How the original basic columns are constructed and how many labels a path must reserve

This chapter explains the construction in Proposition 5.6 on pp. 35–36 of H. We use the fixed October 5, 2026 version. All page references below refer to this version.

Chapter 9 requires a bound on the number of raw labels, whereas Chapter 10 bounds the number of active cells. These count different objects. We begin with the greedy construction of the old projections, then insert the prediction columns and the more accurate fitting columns in the high slots. Only then do we combine their counts into the raw budget. The main counting inequality is

It uses neither the number of active cells, Proposition 6.8, nor the termination conclusion of Lemma 6.7.

Spatial compression, delayed replacement, and the count of raw labels are proved below and in the appendices. The discrete quadratic inverse theorem used for detection is an external input with its source specified. The proofs of the all-scale Fourier tail bound and the scale summation are in the fixed prediction catalog appendix. Our conclusions concern only the construction of the basic catalog and the inputs it supplies to later arguments.

1. Fixed objects and three different counts

H, p. 4 · H, p. 37 · H, p. 47 · H, p. 48 · H, p. 49

Fix a finite dyadic tree with root $R$. For each actual input $f_e$, set $F_e=(f_e,0)$. The original normalization is

This condition is required on the cells of the original normalized tree. It does not automatically hold on the finer cells later introduced for delayed projections.

The normalized quadrilinear form associated with the kernel is

The kernels have uniform interior support, scaled derivative bounds, and vanishing integrals in all four marginal directions. In particular, after any two slots are chosen as coordinates, the marginal integral along either coordinate is zero. Only the physical coordinates enter this form.

We must keep three counts separate. $N_{\mathrm{base}}$ counts all births of raw labels along a path; $M$ counts the active cells on that path; $n_r$ counts all labels of rank at most $r$ after the columns of the various ranks from Section 6 of H have been added. Equation $(1)$ produces the first count. Only Chapter 10 derives the second count from the first. Catalog termination in Chapter 9 then takes these two counts as inputs and produces the third.

2. A label may be born before it becomes available for projection

H, p. 7 · H, p. 35 · H, p. 36

A raw label $v$ has a support birth cell $J_v$, a scalar atom $\phi_v$, a private weight $\epsilon_v>0$, and the column

The vectors $e_v$ for distinct labels are orthogonal. Even identical scalar atoms do not make two labels the same. Occurrences in different slots are also counted separately.

On a cell $I$, the list $\mathcal V_e(I)$ used by the projection must satisfy

whenever $I'$ is a subcell of $I$. This does not require every label whose support has been born to enter the projection immediately. The high-slot construction needs precisely this distinction: the support of a label is born on a coarse cell at depth $s$, but the label becomes available only for projections on its descendants at depth $s+K_0$.

Once the list is specified, $P_{e,I}$ is the orthogonal projection in $L^2(I;\mathcal H)$ onto the constant-coefficient combinations of these columns. Restricting an old combination to a subcell still gives an admissible combination. Thus the direct-sum projection spaces increase as the partition is refined. Refitting the coefficients on subcells creates no new labels.

If

then the union of all labels that ever become available along any complete path has at most $N$ elements. All raw births are counted here, including columns born on connecting cells and columns that enter a refined projection only after the last summation cell. The list at the current active cell alone is not enough.

3. Two uses of the private coordinates

H, p. 7 · H, p. 8 · H, p. 33

Suppose a cell has at most $N$ labels, $|\phi_v|\le1$, and $\epsilon_v\ge\epsilon_*$. For arbitrary coefficients,

Hence the coefficients are unique. If $P_IF=\sum_vc_v\Phi_v$, contractivity and Cauchy–Schwarz imply

This controls both the sum of the absolute values of the constant coefficients and the supremum norm of the physical part.

The second use is to obtain an energy increment from each new insertion. Let $W$ be the old space, let $r=F-P_WF$, and let the new column be $\Phi=(\phi,e_v)$, where $e_v$ occurs in neither the old space nor $F$. If

then $\langle r,\Phi\rangle_I$ is exactly this physical moment. Set $q=(1-P_W)\Phi$. Since $r\perp W$ and $\|q\|_{2,I}\le\sqrt2$, the enlarged space $W'$ satisfies

If the moment is nonzero, then $q\ne0$, so the division is valid. This argument generally fails if the same private coordinate is reused.

4. What detection supplies

H, p. 27 · H, p. 28 · H, p. 29 · The two detection decompositions

For slot $e$ and a physical residual $r$, let $d_{e,I}(r)$ denote the supremum of the absolute value of the form: among the remaining three slots, one contains a normalized $L^q(I)$ input and the other two contain normalized $L^2(I)$ inputs. The supremum also runs over the choices of these three positions.

Lemma 4.12 on pp. 27–29 of H states that, when $\|r\|_{2,I}\le C_1$ and $d_{e,I}(r)>\delta$, there is a bounded quadratic chart atom $\phi$ such that

The exponent $c$ is independent of $q$; the constants and the initial parameter may depend on $q$. If the supremum of the absolute value of the form is not attained, first choose test inputs exceeding half the required threshold. This changes only a fixed constant.

The local residual detection appendix traces and derives this input. It uses, in order, a bounded continuous quadratic inverse theorem, a Hahn–Banach decomposition, two bounded-input decompositions, and single-scale separation. Only a fixed number of quasipolynomial bounds are composed. The discrete inverse theorem is an external input; it depends on neither the basic catalog nor the same-label detection in Section 6 of H.

5. Old fitting columns: the energy bound precedes the path bound

H, p. 33 · H, p. 34

Fix an update accuracy $\delta=\mathrm e^{-k}$. Start with the empty list at the root and process the normalized cells by depth. On a cell, if the residual fails $d_{e,I}\le\delta$, apply $(5)$, insert a new column of weight one, and refit. Repeat until the test passes or the request limit is reached.

Since $\|F_e\|_{2,I}\le1$, every residual tested has $L^2$ norm at most two. Thus the same $\rho=\exp(-k^{O(1)})$ and the same complexity bound can be fixed in advance. Repeated fitting and fitting on refined partitions are projections onto increasing subspaces of the full-root space $L^2(R;\mathcal H)$. By $(4)$,

This integrated count does not directly imply a bound on each path. A stopping argument is still needed.

Take the integer $N=\lceil A_0\rho^{-2}\rceil$, where $A_0$ is a sufficiently large fixed constant. Stop at the first cell where a path would exceed $N$ requests, and assign the summation over that cell and its descendants to a new attempt. Let $E$ be the union of the stopped cells. They are disjoint, and each $y\in E$ has already passed through at least $N$ requests. Hence

Taking $A_0\ge8$ yields $|E|\le|R|/4$. If the implementation inserts the final column that exceeds the limit before stopping, use the budget $N+1$; the conclusion is unchanged.

Repeat at the new roots. The sum of all root lengths is at most $4|R|/3$. On each retained piece,

When the private weights are one, $(3)$ also bounds the physical supremum norm by $\sqrt{N+1}$. This is the construction in Lemma 5.4. It does not begin by assuming a finite catalog and then explaining how to project onto it.

6. Why a quadruple involves only boundedly many old catalogs

H, p. 34 · H, p. 35

Update the four slots in a fixed cyclic order, and set

$g_{t,I}$ denotes the fit at update level $t$ in the corresponding slot, with value zero before that slot is updated. Fix the earliest two increment levels $u<t$ in distinct slots. Write the corresponding slots as $i,j$ and the remaining slots as $a,b$. The four full arguments are

First, at a finite update cutoff, write each actual input as the sum of all earlier increments and the terminal residual. Expand the quadrilinear form. Group every term containing at least two increments by its earliest two levels $u<t$. In each remaining slot, sum all later increments and the terminal residual back together. This yields exactly the argument in $(7)$, namely $F_e-g_{e,\mathrm{prev}}$. Thus no such term is lost or counted twice. The remaining terms consist either of four terminal residuals or of one full terminal projection and three terminal residuals. By the comparison of earlier and later accuracies in Section 15, the form norm for terminal residuals in two distinct slots tends to zero, while the $L^2$ norms of the other two arguments remain uniformly bounded. The remaining terms therefore tend to zero on any fixed finite set of cells. We first fix the finite tree and then let the update cutoff tend to infinity; we do not interchange the infinite-tree limit with the update limit in advance.

For a fixed quadruple, at most six old projections are needed. Each projection has its own labels and inherited pieces. In the common refinement of these boundedly many families of pieces, each new root is either a root of an original piece or the original full root. The sum of root lengths is still bounded by a fixed multiple of the original root length.

Within a common piece, count the slot occurrences in these six catalogs separately. Write the total path budget as $N_{\mathrm{old}}$, and use $D_{\mathrm{old}}$ to bound their number, their complexity, and their sums of absolute coefficients. Since the parameters of all these levels are at most $k=k_t$,

There is no need to take the union of all catalogs from the first level through level $t$.

7. For a fixed pair of low columns, the prediction catalog must be fixed across all scales

H, p. 29 · H, p. 30 · H, p. 31 · H, p. 32 · H, p. 35 · Full derivation of the fixed catalog · The two precision conventions

Expand the two low arguments on the current coarse cell as constant-coefficient combinations of raw columns. Fix a pair of labels $(v,w)$ and consider them only on the subtree where both are available. Their earliest common availability cell is the deeper birth cell, or the piece root if both columns are inherited there. A more general inheritance mask may have several disjoint earliest common availability cells, but each path meets at most one of them.

Corollary 5.2 does not merely assert a finite expansion on each cell. It requires a single fixed finite list of triples on the subtree of a given pair:

Here $\|\psi_{e,\nu}\|_\infty\le1$, the coefficients may depend on scale, and the retained terms are

The length of the list, the atom complexity, and the sum of absolute coefficients are all bounded by $D_{\mathrm{pred}}$. If the accuracy is expressed by an error exponent $B\ge1$, then

The error in the absolute length-weighted scale sum is

Here $D_{\mathrm{high}}$ is the high-slot projection catalog budget; it enters only through a fixed power of its logarithm. $L$ is the square-sum constant for energy-type test inputs.

To obtain a single list, first include all rational splittings of admissible height, all lattice residue classes, and all Fourier indices. Only then choose the actual expansion at each scale. Large real frequencies remain rational evaluations of the same fixed parameter vector throughout. Truncate by a coefficient tail that is summable across all scales. Uniform approximation on each cell alone does not imply $(9)$. The full proof is in the fixed prediction catalog appendix.

Page 35 of the manuscript says “precision log B”, whereas p. 31 defines $B$ itself as the error exponent. We consistently use the latter notation and choose $B$ to be a fixed polynomial in $k$. This direct choice supplies the quantitative step needed in the application; it does not show that the original paragraph needs no further explanation. If $B$ on the two pages is read literally as the same parameter, the available complexity bound is insufficient to deduce the required bound $\log D_{\mathrm{pred}}\le k^{O(1)}$. This does not prove that the actual catalog must exceed the budget. It is not clear here whether the manuscript intended the reciprocal error instead. The source pages and an explicit choice satisfying both budgets are discussed in the parameter selection note.

8. Prediction-column births are counted by pairs of low columns

H, p. 35

For each $(v,w,\nu)$, insert one column in each of the two high slots:

The conjugate is chosen so that the projection residual pairs bilinearly with $\psi_{e,\nu}$. Insert the columns when the low-column pair first becomes jointly available, and inherit them thereafter. A change in a scale coefficient does not create another birth.

Along a path, the unions of low-slot labels have at most $N_i,N_j$ elements, respectively, where $N_i+N_j\le N_{\mathrm{old}}$. Hence there are at most $N_iN_j\le N_{\mathrm{old}}^2$ low-column pairs. Each pair creates at most $D_{\mathrm{pred}}$ labels in each high slot. Thus the total number of prediction-column births along a path is at most

The whole tree may have arbitrarily many labels on different branches. $(10)$ never uses the total number of labels in the tree.

9. Why spatial compression requires the full Lemma 3.4

H, p. 9 · H, p. 10 · H, p. 11 · H, p. 12 · H, p. 13 · Full-depth spatial compression

The absolute scale sum for the prediction catalog requires applying Bessel rows to the current projection on each cell. The cells supporting these rows may have different depths. Dyadic grouping by the number of insertions does not directly resolve this issue.

The spatial compression appendix reconstructs the full proof on pp. 9–13. Its conclusion is as follows. Fix a column system with path birth budget $D$, $m$ inherited projection families, and local rows satisfying

Then

The constant is independent of spatial depth and of the total number of labels on different branches. The proof first replaces the input by a bounded representative with the same first-layer moments, then stops according to changes in the last-layer coefficients. Both the sum of lengths and the representative energy contract, allowing the estimates over all attempts to be summed. The coefficient error is estimated using the integrated form of the birth budget.

$(11)$ starts only from a finite column system already fixed. Here it verifies the replacement error after any finite enlargement of a high catalog. The eventual finiteness of the high catalog follows from energy stopping based on $(4)$, not from this replacement error. Thus spatial compression is not proved by assuming final catalog termination.

10. The exact moment identity for delayed replacement

H, p. 32 · H, p. 33

Fix a prediction column and its multiplier $\psi$. Let $P_h$ be the projection on the partition at depth $h$, let $E_h$ be conditional expectation, and set

If the coefficient of the prediction column is $a_{h,p}$, orthogonality implies

Thus $\mu_{h+1}-\mu_h$ is $\sqrt\epsilon$ times the $p$-th private coordinate of the projection increment. Inserting any other new private columns preserves this identity.

At depth $s$, fix a cell $I$ and a smooth normalized bump $\omega_I$, and take $H=s+K_0$. Direct telescoping yields

All integrals are with respect to $\mathrm dy$. The remainder after a finite truncation is $\int_I \mathcal D_{N+1}(\omega_I-E_{N+1}\omega_I)$. Projections are contractions on $L^2(I)$, so $\mathcal D_{N+1}$ is uniformly bounded. The conditional-expectation error of the smooth bump tends to zero. Hence the remainder tends to zero, without assuming $P_hF\to F$.

The bump errors in both infinite sums are $O(2^{s-h})$. For a fixed offset $h-s$, the square sum of local rows on disjoint cells and the orthogonality of projection increments across depths yield the row norm $O(2^{s-h})\|1_JF\|_2$. Summing over offsets gives the total bound $O(2^{-K_0})\|1_JF\|_2$.

If $\int_I\omega_I=0$, in the first term we may use $\mu_H-\mu_s$ in place of $\mu_H$. Split this into $K_0$ private-coordinate increments. Their total row norm is at most

The mean-zero condition here cannot be omitted.

11. From the moment identity to the replacement error for the whole form

H, p. 32 · H, p. 33

First replace the quadratic phase in $(8)$ by one. In the coordinates of the two high slots, the kernel has zero marginal integral in each variable. Expand its double antiderivative in a Fourier series and then differentiate in each variable. This yields an absolutely summable family of mean-zero tensor bumps. Pair the error rows in $(13)$ with Bessel rows from the other slot. If the other slot contains a current projection, use $(11)$; if it contains an energy sequence, use its square sum over cells.

For the difference kernel $w_I(\mathrm e^{2\pi\mathrm i\lambda t^2}-1)$, the tensor bumps need not have mean zero. Derivatives of fixed order cost a factor $|\lambda||I|^2$ times a fixed power of $D_{\mathrm{pred}}$. By $(12)$ and projection contractivity, the first term has norm $O(\sqrt\epsilon)\|1_JF\|_2$ at each depth. For fixed $\lambda\ne0$,

The difference kernel vanishes when $\lambda=0$. Thus the additional first term is still summable.

We obtain the actual bound in Lemma 5.3:

After one high slot has been replaced, the test input for the second replacement is

The second term satisfies the conservative squared-energy budget $L\le C(1+K_0)$. On the full increasing sequence of depth partitions, orthogonality directly bounds its square sum by $K_0\|F\|_2^2$. We do not need this better constant here.

For each common birth root $J$ of a fixed low-column pair, apply $(9)$ and $(14)$, then multiply by the uniform bound for the current low coefficients. The coefficients may vary with the cell, but their upper bound suffices for the absolute error sum. Over all low-pair roots, Cauchy–Schwarz and path multiplicity imply

This step does not treat varying coefficients as fixed Fourier coefficients.

The two replacements are organized by a multilinear identity. Write $P^d_eF_e=P^{\mathrm{new}}_{e,s+K_0}F_e$, and let $\mathcal H_I(A,B)$ denote the form whose original slots $i,j$ contain the fixed arguments $X_i,X_j$ and whose original slots $a,b$ contain $A,B$. Then

The first test input, $X_b$, is a fixed input minus an old current projection. The second, $Y_a$, is a sum of a new current projection, a moving difference, and an old current projection. Both belong to the admissible classes above. Apply $(9)$ before and after replacement, apply $(14)$ to the retained terms, and finally sum using $(15)$. The choices in Section 14 make the total error at most $C\mathrm e^{-k}|R|$. Summing over the piece roots, whose lengths satisfy a uniform packing bound, yields the absolute length-weighted error asserted in Proposition 5.6.

12. How the extra high-slot columns are inserted jointly on fine cells

H, p. 35 · H, p. 36

The prediction columns control only the absolute replacement error. They do not yet make every pair of full arguments small. More accurate residual fitting is therefore needed in each high slot.

Process the projection depths $h$ in order. First insert all prediction columns already eligible at that depth. Then, at coarse depth $s=h-K_0$, test the residual on a cell $I$:

Here $P^{\mathrm{new}}_{e,h}$ is an orthogonal projection on the refined partition of $I$, so $\|r_I\|_{2,I}\le2$. No $L^q$ normalization on individual fine cells is needed.

If the test fails, detection produces $|\phi|\le1$ and a coarse-cell moment bound $\rho_e>0$. On all descendants at depth $h$, written as $Q\subset I$, insert the same label $(\phi,e_v)$. Set its support birth cell to $J_v=I$ and inherit it on the descendants of these fine cells. Different fine cells may use independent coefficients, but the private label remains the same.

Write $m_Q=|Q|^{-1}\int_Qr_I\overline\phi\,\mathrm dy$. By the proof of $(4)$, the energy increment on each fine cell is at least $|Q||m_Q|^2/2$. Since

weighted Cauchy–Schwarz yields the total increment

Thus each coarse-cell request can still be charged according to the coarse-cell length. One cannot select only one fine cell, nor should the path label count be multiplied by the number $2^{K_0}$ of fine cells.

Increasing the depth, making successive insertions at the same depth, and adding prediction columns together form an increasing sequence of projections. Prediction columns only increase the energy; they do not reduce the increments already accumulated in $(16)$. As in Section 5, take

Stop at the first coarse cell where the path exceeds the budget. The proportion of length passed to new roots can be made sufficiently small in advance.

13. Why stopping on coarse cells preserves the finer projections

H, p. 36

When a coarse cell $I$ is stopped at projection depth $h=s+K_0$, the delayed depths required by all its strict coarse ancestors are less than $h$. Their summations have already been assigned to the old attempt. Below $I$, the old projection family continues to retain the extra columns already inserted and the prediction columns, and to refit locally. It performs no further extra detection for coarse cells assigned to the new attempt. This is exactly the full descendant projection family required by $(13)$.

The new attempt may start a new extra catalog for descendant coarse cells. The extra labels of the old and new attempts must not be merged into one indefinitely accumulating catalog and then claimed to satisfy a single-attempt path bound. Each estimate uses its own fixed projection family, continued downwards. The attempts are summed using the root-length packing bound.

Take a common refinement of the coarse pieces of the two high slots and the pieces of at most six old catalogs. The root-length sum remains uniformly bounded. On restriction to a common subroot, existing labels are inherited with their original functions and private coordinates. Future labels retain their original availability stages and are not inserted into the root projection early. This restriction increases no path birth budget.

14. All parameters can be chosen in order

H, p. 31 · H, p. 35 · H, p. 36 · Explicit parameter calculations

First fix $q$, the kernel class, and the absolute exponents in detection and the Fourier estimates. Then fix the cyclic growth exponent $A$ larger than the exponent in the physical supremum-norm budget for the old fits, with a fixed margin. For a fixed quadruple, proceed in the following order.

First determine $k=k_t$ and $D_{\mathrm{old}}$. If the total cost of old labels and coefficients is $D_{\mathrm{old}}^{p_o}$, explicitly take $B=\lceil8k+2(p_o+1)\log(2D_{\mathrm{old}})\rceil$. This leaves a prediction-error margin of $\mathrm e^{-8k}$ before the later fixed polynomial costs. The prediction catalog now determines $D_{\mathrm{pred}}$. Set

Choose

If the prediction cost in $(14)$ is $D_{\mathrm{pred}}^{p_d}$, we may take $C_*=p_o+p_d+20$. This leaves a margin of $\mathrm e^{-20Q_0}$ before later fixed polynomial costs. The full calculation is in the parameter selection note. No indefinite sequence of unspecified reductions is used here.

Next choose the first high-slot accuracy $\delta_a=\exp(-p_a(k))$ so that $p_a$ dominates the budgets for all old projections, prediction columns, and $(18)$. Detection and $(17)$ determine the size of its new catalog. At depth $s+K_0$, each fine cell $Q\subset I$ satisfies

Together with $(3)$, this bounds the physical supremum norm of the first high projection on the coarse cell by

Here $N_a^{\mathrm{all}}$ includes its prediction columns. The bound depends only on quantities already fixed. Then choose the second high-slot accuracy $\delta_b=\exp(-p_b(k))$ so that its accuracy exponent dominates the logarithm of $(19)$ and all preceding costs.

These choices involve only a fixed number of polynomial compositions. Hence the final $\log D_{\mathrm{high}}$ remains a fixed polynomial in $k$. The later costs not yet known in $(9)$ and $(14)$ are only fixed powers of $\log D_{\mathrm{high}}$, hence fixed powers of $k$. The exponential margins reserved earlier absorb them once the initial $k$ is sufficiently large. We do not need to know the actual size of the final catalog in advance, and we have not used Lemma 6.7.

15. Why the two accuracies make pairs of full arguments small

H, p. 34 · H, p. 35 · H, p. 36

Suppose the fitting accuracy levels in two distinct slots are $h>\ell$. First replace the more accurate projection by the actual input, then replace the earlier projection. In the first step, use the physical supremum norm of the earlier projection as the $L^q$ size of the test input. In the second step, the test input is already a normalized actual input. Thus the difference in form norm between the pair of projections and the pair of actual inputs is at most

When $A>d$, the later accuracy dominates the supremum norm of the earlier projection. The new high-slot accuracies satisfy the same ordering by Section 14.

The final arguments are

Each old projection is contractive on the original coarse cell, and each new delayed projection is contractive on the full refined partition of that coarse cell. Hence $\|Y_e\|_{2,I}\le2$. Again, normalization on individual fine cells is not needed.

Choose any pair of full arguments. If $u<t-4$ and the earlier low argument is selected, directly compare the fits in the other, later difference or residual with its actual input. Its actual-input terms cancel; there is no need to use the small error of the earlier low argument itself. Its smallest old accuracy level is at least $t-4$, strictly later than every level appearing in this low argument.

In all other cases, every relevant nonzero old fitting level is at least $t-7$. Apply $(20)$ term by term. At least one of the two selected full arguments becomes zero when all fits are replaced by actual inputs. The terms containing the pair of actual inputs therefore cancel. Initial zero projections corresponding to updates not yet made do not affect this conclusion: when $t\ge12$, the other selected, later full argument still has this cancellation.

Since $k_{t-7}=k_t^{A^{-7}}$, we can choose a fixed $0<\theta<A^{-7}$ and increase the initial parameter to obtain

This estimate concerns the full $Y$ arguments. It does not assert the same smallness for an individual pair of basic columns, an arbitrary subcollection obtained by deleting columns, or every later expansion pattern.

16. The exact inputs to coloring and catalog termination

H, p. 37 · H, p. 39 · H, p. 41 · H, p. 42 · H, p. 43 · H, p. 47 · H, p. 48 · H, p. 49 · From basic data to the active path count · Inputs to later catalog termination

Combine the old labels, prediction labels, and extra labels of the two high slots. Equations $(10)$ and $(17)$ yield $(1)$. Choose an integer at least one such that

If the support birth budget gives an empty catalog, taking a bookkeeping budget of one is harmless. Then choose $D_{\mathrm{base}}\ge\mathrm e^k$ large enough to cover $N_{\mathrm{base}}$, all column complexities, inverse private weights, the coefficient sums in $(3)$, and the coarse-cell supremum norm in $(19)$. This yields

Here $C_0$ is a fixed exponent that may be enlarged. The initial $k$ may depend on $q$.

Page 37 splits a high argument into the current basic part and the delayed difference:

Only the constant-coefficient raw columns of the current $B$ enter the raw vertex list. $U$ is kept intact; it is not expanded on every fine cell with those occurrences then mistaken for current vertices of the coarse cell. But $(23)$ reserves raw labels for the entire path, so labels that become current columns later have already been charged.

The coloring in Lemma 6.3 needs only these raw vertices, their fixed curvature labels, the inherited lists of labels actually available, and $D_{\mathrm{base}}$. The total number of vertices across different branches need not be bounded. The main and helper catalogs need not already have been closed.

Lemma 6.7 takes $(24)$ as the rank-zero raw budget and separately takes the active-cell path budget $M$ from Chapter 10. New main-catalog and helper columns are attached to raw vertices and create no new graph vertices, but they still add to their own label counts. Actual history names, mask choices, and the full helper basis must be supplied individually by the later localization and transfer arguments. This chapter does not replace that work.

This is the extent of the conclusion here. $(24)$ uses the specified upstream inputs, including the discrete quadratic inverse theorem, rational splitting, and all-scale prediction. If any of these is omitted, the later recurrence for $n_r$ alone cannot recover the budget for the original $n_0$. The later actual histories, the full helper basis, error absorption, and the main theorem of H as a whole are outside the conclusions of this chapter.

The basic bound proved here applies to the actual history list in H12. That chapter proves current-group localization and local far-band transfer given fixed complete side bases, retaining the original-cell minimum and complete group sums. Actual construction of complete side bases, linear side energy, and global near-band estimates are still not included.