile \(f\) on (73); write \(S\) for the limiting time profile. The finite root suprema are bounded, so pass to a further subsequence and write \(\tau\) for the limit of \(\tau_L(T)\). The limiting score along the path is continuous: \(b,S\) are Lipschitz, and the box comparisons give continuity of \(f\) along \(\Pi_m(b(u),u)\). Saturation at the dense dyadic times therefore extends to every \(0<u<1\). The positive constant in (50) is obtained for this selected component by the least-aspect argument; it need not be the constant before the split. The symbols \(f,S,\tau\) below refer to this component, whose profiles may differ from those before the split. Later, a trace point and a positive block length will be chosen for this fixed \(f\) before the required real-depth bins and finite precision are selected.

The information of a matched aperture is therefore the restriction \[ \Phi(i,u)=f\bigl(\Pi_m(i,u)\bigr). \tag{76}\] The weight of the aperture at \((i,u)\) is \(2i+(1-\gamma)mu\). Rearranging the universal score bound and using saturation on the path gives \[ \Phi(i,u)\ge 2i+(1-\gamma)mu+2S(u)-\beta u-\tau, \qquad\text{with equality when }i=b(u). \tag{77}\] We call this equality the contact relation. The local projection arguments will constrain the information rates at almost every plane point with \(i>0\), while saturation is known on this particular contact curve. We therefore need both local affine approximations on the plane and control of how the rates cross the curve.

At a differentiability point of \(b\) put \(p=-b'(u)\). A direction \(V_c=\partial_u-c\partial_i\) preserves the data depth \(i+cu\), and its crossing factor at the contact curve is \(p-c\). Here the clocks are \(0,1,m,1+m\), with the two middle clocks grouped when \(m=1\). Figure 2 shows these relations.

Ambient almost-everywhere differentiability does not solve the local problem: the entire matched plane lies on the boundary of (73), and could lie in the ambient exceptional set. We will obtain the needed traces from the signs of the mixed derivatives. Their affine approximation will produce finite entropy blocks for Section 7; their measure identities will later pass those rate constraints to the contact curve.

Here are the properties of \(f\) supplied by the finite observations. It is locally Lipschitz and nondecreasing in every coordinate; its data derivatives lie between zero and the capacity of that coordinate. Its label derivative is nonnegative and locally bounded. Every mixed derivative involving two different coordinates, including a data coordinate and \(v\), is a nonpositive distribution in the interior. We verify the signs, especially the one involving the label.

Advance \(v\) to \(v'>v\). Write \(a=C_{v'}-C_v\) and \(r=R_{mv}-R_{mv'}\). Nested labels give \(|a|\lesssim\epsilon^v\), \(|r|\lesssim\epsilon^{mv}\), and the change of data is \[ X'=X,\quad Y'=Y+aX,\quad W'=W+rX,\quad Z'=Z+aW+rY+arX. \tag{78}\] In a data box, the uncertainty in \(Y'\) is bounded by a constant times \(\epsilon^{L_Y}+\epsilon^{v+L_X}\), and that in \(W'\) by a constant times \(\epsilon^{L_W}+\epsilon^{mv+L_X}\). The uncertainty in \(Z'\) is bounded by \[C\bigl(\epsilon^{L_Z}+\epsilon^{v+L_W} +\epsilon^{mv+L_Y}+\epsilon^{(1+m)v+L_X}\bigr).\] The four inequalities in (73) bound these by the respective target widths. The inverse shear has the same bounds. Consequently, conditional on \(E_{v'}\), the old and new data descriptions at fixed admissible depths determine each other with bounded ambiguity. This assertion also holds at each corner of any coordinate rectangle contained in the domain.

At fixed labels, a data refinement is determined by the particle and labels. Its increase in particle information is therefore exactly its conditional entropy, up to the bounded bin ambiguities. A further data refinement can only decrease this conditional entropy. For a label refinement, express both ends of the data refinement in the old frame conditional on \(E_{v'}\), using (78). The two information increments being compared are then \[\mathsf H(D_{\rm fine}\mid E_v,D_{\rm coarse}) \quad\hbox{and}\quad \mathsf H(D_{\rm fine}\mid E_{v'},D_{\rm coarse}),\] up to \(O(1/\log(1/\epsilon))\); the latter is no larger than the former. Here \(D_{\rm coarse}\) includes every other data coordinate. Thus each mixed rectangle difference of \(f_\epsilon\) is nonpositive up to an error vanishing as \(\epsilon\downarrow0\). Passing to the limiting profile proves the mixed signs, first for rectangle differences and then for distributions, by integrating the rectangle inequalities against nonnegative smooth test functions and shrinking the rectangles.

Refining a scalar datum by \(d\) costs at most \(d+O(1/\log(1/\epsilon))\) in normalized entropy; for \(U\) the bound is \(2d+O(1/\log(1/\epsilon))\). Advancing the labels by \(d\) costs at most \((1+m)d+O(1/\log(1/\epsilon))\). The old data are recoverable from the new observation, and the change of data conditional on the new labels has bounded ambiguity. These facts give monotonicity and the asserted Lipschitz bounds. For two nearby points on flat faces, move first to a common slightly finer label so that coordinatewise comparison is admissible. This proves the same bounds up to those faces. The argument uses vector conditional entropy unchanged when \(Y,W\) are grouped.

These comparisons also upgrade the dense diagonal to locally uniform convergence. Fix a compact subset \(K\) of (73) and a compact relative neighborhood \(K^+\) of it. Uniformly for nearby \(z,z'\in K^+\), the box bounds give \[|f_\epsilon(z)-f_\epsilon(z')| \le C_K\|z-z'\|_1+O_K(1/\log(1/\epsilon)).\] The same estimate holds at the faces: the common finer label needed there advances time by \(O_K(\|z-z'\|_1)\), and stays in the domain for nearby points because \(K^+\) has label times bounded away from the endpoints. For a fixed sufficiently small \(\delta>0\), choose a finite \(\delta\)-net \(z_1,\ldots,z_N\) for \(K\) from the dense family in \(K^+\). The same Lipschitz bound for \(f\) then gives \[\sup_{z\in K}|f_\epsilon(z)-f(z)| \le \max_{1\le j\le N}|f_\epsilon(z_j)-f(z_j)| +2C_K\delta+O_K(1/\log(1/\epsilon)).\] First let \(\epsilon\to0\) along the selected sequence, using convergence at these finitely many net points, and then let \(\delta\downarrow0\). Thus \(f_\epsilon\to f\) locally uniformly on the full closed-sided domain, including its faces.

An abstract trace theorem

We now isolate the consequence of these mixed signs for a plane whose data depths move at distinct speeds. The grouped middle datum at \(m=1\) is what makes the speeds distinct in that case.

Let \(n\ge2\), let \(c_1<\cdots<c_n\) be distinct real numbers, and let \(\mathcal E\) be a nonempty set of ordered pairs \((j,k)\) with \(j<k\). For an open interval \(I\) define the closed-sided region \[\mathcal D=\{(\mathbf L,v)\in\mathbb R^n\times I: L_k-L_j\le(c_k-c_j)v\text{ for }(j,k)\in\mathcal E\}.\] Its interior is obtained by making these inequalities strict. Define \[ \begin{split} \Pi(i,u)&=(i+c_1u,\ldots,i+c_nu,u),\\ \mathcal K&=\{\xi\in\mathbb R^{n+1}: \xi_k-\xi_j\le(c_k-c_j)\xi_v \text{ for }(j,k)\in\mathcal E\},\\ V_j&=\partial_u-c_j\partial_i. \end{split} \tag{79}\] The cone \(\mathcal K\) has nonempty interior and is invariant under translation by either tangent vector \(e=(1,\ldots,1,0)\) or \(t=(c_1,\ldots,c_n,1)\). In particular, for \(h\ge0\), \(\Pi(i,u)+h\xi\) belongs to \(\mathcal D\) for \(\xi\in\mathcal K\), provided its label time belongs to \(I\).

Finite approximation after selecting the block scale

The next consequence specifies exactly what is required of the finite entropy systems. Its exceptional set is a set of base points on the plane; probabilistic trimming of conditional cells is a separate use of Lemma 5 and Theorem 8.

There is no uniform choice of \(h\) for all Lipschitz profiles in this corollary. The translation moduli in (80), and hence the block scale, depend on the limiting profile. At a fixed nonexceptional plane point one may equivalently use its sequence in (85), choose a block from that sequence, and then make the finite-system errors small compared with that block.

For the profile (73), the clocks are \(c_X=0,c_Y=1,c_W=m,c_Z=1+m\), ordered increasingly when applying the theorem. Write the trace fields as \(x,y,w,z,D\). At \(m=1\) the clocks are \(0,1,2\), the fields are \(x,q,z,D\), and \(q=a_U\); notation such as \(y+w\) means \(q\) and asserts no separate traces for \(Y,W\). Thus (83) reads \[ (\partial_u-(1+m)\partial_i)z\ge\partial_iD. \tag{92}\] The distinction between grouped and separate depths is necessary: \(f(L_Y,L_W)=\max(L_Y,L_W)\) has nonpositive mixed derivatives, but its separate first derivatives have different traces from the two sides of \(L_Y=L_W\). On the grouped domain it is simply \(f(L_U)=L_U\).

Traces on a contact curve

We need a second trace statement within the plane. It uses bounded variation in one direction at a time; it does not require the entire gradient of each rate field to be a measure.

The selected profiles and the parameter children

We first verify what the already selected sequence \(p_\epsilon\) supplies at the depths needed here. Its initial homogeneous list contains all finite tuples from a countable dense family of admissible configurations, including the indicated conditional probabilities given \(T\). For two nearby configurations, give both descriptions the finer label clock. The box comparisons in Section 6 give bounded ambiguity for changing the data frame at a fixed admissible resolution. A depth change of size \(a\) introduces at most \(\epsilon^{-Ca+o(1)}\) possibilities, also conditional on \(T\), on each fixed compact depth range. Applying the conditional-surprise counting bound from Lemma 10 to both descriptions extends the homogeneous exponents from the dense family to every fixed admissible configuration. The same argument applies to finite tuples. It does not require independent data refinements to remain admissible at the coarse label clock.

Write \(\mathfrak h(A)\) for the limiting normalized entropy of a listed observation. For a finite pair \(A,B\), the selected homogeneous probabilities give, in probability, \[ -\log_{1/\epsilon}\!\left(p_\epsilon(B\mid A)\right) =\mathfrak h(A,B)-\mathfrak h(A)+o(1). \tag{96}\] The same statement holds for listed probabilities conditional on \(T\); the probability of \(T\) itself is never binned. After a trace point, a positive block length \(h\), and a finite list of real configurations have been chosen, take one event \(\mathcal E_\epsilon\) on which all their original conditional surprises differ from their limiting exponents by at most \(\omega_\epsilon=o(h)\). Its probability tends to one. On this event, ratios of the joint and parent masses give uniform upper conditional masses and upper retained support counts. The support count follows by dividing the upper mass of a parent by the lower mass of each retained joint value.

This finite real-depth event is imposed before any restriction which uses its bounds. If a later law satisfies \(\mathsf Q\le Kp_\epsilon(\,\cdot\mid\mathcal E_\epsilon)\), with \(\log K=o(h\ell)\) and \(\ell=\log(1/\epsilon)\), its conditional entropy of each listed child given its listed parent differs from the listed exponent by at most \(\omega_\epsilon+\log(K/p_\epsilon(\mathcal E_\epsilon))/\ell\), by conditional relative entropy and the support count. The parent likelihood and child-support thresholds in the proof of Lemma 5 give the same absolute error bound for realized surprise, with \(\log(1/\eta)/\ell\) added, outside \(\mathsf Q\)-mass at most \(2\eta\); choose \(\eta\downarrow0\) with \(\log(1/\eta)=o(h\ell)\). The dense-list profiles remain the same under this high-probability restriction by Lemma 10. This verifies the finite inheritance for the component selected in Section 6; it does not select a new \(f\) or assert uniform bins at all real depths.

For this component the prediction identities and the positive constant obtained from its least-aspect argument are \[ \mathsf H(R_{mv}\mid T,C_v)=0,\qquad \mathsf H(R_{m(v+a)}\mid T,C_v)\ge\kappa a \quad(0<v<v+a<1). \tag{97}\] These are limiting normalized statements for this same \(p_\epsilon\).

Fix a regular plane point \((i,u)\) and a trace block length \(h>0\). Write \(E\) for the coarse observation at \(\Pi_m(i,u)\) and put \(\rho=\epsilon^h\). Given the coarse label, \(E\) is a function of \((T,C_u,R_{mu})\) up to bounded grid choices, which we record when needed. Choose \(K>\max(1,1/m)\), fixed throughout this construction, and define the normalized parameter increments \[c=\epsilon^{-u}(C_{u+Kh}-C_u),\qquad r=-\epsilon^{-mu}(R_{m(u+Kh)}-R_{mu}).\] The full finite label \(\theta\) records the advanced label and the bounded choices just mentioned. Before all tests, fix one half-open dyadic tree on a fixed bounded box containing the normalized pair \((c,r)\). For every real \(0\le d\le1\), let \(\Theta_d\) be its atom at level \[n_\rho(d)=\left\lceil d\log_2(1/\rho)\right\rceil.\] These atoms are exactly nested for all real depths: \(\Theta_a\) is determined by \(\Theta_b\) when \(a\le b\), and each \(\Theta_a\) atom has at most \[4^{n_\rho(b)-n_\rho(a)}\le4\rho^{-2(b-a)}\] \(\Theta_b\) children. The time prefix records \(C_{u+dh}\) and the normal prefix records \(R_{mu+dh}\) up to bounded ambiguity over \(E\). For other finite observations whose nesting holds only up to bounded choices, we append the parent to the child when needed. Here \(h\) and the finite prefix list are fixed before \(\epsilon\to0\).

The chain rule gives \[\begin{align*} \mathsf H(C_{u+dh}\mid E) &\ge\mathsf H(C_{u+dh}\mid T,E)\\ &\ge S(u+dh)-S(u)-\mathsf H(R_{mu}\mid T,C_u)-o(1), \tag{98}\\ \mathsf H(R_{mu+dh}\mid E) &\ge\mathsf H(R_{mu+dh}\mid T,C_u,R_{mu})-o(1)\\ &\ge(\kappa/m)dh-o(1). \tag{99}\end{align*}\] For the first inequality, the data in \(E\) are determined by the particle and coarse labels, and adjoining \(R_{mu}\) costs at most its conditional entropy. The second uses (97) at time increment \(dh/m\). Prediction also gives \[\kappa a\le\mathsf H(R_{m(u+a)}\mid T,C_u) \le S(u+a)-S(u)+o(1).\] Consequently \(s=S'(u)\ge\kappa>0\) at the regular points used below. Fix any \(0<\sigma<\min(1,\kappa/m)\).

At tied clocks there is also a bound between any two tested parameter prefixes. If \(m=1\) and \(0\le a<b\le1\), then conditional on \(E\) the symbol \(\Theta_a\) is equivalent to \((C_{u+ah},R_{u+ah})\) up to the recorded bounded choices. Therefore \[\begin{align*} \mathsf H(\Theta_b\mid E,\Theta_a) &\ge\mathsf H(C_{u+bh}\mid T,E,\Theta_a)\\ &\ge S(u+bh)-S(u+ah) -\mathsf H(R_{u+ah}\mid T,C_{u+ah})-o(1)\\ &=S(u+bh)-S(u+ah)-o(1). \tag{100}\end{align*}\] After \(T,C_{u+ah}\) have been given, the extra conditioning in \(E,\Theta_a\) is a function of \(R_{u+ah}\) and the bounded choices. This is the stated entropy cost and does not use an entropy bound for \(T\).

The homogeneous joint symbols \((E,\Theta_a)\) and \((E,\Theta_b)\) turn this last entropy inequality into a bound on their probability ratio. On the common real-depth event, \[ p_\epsilon(\Theta_b\mid E,\Theta_a) \le\epsilon^{S(u+bh)-S(u+ah)-o(1)} \quad(m=1) \tag{101}\] at every retained child. The time and normal inequalities give the corresponding marginal prefix bounds by the same ratio calculation. Lemma 3, with the coarse observation as \(C\) and each listed parent containing \(C\), selects base cells of adequate relative retention without an inverse original cell mass. We will keep the event as an unnormalized allowed set inside the chosen base cell until both product comparisons have been imposed.

One finite affine-block law

Center the common pre-label data in the boxes of \(E\), dilate them to bounded coordinates, and round the resulting point \(P\) at precision \(\rho\). For a full label with normalized increment \((c,r)\) define its canonical post-coordinate symbols by applying the shear to this same rounded point and then taking the indicated grid bins: \[ X_\theta=X,\qquad Y_\theta=Y+cX,\qquad W_\theta=W+rX,\qquad Z_\theta=Z+cW+rY+crX. \tag{102}\] The same origin is transformed at every label, so the shears compose by adding \((c,r)\) and the map from frame \(0\) to frame \(1\) uses \(\theta_1-\theta_0\). Applying the shear before or after rounding changes each real output by \(O(\rho)\). The canonical bins and the physically recorded bins consequently determine one another with bounded conditional ambiguity. We use the canonical symbols in the finite law below: each is a genuine function of \((P,\theta)\). Their prefixes and joint numerator and denominator symbols are included in the finite uniformization list. Bounded ambiguity transfers the entropy identities to these symbols; their own uniform probability bins, rather than a pointwise assertion from that ambiguity alone, will supply the retained mass bounds.

Let \(\mathcal I=\{X,Y,W,Z\}\) when \(m\ne1\), and \(\mathcal I=\{X,U,Z\}\) when \(m=1\), with \(U=(Y,W)\). Write \(Q_{j,\theta}\) for the canonical coordinate in group \(j\in\mathcal I\), and \((Q_{j,\theta})_d\) for its prefix at precision \(\rho^d\). Put \((a_X,a_Y,a_W,a_Z)=(x,y,w,z)\) in the distinct-clock case and \((a_X,a_U,a_Z)=(x,q,z)\) in the tied case. Choose the fixed \(K\) large enough that raising any individual data depth by at most \(h\) is admissible after the label advances by \(Kh\). For any fixed finite list of \(t_j\in[0,1]\), the affine trace gives \[ \mathsf H\bigl(((Q_{j,\theta})_{t_j})_{j\in\mathcal I} \mid E,\theta\bigr) =h\sum_{j\in\mathcal I} a_jt_j+o(h). \tag{103}\] Indeed, at the advanced label the unrefined pre- and post-data determine one another to bounded ambiguity by (73). The difference between the information with the selected refinements and this common base information is their conditional entropy, because the particle and label determine the refinement. The affine trace gives the displayed increment, and the canonical rounding costs only bounded ambiguity.

Raising all data depths by \(h\) is also admissible at the original label and is preserved under the label-conditioned shear. The same calculation gives \[ \mathsf H(P\mid E)=h\sum_j a_j+o(h),\qquad \mathsf H(P\mid E,\theta)=h\sum_j a_j+o(h),\qquad \mathsf I(P;\theta\mid E)=o(h). \tag{104}\] In particular, the conditional coordinate multiinformation obtained by subtracting the joint entropy in (103) from the sum of its single-coordinate entropies is \(o(h)\). Both information quantities are nonnegative averages over the base observation \(E\).

We now form one sequence of finite blocks. This order is needed because the trace error divided by a fixed \(h\) need not vanish as only \(\epsilon\) tends to zero. Choose \(h_n\downarrow0\) on the trace subsequence for this fixed profile and plane point, so that its affine error divided by \(h_n\) tends to zero uniformly on the bounded cone of displacements just used. Differentiability of \(S\) gives the analogous convergence for its time increments. At stage \(n\), include the first \(n\) tuples from a countable list of rational block prefixes, their canonical joint numerator and denominator symbols, and the finitely many information differences above. These are real original depths fixed by the already chosen \(h_n\).

Choose \(\epsilon_n\) within the converging sequence \(p_\epsilon\) so that all finite probability, prediction, information, and box-ambiguity errors are at most \(\eta_n h_n\), where \(\eta_n\downarrow0\), and \(h_n\log(1/\epsilon_n)\ge n\). Put \[\rho_n=\epsilon_n^{h_n},\qquad L_n=\log(1/\rho_n).\] For this extraction, let \(\mathcal H_n\) contain only the tested prefix conditions whose truth, after \(E\) is fixed, is determined by the canonical pair \((P,\theta)\): the parameter prefixes and canonical coordinate numerator and denominator symbols. The earlier bins conditional on \(T\) need not be pair-measurable. They are used to justify the entropy estimates, but are not included in \(\mathcal H_n\). The two nonnegative information quantities and the failure probability of the common prefix event can be selected simultaneously on the base cells by Markov’s inequality. Choose deterministic \(d_n,\zeta_n,\omega_n\downarrow0\) so that, on every selected base cell, the two natural-log quantities are at most \(d_nL_n\), the prefix event fails with probability at most \(\zeta_n\), and its normalized surprise errors are at most \(\omega_n\) in block units. The finite lists and thresholds are chosen before the precision; increasing them slowly gives these common envelopes. Select one such cell \(E=e_n\) and let \(p_n^0\) be the original conditional law of the canonical pair \((P,\theta)\) on that cell, before normalizing the prefix event \(\mathcal H_n\). Thus \(p_n^0(\mathcal H_n)\ge1-\zeta_n\). The cell may have small original mass: the selected bounds are bounds for its conditional law and contain no inverse mass of \(E=e_n\).

For clarity, the two comparisons to be imposed on this law have different spaces and factors. Put \[\mu_n^0=(p_n^0)_P,\qquad \lambda_n^0=(p_n^0)_\theta,\qquad \xi_{j,n}^{0,\theta}=(p_n^0)_{Q_{j,\theta}\mid\theta}.\] The first comparison is with \(\mu_n^0(P)\lambda_n^0(\theta)\). The second is with \[\lambda_n^0(\theta) \prod_{j\in\mathcal I}\xi_{j,n}^{0,\theta}(Q_{j,\theta});\] it uses the coordinate factors conditional on that label. The divergences from these two probability laws are respectively the pair information and the averaged conditional coordinate multiinformation, each bounded by \(d_nL_n\). We do not assume a uniform multiinformation estimate on individual labels.

Apply Lemma 30 to \(p_n^0\) and \(\mathcal H_n\). Its list \(\mathcal F_n\) contains the constant observation, \(P\), \(\theta\), every \((\theta,Q_{j,\theta})\) at full block precision, and every tested canonical prefix and joint numerator or denominator symbol. The parameter children use the fixed dyadic tree; every other child includes its parent when needed, so the nested-symbol conclusion applies exactly. Write \(N_n=|\mathcal F_n|\). Set \[t_n=\sqrt{d_n+L_n^{-1}},\qquad v_n=t_nL_n,\qquad M_{*,n}=1-\zeta_n-2t_n,\qquad \delta_n=\frac{M_{*,n}}{2N_n}.\] For all sufficiently large \(n\), \(M_{*,n}>0\) and \[\frac{D_{\mathrm{pair}}+D_{\mathrm{coord}}+2/e}{v_n} \le 2t_n.\] The lemma therefore gives a restriction \(r_n\le p_n^0\) of mass \(M_n\ge M_{*,n}/2\), supported on the simultaneous prefix and likelihood trims. Define its normalized pair law and actual marginals \[\pi_n=r_n/M_n,\qquad \mu_n=(\pi_n)_P,\qquad \lambda_n=(\pi_n)_\theta,\qquad \xi_{j,n}^{\theta}=(\pi_n)_{Q_{j,\theta}\mid\theta}.\] The deletion proof acts on the common pair space; thus a retained pair is an incidence for the canonical shear, and its support can be used when labels are sampled again.

The finite lists can be increased slowly enough that \(\log N_n=o(L_n)\), since the precision is chosen after each list. The one deterministic envelope \[\kappa_n= \omega_n+t_n+ \max\{2,|\mathcal I|\}\frac{\log(2N_n/M_{*,n})}{L_n} +\frac{\log(2/M_{*,n})}{L_n} \longrightarrow0\] bounds every normalized loss just obtained. Suppress \(n\) and put \(K_0=\rho^{-\kappa_n}\) in the following formulas. The output is the single finite law \[ \begin{aligned} \pi(P,\theta)&\le K_0\mu(P)\lambda(\theta),\\ \pi\bigl((Q_{j,\theta})_{j\in\mathcal I}\mid\theta\bigr) &\le K_0\prod_{j\in\mathcal I}\xi_j^\theta(Q_{j,\theta}). \end{aligned} \tag{109}\] where every factor is an actual marginal of \(\pi\), and the second bound holds at every retained full label. It also holds after pushing any coordinate to a tested prefix.

For every retained value of each tested nested parent and child, the ratio between its conditional mass in \(\pi\) and in \(p^0\) lies in \([\delta_n,\delta_n^{-1}]\). In particular the parameter and coordinate prefix bounds include \[ \begin{aligned} \lambda(c_d\in I)&\le\rho^{sd-\kappa_n},& \lambda(r_d\in I)&\le\rho^{\sigma d-\kappa_n},\\ \lambda(\Theta_b\mid\Theta_a) &\le\rho^{s(b-a)-\kappa_n}\quad(m=1),& \xi_j^\theta((Q_{j,\theta})_d=q) &\le\rho^{a_jd-\kappa_n}. \end{aligned} \tag{110}\] Here the bounds refer to the finite tested list; \(I\) is an interval bin at the stated depth. The first two inequalities are absolute parameter masses; the conditional parameter and coordinate inequalities are on retained parent fibers. The corresponding original lower masses on \(\mathcal H_n\) give \[ \#\{q:(Q_{j,\theta})_d=q \text{ for some retained neighbor of }\theta\} \le\rho^{-a_jd-\omega_n} \tag{111}\] for every retained full label and tested depth. Analogous counts hold for every listed joint child. These support bounds are in place before any label is replicated. The upper mass bounds can be summed over intervals or balls; a finite mesh of tested prefix depths gives the bounds for intermediate radii with the mesh error. The mesh is fixed before its precision is selected.

All later scalar and direction profiles are extracted from this already chosen sequence \(\pi_n\) at \(\rho_n\to0\). They do not choose new \(h_n\). If a later profile selects a real parent depth and a positive derivative increment, extend the nested surprises from the rational prefixes and impose a fresh uniform event for that finite real list before the restrictions which use it. For each fixed increment, errors tending to zero in block units also tend to zero after division by that increment. Only then may the increment shrink. This is a subsequence of the existing block sequence, not a new trace-scale choice.

An arbitrary later restriction does not automatically preserve products of its new actual marginals. If \(\widehat\pi=w\pi/m_R\) with \(0\le w\le1\), then (109) bounds its pair and coordinate laws by \(K_0/m_R\) times the old named products. Decomposing the two relative entropies against those products shows \[\mathsf I_{\mathrm{nat}}^{\widehat\pi}(P;\theta)\le\log(K_0/m_R),\qquad \sum_j\mathsf H_{\mathrm{nat}}^{\widehat\pi}(Q_j\mid\theta) -\mathsf H_{\mathrm{nat}}^{\widehat\pi}((Q_j)_j\mid\theta) \le\log(K_0/m_R).\] Thus a subpower restriction retains small information, while its own product bounds are obtained by applying Lemma 30 again after the required finite prefix refresh. The old upper prefix masses are first preserved on typical parent fibers by Lemma 3; support counts persist by inclusion. This is the procedure used when the tied-clock argument later selects a homogeneous \(Y\) profile. In every later use of this one-label recoring procedure, reset \(\pi,\mu,\lambda\) and \(\xi_j^\theta\) to its new law and actual factors. Also renew \(K_0\) and one common deterministic finite-test subpower envelope for the products and tested prefix losses in the current working units. A fixed positive shortening by a factor \(a\) preserves subpower loss, since \(o(L)=o(aL)\) when both are expressed in the shortened units.

Replication and the first projection tests

For \(1\le k_{\mathrm{rep}}\le3\), sample labels \(\theta_0,\ldots,\theta_{k_{\mathrm{rep}}-1}\) independently given \(P\) with the conditional kernel of \(\pi\). Their exact star law is \[\Pi(P,\theta_0,\ldots,\theta_{k_{\mathrm{rep}}-1}) =\mu(P)\prod_{j=0}^{k_{\mathrm{rep}}-1}\pi(\theta_j\mid P).\] Since \(\pi(\theta\mid P)\le K_0\lambda(\theta)\), it satisfies \[ \begin{aligned} \Pi&\le K_0^{k_{\mathrm{rep}}}\mu\prod_j\lambda_j,\\ \Pi&\le K_0^{k_{\mathrm{rep}}-1} \pi(P,\theta_i)\prod_{j\ne i}\lambda_j \quad(0\le i<k_{\mathrm{rep}}). \end{aligned} \tag{112}\] where every \(\lambda_j\) is a copy of \(\lambda\). These are the named comparison laws used after replication.

Let \(R=w\Pi/m_R\) be a graph restriction chosen before conditioning. Fix a set \(J\) of full labels and leave \(J^c\) free. The conditioning calculation in Theorem 8, applied to (112), has two consequences on typical fixed-label values. First, for each \(i\in J\), the point law is dominated by a subpower times \(\pi(P\mid\theta_i)\); pushing this comparison through the coordinate map at label \(i\) retains its product against the named factors \(\xi_j^{\theta_i}\). Second, after the theorem’s point-likelihood trim, the joint law of the point and the free labels is compared with its actual conditional point marginal times the copies \(\lambda_j\) for \(j\notin J\).

Feed all coordinate products needed at these fixed labels and the point–free-label comparison into one application of Lemma 7. Repeated occurrences of a coordinate with different named factors are counted separately: for example, \(\xi_X^{\theta_0}\) and \(\xi_X^{\theta_1}\) may differ even though the map \(X_\theta=X\) is invariant. The simultaneous core replaces both by the same current actual point marginal of \(X\). If \(J=\{0,1\}\) in the tied case, the resulting point law \(\nu\) satisfies \[ \nu_{X,U_0}\le K_1\nu_X\nu_{U_0},\qquad \nu_{X,U_1,Z_1}\le K_1\nu_X\nu_{U_1}\nu_{Z_1}, \tag{113}\] and the full coordinate product at either label is available when required. Its coordinate marginal prefix bounds retain the rates \(x,q,z\) with subpower losses. If label \(2\) is free, the same core gives a point–label-2 comparison with the actual point marginal and a retained label comparison whose absolute prefix masses are bounded by a subpower times those of the named pre-core label law. Conditional child caps for the new label marginal also require the tested parent floors or typical-parent likelihood thresholds; these are imposed for the finite list before those caps are used.

Here and in later uses, the number of replicas is fixed and the finite factor list, graph retention \(m_R\), likelihood thresholds, and core retention are chosen with common deterministic subpower envelopes. The constants in Theorem 8 are fixed products of \(K_0\), their inverse retentions and thresholds, and the finite factor count. Their normalized logarithms therefore tend to zero together on every retained conditional cell. The bound does not depend on the number of possible fixed-label values. For example, if \[\log(1/m_R),\ \log(1/\eta),\ \log(1/u),\ \log(2N_{\mathrm{fac}})\le b_nL_n \qquad\text{with }b_n\to0\] throughout a chosen conditional array, then its fixed-label comparisons and fixed-size cores have constants at most \(\exp(C(\kappa_n+b_n)L_n)\), with \(C\) depending only on the fixed numbers of replicas and factors. This supplies one common subpower envelope for those cells.

A subsequent point observation is fixed only if it is a function of \(P\) and the labels already in \(J\). Because the point factor after the core is actual, Corollary 9 preserves the same pre-slice free-label comparison on every positive point slice. Any coordinate product needed within such a slice is separately conditioned by Lemma 3, on typical values of the specified coordinate parent. An observation involving a free label must instead be included among the fixed labels before this argument, or be handled by a different verified comparison.

Suppose first that \(m\ne1\). Use two replicas and fix a typical label \(0\), leaving label \(1\) free. The fixed-label core just described gives an actual point law for \((X,Y_0)\) with ball dimension \(x+y\): its coordinate product and the tested prefix upper masses bound a square of radius \(\rho^d\) by \(\rho^{(x+y)d-o(1)}\). The free comparison is the retained pre-slice label-\(1\) law, whose time marginal has dimension \(s\). The dominated incidence law has graph product mass \(\rho^{o(1)}\). For each full label \(1\) the projected coordinate \[Y_1=Y_0+(c_1-c_0)X\] has at most \(\rho^{-y-o(1)}\) bins on its retained neighbors by (111). The other component of the label is kept as a tag. Theorem 12 gives \[ y\ge\min\{1,x+y,(x+y+s)/2\}. \tag{114}\]

For the chain from \(w\) to \(z\), fix also the full block symbols \(X,Y_0\). The label-\(0\) coordinate product, conditioned on typical such values, gives the point population \((W_0,Z_0)\) dimension \(w+z\). Corollary 9 keeps the same free label-\(1\) comparison. The last coordinate is \(Z_0+(c_1-c_0)W_0\) plus a translation determined by the fixed symbols and the full free label, and its original support count is still available. The projection theorem gives \(z\ge\min\{1,w+z,(w+z+s)/2\}\). Interchanging the two shears gives the chains from \(x\) to \(w\) and from \(y\) to \(z\) with \(\sigma\) in place of \(s\); for the latter the fixed point symbols are \(X,W_0\). All slices are finite grid symbols, and the typical-conditioning loss has no factor depending on their alphabet sizes.

For any of these inequalities, if the source \(d\) is positive and the target \(e\) is below \(1\), the alternative \(e\ge d+e\) is impossible. The remaining alternative gives \(e\ge d+s\) or \(e\ge d+\sigma\), as required. This proves the distinct-clock part of Proposition 29.

Now let \(m=1\). Equation (102) becomes \[ U_1=U_0+(\theta_1-\theta_0)X,\qquad Z_1=Z_0+\operatorname{flip}(\theta_1-\theta_0)\cdot U_0 +\operatorname{prod}(\theta_1-\theta_0)X, \tag{115}\] where \(\operatorname{flip}(c,r)=(r,c)\) and \(\operatorname{prod}(c,r)=cr\). The trace controls the joint block \(U\), so it supplies no separate affine rates for \(Y,W\). Homogenize the conditional \(Y\)-prefix probabilities on a subpower part of the current one-label law, and write \(g(d)\) for their limiting entropy in block units. This is a selection within the already fixed block sequence. At each finite stage take the post-split law as the new \(p^0\). Relative to the old one-label law, select typical values of every tested parent needed for the parameter and \(X\) upper masses. For each tested nested parameter pair \(\Theta_a,\Theta_b\), also impose the capacity tail \[p^0(\Theta_b\mid\Theta_a) \ge\tau C^{-1}\rho^{2(b-a)}.\] Here \(C=4\) is the capacity constant of the fixed parameter tree. There are at most \(C\rho^{-2(b-a)}\) grid children over each parent, so the failure of this condition has \(p^0\)-mass at most \(\tau\). Choose \(\tau\) with \(N_{\mathrm{test}}\tau\to0\) and \(\log(1/\tau)=o(L)\), where \(N_{\mathrm{test}}\) is the number of these tested pairs and \(L=\log(1/\rho)\). Intersect all good-parent filters, capacity tails, and fresh \(Y\) and coordinate surprise bins to form one allowed event \(\mathcal H\) for \(p^0\), and keep \(\boldsymbol1_{\mathcal H}p^0\) unnormalized until the repeated application of Lemma 30. Its list contains all the tested numerator and denominator observations. The old named products give the small information needed for this extraction; the simultaneous floors against \(p^0\) preserve the inherited upper masses and selected \(Y\) exponents on the same retained fibers. The new products use the actual marginals of the output, and its support is a subset of the original coordinate support.

Relabel this final one-label law as \(\pi\), with actual point and parameter marginals \(\mu,\lambda\). If \(\delta\) is the repeated core’s threshold, its simultaneous prefix floors give, on every retained tested parameter child, \[ \lambda(\Theta_b\mid\Theta_a) \ge\delta\tau C^{-1}\rho^{2(b-a)} =\rho^{2(b-a)+o(1)}. \tag{116}\] This uses only the planar child capacity and supplies a lower mass for the short mesh denominators used below. Put \[g_*=\inf_{0<d\le1}\frac{g(d)}d.\] The law \(\pi\) has the \(X\)–\(U\) product, the \(X\) and time upper masses, and the homogeneous \(Y\) prefix masses and support counts on the same retained incidences. Each \((P,\theta_i)\) marginal of its exact star is this same law. If a later graph restriction is \(R=w\Pi/m_R\), that marginal is bounded by \(\pi/m_R\). Typical label likelihood thresholds therefore preserve its named prefix upper bounds, and support counts can only decrease. Pushing the \(X\)–\(U\) product through \(U\mapsto Y\) bounds a square of radius \(\rho^r\) in \((X,Y_0)\) by \(\rho^{xr+g(r)-o(1)}\) on the retained typical labels. Both replicas use the profile of this one law.

At precision \(\rho^d\), the \((X,Y_0)\) point population has ball dimension at least \(x+g_*\), whereas its \(Y_1\) projection count has exponent \(g(d)/d\). Indeed \(g(r)\ge g_*r\) at every smaller prefix; the requisite ball bounds follow from the homogeneous conditional masses. Fix label \(0\), leave label \(1\) free, and use the same product comparison as for (114). At each fixed \(d>0\), Theorem 12 gives \[\frac{g(d)}d\ge\min\{1,x+g_*,(x+g_*+s)/2\}.\] Taking the infimum does not assume it is attained: test successively fixed \(d\) with \(g(d)/d\) approaching the infimum, choosing all losses after \(d\). If \(x>0\), the resulting inequality implies \(g_*\ge\min(1,x+s)\ge s\). Since \(g(1)\le q\), we have proved \(q\ge s\).

Directions between parameter samples

The reciprocal use of pinned directions and the local linearization below have close precedents in the radial-projection bootstrap of Shmerkin and Wang and the thin-tube bootstrap of Orponen, Shmerkin, and Wang (Shmerkin and Wang 2025; Orponen et al. 2024). We prove the version needed here for the retained conditional pair law, including its selection order and the resulting improvement of the directional rate.

Throughout this subsection and its applications, set \(\mathop{\mathrm{dir}}(0)=(1,0)\) and give this fixed unit vector the ordinary nested direction-grid prefixes. Thus the direction and scalar observations are defined on the entire finite space, including coincident labels.

We give the needed directional statement with its assumptions in finite probability form. A child bound of rate \(s\) means the conditional parameter-prefix bound for \(\lambda(\Theta_b\mid\Theta_a)\) in (110) at every fixed tested pair of depths, on uniform trims and typical parent fibers. On each tested good parent, it applies to the entire named conditional population, hence to all its positive child values. This includes all intermediate radii after using a slowly growing finite mesh.

Three labels and the full-precision slices

Assume first that (117) holds for the one-label parameter population. Use the globally defined parameter-tree atoms from the one-label construction. Their simultaneous bins, parent filters, and prefix floors give the rate-\(s\) upper child cap in (110) and the lower capacity bound (116) for the same current \(\lambda\) on every retained tested child. These are the conditional masses of the named one-label population used below.

They cover a later fixed real parent \(j\) and increment \(a\) by a finite mesh. Choose tested depths \(j_-<j<j_+<b_-<j+a\) within mesh size \(\delta_{\mathrm{mesh}}\) of \(j\) and \(j+a\). For the compatible nested prefixes of a retained child, use the globally defined tree atoms at all five depths below. In particular the denominator uses the already floored \(\Theta_{j_+}\) atom, unchanged by the choice of the real depth \(j\). Thus \[\lambda(\Theta_{j+a}\mid\Theta_j) \le\frac{\lambda(\Theta_{b_-}\mid\Theta_{j_-})} {\lambda(\Theta_{j_+}\mid\Theta_{j_-})} \le\rho^{s(b_--j_-)-2(j_+-j_-)-o(1)}.\] Choose \(\delta_{\mathrm{mesh}}\ll a\) after \(j,a\) are fixed, and then the precision. This gives the required rate-\(s\) child cap with the mesh error and one common envelope, within the existing block sequence.

Sample three labels over \(P\) using (112). At each finite stage apply Lemma 5 to the direction-grid children of the first pair conditional on each full pin, and to the scalar-grid children below conditional on the full pair. Their child alphabets have size at most a fixed power of \(\rho^{-1}\) for each finite list, while no bound on the parent alphabets is needed. Choose the lemma’s parent and child thresholds together for this finite list, with vanishing total exceptional mass under the common envelope. The diagonal in Lemma 10 gives a subpower retained triple law carrying the actual conditional profiles. Its first-pair marginal is dominated by a subpower times the unchanged named product \(\lambda_0\lambda_1\). These copies retain the tested child and no-strip bounds just verified, and the two direction profiles belong to this same first-pair law. Thus Proposition 31 applies. Lift its common pair event to this same point–label law and normalize it as \(\mathcal R\). Its pair marginal is the normalized law in the proposition, so the actual conditional direction caps hold on the respective good-pin values. Keep this pre-average law for the two projection tests below. Write \[d=\mathop{\mathrm{dir}}(\theta_1-\theta_0),\qquad \zeta_2=\operatorname{flip}(d)\cdot\theta_2.\] For the scalar test, retain one good label-\(0\) set from the proposition, of \(\mathcal R\)-mass at least \(1-\eta_{\mathrm{pin}}\). The label marginal satisfies \(\mathcal R_{012}\le K_{\mathrm{lab}}\lambda_0\lambda_1\lambda_2\) with subpower \(K_{\mathrm{lab}}\). Also retain the pair values \[\mathcal R_{01}(b)\ge \eta_{\mathrm{pair}}(\lambda_0\lambda_1)(b).\] Their complement has \(\mathcal R\)-mass at most \(\eta_{\mathrm{pair}}\) by (10). Normalize the intersection of these two pair-only filters as \(\mathcal R^0\). Averaging the conditional direction cap over the retained good label-\(0\) values gives an absolute rate-\(s\) cap for \(d\) under \((\mathcal R^0)_{01}\), with full-pair tags and only the inverse retained mass loss. Since both filters depend on the pair, their normalization gives the exact comparison \[(\mathcal R^0)_{012} \le\frac{K_{\mathrm{lab}}}{\eta_{\mathrm{pair}}} (\mathcal R^0)_{01}\lambda_2.\] The third population \(\lambda_2\) has point dimension at least \(s\). The thresholds tend to zero with subpower inverses in the common finite-test envelope.

Let \(l(r)\) be the inherited scalar profile of \(\zeta_{2,r}\) given the full pair under \(\mathcal R^0\). The pair-only filters leave its conditional law on each retained pair unchanged, and the earlier common bins give its profile under this subpower restriction. For a finite list of \(r\), choose one event \(\mathcal H_{\mathrm{sc}}\) of mass \(m_{\mathrm{sc}}=\mathcal R^0(\mathcal H_{\mathrm{sc}})\) on which \[\left|-L^{-1}\log\mathcal R^0(\zeta_{2,r}\mid b)-l(r)\right| \le\omega_{\mathrm{sc}},\qquad \omega_{\mathrm{sc}}\to0,\] and put \(\mathcal R^{\mathrm{sc}} =\mathcal R^0(\,\cdot\mid\mathcal H_{\mathrm{sc}})\). The inverse mass is subpower. This law is dominated by \(K_{\mathrm{lab}}/(\eta_{\mathrm{pair}}m_{\mathrm{sc}})\) times the named product \((\mathcal R^0)_{01}\lambda_2\), so the pre-event pair population supplies the direction cap. The original lower scalar masses on \(\mathcal H_{\mathrm{sc}}\) give at most \(\rho^{-l(r)-\omega_{\mathrm{sc}}}\) retained scalar bins per full pair. Theorem 12, with full-pair tags and dimensions \(s,s\), therefore gives \(l(r)\ge sr\).

Put \(r_b=\mathcal R^0(\mathcal H_{\mathrm{sc}}\mid b)\) and \(\lambda_b^0=(\mathcal R^{\mathrm{sc}})_{\theta_2\mid b}\). Lemma 3 gives \(r_b\ge m_{\mathrm{sc}}\eta_{\mathrm{sc}}\) outside \((\mathcal R^{\mathrm{sc}})_{01}\)-mass at most \(\eta_{\mathrm{sc}}\). For each retained scalar bin on such a pair, \[\lambda_b^0\{\zeta_{2,r}\in I\} \le r_b^{-1}\mathcal R^0(\zeta_{2,r}\in I\mid b) \le(m_{\mathrm{sc}}\eta_{\mathrm{sc}})^{-1} \rho^{l(r)-\omega_{\mathrm{sc}}} \le\rho^{sr-o(1)}.\] Choose \(\eta_{\mathrm{sc}}\) with a subpower inverse and intersect these pairs with the typical fixed-pair values from the star comparisons for the graph \(\mathcal R^{\mathrm{sc}}\). Also impose the pair-only filter \(|\theta_1-\theta_0|\ge\rho^{\vartheta_n}\), with \(\vartheta_n\downarrow0\) chosen slowly for this graph’s envelope. Indeed, for each fixed \(0<t\le1\), its pair domination by a subpower times \(\lambda_0\lambda_1\) and the absolute parameter ball cap give \[(\mathcal R^{\mathrm{sc}})_{01} \{|\theta_1-\theta_0|<\rho^t\} \le\rho^{st-o(1)}.\] A slow diagonal makes the excluded mass vanish at \(t=\vartheta_n\). This pair-only filter leaves conditional laws at retained pairs unchanged. With \(\alpha=|\theta_1-\theta_0|\), it gives \(\alpha^{-1}\le\rho^{-\vartheta_n}=\rho^{-o(1)}\). On each remaining pair \(b\), feed the point–third-parameter comparison and the two coordinate products supplied by the \(i=0,1\) star comparisons into one simultaneous core. Let \(M_b\) be the total raw retained mass relative to \(\mathcal R^{\mathrm{sc}}(\,\cdot\mid b)\), including the point-likelihood trim and all core deletions. Then the new third marginal satisfies \(\lambda_{01}\le M_b^{-1}\lambda_b^0\). All inverse retentions use the same subpower envelope. Hence \[ \lambda_{01}\{\zeta_{2,r}\in I\}\le\rho^{sr-o(1)}. \tag{119}\] The two coordinate products now use actual marginals of that same retained point law. This establishes their coexistence with the scalar slope bound before any point-coordinate slice.

Fix such a pair, and write \(\theta_1-\theta_0=\alpha d\), where \(\alpha^{-1}\) has subpower size. Before the last trim form the real linear coordinates \[N=d^\perp\cdot U_0=d^\perp\cdot U_1,\qquad L=d\cdot U_1.\] Bin \(N,L\) at precision \(\rho\). Enlarge the \(U_0\) symbol to include the \(N\) bin, which has boundedly many possibilities given it, and use \((N,L)\) as the \(U_1\) symbol. Rotated and original square bins determine one another with bounded ambiguity.

Let \(\widetilde\nu^\star(P,\theta_2)\) denote the joint law just obtained from the fixed-pair core, with actual point marginal \(\widetilde\nu\). It has \[\widetilde\nu^\star(P,\theta_2) \le A\,\widetilde\nu(P)\lambda_{01}(\theta_2)\] with subpower \(A\), and its point marginal carries both coordinate products and the \(X,Z_1\) prefix upper masses. The bounded new symbols change the coordinate multiinformations by \(O(1)\) in natural-log units. Apply Lemma 6 and the simultaneous marginal core to these two coordinate products on the point law \(\widetilde\nu\) alone. The likelihood events and all core deletions are functions of the point and the already fixed pair. They therefore give a point-only weight \(0\le w(P)\le1\) of subpower retained mass \(M=\mathbb E_{\widetilde\nu}w>0\). Lift that weight to the joint law and put \[\nu^\star(P,\theta_2)=\frac{w(P)\widetilde\nu^\star(P,\theta_2)}M, \qquad \nu(P)=\frac{w(P)\widetilde\nu(P)}M.\] The point–free-label comparison is then exactly \[ \nu^\star(P,\theta_2) \le A\,\nu(P)\lambda_{01}(\theta_2). \tag{120}\] There is no new factor \(M^{-1}\) in this comparison: the same point weight and normalization occur in its actual point marginal. The coordinate core gives (113) for the new finite symbols on \(\nu\), and the \(X,Z_1\) upper masses lose at most the subpower factor \(M^{-1}\).

The fixed labels are still \(J=\{0,1\}\) and label \(2\) remains free. Both \(N\) and \(X\) are finite functions of the point and these fixed labels. By Corollary 9, every positive \((N,X)\) slice of (120) keeps the same pre-slice comparison \(\lambda_{01}\) and the actual sliced point marginal. In particular its scalar slope bound remains available. No exact real coordinate is conditioned on.

We prove carefully that the population \((L,Z_1)\) conditional on \((N,X)\) has ball dimension at least \(x+z\). If a \(U_0\) cell with center \(u_0\) is fixed, then \(L\) is within \(O(\rho)\) of \(d\cdot u_0+\alpha X\). The first product in (113) and the marginal \(X\) bound give, for an interval \(I\) of length \(r\ge\rho\) (absorbing the constant enlargement), \[ \nu(L\in I\mid U_0=u_0) \le K_1 C\alpha^{-x}r^x, \tag{121}\] where \(C\) is subpower. Average this inequality over \(u_0\) conditional on \(N=n\). This proves the same bound for \(L\mid N=n\).

The second product in (113), after rotating \(U_1\) to \((N,L)\), gives the exact finite inequality \[ \nu(l,z\mid N=n,X=t) \le K_1\frac{\nu_X(t)\nu_N(n)}{\nu_{X,N}(t,n)} \nu(l\mid N=n)\nu_{Z_1}(z). \tag{122}\] The \(\nu\)-probability of the values \((t,n)\) where the displayed ratio exceeds \(A>1\) is at most \(A^{-1}\): sum \(\nu_{X,N}(t,n)<A^{-1}\nu_X(t)\nu_N(n)\) over those values. On the remaining slices, (121) and the \(Z_1\) marginal bound imply \[ \nu((L,Z_1)\in I\times J\mid N=n,X=t) \le K_1^2 A C^2\alpha^{-x}r^{x+z} \tag{123}\] for intervals of length \(r\). Take \(A\to\infty\) with subpower growth. Thus there is no loss proportional to the number of full-precision \(X\) or \(N\) bins.

The free label-\(2\) law on these slices uses the same comparison \(\lambda_{01}\) from (119): the point-only identity (120) and its \((N,X)\) slicing were verified above. Hence the free scalar slope bound is available on these slices.

In (115), the change from frame \(1\) to frame \(2\) is a scalar projection of \((L,Z_1)\): its slope is \(\operatorname{flip}(d)\cdot(\theta_2-\theta_1)\), and the remaining terms are translations determined by \((N,X)\) and the free label. The support count of \(Z_2\) per label is still at most \(\rho^{-z-o(1)}\), because every restriction only removed original neighbors. Apply Theorem 12 to (123) and (119). We obtain \[ z\ge\min\{1,x+z,(x+z+s)/2\}. \tag{124}\] For \(x>0\) and \(z<1\) this is exactly \(z\ge x+s\).

For the second test, return to a fresh copy of the pre-average point–label law \(\mathcal R\), restrict it to its good label-\(0\) values from Proposition 31, and marginalize label \(2\). This lifts the pair-only filter to the same incidence law; the star comparisons and label-\(0\) coordinate support remain on one graph. The conditional direction law of label \(1\) at each retained label \(0\) is unchanged and has rate \(s\). Fix a typical label \(0\) and apply the simultaneous core to the point–label-\(1\) comparison and the label-\(0\) coordinate product. Its new label-\(1\) marginal retains the direction cap with a subpower inverse mass loss. Then fix the point coordinates \(X,Z_0\); label \(1\) is free. The point population is \(U_0\), of dimension \(q\) on typical slices by that coordinate product. Its direction comparison is this retained pre-slice label-\(1\) marginal, for the direction \(\operatorname{flip}(\theta_1-\theta_0)\). The fixed-label core uses the \(i=0\) comparison in (112); Corollary 9 preserves this same direction population on the subsequent \((X,Z_0)\) slice. Equation (115) projects \(U_0\) to the \(Z_1\) support, giving \[ z\ge\min\{1,q,(q+s)/2\}. \tag{125}\] If \(q>2-s\), both other terms exceed or equal \(1\), so \(z\ge1\). This proves the last assertion in the absence of strip concentration.

All rotations and scalar slices just used have a finite-grid meaning. Rotated finest-scale squares meet boundedly many original squares and conversely. Add the two descriptions and their bounded neighbor tags when conditioning. Equation (115) then holds up to \(O(\rho)\); all interval bounds use constant enlargements. The factors \(\alpha^{-1}\) and the harmless angular separations can be bounded by \(\rho^{-\eta}\) for each fixed \(\eta>0\) before the limit. Replacing a finest scale by a \(\rho^{-O(\eta)}\) enlargement changes the displayed exponent inequalities by \(O(\eta)\), which is sent to zero afterwards.

Concentration in a strip

It remains to justify the alternative omitted above. Test the original retained one-label marginal \(\lambda\) before sampling replicas. If (117) fails along some subsequence at a fixed depth \(t>0\), pass to a subsequence and choose a strip of width \(\rho^t\) whose \(\lambda\)-mass is \(\rho^{o(1)}\). This follows by choosing a sequence on which the nonnegative exponent of the maximal strip mass tends to zero. Otherwise a positive lower bound for that exponent supplies (117) at each fixed depth, and a diagonal supplies all the depths used above. Subpower product domination then prevents concentration of a replicated marginal that was absent from the one-label population.

Restrict the one-label graph to the chosen strip and normalize it as the new \(p^0\). Its mass in the previous law is subpower, so the old named products give the small information required by Lemma 30. Shorten the working block to a fixed positive depth no greater than \(t\), if necessary, and fix its prefix list before taking the small-parameter limit. Intersect the fresh bins with all typical-parent filters needed for the inherited conditional bounds, keep this allowed restriction unnormalized, and apply the lemma once for that finite list. Relabel its output \(\pi\) and its actual pair marginals \(\mu,\lambda\). Its pair and coordinate products now use actual factors before any new replicas are sampled, and its support still lies in the chosen strip. In these shorter block units the coordinate rates remain \(x,q,z\), by (103); the time and joint parameter rates remain at least \(s\). The strip is now as thin as the finest precision. Let \(d_0\) be its unit direction. Both coordinates of \(d_0\) have exponent zero. Indeed, if (say) its first coordinate had magnitude at most \(\rho^b\) along a subsequence with \(b>0\), the bounded strip would lie in a time-coordinate interval of a fixed positive depth. Its subpower retained mass contradicts the time marginal bound in (110). The second coordinate is treated using the positive normal marginal bound. In particular \[|\operatorname{flip}(d_0)\cdot d_0| =2|(d_0)_1(d_0)_2|=\rho^{o(1)}.\]

The scalar position along the strip has prefix ball dimension at least \(s\): projection to the time coordinate has this bound, and its slope along the strip has exponent zero. After sampling three labels over the restricted graph, typical pairs have separation \(\rho^{o(1)}\) by this scalar bound. Their differences are parallel to \(d_0\) up to finest-scale errors, so the scalar third slope \(\operatorname{flip}(d_0)\cdot\theta_2\) has dimension at least \(s\) directly. Typical fixed-pair comparisons preserve it. The proof of (124), including both products at labels \(0,1\), now applies with \(d_0\) in place of \(d\): perpendicular invariance and parallel transfer have only finest-scale errors. Thus \(x>0,z<1\) again gives \(z\ge x+s\).

For the remaining test, start from a fresh copy of the strip star before the preceding fixed-pair restrictions, then fix label \(0\) and \(X\), leaving label \(1\) free. Set \(B_0=\operatorname{flip}(d_0)\cdot U_0\). If \(q>2-s\), then \(q>1\). A strip of width \(r\) for \(B_0\) is covered by \(O(r^{-1})\) balls of radius \(r\) in the bounded \(U_0\)-plane. Hence its mass is at most \(C r^{q-1}\), with subpower \(C\). The label-\(0\) coordinate product, after the typical \(X\) slice, gives ball dimension at least \(q-1+z\) for \((B_0,Z_0)\). The free comparison law is the retained label-\(1\) marginal; its scalar position along the strip has dimension \(s\). Equation (115) projects this pair to \(Z_1\) with that scalar slope, up to a known translation and finest-scale errors. Theorem 12 gives \[z\ge\min\{1,q-1+z,(q-1+z+s)/2\}.\] If \(z<1\) and \(q>2-s\), each expression inside this minimum is strictly greater than \(z\): for the last one use \(q-1+s>1>z\). This is impossible. Consequently \(z<1\) implies \(q\le2-s\) also in the strip case.

This completes the proof of Proposition 29. The order is the block sequence \((h_n,\epsilon_n)\) first, the homogeneous scalar and direction profiles second, and their tested depths and positive derivative increments third. At that third stage every test fixes its finite list before taking a subsequence limit in \(n\), and uniform trims precede all restrictions using their bounds. The derivative increment is sent to zero only afterwards. Each prescribed accuracy in the directional bootstrap requires finitely many iterations; a diagonal over those finite tests proves the exact inequalities. Thus neither a new \(h_n\) chosen from a later direction profile nor a uniform differentiability modulus across profiles is required.

Rate tuples at a contact point

For distinct clocks put \[\begin{align*} B&=x+y+w+z-2,\tag{126}\\ F(P)&=2P-m-Px+(1-P)y+(m-P)w+(1+m-P)z. \tag{127}\end{align*}\] For tied clocks these definitions mean \[ B=x+q+z-2,\qquad F(P)=2P-1-Px+(1-P)q+(2-P)z. \tag{128}\] In either case \(F'(P)=-B\), with the rates held fixed.

Aperture refinement forces a positive tangent rate

The finite sign calculation

Contradiction and completion

The phase class and the growth supremum

The longitudinal coordinate is \(c\in[0,1]\), the input variable is \(v\in\mathbb R^2\), and the observation position is \(x\in\mathbb R^2\). The phase is defined on \([0,1]\times\mathbb R^2\times U_v\), for an open neighborhood \(U_v\) of the bounded input patch. We use real phases satisfying \[ \partial_v\Phi(c,x,v)=\zeta(v,x-cv). \tag{143}\] The function \(\zeta\) is defined for every \(p\in\mathbb R^2\) and on all bounded \(v\)-patches needed in the argument. On each such patch, \(\partial_p\zeta(v,p)\) is symmetric and either \[ a_0 I\le \partial_p\zeta(v,p)\le a_1 I \quad\hbox{for all }(v,p), \tag{144}\] or the negative of this inequality holds. Here \(a_0>0\). For some fixed \(A_\Phi\ge1\), all derivatives of order \(j\) of this matrix are bounded by \(A_\Phi^{j+1}(j!)^3\). Bounds on domains, \(a_0^{-1}\), \(a_1\), and \(A_\Phi\) may vary between applications; they are uniform within each sequence in which \(\epsilon\to0\). Input-only terms in \(\Phi\) are absorbed into the input.

Write \[ \lambda^{-1}=\epsilon^{H_{\mathrm P}},\qquad u_t=\frac{H_{\mathrm P}-t}{2},\qquad \rho_t=\epsilon^{u_t},\qquad \sigma_t=(\lambda\rho_t)^{-1}=\epsilon^{u_t+t}. \tag{145}\] Time boxes are nested dyadic intervals in \([0,1]\), with root center \(C_0=1/2\). A depth-\(t\) box has length comparable to \(\epsilon^t\). Its center is denoted by \(C_t\). We take actual dyadic lengths in changes of coordinates; this changes depths by at most \(\log 2/\ell\).

An observation \(O=(C_t,V,X)\) uses velocity and position lattices of spacings \(\rho_t\) and \(\sigma_t\), respectively. Bounded multiplicities and translated lattices are allowed. Its mass is \[ m_O(f)=\left|\rho_t^{-1}\int f(v) \psi_O\left(\frac{v-V}{\rho_t}\right) e^{i\lambda\Phi(C_t,X,v)}\,dv\right|^2. \tag{146}\] The input belongs to \(L^2\) and has bounded support. The amplitudes have fixed compact supports in their displayed variables, and every derivative has a uniform bound. The centers \(X,V\) range over bounded sets when \(t>0\). Suprema over any specified amplitude class with these uniform bounds are also permitted. Lemma 36 justifies these suprema by simultaneous choices at the sites.

At the root, the canonical mass is the supremum in (146) over amplitudes supported in \(\{|z|\le100\}\) with \(\|\partial^\alpha\psi\|_\infty \le100^{|\alpha|+1}(|\alpha|!)^{10}\). The root grids are fixed. The position cutoff is \(|X|\le1+\log\lambda\), with a natural logarithm here, and velocity centers lie within \(100\rho_0\) of the input support, or in a fixed sufficiently large enlargement of that support.

We use the aperture convention of Definition 14. Thus a label \(D\) includes \(C_t\), an orientation, a rectangle locating \(V\), and a rectangle locating \(X\), with tangent/normal widths \((\epsilon^b,\epsilon^a)\) and \((\epsilon^{b+t},\epsilon^{a+t})\). Labels may contain additional information. For packets the admissible depths satisfy \[ 0\le b\le a\le\min(L-t,u_t),\qquad L\ge t>0, \tag{147}\] and \(w(b,a)=(1+\gamma)b+(1-\gamma)a\). For any probability law on observations and aperture labels, supported where \(m_O>0\), define \[ P_t=\mathsf H(O)+\frac12\mathsf H(O\mid D) +\frac32\mathbb E\log_{1/\epsilon}\!\left(m_O\right) +\frac12\mathbb Ew(D). \tag{148}\] Let \(K_0(f)\) be the supremum of the corresponding root scores for the canonical masses, with \(0\le b\le a\le\min(L,u_0)\), enlarged to include the value \(\frac32\log_{1/\epsilon}\!\left(\|f\|_2^2\right)\). The last option is part of the definition, including for rescaled local inputs.

The entropy terms connect the score to the desired \(L^3\) estimate. For a finite family with at least one positive packet mass, omit the zero masses. The log-sum identity gives \[\sup_{\pi}\left\{\mathsf H_\pi(O) +\frac32\mathbb E_\pi\log_{1/\epsilon}\!\left(m_O\right)\right\} =\log_{1/\epsilon}\!\left(\sum_Om_O^{3/2}\right),\] where the supremum is over probability laws on the remaining sites. Because \(m_O\) is a squared packet amplitude, the right side is the normalized logarithm of the cubic packet sum. At \(t=H_{\mathrm P}\), admissibility forces \(b=a=0\), hence \(w=0\). For the maximizing observation law, every compatible endpoint score is at least the displayed value, because \(\mathsf H(O\mid D)\ge0\). For a given aperture score, put \(M=\sum_Om_O\), and let \(M_D\) be the sum over all sites in the cap specified by \(D\). Applying log-sum once without conditioning and once conditional on \(D\) gives \[P_t\le \log_{1/\epsilon}\!\left(M\right) +\frac12\mathbb E\bigl[\log_{1/\epsilon}\!\left(M_D\right)+w(D)\bigr].\] Thus total and cap masses control the root aperture scores; the additional norm option in \(K_0\) is estimated separately.

Multiplying \(f\) by a scalar shifts both \(P_t\) and \(K_0\) by the same quantity. We may therefore normalize \(\|f\|_2=1\) when considering their difference. Define \(\beta_{\mathrm q}\) to be the supremum of \[ \limsup_{\epsilon\to0}\frac{P_t-K_0}{t} \tag{149}\] over all the preceding systems, allowing convergent bounded depth parameters with positive limiting duration. In each sequence all domain, phase, amplitude, and multiplicity bounds are fixed. The input and the individual phase and amplitude choices may depend on \(\epsilon\). The supremum over sequences also allows any choices of these fixed bounds.

The proof is completed in Section 11. The present section proves all the analytic decomposition statements and the preliminary bound \(\beta_{\mathrm q}\le1\) used in that proof. Uniformity in (150) follows from the supremum definition: a failure for fixed bounds would select a sequence contradicting the claimed bound on \(\beta_{\mathrm q}\).

Bessel estimates and changes of scale

Fourier localization and almost orthogonality are the standard analytic ingredients of wave-packet decomposition; see, for example, Tao (2003) and Guo et al. (2025). We give the estimates with the phase class and uniform amplitude bounds required by our packet score, including changes of scale and quantified tails.

Moving a center by a bounded number of its lattice spacings changes only an admissible amplitude factor, after removal of a scalar phase. The same is true when the time anchor moves inside its box and \(X\) is transported by that displacement times \(V\). Indeed, after a scalar phase is removed, the first derivative is bounded in packet units; the higher derivatives of the phase change are bounded by \(C_j\lambda\rho_t^2\epsilon^t=C_j\). This also proves the bounded ambiguity assertions used with shifted grids below.

Fix \(0\le r<t\), fixed widths \(b,a\), and \(0\le b_*\le b\). The geometric localization label is \[ J_0=\left(C_r,V_{b_*}, (X+(C_r-C_t)V)_{r+b_*}\right). \tag{154}\] Subscripts on vectors denote grid cells. The endpoint observation and its aperture each determine this label to bounded ambiguity; adjoining the bounded neighboring-cell choices makes the determination exact. Write \(V_J,X_J\) for its velocity center and its trajectory position at \(C_r\). Let \(s_v=\epsilon^{b_*}\), \(s_c=\epsilon^r\), \(s_x=s_cs_v\). In exact dyadic coordinates take the actual length of the time box for \(s_c\). Set \[\begin{align*} v&=V_J+s_vv',& c&=C_r+s_c(c'-1/2),\tag{155}\\ x&=X_J+(c-C_r)V_J+s_xx',& p'&=x'-c'v'. \tag{156}\end{align*}\] Then \(x-cv=X_J-C_rv+s_x(p'+v'/2)\). Subtract \(\Phi(C_r,X_J,v)\) and divide the phase by \(s_cs_v^2\). The new gradient function is \[ \zeta'(v',p')= \frac{\zeta(v,X_J-C_rv+s_x(p'+v'/2)) -\zeta(v,X_J-C_rv)}{s_x}. \tag{157}\] In particular, \(\partial_{p'}\zeta'=\partial_p\zeta(v,X_J-C_rv+s_x(p'+v'/2))\). Symmetry, definiteness, and uniform derivative bounds persist, with changed fixed constants. Differentiating this formula gives all the required derivative bounds; the affine substitutions have bounded derivatives, and \(s_v,s_x\le1\). The new parameters and input are \[ H_{\mathrm P}'=H_{\mathrm P}-r-2b_*,\quad L'=L-r-b_*,\quad f'(v')=s_v f(V_J+s_vv')e^{i\lambda\Phi(C_r,X_J,V_J+s_vv')}. \tag{158}\] The input change preserves its \(L^2\) norm and the pairing (146). At local endpoint time \(t-r\), the velocity and position depths are \(u_t-b_*\) and \(u_t+t-r-b_*\). Also \(H_{\mathrm P}'\ge t-r>0\), and the new aperture widths are \((b-b_*,a-b_*)\). Their weight is \(w(b,a)-2b_*\). Conversely, a local root aperture becomes an aperture at \(r\) after adding \(b_*\) to both widths. Thus returning a score to the original coordinates adds \(b_*\) to its weight contribution. The local inputs used below are sums of windows with centers in a fixed enlargement of the velocity cell of \(J_0\). Their supports are therefore bounded in \(v'\), uniformly in the localizer.

A finite Fourier decomposition with quantified tails

We state error sizes before taking any limits. At fixed depths and bounded domains all the observation lattices, retained atom sets, and cap menus below have at most \(C\epsilon^{-Q}\) elements for some fixed \(Q\). The logarithmic root cutoff contributes a factor which can also be absorbed in this bound by increasing \(Q\). This exponent does not bound the number of arbitrary extra labels; those are accounted for separately.

The tolerance in (161) is \(\sigma_r\), the earlier position spacing. At the endpoint uncertainty scale it can be larger than \(\sigma_t\) by a power. Thus this lemma alone does not identify coherent packet masses with a positive law of straight trajectories at that finer precision.

Classes with cap count bounds

Here are explicit conventions for the finite errors in the remaining statements. Let \(\xi>0\) be the mesh for coefficient and count exponents, and let \(S\ge1\) bound the number of extra subdivision and winner tags at each geometric localizer. Constants implicit in \(S\) may include powers of \(1+\ell\) from dyadic shape menus. With fixed data, \(\eta,\xi,A\), an admissible packet error is \[ e_{\rm pkt}(\epsilon)= C(\eta+\xi+\log_{1/\epsilon}\!\left(S\right)) +C_{\mathrm{data},\eta,\xi,A} \frac{1+\log(2+\ell)}{\ell}. \tag{167}\] The constants may be increased a finite number of times. There is no convergence assertion uniform as \(\eta\) tends to zero: the order is first \(\epsilon\to0\), then \(\eta,\xi\to0\), with \(\log_{1/\epsilon}\!\left(S\right)\to0\). If a prescribed final error is needed, fix its small \(\eta,\xi\) first and choose all integration orders and \(\epsilon_0\) afterwards.

For an endpoint observation in a localizer, choose a winning class by the largest absolute pairing with its class superposition. Here and later it is enough to keep endpoint masses satisfying \[ m_O\ge\epsilon^{A_0}\|f\|_2^2 \tag{171}\] for a sufficiently large fixed \(A_0\). To see this without any assumption on the law, normalize \(\|f\|_2=1\). The number of endpoint sites is at most \(C\epsilon^{-Q}\), and \(w\) is bounded by a fixed constant \(W\). A law on masses smaller than the displayed threshold has score at most \(\frac32 Q+\frac12W-\frac32A_0+O(1/\ell)\), whereas \(K_0\ge0\). Choose \(A_0>Q+W+2\). Splitting any law into this part and its complement changes its averaged score by at most \(3\log 2/(2\ell)\). The low-mass part consequently cannot increase a nonnegative upper growth bound.

Choose the tail order with \(A>A_0/2+1\). Write \(q=C_A\epsilon^{A-A_0/2}\), and reduce \(\epsilon_0\) so \(q<1/2\). If there are at most \(S\) possible winning parts, the triangle inequality gives the finite bound \[ m_O(f)\le (1-q)^{-2}S^2m_O(h_J^{\rm win}) \le4S^2m_O(h_J^{\rm win}). \tag{172}\] Its loss in the mass term of the score is at most \(3\log_{1/\epsilon}\!\left(S\right)+3\log_{1/\epsilon}\!\left((1-q)^{-1}\right)\). Thus no power-sized tail is discarded without comparison to an output mass floor. The label \(J\) below consists of \(J_0\), the mass and cap bins, and any extra subdivision and winning tags. The geometric part is determined by both endpoint data and aperture; the additional tag entropy is bounded by \(\log_{1/\epsilon}\!\left(S\right)\).

The local root and the earlier test

The preliminary growth bound

An error budget and a finite composition inequality

Write \(\ell=\log(1/\epsilon)\); all entropies and all logarithms denoted by \(\log_{1/\epsilon}\!\left(\cdot\right)\) in this section are divided by \(\ell\). A fixed collection of bounds \(\mathcal B\) will specify bounded ranges for \(H_{\mathrm P},L,t,r\), bounded spatial and velocity domains, the phase and amplitude bounds, and a positive lower bound for each time gap to which a growth estimate is applied. In particular, \(\mathcal B\) is fixed before \(\epsilon\) tends to zero. Its size may change in a later outer sequence.

For an array of conditional systems these bounds, all grid and observation-count bounds, and every vanishing-error envelope are common deterministic bounds for the whole array. This condition is preserved here: the translated phase bounds are uniform over the localizers, the velocity supports are cut into a fixed bounded domain, and a single truncation order, coefficient threshold, depth mesh, and tag-count bound are used for every retained \(J\). Thus every selection of one conditional system at each precision is an admissible sequence with the same bounds. Individual extendibility to unspecified vanishing-error envelopes would not suffice.

There is a useful precise way of using the definition of \(\beta_{\mathrm q}\). For systems obeying \(\mathcal B\), let \[ \omega_{\mathcal B}(\epsilon) =\sup\bigl(P_d-K_0-\beta_{\mathrm q}d\bigr)_+, \tag{182}\] where the supremum also includes the bounded translated systems in Lemma 37, and \(d\) ranges over the prescribed compact subset of \((0,\infty)\). Enlarging the fixed bounds slightly to accommodate rounded grids is understood. Then \(\omega_{\mathcal B}(\epsilon)\to0\). Indeed, a failure would select a sequence of admissible systems with a fixed positive excess; a subsequence of their depth parameters converges, contradicting the definition of \(\beta_{\mathrm q}\). This is a modulus for a fixed class, not a modulus uniform over growing values of \(\mathcal B\). At duration zero the corresponding estimate is the definition of \(K_0\), with zero error.

Here is one error budget that will be used below. Choose a depth mesh \(\xi>0\), a packet enlargement exponent \(\eta>0\), and let \(S_\epsilon\ge2\) bound the number of auxiliary choices at a localizer: shape bins, mass bins, cap-class tags, winning tags, and any prescribed further subdivisions. The geometric localizer itself is not one of these choices; its entropy will be kept. On the output tests under consideration assume \[ m_O\ge\epsilon^{A_0}\norm{f}_2^2. \tag{183}\] Take the order of the packet truncation large enough that its amplitude error is at most \(C_A\epsilon^A\norm{f}_2\), with \(A>A_0/2\), and put \[q_\epsilon=C_A\epsilon^{A-A_0/2}<\tfrac12.\] For a sufficiently large fixed \(C_{\mathcal B}\), we may use \[ \mathcal R= C_{\mathcal B}(\eta+\xi+\log_{1/\epsilon}\!\left(S_\epsilon\right)) +C_{\mathcal B,\eta,\xi,A} \frac{1+\log(2+\ell)}{\ell} +3\log_{1/\epsilon}\!\left(\frac 1{1-q_\epsilon}\right) +\omega_{\mathcal B}(\epsilon). \tag{184}\] Each occurrence below of a fixed enlargement of \(\mathcal B\) is included in this choice. One may instead use the sum of the separate error bounds in Lemmas 37 and 38 and Proposition 39; only their upper bound by \(\mathcal R\) is needed.

With \(S=S_\epsilon\) and \(q=q_\epsilon\), the winner estimate (172) gives the two contributions \(3\log_{1/\epsilon}\!\left(S_\epsilon\right)\) and \(3\log_{1/\epsilon}\!\left(1/(1-q_\epsilon)\right)\) to the loss in the score’s mass term. Its argument uses only \(q_\epsilon<1/2\), so the present choice \(A>A_0/2\) suffices. The other terms in (184) cover the tag entropies, cap enlargement and binning, bounded grid ambiguities, and the logarithmically growing root cutoff.

The floor also preserves saturation. After normalizing \(\norm{f}_2=1\), the calculation at (171), with the fixed weight bound from \(\mathcal B\), lets us choose \(A_0\) so that every low-mass law has score below \(-1\), whereas \(K_0\ge0\). The two-part split costs at most \(3\log 2/(2\ell)\). For \(\beta_{\mathrm q}>0\), the low-mass probability therefore tends to zero on a saturating sequence, and the complementary conditional law still saturates.

What saturation permits one to select

In the remainder of the packet argument, saturation and its successive limits have this meaning. The class of arrays includes the arrays obtained by the finite selections and fixed positive duration normalizations below. Here is the relevant closure justification. At a fixed cut, the four nonnegative composition deficits have total mean at most \(\Delta_j+o_n(1)\) plus the chosen finite error budget. On any event of mass at least a fixed \(p_0>0\), their conditional mean is bounded by this quantity divided by \(p_0\). Markov selection therefore gives conditional local families whose outer deficits tend to zero. They obey fixed enlarged bounds \(\mathcal B'_j\) within each row, independently of which localizer is selected. The earlier test retains its full selected localizer marginal and has the same conclusion by the retained-score inequality.

For each row, make only finitely many cuts with fixed positive gaps. Depth and score units have been divided by the initial duration. Then choose the depth meshes, packet enlargement exponents, and the permitted exponent of the additional tag count as small as required. Choose the truncation order after the mass floor and these parameters; finally choose \(\epsilon\) small enough for (184) and all of the finitely many universal moduli to have the required size. Equivalently one may take the inner \(\epsilon\to0\) limits first, then send the mesh and enlargement parameters to zero, and only afterwards take the outer extremizing sequence. No estimate here requires uniformity as \(H_{\mathrm P}\), the aspect, or the phase bounds grow in that outer sequence.

If \(\Delta_t\to0\) in this order, all four nonnegative deficits vanish in the sense of (188). In particular \(K_J=A_J+o(1)\) and \(P_J=K_J+\beta_{\mathrm q}(t-r)+o(1)\) on selected typical conditional systems. The retained earlier law has deficit tending to zero by (189). To be explicit, its deficit is at most \[d_0+\tfrac32\mathbb Ed_1+\mathcal R \le\tfrac32 R_t+\mathcal R.\] For this earlier test one keeps the whole marginal on \(J\). Selecting a single value of \(J\) is appropriate when testing a local growth bound, and does not replace that marginal in the earlier test.

Further splitting has an equally finite justification. If a label \(B\) takes at most \(S\) values, the score of a law and the average scores of its conditional laws differ by at most \(\tfrac32\mathsf H(B)\le3\log_{1/\epsilon}\!\left(S\right)/2\); this follows directly from the two entropy terms in the packet score. The input and its canonical root score are the same in these comparisons. Add the uniform growth error to make the conditional deficits nonnegative, and use Markov’s inequality. Thus a split made before composition preserves saturation on typical branches, also within any event whose probability is bounded below. For an event of probability \(p\) tending to zero, the required condition is explicitly that the available deficit and error be \(o(p)\); subpower probability by itself is not this condition. More generally, if a sum of nonnegative shifted deficits has mean \(D\), its mean on a current event of probability \(p\) is at most \(D/p\); outside a set of relative probability at most \(q\) there, the sum is at most \(D/(pq)\) pointwise. Polynomially many geometric localizers are handled by retaining their entropy, rather than by declaring the cost of fixing one negligible. The conditional local estimate, on the other hand, is pointwise and uniform in the inner limit, so a violating local system can always be selected without a probability loss argument.

One can also specify the probability of a chosen tag. If a union of tag branches has probability \(p\), discard tags of probability below \(p/(4S)\), losing at most \(p/4\). If their total nonnegative shifted deficit is at most \(R\), discard branches with conditional deficit greater than \(4R/p\), losing at most another \(p/4\). Some remaining tag has probability at least \(p/(4S)\) and conditional deficit at most \(4R/p\). Its logarithmic retention cost is at most \(\log_{1/\epsilon}\!\left(4S/p\right)\). For fixed \(p>0\) and \(\log_{1/\epsilon}\!\left(S\right)\to0\), this is a subpower retention with a quantified deficit. Apply composition anew on that tag, keeping its marginal on \(J\) for the earlier test. Repeating this argument a fixed number of times gives, for any prescribed accuracy \(\vartheta>0\), a branch on which the endpoint deficit, the mean retained-count loss, the local root comparison deficit, the earlier retained-law deficit, and the upper error in each specified admissible score-replacement comparison are all at most \(\vartheta\). For the last assertion, the universal bound for a competing endpoint test gives \(P_t^{\rm new}-P_t\le\Delta_t+\omega_{\mathcal B}\). Here \(p\) is the probability of the relevant event in the current normalized law. The displayed simultaneous selection divides the mean deficit by \(p\), not by the probability of the individual tag; it requires no estimate of the form \(R=o(1/S)\). For a subsequent event, apply the same rule to its probability in the then current law and choose the incoming errors accordingly. Thus a finite subsequent argument may use \(\vartheta\) to denote the maximum of these selected errors; it need not identify it with the original unsplit deficit.

These rules also apply to any fixed finite list of observations and cuts, using the sum of their errors and a union bound. They concern score differences. They do not require the separate scores to have finite outer limits, or a uniform differentiability modulus for any limiting entropy profile.

Faithful classical tests away from the boundary

The two ways to continue backwards

For a packet system write \(K_0^{\ge u_0-\nu}\) for the supremum of its actual root aperture scores with smaller width \(b\ge u_0-\nu\), excluding the total norm option. Call a duration-one saturation array in the sense of Definition 42 forced at the root if there are fixed \(\nu,\kappa>0\) and an outer subsequence of rows on which \[ K_0^{\ge u_0-\nu}\le K_0-\kappa \tag{195}\] in each row’s inner limit. The quantifier ranges over all such saturation arrays, including arrays of normalized conditional systems. Thus the negation excludes a forced array obtained by selecting and rescaling local systems as well as an originally chosen one.

The constants \(\nu,\kappa>0\) must be fixed across an outer subsequence of rows, with the asserted gap holding in each row’s inner limit (and with a smaller fixed gap if necessary). A window or gap tending to zero with the row index does not constitute a forced array. In the forced case the backward step \(h\) is chosen once from these fixed constants and the fixed number of steps. In the absence of a forced array, a fixed cap excess at a fixed cut would produce one by the local-root comparison; hence the isotropic alternative applies with a step chosen once from its fixed time interval and the number of steps. In either case \(h\) is fixed before \(j\to\infty\), so the outer deficits are \(o(h)\).

The incoming fine-velocity information

Here is the precise meaning of the velocity information carried between steps. At time \(t\), let \[u_t=(H_{\mathrm P}-t)/2,\qquad P_f=(C_t,\text{isotropic velocity and position cells of width } \epsilon^{u_t-h}).\] The position width in this formula is \(\epsilon^{t+u_t-h}\). Fixed dilations and translations of the grids are allowed. Additional tags are allowed only if their conditional number, given the geometric part of \(P_f\), is at most \(\epsilon^{-\alpha}\); their cost is included in \(\alpha\).

In the elongated case \(n_f(P_f)\) is a unit normal, modulo sign, and the normal of the aperture \(D\) is within \(C\epsilon^h\) of \(n_f(P_f)\). The label \(W_f\) locates the velocity component along \(n_f(P_f)\) in intervals of length \(\epsilon^{u_t}\). In the isotropic case \(W_f\) locates both velocity components at that precision. An incoming credit \(k_f\geq0\), with error \(\alpha_f\), means that on the normalized law being used, \[ \sup_w\mathbb P(W_f=w\mid P_f=p)\leq \epsilon^{\,k_f-\alpha_f} \quad\text{for every \(p\) of positive probability}. \tag{198}\] In particular \(\mathsf H(W_f\mid P_f)\geq k_f-\alpha_f\). On the first step we set \(k_f=0\) and impose no incoming direction or fine-velocity condition.

We will use the following exact retention rule. It specifies how (198) is maintained when selecting branches.

Every removal below is also a score split under Section 10; its probability alone does not control the remaining score.

Constructing the finite cut

A prepared cut starts with a boundary step from \(r=t-2h>0\) to \(t\), with \(u_t>h\), as supplied by Proposition 44. The current endpoint law will be denoted by \(\pi^{\rm in}\). It is a probability law on a finite sample space \(\Omega\). A sample \(\omega\) records \(O(\omega),D(\omega)\), all incoming labels \(P_f,W_f\) and their extra tags when present, and any auxiliary randomness already used to choose those labels. In particular, an incoming label need not be a deterministic function of \((O,D)\). For any endpoint law \(\nu\) on \(\Omega\), write \(P_t(\nu)\) for the score of the original input with its original masses. If \(k_f>0\), the bound (198) holds under \(\pi^{\rm in}\), with its current error. On the first step \(k_f=0\).

The finite geometry is the following. The endpoint shape has \(a=u_t\) to depth error at most \(\eta\). In the isotropic alternative the apertures are isotropic and \(b_*=0\). In the elongated alternative \(b_*=b\), the endpoint aspect \(e=a-b\) is at least \(e_0>0\). In that case require \[ C_*(h+\eta)<\gamma e_0 . \tag{200}\] The grids and aperture menu admit all coarsenings below. We take \(C_*\ge2\). Thus in the elongated case \(b\le u_t+\eta-e_0\le u_t-h\), after increasing \(C_*\) if necessary; in the isotropic case \(b_*=0<u_t-h\). We will use \[ b_*\le u_t-h \tag{201}\] when comparing the incoming coarse label with the localizer.

Use the fixed analytic, domain, depth, grid, and count bounds of Section 10. For supremum masses, fix at each site a test within a fixed factor of the supremum and call its mass \(m_O(f)\). This changes any endpoint score by at most \(C/\ell\), included in \(\mathcal R\), and the Bessel bounds are uniform in these choices. Apply the floor \(m_O(f)\ge\epsilon^{A_0}\|f\|_2^2\) of (171) and the fixed-shape score splits, recording their probabilities for credit retention.

Fix the depths, then \(\theta<h/10\) and \(\xi\). By (201), \(\theta<(u_r-b_*)/4\), as required by Lemma 37. Choose its amplitude error \(C_A\epsilon^A\|f\|_2\) with \(A>A_0/2+1\). If \(Q_0\) bounds all relevant count exponents, choose the coefficient cutoff \[ B>2A+3Q_0+10 . \tag{202}\] uniformly over geometric localizers. The later transport order may be increased after this coefficient family is fixed.

For one geometric localizer \(J_0\), let \(\iota\) range over its coefficient bins and greedy cap classes. The associated whole class is a finite set \(\mathcal T_\iota^{\rm all}\) of earlier atoms, with coefficient mass bin \(m(\iota)\), cap maximum bin \(g(\iota)\), and assigned cap \(E_\iota(T)\). Lemma 38 gives the cap upper bound on this set and the lower bound on each of its whole assigned fibers. At this point \(\iota\) indexes available classes; no class has yet been assigned to an endpoint sample as its final winner. The buffer used next is chosen for all possible contributors in this finite family, before any of its classes is subdivided.

During preparation, a replacement loss is the increase in score when the aperture is changed at fixed masses, evaluated on the specified current marginal. Section 10 supplies these losses from the current deficits and charges each split. This also applies to the two-copy marginals in the next lemma, since adjoining the unsplit conditional copies preserves the old marginal.

For every possible contributor, the velocity difference from its output center is \(O(\epsilon^{u_t})\). After referring the positions to \(C_r\), their difference is \[O(\epsilon^{r+u_t})+ O(\epsilon^{-\theta}\epsilon^{r+u_r}) =O(\epsilon^{r+u_t}), \qquad u_r=u_t+h,\quad \theta<h/10.\] Use isotropic cells with side \(L_{\rm grid}\) times \(\epsilon^{u_t}\) in velocity and \(\epsilon^{r+u_t}\) in position, where \(L_{\rm grid}\) is a sufficiently large fixed constant. A uniformly random translation places any output center farther than these possible differences from every face with probability at least \(3/4\). Averaging over translations therefore gives one translation for which a part of at least that probability has the coarse label \[P=(C_r,V_{u_t},(X+(C_r-C_t)V)_{r+u_t})\] in common with every possible contributing earlier atom. Retain this part as a score split and denote its normalized endpoint law by \(\pi^{\rm buf}\). Its probability and deficit are charged by the finite selection rule above. This completes the geometric preparation in the isotropic case.

The subclasses and the current endpoint law

Retain the buffered and, in the elongated case, predicted-normal parts just constructed. The table \(n(P)\) and the translated grid are now fixed. Their selection probabilities, together with the earlier mass-floor and shape selections, are included in one preparation event \(A_{\rm pre}\) in the original sample space \(\Omega\). We will restore the incoming conditional probability bound after the atom partition and the winning map have been fixed.

For an earlier atom \(T\), let \(P(T)\) be its buffered coarse cell. In the elongated case first mark the atom according to whether the normal of its assigned aperture \(E_\iota(T)\) lies within \(2C\epsilon^h\) of \(n(P(T))\). This divides each whole class into aligned and misaligned atoms; both parts remain available when the winner is chosen. In the isotropic case no such mark is needed.

Let \(W(T)\) record the velocity at depth \(u_r=u_t+h\). It records the component along \(n(P(T))\) in the elongated case and both components in the isotropic case. These are the original depth units; after subtracting \(b_*\), the fine spacing is \(\epsilon^{u_r-b_*}\). Within each marked part of \(\mathcal T_\iota^{\rm all}\), and for each \(P\), bin \[\log_{1/\epsilon}\!\left(\#\{T:(P(T),W(T))=(p,w)\}\right)\] with mesh \(\xi\). Among fibers in one such size bin, bin the logarithm of the number of represented \(W\)’s at \(P\), again with mesh \(\xi\). Write \(k\ge0\) for this second numerical tag and \(d=1\) in the elongated case, \(d=2\) in the isotropic case. For each resulting subclass and every represented \(P\), \[ \epsilon^{-k+\xi}\leq \#\{W\mid P\}\leq\epsilon^{-k-\xi}, \qquad k\leq dh+C\xi+\frac{\log C}{\ell}. \tag{207}\] The last bound covers a velocity cell of width \(C\epsilon^{u_t}\) by cells of width \(\epsilon^{u_r}\) in \(d\) dimensions. The sizes of two nonempty \((P,W)\) fibers in this subclass differ by at most \(\epsilon^{-\xi}\). Hence its uniform atom law satisfies \[ \mathbb P(W=w\mid P,J)\leq\epsilon^{k-2\xi}. \tag{208}\] Here \(J\) will be the label of the subclass when it is chosen. The subdivision groups the numerical fiber-size and \(k\) tags across different \(P\)’s; \(P\) itself remains random in the subclass and is not appended as an auxiliary value of \(J\). This distinction keeps the later conditional cost of \(J\) small.

These are the final subdivisions that change atom sets. Later tags, such as total size, label a whole subclass. Let \(S_\epsilon\) bound the number of available subclasses at a geometric localizer; at fixed depths \(\log S_\epsilon=o(\ell)\). Their labels contain \(J_0\), the \(\iota\) tags, the alignment mark when present, the two numerical velocity tags, and any whole-subclass tags. The winner map will choose one such tuple \(J(\omega)\). For its atom set \(\mathcal T_J\), set \(\mathcal T_J^{\rm all}:=\mathcal T_\iota^{\rm all}\), \(E(T):=E_\iota(T)\), and \[N(J)=\#\mathcal T_J,\qquad N_{\rm all}(J)=\#\mathcal T_\iota^{\rm all},\qquad m(J)=m(\iota),\qquad g(J)=g(\iota).\] Thus \(N_{\rm all}\) still counts the whole greedy class before alignment and velocity subdivision. Its cap upper bound persists on \(\mathcal T_J\); its assigned-fiber lower bound continues to refer to the whole class.

Write \(\mathfrak a_O(\varphi)\) for the normalized pairing with the fixed site test, so \(m_O(\varphi)=|\mathfrak a_O(\varphi)|^2\), and \(h_J=\sum_{T\in\mathcal T_J}c_Tg_T\). Put \[q_0=C_A\epsilon^{A-A_0/2}<\tfrac12.\] Choose at each retained sample a subclass maximizing \(|\mathfrak a_O(h_J)|\), breaking ties by a fixed rule. All atom sets, superpositions, and this map to \(J\) are now fixed. Applied to these final subclasses, (172) gives \[\begin{align*} |\mathfrak a_O(h_J)| &\ge\frac{1-q_0}{S_\epsilon}\sqrt{m_O(f)} \ge\frac{1-q_0}{S_\epsilon}\epsilon^{A_0/2}\|f\|_2, \tag{209}\\ m_O(f)&\le (1-q_0)^{-2}S_\epsilon^2m_O(h_J) \le4S_\epsilon^2m_O(h_J). \tag{210}\end{align*}\] The mass-term loss \(3\log_{1/\epsilon}\!\left(S_\epsilon\right)+3\log_{1/\epsilon}\!\left(1/(1-q_0)\right)\) is included in \(\mathcal R\). In the elongated case both alignment marks participated in this choice.

After this finite menu, winner map, and coefficient cutoff are fixed, Lemma 37 allows one common transport power \[ A_{\rm tr}>(B+Q_0+1)/2 \tag{211}\] for every retained subclass, without a new coefficient cutoff. Its constants depend only on the fixed data and \(Q_0\). This order controls the alignment collapse and the later candidate truncation, and is unchanged by subsequent subclass selections.

Take the endpoint law conditioned on \(A_{\rm pre}\). If \(k_f>0\), apply Lemma 45 to this event, using its probability under \(\pi^{\rm in}\), and remove the exceptional \(P_f\) fibers. This changes the endpoint law and its full \(J\) marginal, while leaving the atom sets, counts, \(g,m\), and winner map fixed. The pointwise geometric and mass comparisons persist. Apply Proposition 41 anew to this law; its new deficit inequality controls the mean \(\mathbb E\log_{1/\epsilon}\!\left(N_{\rm all}/N\right)\) under its new \(J\) marginal.

For each later numerical-tag or alignment restriction \(A\), use its probability \(p_A\) in the current normalized law. When \(k_f>0\), repeat the retention lemma and the exceptional-fiber score split. In either case reapply composition with the same frozen data. Each retention application adds \[ \zeta+\log_{1/\epsilon}\!\left(1/p_A\right) \tag{212}\] to the incoming error. Use the weighted tag rule of Section 10 with its current union mass \(p\) and shifted deficit \(R\): it supplies probability at least \(p/(4S_\epsilon)\) and conditional deficit at most \(4R/p\). Only a prescribed finite number of these operations is used.

The following exclusion determines which marked subclasses can remain.

For the elongated application, first select a branch with qualifying common earlier shape bins by the weighted rule, perform credit retention when needed, and recompose. Let \(\nu\) be this current law and let \(B\) be all its misaligned winners. This event is \(J\)-measurable. If \(\nu(B)=0\), the law is already aligned; otherwise (213) applies to its entire current mass. At sufficiently small current deficit, the aligned complement has positive probability, tending to one with the positive gap fixed. Restrict to that complement, then apply the current-law retention and recomposition rule. In either alternative, apply the same rule to any remaining common numerical-bin selections and recompose after the last restriction. Denote the resulting endpoint law on \(\Omega\) by \(\pi\), and now set \(P_t=P_t(\pi)\), the score of the original input with its original masses. Every final elongated winning class is aligned. The buffered property, and prediction when present, persist pointwise. The incoming bound is \[ \sup_w\pi(W_f=w\mid P_f=p) \le \epsilon^{k_f-\alpha_f} \quad\text{for every \(p\) with \(\pi(P_f=p)>0\)}, \tag{218}\] where \(\alpha_f\) includes the initial error and the sum of (212). When \(k_f=0\), this estimate will be replaced by nonnegativity of conditional mutual information.

Every expectation in the rest of this cut uses the selected marginal \(\pi_J\), unless a different law is explicitly displayed. In particular, set \[\begin{align*} d_1(J)&=\log_{1/\epsilon}\!\left(N_{\rm all}(J)/N(J)\right),\\ A_J&=\log_{1/\epsilon}\!\left(N(J)\right)+\tfrac32\log_{1/\epsilon}\!\left(m(J)\right)+\tfrac12g(J),\\ K_r^*(\pi)&=\mathsf H_\pi(J)+b_*+ \mathbb E_\pi\left[\log_{1/\epsilon}\!\left(N_{\rm all}(J)\right) +\tfrac32\log_{1/\epsilon}\!\left(m(J)\right)+\tfrac12g(J)\right], \\ B_r&=\mathsf H_\pi(J)+b_*+\mathbb E_\pi A_J =K_r^*(\pi)-\mathbb E_\pi d_1 . \tag{219}\end{align*}\] The last equality is exact. The number \(B_r\) is a benchmark formed with retained counts; it is not asserted to be an earlier score. For the final composition law, put \(\Delta_t=K_0+\beta_{\mathrm q}t-P_t\). Its current \(d_0\) gives \[ B_r+2\beta_{\mathrm q}h-P_t =\Delta_t+2\mathcal R-d_0-\mathbb E_\pi d_1 \le\Delta_t+2\mathcal R . \tag{220}\]

The earlier whole-class law and the returned retained-class law are, respectively, \[ \mu^{\rm all}(J,T)=\pi_J(J) \frac{\boldsymbol1_{\{T\in\mathcal T_J^{\rm all}\}}}{N_{\rm all}(J)}, \qquad \mu(J,T)=\pi_J(J) \frac{\boldsymbol1_{\{T\in\mathcal T_J\}}}{N(J)}. \tag{221}\] Both use the selected marginal \(\pi_J\). Their observations are the sites of \(T\) at time \(r\), and their masses are coefficient tests of the original input \(f\). With assigned caps \(E(T)\), finite composition gives \[\begin{align*} P_r^{\rm all}&\ge K_r^*(\pi)-\mathcal R,\\ P_r^{\rm ret}&\ge K_r^*(\pi)-\tfrac32\mathbb E_\pi d_1-\mathcal R =B_r-\tfrac12\mathbb E_\pi d_1-\mathcal R . \tag{222}\end{align*}\] In the elongated case \(\mu\) continues the aligned assigned cap. In the isotropic case the same atom law uses the exact isotropic boundary aperture of width \(u_r\). The horizon and packet menu include this cap by Proposition 44. With \(b_*=0\), the isotropic score satisfies the separate bound \[ P_r^{\rm iso}\ge B_r-\tfrac12\mathbb E_\pi(g-2u_r)-C\mathcal R . \tag{223}\] This uses weight \(2u_r\) and nonnegative conditional entropy, independently of the assigned-fiber gain. A charged selection fixed the common \(g\) bin, and the final credit trim, when needed, preserved its pointwise support before \(\pi\) was named. Thus \(g-2u_r\) has the required small error under the displayed expectation.

Finally, (208) applies under \(\mu(\,\cdot\mid J)\). Since the numerical \(k\) tags are common bins under \(\pi_J\), averaging in \(J\) gives \[ \mu(W=w\mid P=p)\le\epsilon^{k-C_*\eta} \quad\text{for every \(p\) with \(\mu(P=p)>0\)}. \tag{224}\] This statement holds for either earlier aperture assignment under \(\mu\), which is uniform within each retained class.

From winning masses to candidate masses

Before stating the repayment estimate, we finish the finite comparison of masses that its proof will use. Work in the translated coordinates of a selected localizer \(J\), put \(u=u_t-b_*\), and write \[\rho=\epsilon^u,\qquad s=\epsilon^{u+h},\qquad \sigma=\epsilon^{u+2h}=s^2/\rho,\qquad \lambda_{\rm loc}=s^{-2}.\] The buffered \(P\) cell has width \(\rho\). Let \(Q=Q(J,O)\) record \(P,C_t\), and the position cell of side \(s\) at \(C_t\). For each such \(Q\), let \(\mathcal T_{J,Q}\subset\mathcal T_J\) consist of the atoms at this \(P\) whose transported positions are within \(C_0 R s\) of that cell, where \(R=\epsilon^{-\theta}\) and \(C_0\) is a fixed constant large enough for the transport and cell-boundary enlargements. Absorb \(C_0\) into the fixed constants in the coherence estimates. Write \(N_{J,Q}=\#\mathcal T_{J,Q}\) and \[h_{J,Q}=\sum_{T\in\mathcal T_{J,Q}}c_Tg_T.\] Let \(Z=Z(J,O)\) add to \(Q\) the output position at precision \(\sigma\) along \(n(P)\) in the elongated case and in both directions in the isotropic case. These labels are functions of the geometric endpoint observation and the fixed grid and normal table; they do not include the arbitrary extra labels carried by \(\omega\). On the full sample space write \(Q(\omega)=Q(J(\omega),O(\omega))\) and likewise for \(Z(\omega)\).

The winning superposition \(h_J\) is fixed, while \(h_{J,Q}\) depends on \(Q\). On every retained endpoint site, the buffered property and the transport order fixed after the winning map give \[ |\mathfrak a_O(h_J)-\mathfrak a_O(h_{J,Q(J,O)})| \le C_{\rm tr}\epsilon^{A_{\rm tr}}\|f\|_2 . \tag{225}\] The coefficient and count comparison in the original construction is still available: the squared summed tail has exponent beyond \(B+1\) after the \(Q_0\) count loss. For the score we use the stronger information that the winning amplitude itself has the floor (209). Set \[q_{\rm tr}= \frac{C_{\rm tr}S_\epsilon}{1-q_0} \epsilon^{A_{\rm tr}-A_0/2}.\] The choices \(B>2A+3Q_0+10\), \(A>A_0/2+1\), and (211), together with the subpower bound on \(S_\epsilon\), make \(q_{\rm tr}\to0\) at fixed finite parameters. Choose \(\epsilon\) small enough that \(q_{\rm tr}<1/2\). Equations (209) and (225) then imply \[ m_O(h_J)\le(1-q_{\rm tr})^{-2}m_O(h_{J,Q(J,O)}). \tag{226}\] In particular \(m_O(h_{J,Q(J,O)})>0\) and \(N_{J,Q(J,O)}>0\) on every retained sample. Replacing the winning mass by this candidate-truncated mass costs at most \[ 3\log_{1/\epsilon}\!\left(1/(1-q_{\rm tr})\right) \tag{227}\] in the score’s mass term. This cost is included in \(\eta\).

For a fixed \(J,Q\), define \[ m_Z=\sum_{\substack{O\mapsto Z\\ O\ {\rm in\ the\ localizer}}} m_O(h_{J,Q}) . \tag{228}\] The sum uses every geometric output site mapping to \(Z\) once, with the allowed bounded lattice multiplicity and its chosen site-dependent amplitude; it does not sum over copies of a site in \(\Omega\). Because \(Z\) includes \(Q\), the entire fiber uses one input \(h_{J,Q}\). Thus \(m_Z\) is exactly the candidate-truncated mass to which the next lemma applies. The input \(h_{J,Q}\) varies with \(Q\). These masses are used only to bound an entropy expression under \(\pi\); they are not treated as a new admissible packet system for a growth estimate.

The repayment estimate

The prepared cut supplies \(\mu\) and the capacity bound (207). The remaining estimate shows that a near-maximal endpoint score increases the fine-velocity credit.

A coherent estimate on the full set of output sites

We next prove the mass estimate before introducing any positive law on candidate packets. The output amplitudes may be chosen separately at every site.

The candidate law and its entropy bound

We now prove Proposition 48. For a value \(J=j\) of positive \(\pi_J\)-probability, let \(\pi_j\) denote the selected endpoint law conditional on \(J=j\). Define the finite entropy expression \[\mathcal E_j= \mathsf H_{\pi_j}(Z)+\tfrac12\mathsf H_{\pi_j}(Z\mid D) +\tfrac32\mathbb E_{\pi_j}\log_{1/\epsilon}\!\left(m_Z\right) +\tfrac12\mathbb E_{\pi_j}w_{\rm loc}(D).\] All entropies here concern the displayed geometric observations, rather than the entire sample \(\omega\). The latter carries extra labels only so that later augmentation preserves them. Finite composition first compares the original score with the conditional winning scores. Equation (226) then replaces their masses by the candidate-truncated masses. Finally, conditional log-sum on each \(O\mapsto Z\) fiber bounds the two entropy–mass terms by those in \(\mathcal E_j\). The unconditional fiber calculation uses all sites mapping to \(Z\); conditional on \(D\), its support is a subset of those sites. Thus \[ P_t\le \mathsf H_\pi(J)+b_*+\mathbb E_{\pi_J}\mathcal E_J +\mathcal R+3\log_{1/\epsilon}\!\left(1/(1-q_{\rm tr})\right). \tag{234}\] This is a comparison of finite expressions under the same endpoint law. No growth bound has been applied to the \(Q\)-dependent inputs.

Lemma 49 now gives its two exact bounds for the masses \(m_Z\) in (228). Only after these analytic inequalities do we introduce a positive candidate law. Adjoin an earlier atom to the full endpoint sample by \[ \widetilde\pi(\omega,T)=\pi(\omega) \frac{\boldsymbol1_{\{T\in\mathcal T_{J(\omega),Q(\omega)}\}}} {N_{J(\omega),Q(\omega)}} . \tag{235}\] The denominator is positive on every sampled value, as proved above. This law has exactly the endpoint marginal \(\pi\), including \(P_f,W_f\) and every stochastic incoming tag. Conditional on \((J,Q)\), the atom is uniform on the candidate set and independent of every other endpoint label. Its marginal on atoms is generally biased; it is different from \(\mu\), which is uniform within each retained class.

For the next calculation fix \(J=j\), and write \(\mathsf H_j,\mathbb E_j\) for entropy and expectation under \(\widetilde\pi(\,\cdot\mid J=j)\). Put \[B_{1,j}=\mathbb E_j\log_{1/\epsilon}\!\left(N_{J,Q}\right),\qquad S_{t,j}=\mathsf H_j(C_t\mid T),\qquad \mathcal M_j=\mathbb E_j\log_{1/\epsilon}\!\left(m_Z\right).\] Since \(Z\) includes \(Q\), the sampling rule gives exactly \[\mathsf H_j(T\mid Z,D)=\mathsf H_j(T\mid Z)=B_{1,j}.\] Candidate proximity gives \(\mathsf H_j(Q\mid T,C_t)\le C_*\eta\). It also gives \(\mathsf H_j(Z\mid T,D)\le C_*\eta\). For the latter bound, \(T,C_t\) locate the position to precision \(s\), while \(D\) supplies its normal component at precision \(\sigma\). In the elongated case the current prediction \(\angle(n(D),n(P))\le C\epsilon^{4h}\) moves a residual of size at most \(\epsilon^{-\theta}s\) by at most \(C\epsilon^{4h-\theta}s< C\sigma\). In the isotropic case the aperture supplies both fine components. The remaining costs are the stated lists and boundary-depth errors.

The chain rule gives \[\begin{align*} \mathsf H_j(Z)&\le \mathsf H_j(T)-B_{1,j}+S_{t,j} +\mathsf H_j(Z\mid Q)+C_*\eta, \tag{236}\\ \mathsf H_j(Z\mid D)+B_{1,j} &\le \mathsf H_j(T\mid D)+C_*\eta. \tag{237}\end{align*}\] Conditional log-sum applied to the aggregate coherence bound, and the logarithm of its pointwise bound, respectively give \[\begin{align*} \mathsf H_j(Z\mid Q)+\mathcal M_j &\le B_{1,j}+\log_{1/\epsilon}\!\left(m(j)\right)+C_*\eta,\\ \mathcal M_j&\le B_{1,j}+\log_{1/\epsilon}\!\left(m(j)\right)+k-dh+C_*\eta . \end{align*}\] Use the first inequality with coefficient \(1\) and the second with coefficient \(1/2\) in (236)–(237). Every occurrence of \(B_{1,j}\) cancels. The result is \[ \mathcal E_j\le \mathsf H_j(T)+\tfrac32\log_{1/\epsilon}\!\left(m(j)\right) +\tfrac12\{\mathbb E_jw_{\rm loc}(D)+\mathsf H_j(T\mid D) +2S_{t,j}+k-dh\}+C_*\eta . \tag{238}\] Here the common \(k\) bin absorbs its stated finite bin error.

We assemble these conditional kernels under \(\pi_J\) before using incoming information. Averaging (238) gives \[\begin{align*} \mathbb E_{\pi_J}\mathcal E_J &\le \mathsf H_{\widetilde\pi}(T\mid J) +\tfrac32\mathbb E_\pi\log_{1/\epsilon}\!\left(m(J)\right)\\ &\quad+\tfrac12\bigl\{ \mathbb E_\pi w_{\rm loc}(D) +\mathsf H_{\widetilde\pi}(T\mid D,J) +2\mathsf H_{\widetilde\pi}(C_t\mid T,J) +k-dh\bigr\}+C_*\eta . \tag{239}\end{align*}\] The incoming conditional probability bound will be used under this assembled law, not separately on each \(J\) fiber.

A faithful aperture and the incoming information

The candidate atoms carry exact straight lines through their earlier positions with their assigned velocities. In a localizer \(J\), let \(F_T\) record \(C_t\) and both columns of this line at isotropic width \(u-h\) at the endpoint. Its local position precision is \(\epsilon^{u+h}=s\). Thus \(F_T\) is a function of \((T,C_t,J)\); conditional on \(J\), it need not be a function of \(T\) alone.

In the elongated case coarsen the orientation of \(D\) and its aperture data to widths \(0,u-h\), computing the latter from the center given by \(F_T\). In the isotropic case use widths \(u-h,u-h\). Denote this intermediate observation by \(Y\). Transport proximity makes \(Y\) determined by \(D\), conditional on \(J\), up to a list of size at most \(\epsilon^{-C_*\eta}\).

Enlarge each rectangle of \(Y\) by one fixed factor so that it contains the entire corresponding isotropic \(F_T\) cell. The factor works for every orientation: projections of a square have length at most its diagonal, and rounding the center adds only a fixed grid spacing. This costs at most \((\log C)/\ell\) in depth. Call the enlarged label \(\bar Y\). It is a function of \(Y\), so it has the same conditional list bound from \(D\). Let \(\kappa(g\mid F_T,J)\) be the conditional law of \(\bar Y\) under \(\widetilde\pi\), and adjoin \(G\) by that kernel: \[\widetilde\pi(\omega,T,G) =\widetilde\pi(\omega,T)\kappa(G\mid F_T,J).\] We keep the same notation for the augmented law. It preserves the entire old law of \((\omega,T)\) and the joint law of \((F_T,J,\bar Y)\) with \(\bar Y\) replaced by \(G\). The new \(G\) is independent of \(T\) conditional on \((F_T,J)\), contains \(C_t\), and is faithful for every line with that \(F_T,J\) value. No \(T\)-dependent cell is appended to its label.

Apply Theorem 15 in each localizer with duration \(2h\) and horizon \(u+h\). The strict eccentricity condition, or \(b_*=0\) with \(u_t>h\), gives \(u>h\). Its root caps have widths at most \(u+h=u_r-b_*\), and the endpoint horizon \(a+t\le L\), with its stated depth error, also places them inside the local packet menu. For every such root test \(F\), the support upper bound gives \[\mathsf H_j(T\mid F)+\mathbb E_j w(F)\le g(j)+C_*\eta.\] This uses only the retained support, so it holds for the biased candidate marginal. The positive root supremum is consequently at most \(g(j)-\mathsf H_j(T)+C_*\eta\). Positive growth yields, on each \(J=j\) fiber, \[ \mathbb E_j w(G)+\mathsf H_j(T\mid G)+2S_{t,j} \le g(j)+2\gamma h+C_*\eta . \tag{240}\] Averaging these inequalities under the same \(\pi_J\) as before, \[ \mathbb E_{\widetilde\pi}w(G)+\mathsf H_{\widetilde\pi}(T\mid G,J) +2\mathsf H_{\widetilde\pi}(C_t\mid T,J) \le \mathbb E_\pi g(J)+2\gamma h+C_*\eta . \tag{241}\]

All information in the remainder of this subsection is under the assembled augmented law \(\widetilde\pi\). Both \(D\) and \(G\) contain \(C_t\), and \(F_T\) is a function of \((T,C_t,J)\). The common time information cancels in the exact chain rule \[\begin{align*} \mathsf I(T;D\mid J)-\mathsf I(T;G\mid J) &=\mathsf I(F_T;D\mid C_t,J)-\mathsf I(F_T;G\mid C_t,J) \\ &\quad+\mathsf I(T;D\mid F_T,J)-\mathsf I(T;G\mid F_T,J). \tag{242}\end{align*}\] The last term is zero by the resampling kernel. Data processing for \(\bar Y\), followed by preservation of its joint law with \((F_T,J)\), bounds the first line below by \(-C_*\eta\). Therefore \[ \mathsf I(T;D\mid J)-\mathsf I(T;G\mid J) \ge \mathsf I(T;D\mid F_T,J)-C_*\eta . \tag{243}\]

When \(k_f>0\), the preserved endpoint marginal supplies an averaged information lower bound \[ \mathsf I(T;D\mid F_T,J)\ge k_f-C_*\eta . \tag{244}\] Here is the finite-list verification. In absolute units, \(F_T\) and \(P_f\) have identical velocity depth \(u_t-h\) and position depth \(t+u_t-h\). The candidate position differs from the observed one by at most \(\epsilon^{-\theta}\epsilon^{r+u_r}\), exactly an \(\epsilon^{-\theta}\) multiple of that position unit, and the velocity difference is \(O(\epsilon^{u_t})\). The localizer has the coarser width in (201); its remaining tags have the charged conditional count. It follows that \[\mathsf H(F_T,J\mid P_f)\le C_*\eta,\qquad \mathsf H(P_f\mid F_T,J)\le C_*\eta .\] Because \(\widetilde\pi_\Omega=\pi\), (218) still holds for its endpoint labels. It is not asserted after conditioning separately on \(J\). The chain rule gives \[\mathsf H(W_f\mid F_T,J) \ge \mathsf H(W_f\mid P_f)-\mathsf H(F_T,J\mid P_f) \ge k_f-C_*\eta .\]

Neither coarse label alone determines \(W_f\). The pairs \((D,F_T,J)\) and \((T,F_T,J)\), however, each determine a list of at most \(\epsilon^{-C_*\eta}\) possibilities. First list the possible \(P_f\)’s supplied by \(F_T,J\). For each listed value, \(n_f(P_f)\) is known, with no continuity assumption on this function. Use that value’s coarse velocity center as reference. The remaining velocity has size \(O(\epsilon^{u_t-h})\). The incoming alignment of the normal of \(D\) with \(n_f(P_f)\) is \(O(\epsilon^h)\), so changing between them changes the measured component by \(O(\epsilon^{u_t})\), one \(W_f\) unit. Thus \(D\) resolves that component to the stated list. The exact velocity of \(T\) does so as well, since it differs from the output velocity by \(O(\epsilon^{u_t})\). For isotropic tests use both velocity coordinates. Consequently \[\mathsf H(W_f\mid D,F_T,J)+\mathsf H(W_f\mid T,F_T,J)\le C_*\eta .\] The elementary information inequality \[\mathsf I(T;D\mid F_T,J)\ge \mathsf H(W_f\mid F_T,J)-\mathsf H(W_f\mid D,F_T,J) -\mathsf H(W_f\mid T,F_T,J)\] proves (244). This is a conditional information averaged over \(J\) under \(\widetilde\pi\), which is exactly what the averaged bounds need. If \(k_f=0\), use nonnegativity of conditional mutual information instead.

Combining (243) and (244) with (241) gives \[\mathbb E_{\widetilde\pi}w(G)+\mathsf H_{\widetilde\pi}(T\mid D,J) +2\mathsf H_{\widetilde\pi}(C_t\mid T,J) \le \mathbb E_\pi g(J)+2\gamma h-k_f+C_*\eta .\] The weight change on each localizer is \[w_{\rm loc}(D)-w(G)= \begin{cases} (1-\gamma)h+O(\eta),&d=1,\\ 2h+O(\eta),&d=2, \end{cases} \qquad w_{\rm loc}(D)-w(G)\le dh+C_*\eta .\] Also \(\mathsf H_{\widetilde\pi}(T\mid J)\le \mathbb E_\pi\log_{1/\epsilon}\!\left(N(J)\right)\), since every candidate lies in its retained class. Insert these bounds into (239), then use (234). The tail comparison costs are included in \(\eta\), so after returning to the original units, \[P_t\le B_r+\gamma h+\tfrac12(k-k_f)+C_*\eta .\] Comparison with (229) proves the lower bound in (230). Its upper bound is (207).

Under \(\mu\) at time \(r\), \(P\) has depth \(u_t=u_r-h\) and \(W\) has depth \(u_r\). Thus (224) and the earlier aperture and score bounds supply the next cut’s incoming data. That cut may use a new atom decomposition: the coarse-label lists carry the observable \(W\mid P\) bound, without identifying the two temporary candidate populations. This completes Proposition 48.

A finite backward chain

Suppose, for a contradiction, that \(\beta_{\mathrm q}>\gamma/2\), and set \[c=4\beta_{\mathrm q}-2\gamma>0.\] Choose an integer \(q\) with \(qc>2\). Proposition 44 supplies \(q\) backward steps of a common positive length \(2h\), with all times positive and \(u_t>h\). In the forced alternative first take \(h\) sufficiently small compared with the fixed aspect gap times \(\gamma\); this verifies (200). In the other alternative all the back tests are exactly isotropic.

Fix this chain and start with \(k_0=0\). At cut \(i\), apply the rule for a prepared cut and call its final endpoint law \(\pi_i\); the returned \(\mu_i\) is the next endpoint law. Let \(\Delta_i,\eta_i\) be the actual selected errors for \(\pi_i\), with \(J\)-averages under \((\pi_i)_J\). They include the exceptional-fiber score splits and the accumulated \(\zeta+\log_{1/\epsilon}\!\left(1/p_A\right)\) with every \(p_A\) measured in its current law. Choose the finitely many retention parameters and current-law branches so this finite cost sum is as small as required.

Within each fixed outer row \(j\), choose the finite inner parameters \(\theta,\xi\) and the finitely many \(\zeta\)’s, then the initial packet tail orders and coefficient cutoff, the later transport order, and finally the precision \(\epsilon\). These choices make the inner errors, including both mass-comparison costs, arbitrarily small in that row. Finite composition and (220) leave the selected deficits bounded by that row’s \(\Delta_j\), with the fixed finite selection factors, plus those inner errors. With the target shape and \(g\) accuracies fixed before choosing the late row, these factors are bounded across the subsequence: only uniformly positive current masses of qualifying unions and retained complements divide deficits, while individual tag probabilities enter only logarithmic costs. The selected deficits are not sent to zero inside the row. Only then choose a sufficiently late outer row, with \(q\) and \(h\) already fixed across the subsequence, and choose finite inner precision in that row so that the total error bound below holds. No transport order uniform in unbounded outer aspect parameters is required.

Apply Proposition 48 with \(b_{\mathrm q}=\beta_{\mathrm q}\). Iteration of (230) gives \[k_q\geq qch-\sum_{i=1}^q(2\Delta_i+C_*\eta_i), \qquad k_q\leq2h+C_*\eta_q.\] Choose the finite errors so that \[\sum_{i=1}^q(2\Delta_i+C_*\eta_i)+C_*\eta_q <(qc-2)h.\] The two inequalities are incompatible. Consequently \(\beta_{\mathrm q}\leq\gamma/2\), completing the proof of Theorem 35.

The radial kernel and its dyadic pieces

Use the Fourier convention \(\widehat f(\xi)=\int e^{-2\pi i x\cdot\xi}f(x)\,dx\). For real \(\delta>0\) put \[m_\delta(\xi)=(1-|\xi|^2)_+^\delta, \qquad K_\delta=\mathcal F^{-1}m_\delta.\]

Take a smooth radial partition of unity into a bounded region and shells \(|x|\asymp\lambda\), \(\lambda=2^j\ge2\). The bounded piece of \(K_\delta\) is integrable. By Lemma 51, each shell is the sum of two kernels \[\lambda^{-2-\delta}e^{\pm2\pi i|x|} a_{\lambda,\pm}(x/\lambda),\] where the amplitudes are supported in a fixed annulus and all their derivatives are bounded uniformly in \(\lambda\). In the convolution operator set \(x=\lambda z\), \(y=\lambda y'\). The factor from volume is \(\lambda^3\), so the rescaled operator is \[ \lambda^{1-\delta}A_{\lambda,\pm}f(z),\qquad A_{\lambda,\pm}f(z)= \int e^{\pm2\pi i\lambda|z-y|}a_{\lambda,\pm}(z-y)f(y)\,dy. \tag{251}\] The input and output dilation factors cancel in its \(L^3\) operator norm. We next prove \[ \|A_{\lambda,\pm}\|_{L^3(\mathbb R^3)\to L^3(\mathbb R^3)} \le C_\nu\lambda^{-1+\nu} \qquad(\nu>0). \tag{252}\]

The distance phase

Partition input and output space into unit cubes with smooth cutoffs of bounded overlap. The annular support in (251) allows only a bounded number of input neighbors for each output cube, and conversely. For one such pair, translate both cubes together to a fixed bounded region. A finite smooth angular partition and fixed rotations reduce the phase to regions where \[0<d_0\le z_3-y_3\le d_1<\infty,\] with fixed constants. The dyadic shell operators have the Carleson–Sjölin distance-phase form treated by Guo et al. (2025); we use their subsequent pseudoconformal change of variables. Fix the slice variable \(y_3\), and set \[s=(z_3-y_3)^{-1},\qquad Z=s(z_1,z_2),\qquad v=(y_1,y_2).\] Then, for \(g(q)=\sqrt{1+|q|^2}\), \[|z-y|=s^{-1}g(Z-sv),\qquad \left|\det\frac{\partial(Z,s)}{\partial z}\right|=s^4.\] Choose a fixed positive interval \([s_*,s_*+D]\) containing the range of \(s\), and put \(s=s_*+Dc\), \(Z=Dx\). Both the coordinate change and its inverse have bounded derivatives and Jacobians on the relevant regions, uniformly in the slice. Direct differentiation gives \[\begin{align*} \Phi(c,x,v)&=\pm2\pi(s_*+Dc)^{-1} g\bigl(Dx-(s_*+Dc)v\bigr),\tag{253}\\ \partial_v\Phi(c,x,v)&=\zeta(v,x-cv),\qquad \zeta(v,p)=\mp2\pi\nabla g(Dp-s_*v),\tag{254}\\ \partial_p\zeta(v,p)&=\mp2\pi D\nabla^2g(Dp-s_*v). \tag{255}\end{align*}\] The eigenvalues of \(\nabla^2g(q)\) are \((1+|q|^2)^{-1/2}\) and \((1+|q|^2)^{-3/2}\). They are bounded away from zero on the fixed argument region of the integral, but the smaller eigenvalue tends to zero at infinity. The packet decomposition uses global definiteness when solving (164) for every Fourier mode. We therefore modify \(g\) outside the argument region to obtain the global hypothesis of Theorem 50.

Choose \(R_0>1\) so that all arguments in (253) have norm less than \(R_0\). Let \(\chi\) be a nondecreasing smooth function, zero on \((-\infty,R_0]\), one on \([R_0+1,\infty)\), with \[|\chi^{(j)}(r)|\le C^{j+1}(j!)^2.\] Such a function is obtained by integrating and normalizing the bump \(\exp[-1/t-1/(1-t)]\) on \(0<t<1\), extended by zero. For completeness, Cauchy’s estimate on a complex disk of radius a small fixed multiple of \(t\) gives a derivative bound \(j!(C/t)^j e^{-c/t}\) near \(t=0\). Maximizing \(t^{-j}e^{-c/t}\) gives \(C^{j+1}(j!)^2\); the estimate at \(t=1\) is identical, and away from the endpoints analyticity gives the bound directly.

Let \(h(r)=0\) for \(r\le R_0\) and, for \(r>R_0\), let \[h(r)=\int_{R_0}^r(r-u)\chi(u)\,du.\] Replace \(g(q)\) by \(\widetilde g(q)=g(q)+h(|q|)\). The added Hessian has radial and tangential eigenvalues \(\chi(r)\) and \(h'(r)/r\), both nonnegative and at most one. For \(r\ge2(R_0+1)\) they are at least \(1/2\). On the remaining compact region the original Hessian is uniformly positive. Thus \[c_0 I\le\nabla^2\widetilde g\le C_0 I\quad\hbox{on }\mathbb R^2.\] The modification vanishes on the argument region. Its Hessian has derivative bounds \(A_0^{j+1}(j!)^3\): on the transition annulus this follows by the chain rule from the displayed Gevrey bounds and the analyticity of \(q\mapsto|q|\) away from zero. More explicitly, grouping the terms in a derivative of order \(j\) by the \(k\) derivatives landing on the outer scalar function bounds their sum by \[C^j j!\sum_{k=1}^j C^k k!\binom{j-1}{k-1} \le C_1^{j+1}(j!)^2.\] The analytic factors \(q/|q|\) in the radial Hessian obey the same bound after the product rule. Outside the transition annulus, \(h(r)\) is a quadratic polynomial in \(r\) plus a linear term and a constant, whose radial Hessian has the same bounds. Near the origin there is no modification. The original \(g\) has uniform analytic Hessian bounds. Formula (255), with \(\widetilde g\) in place of \(g\), therefore satisfies all the required global hypotheses, including the mixed \(v,p\) derivative bounds.

For each fixed \(y_3\), Theorem 50, applied after this change of variables, bounds the sliced integral by \[C_\nu\lambda^{-1+\nu}\|f(\cdot,y_3)\|_{L^3(\mathbb R^2)}\] in \(L^3\) of the output variables. The transformed amplitudes have uniform seminorms, and the output Jacobians are uniformly bounded. Minkowski’s inequality and then Hölder’s inequality on the bounded interval of \(y_3\) prove the corresponding three-dimensional bound for the selected cube pair. Finally, if \(a_k\) is the input \(L^3\) norm in cube \(k\), the output norm in cube \(j\) is at most \(C_\nu\lambda^{-1+\nu}\sum_{k\sim j}a_k\). The inequalities \[\sum_j\left(\sum_{k\sim j}a_k\right)^3 \le C\sum_k a_k^3\] follow from the bounded neighbor counts in both directions. Summing the finite angular partition proves (252).

The full strict range

We record the consequence for other exponents. Its proof uses the positive-order theorem at arbitrarily small, fixed orders; no uniformity as the order tends to zero is needed.

For completeness, the norm-convergence assertion in the introduction follows from the same uniform radius bounds. If \(1\le p<\infty\) and \(\widehat f\in C_c^\infty\), then \(T_{\delta,R}f\to f\) in the Schwartz topology: on the fixed Fourier support, every derivative of \((1-|\xi|^2/R^2)^\delta\) converges to the corresponding derivative of \(1\). Inverse Fourier transforms of \(C_c^\infty\) functions are dense in \(L^p\). The uniform bounds and a density argument therefore give \(T_{\delta,R}f\to f\) in \(L^p\) for all \(f\) in that space whenever (259) holds.