From actual search histories to current-group localization and far-band transfer to side arguments

This chapter explains the filter arrangement, Proposition 6.8, and Lemma 6.9 on pp. 47–52 of H. We use the 93-page version of October 5, 2026. All source page numbers below refer to this version.

The basic columns already constructed describe the components of each argument. Historical masks determine which curvatures may belong to the same group. Finite catalogs ensure that all residual tests in a prescribed finite list can be satisfied. We combine these three facts: first list the projection words actually used by binary search, then prove localization, and finally verify filter transfer in each direction.

One further input remains explicit throughout: the complete side states used as bases for the helper catalog must first be constructed and fixed in Sections 7–8 of H, and their number must satisfy the prescribed bound. We provide all transfer histories needed once these fixed states are supplied. Finite catalog termination alone does not establish that the complete side states already exist.

1. Objects and inputs on a fixed cell

H, p. 37 · H, p. 38 · H, p. 39 · H, p. 42 · H, p. 43 · H, p. 48 · H, p. 49 · Basic columns and the original path bound · Fixed inputs for finite catalogs

Fix a finite dyadic tree inside a normalized root $R$. Let $\mathcal I$ denote the original active summation cells of the current successful attempt: the first roots at which coloring fails have already been assigned to new attempts by Lemma 6.3. Every descending path meets at most $M$ such active cells. The length bound for failed roots is an input from the preceding chapters. We neither reprove it here nor recut the input functions during localization. Write $I$ for a cell of depth $s$, with normalized norm

Only the physical coordinate of $F$ enters the quadrilinear form $H_I$. The slots are always $0,1,2,3$. The later notation $(a,b,c,d)$ indicates the roles of the arguments in a formula; it does not assume symmetry of the kernel.

A basic pattern $z=(z_0,z_1,z_2,z_3)$ has at least three coarse arguments $B_e$ and at most one basic side argument $U_e$. Each $B_e$ is a difference of at most two basic projection outputs. For the arguments $B,U$, the local normalized $L^2$ norms are bounded uniformly by $C_b$. The basic columns, their private coordinates, their current availability, and the entire family of active cells are fixed. We do not further subdivide these original summation cells.

We use the bounds from pp. 37–43 of H and the preceding chapters:

The input normalization exponent $q>2$ is fixed. Here $L$ is the number of colors, and $P=\lceil\eta^{-\alpha}\rceil\ge1$ is the averaging length. All fixed exponents are chosen before the later parameter optimization. A path contains at most $D$ original graph vertices. New main and helper columns are attached only to original vertices; they introduce no new vertices, colors, or graph contexts.

Successful coloring, inclusion and separation of historical masks, and separation estimates for pure columns come from Lemmas 6.3, 6.4, and 6.1 of H, respectively. We construct the finite catalogs using Lemma 6.7. Section 6 below lists the actual histories it requires. A finite list of names and the concrete columns subsequently inserted are distinct objects.

2. Two types of masks and the rank and pad choices

H, p. 43 · H, p. 44 · H, p. 45 · H, p. 47 · Neighbor closure of historical masks

Let $s$ be the depth, and let $d_s$ be the graph distance at that depth. This distance is nonincreasing with depth and may be infinite. Colors are nonincreasing with depth, and vertices are only added. Thus

increases with depth and, at fixed depth, also with $h$. Distance to the empty target set is defined to be infinite. If a current vertex is adjacent to a point of $U_{m,h}(I)$, the triangle inequality places it in $U_{m,h+1}(I)$.

The other mask is the historical ball $M_{G,h}(I)$. It retains historical centers of the same group $G$ along the current ancestor path. The distinction in Chapter 2 between actual current use and inherited endpoints needed for nesting is used in three specific ways. At an actual current use of a group, the graph ball around a current center lies in $M_{G,h}$; current graph neighbors lie in $M_{G,h+1}$; and any two mask labels of distinct same-color groups have distance strictly greater than one at overlapping usable endpoint cells. All vertices here must already exist at the current time.

Given a slot $e$, rank $r$, and mask $A$, write

for the ordinary orthogonal projection on cell $I$ whose range consists of constant-coefficient combinations of all currently available columns of rank at most $r$ attached to original vertices in $A$. Original columns have rank zero. This is a projection in a space of vector-valued functions, not a projection of the physical coordinate alone.

For a split at color $m$, the cumulative filter uses

The four terminal ball filters, in insertion order $j=1,2,3,4$, use

Within one filter, the projections are nested as $p$ increases. A positive projection with fixed pad is also nested on inherited cells. The averages and their complements are contractions. Projections from different masks need not commute, and their products need not be projections.

The largest ball pad, together with one enlargement to include neighbors, is bounded by

The strict inequality holds for all $L,P\ge1$. Thus the later one-step neighbor closure remains within the domain of the historical masks.

3. The residual test and the coefficient sum

H, p. 47 · H, p. 48 · H, p. 49 · Normalization of the residual threshold

Use the cumulative bounds from Chapter 9 and prescribe $1\le n_0\le n_1\le\cdots\le n_{L+5}$ to bound the total number of columns of rank at most $r$ on a path. Even if the actual original catalog is empty, the convention $n_0=1$ is used only for counting; no fictitious column is added to a projection. Let $g$ be a fixed basic $B$ or $U$, let $Wg$ be a listed history, and let $Q=Q_{e,r,A,I}$ be its target projection. Equation (13) of H requires the following estimate for every available lower-rank pure source column $\Phi_v$ whose label belongs to $A$, and for any two other vector-valued functions $f_1,f_2$:

The ellipses preserve the original slot positions. The right-hand side uses the norm of the fixed base $g$, not that of $Wg$, which may change as the catalog changes. A base of norm zero has zero history, so its test holds automatically. The denominator is exactly $D\sqrt{n_{r-1}}$.

Why is this the appropriate denominator? Suppose that $X=\sum_v c_v\Phi_v$ lies in the range of a positive projection of rank at most $r-1$, and that the inverse square root of each private weight is at most $D$. Then

The first inequality follows from orthogonality of the private coordinates, and the second from Cauchy–Schwarz. If $X=Q'Y$, or if $X$ is an orthogonal band of one nested family applied to the input, then $\|X\|_{2,I}\le\|Y\|_{2,I}$. Multiplying (3) by the coefficients and summing yields exactly

No square root of the full column count remains. A fixed power of the main and helper coefficient bounds is allowed separately only when estimating small pairings between different labels.

For a basic $B$, one may apply the private-coordinate estimate directly in the span of the union of its basic columns. Alternatively, expand its at most two basic projections separately, at the cost of a fixed constant. A physical component restricted to some labels cannot be treated as the output of a contraction without proof. The later grouping is used only in linear coefficient calculations.

4. The actual operator words produced by binary search

H, p. 49 · Full invariants of the search rule

Start with the color interval $[1,L]$. The initial history word $W_e$ in each slot is the identity. Choose a slot that is genuinely a basic $B$ as the upper retained slot $u$, for example the first such slot in numerical order. Initially there is no lower retained slot. The upper retained slot supplies a source that can be expanded using positive projections. Once a lower retained slot is introduced, its outer complementary projection is kept unchanged to remove earlier colors.

For a current nonsingleton interval $[\ell,h]$, set $m=\lfloor(\ell+h)/2\rfloor$. By a fixed rule, choose a slot $t$ that is neither the upper retained slot nor the lower one, if present. There is always such a choice among four slots. The identity

produces two branches.

In the positive branch, left-multiply $C_{t,m,I}$ onto $W_t$, make $t$ the new upper retained slot, release the old upper slot, and replace the color interval by $[\ell,m]$. In the residual branch, left-multiply $1-C_{t,m,I}$ onto $W_t$, make $t$ the new lower retained slot, release the old lower slot, and replace the interval by $[m+1,h]$. The other retained slot is unchanged. No further factor is left-multiplied onto a retained slot before it is released.

Repeated application of (6) to the quadrilinear form gives an exact identity on every original cell:

Here $\Lambda$ is the leaf set of the fixed color-search tree. It has at most $L$ leaves, and exactly $L$ if all colors are retained. Each path has at most $m_* =\lceil\log_2 L\rceil$ splits, with zero splits when $L=1$. The total number of factors in the four words equals the number of splits on that path. Their order is the actual order of left multiplication. Boundedness of each factor does not permit their order to be changed.

We retain the same search leaf even when its color is absent on a cell. Slot choices and split points do not depend on whether that color occurs there. This allows the histories to have common names across cells and the later spatial compression to use the same words.

5. The rank difference between the retained slots at a leaf

H, p. 49 · H, p. 50

Fix a leaf color $j$. If the upper retained slot came from a positive branch, the split point at which it was last selected was $j$; otherwise the final upper endpoint could not be $j$. Its outer factor is therefore a cumulative average of rank

with individual pads in $[10Pr_++1,10Pr_++P]$. If the upper retained slot is still the initial basic argument, the search has never taken a positive branch, so $j=L$. In this case assign its source rank and upper pad the value zero.

If $j>1$, the last split point at which the lower retained slot was selected is $j-1$. Its outer factor is a complementary cumulative average of rank

After expansion into individual pads, every lower pad exceeds every noninitial upper pad by at least

The target rank is exactly one larger than the source rank. For the initial basic upper slot, the source rank and pad are zero, so the strict rank order and the required inclusion also hold. If $j=1$, there is no lower retained slot and no earlier color to remove.

Fix an individual pad $h_+$ of the upper retained slot and expand its outer positive projection. Call a pure source label old if

For the initial basic upper slot, $h_+=0$. Every old label belongs to every target mask of the lower retained slot, with source rank strictly below target rank. Keeping that lower complementary projection unchanged allows (3) to be applied to each old source; (4)–(5) then cancel the source coefficient sum.

If a nonold source belongs to the upper cumulative mask, the definition supplies a current vertex $v_0$ of color at most $j$ and distance at most $h_+$. The nonold condition forces this witness to have color exactly $j$. For an initial basic source, its own original vertex is a witness of color $j=L$. This proves the existence of a current center; a center that occurred only at an ancestor cannot replace it.

6. Which histories the catalog actually needs

H, p. 47 · H, p. 48 · H, p. 49 · H, p. 50 · H, p. 51 · The individual history list · Finite catalog termination

The termination theorem in Chapter 9 applies to a prescribed finite list. We now construct that list from the search in Section 4. Each name specifies the slot, fixed base, target rank, target mask, ordered inner history, the positive or complementary choice of each factor, and every individual pad. The name does not include the identity of a pure source that triggers failure.

The first class consists of the tests for old sources at a leaf. Record the lower retained slot's target cumulative projection and the complete inner word obtained by removing its outer complementary operator. This inner word may still contain positive and complementary factors. Expanding the upper slot later, or adding a ball filter to another slot, does not change this target history: the other two slots are arbitrary $L^2$ observer arguments.

The second class consists of the four ball-insertion tests. At each leaf, first choose a nonretained slot for the first ball by a fixed rule, then order the other three slots by a fixed rule. For the $j$-th ball target, record its rank $L+j$, every individual ball pad, and the original complete cumulative word $W_ez_e$ in that slot. Only one ball is inserted into each slot. Later balls in other slots are not added to its inner history.

The third class consists of transfer tests in the main catalog. For each leaf and each allowed assignment of distinct roles $(a,b,c,d)$, require the basic inputs in $c,d$ to be $B_c,B_d$. Write each complementary average in these two slots as identity minus positive average. Each choice produces a subword of positive factors in the original order. Each slot also has its own outer ball average. Following the deletion rule in Section 12, record the target of each outgoing filter and the inner word remaining after its removal. Include both target endpoints of each band. They use only the existing $1,\ldots,P$ projections, not a new zeroth or $P+1$-th mask.

The fourth class consists of tests in the later helper stage. Once a fixed complete side base $S_a$ is supplied, follow the same deletion sequence. In slot $a$, record the product of helper positive averages already transferred there and the helper target corresponding to the next outgoing mask. The helper rank is always $R_*=L+5$, and sources may come only from the fixed main catalog. Helper inner histories may contain helper factors of the same rank; main inner histories contain no helper factors.

Each class includes every individual pad choice, rather than only a representative one. For a history with $m$ factors, list at most $P^m$ inner choices and also the target pad. The lengths for the third and fourth classes are at most $m_*+2$: the two slots together have at most $m_*$ cumulative factors and two ball factors.

Suppose that a cell requires at most $D^{C_{\mathrm{ctx}}}$ original mask-context names and at most $\exp(k^{C_{\mathrm{side}}})$ fixed complete side bases. Multiplying the numbers of search leaves, slot assignments, positive/complementary expansions, deletion positions, and target and inner pad choices gives the following sufficient bound for the size of the list:

Its logarithm is bounded by a fixed power of $k$. We count only names, not numerical graph distances or columns generated later. Nor is the count repeated for every possible triggering source.

Thus the main list is generated by the actual localization and transfer steps and satisfies the finiteness hypothesis of Lemma 6.7. The form of the helper list is also determined. Construction of its complete side bases and a bound for their number still require the input from Sections 7–8 of H.

7. Restoring the averages does not introduce the number of pad choices

H, p. 48 · H, p. 49 · H, p. 52 · Pad averages in the catalog

If $\overline Q=P^{-1}\sum_pQ_p$, then

For any fixed ordered collection of $m$ positive or complementary averages, distributivity gives

No commutativity is used. For each history on the right, the bound in (3) uses the same $\|g\|_{2,I}$. Multiply the uniform bounds by $P^{-m}$ and sum. The total weight is one. Averaging the target pad afterward does not change the conclusion.

This is separate from the counting in Section 6. To ensure that every history passes its test, the catalog count does include $P^m$. Once those tests hold, the estimate of the actual form is a convex average and incurs no additional $P^m$ factor. Even when some pads are shared by different arguments, the finite average still has total weight one, so the same reasoning applies.

Later we write $1-\overline Q$ as two signed terms so that only positive filters are transferred. This produces $2^{m_*}$ positive words. Their number is polynomial in $k$ and counts distinct search terms; it is not the pad average.

8. A nonold source matches exactly one current group

H, p. 43 · H, p. 44 · H, p. 45 · H, p. 50 · Current-center ball inclusion

Fix a leaf $j$ and a nonold upper source $v$. Section 5 supplies a center of current color $j$, denoted by $v_0$, with $d_s(v,v_0)\le h_+$. Let $G(v)$ be its current group. The smallest pad of the first ball is $20P(L+1)+1$, larger than every possible $h_+$. By the center-ball inclusion in Lemma 6.4, $v$ belongs to every target mask of the first ball for this group.

If the same source matched another current group of the same color, this vertex would lie in allowed masks of two same-color groups. Both are in actual use on the current cell, hence both are usable endpoints. Taking both labels to be $v$ gives distance zero, contradicting the strict separation by more than one in Lemma 6.4. The matching group is therefore unique, although its witness center need not be unique.

More precisely, for another current same-color group $G'$, the distance from $v$ to every pure source label in the ball mask of $G'$ is greater than one. This is the form of separation required by Lemma 6.1 to make mismatched terms small. Disjointness of two sets alone would not suffice.

If the current cell has no center of the leaf's color, there is no nonold upper source. The entire form at that leaf is then controlled by the old-source error. The search tree and its fixed leaves need not be changed.

9. Inserting the first ball while retaining the lower slot

H, p. 48 · H, p. 49 · H, p. 50

Fix a leaf, write $X_e=W_ez_e$ for its four current arguments, and choose the first ball slot $t$ distinct from the upper and lower retained slots. Let $\mathcal G_j(I)$ be the actual groups of current color $j$. We prove

The group sum is taken only on the current original cell.

First fix one pad in the upper-source average and expand the outer pure sources $\sum_v c_v\Phi_v$ of the upper retained slot. For an old source, the pure-source term on the left of (12) is controlled by the unchanged lower retained slot. Each group term on the right is controlled by the same lower slot, since the ball output is merely a contracted $L^2$ observer argument. Summing over $|c_v|$ and using (5) cancels the source coefficients. In both cases the lower slot is tested by the original lower-rank upper source. A newly inserted higher-rank ball column is never treated as an admissible lower-rank source for that test.

For a nonold source $v$, let $G(v)$ be its unique matching group. Subtracting the term for $G(v)$ from the original pure-source term is exactly the insertion in slot $t$ of

The upper source rank is strictly below $L+1$, and its label belongs to every target mask of this ball. The second class of histories in Section 6 therefore applies, with an error of the form (5).

For $G\ne G(v)$, also expand one positive-projection source of $L_{t,G,I}^{(1)}X_t$. The two pure source labels have distance greater than one. Lemma 6.1 bounds the resulting term by $\tau_2$ times the norms of the other two arguments. The two coefficient sums may introduce a fixed power of the main catalog bound. This is precisely the mismatch error allowed when choosing $\tau_2$.

Finally restore the upper-source and ball-target averages. Section 7 shows that no additional pad-count factor appears. This proves (12). The upper argument has not been permanently restricted to a nonorthogonal subsum for each group: the right-hand side of (12) still contains the original complete $X_e$. Hence the main term in the later estimates does not retain the loss from a full column-coefficient expansion.

10. The other three balls and the precise conclusion of Proposition 6.8

H, p. 49 · H, p. 50

Within a fixed group term of (12), keep the first ball slot $t$ unchanged as the source for the next three insertions. Expanding one of its individual positive projections produces sources of rank at most $L+1$. Insert ball averages of ranks $L+2,L+3,L+4$ into the other three slots in order.

The largest old pad and the smallest new pad of successive balls differ by $9P+1$. Since masks of the same group increase with pad, every pure source label of the first ball belongs to every mask of each of the other three targets. The source rank is strictly smaller than the target rank each time, and the target inner history is still the original $W_ez_e$. Equation (5) controls each replacement residual; the other arguments are arbitrary $L^2$ observers.

These insertions may change the original upper or lower retained slot, since old-source removal and first-ball insertion are already complete. The changed arguments are not subsequently used as the retained residuals for old sources. Each slot ends with exactly one outer positive ball average, and we obtain

This is equation (14) of H. The word $W_e$ depends only on the search leaf and slot, not on $G$. Only the outer factor $L_e$ depends on the current group.

On each cell, the test errors leave only $\tau_3$ times the basic norms and fixed-power counts of leaves, groups, and insertion positions. The number of current groups is at most the number of current centers and hence at most $D$. Combining these fixed powers into $D^{C_0}$ gives

Here $D_m$ controls the main catalog count, coefficient sums, and related bounds; the fixed local basic norms are included in $C$. Thus

We used $\sum_I|I|=\int_R\sum_I1_I(y)\,\mathrm dy\le M|R|$. First choose $C_\tau$ sufficiently large and use $D\ge\mathrm e^k$ to control the first term. Once an a priori main catalog bound is available, choose $\tau_2$ sufficiently small to control the second. The result is $C\mathrm e^{-k}|R|$, without an extra factor for the number of spatial depths.

11. Keep the minimum on original cells and outside the complete group sum

H, p. 9 · H, p. 10 · H, p. 11 · H, p. 12 · H, p. 13 · H, p. 38 · H, p. 50 · Full-depth spatial compression · The active path bound

Proposition 5.8 leaves the expression

By (13), on each original cell separately,

For the last step, if all $|S_\lambda|<\eta$, the right-hand side is their sum; if at least one term is at least $\eta$, the right-hand side is at least $\eta$. The first step also uses that $x\mapsto\min(\eta,x)$ is increasing and 1-Lipschitz on the nonnegative axis.

We may therefore treat the minimum leaf by leaf, but the current-group sum inside each $S_{\lambda,I}$ must be retained. We do not replace it by $\sum_G|H_I(Z_G)|$, redefine the minimum on newly subdivided cells, or take separate minima for future mode components. Such changes may yield a larger upper bound, but they lose the cancellation between groups used by the later core estimate.

For basic coarse inputs, $B_e$ is itself a signed sum of at most two basic projections applied to $F_e$. Thus the entire history before the ball is a fixed word of positive or complementary projections of length $O(\log L)$, with only $2^{O(\log L)}$ terms after expansion. The spatial compression of Lemma 3.4 therefore introduces a loss of order

Here $\log D_m\le k^{O(1)}$. At intermediate cells that are not actual uses, positive masks and histories are still evaluated at inherited endpoints when nesting is needed. Ball outputs may be set to zero before the attempt root. This introduces no new actual uses into the current-group sum.

12. Fix the side states and the deletion order before transfer

H, p. 48 · H, p. 50 · H, p. 51 · Fixed-base conditions for the helper catalog · The actual helper-history list

Fix one actual group use in (13). The two slots $a,b$ contain complete side states $S_a,S_b$, fixed before helper insertion. Extra private coordinates are allowed, but they must differ from every new helper coordinate. The number of complete side bases per cell, across slots, groups, and patterns, must be bounded by $\exp(k^{O(1)})$. A single mode, magnitude bin, or frequency expansion is not treated as a separate side base.

The basic inputs in the other two slots $c,d$ are genuinely $B_c,B_d$. First expand every $1-C$ into the two signed terms of $1-C$, preserving the original multiplication order. This produces at most $2^{m_*}$ words of positive filters. Work with one such word.

At each step, compare only the current outermost positive filters in $c,d$ and remove the one of larger rank. A basic $B$ with no remaining filter has rank zero. All positive ranks are distinct: split points on a search path are distinct, the map $m\mapsto L-m+1$ is injective, and the four ball ranks are distinct and exceed all cumulative ranks. As long as a filter remains, the larger outer rank $r$ is strictly above the outer rank $r'$ of the current source in the other slot.

Only the two current outermost ranks are compared. Ranks within one slot are not asserted to decrease from outside to inside. Removing an outer factor may expose an inner factor of larger rank; compare the two outer factors again at the next step. A filter-free $B$ admits a direct rank-zero basic-source expansion.

If the outgoing slot at this step is $c$, denote its individual projections by $Q_{c,p}$, $1\le p\le P$. In slot $a$, use rank $R_*=L+5$ to construct helper projections $Q_{a,p}$ with the same masks and pads. The main catalog is unchanged. Set

Here $D_a$ is formed by left-multiplying helper positive averages in the order of transfers already performed, starting from the identity. All these histories are products of contractions.

13. The sign in equation (15) of H and the two terminal bands

H, p. 50 · H, p. 51 · Small parameters and terminal bands

For $e=a,c$, add the two algebraic endpoints

These require no new masks. Nesting implies that $A_e^p$ are mutually orthogonal projections and $\sum_{p=1}^{P+1}A_e^p=1$. The weight of the average on band $p$ is

In particular, $w_{P+1}=0$. This does not remove the terminal band from the identity decomposition.

Expand $x_a,x_c$ in their respective bands. The first term receives its weight from the $c$-band, and the second from the $a$-band. Hence

The arguments are written in role order here; their actual slots do not change. This is equation (15) of H, with coefficient $(p-q)/P$. Both terminal bands $A_a^{P+1}$ and $A_c^{P+1}$ are necessary. For example, when $x_a$ lies entirely in its terminal band and $x_c$ lies entirely in its first band, the first term may be nonzero while the second vanishes. Removing the terminal row would destroy the identity.

Each high band has the residual representation

The first band is the positive projection $A_e^1=Q_{e,1}$. We use these two representations differently in the two far-band directions.

14. First direction: the helper band is higher, so expand the lower main band

H, p. 51 · H, p. 52

Suppose $p>q+2$. Then $q\le P$, so $A_c^qx_c$ lies in the range of the main projection $Q_{c,q}$ and admits a direct expansion into pure main columns. Its labels belong to pad $q$ of the outgoing mask, and their ranks are at most the outgoing main rank $r<R_*$. The coefficient sum satisfies

There is no need to split the band into two positive projections and estimate them separately. An orthogonal band is itself a contraction, and its range lies in that of the larger positive projection.

If $p\le P$, equation (19) produces two helper residuals in slot $a$, at pads $p-1,p$. Masks increase with pad, and $q<p-1$, so every main source belongs to both target masks. If $p=P+1$, there is only one helper residual, at pad $P$, and the same inclusion holds.

Apply the helper version of (3) to each main source. Its fixed base is the complete $S_a$, its inner history is the already transferred $D_a$, and its other two arbitrary $L^2$ observer arguments are $S_b,x_d$. Multiplying by the coefficients in (20) and summing yields

We used $\|x_d\|_{2,I}\le\|B_d\|_{2,I}$. The terminal high band requires only one residual, but the factor two is a uniform upper bound. The complete main-source coefficient sum in (20) has been canceled by the denominator in the helper test.

This direction uses only monotonicity within the same family of pad masks. No neighbor closure or helper-source expansion is needed.

15. Second direction: a higher main band requires another lower-rank main source

H, p. 43 · H, p. 44 · H, p. 45 · H, p. 51 · H, p. 52 · Individual-pad neighbor closure

Suppose $q>p+2$. Then $p\le P$, so $A_a^px_a$ is a band in a positive helper range. Expanding it directly as a source for the main residual would use ranks above the outgoing main rank, contrary to (3). The required source instead comes from the other nonside slot $d$.

By the ordering in Section 12, $x_d$ has outer main rank $r'$ strictly below the outgoing rank $r$. If it has no filter, use the rank-zero basic expansion. Thus

Let $h+p$ be the outgoing pad and $M_p$ its vertex mask. This may be a cumulative mask or the ball mask of the current group. Split the pure sources in slot $d$ into near sources, whose labels have distance at most one from $M_p$, and far sources. If $M_p$ is empty, then $A_a^p=0$, and the term is already zero.

For a near source, there is a current vertex in $M_p$ that is adjacent to it or has the same label. The triangle inequality for cumulative masks, or Lemma 6.4 at an actual current use for ball masks, places the source label in $M_{p+1}$. Since $q\ge p+3$,

If $q\le P$, write the high $c$-band as two main residuals at pads $q-1,q$. If $q=P+1$, the relevant inclusion is only $M_{p+1}\subseteq M_P$, and there is just one main residual at pad $P$. Every target contains the near source, whose rank is smaller than the target rank.

The main version of (3) therefore applies to the inner history $x_c$ already in the main list. The whole $A_a^px_a$ is simply an arbitrary $L^2$ observer argument, with norm at most $\|S_a\|_{2,I}$. Multiplying by the coefficients in (22) and summing bounds the near-source part by

Later helper insertion changes only an observer argument that is already allowed to vary arbitrarily. It cannot invalidate this established main test.

Only for far sources do we further expand the helper band. All its labels belong to $M_p$. Thus each far source in slot $d$ has distance greater than one from each helper source in slot $a$. Lemma 6.1 supplies the factor $\tau_2$ in these two distinct slots; the complete $c$-band and $S_b$ in the remaining slots serve as observers. The two coefficient sums introduce at most a fixed power of the main and helper bounds. This part is therefore at most

Here $D_{\mathrm{tot}}$ denotes only the a priori bound for the main and helper catalogs, with $\log D_{\mathrm{tot}}\le k^{O(1)}$. It does not include the mode catalog constructed separately from Section 8 of H onward. This small error is controlled by the prior choice of $\tau_2$.

The two directions are therefore different. The first tests a helper residual against a lower main band. The second tests a main residual using another lower-rank main slot and treats the helper band only as an observer. Only the far-source part of the second direction expands pure helper support for a mismatch estimate.

16. Total far-band error, iteration, and the retained near bands

H, p. 51 · H, p. 52 · Error quantifiers and parameter choices

Fix a prescribed constant $C_3>0$. The coefficients in (18) have absolute value at most one, and there are at most $(P+1)^2$ band pairs. By (21), (24), (25), and the basic norm bounds, the total absolute far-band contribution in one transfer step is at most

Since $P\le D^{C_P'}$, first choose $C_\tau$ to absorb this fixed power, $C_3$, and the polynomial number of transfer positions. Then choose $\tau_2$ using the a priori catalog bound. The total far-band error for each cell and group can then be made at most

This parameter choice is made for a given $C_3$. It does not say that one fixed precision works for every $C_3$. Equation (27) is still a local product-of-norms estimate. Summing it over all side states requires the side square-sum bounds from Sections 7–8 of H.

At each step, continue transferring only the new main term, and retain the near-band defects for separate estimation. Once all positive filters in $c,d$ have been removed, the main term is

The number of helper factors in $D_a$ is at most $m_*+2$. Recombine the earlier signed expansions with their original signs. The convex-average argument of Section 7 also applies to these helper histories, without multiplying again by the number of transferred pad combinations.

The retained near bands must be grouped at each fixed difference $\delta=q-p$:

The coefficient for $\delta=0$ is zero. The other four coupled sums remain at power one. Taking absolute values band by band and then summing at this stage would lose the form required for the joint square estimate in Section 7 of H. Equation (29) is not a near-band estimate already proved in this chapter.

H13 gives the short-block estimate for fixed stacks. Converting side outputs that vary within a block into these stacks, obtaining full side energy, and summing the near bands globally still require later proofs.

17. Terminal ranges and restoration of the complete arguments

H, p. 37 · H, p. 38 · H, p. 52

If the outgoing filter is a ball filter, the terminal band in slot $c$ is

After removing this ball, $x_c$ contains only group-independent cumulative history. The first term is therefore a group-independent identity part, while the second still lies in an available positive ball range. All nonterminal main bands lie in their corresponding positive ball ranges. The signed decomposition must be retained: the entire terminal complementary space does not lie in the ball range.

If the outgoing filter is cumulative, that slot's outermost ball has already been removed. Its entire remaining family of cumulative bands is therefore group-independent. The other nonside slot $d$ has only two possibilities as well: its outermost filter is still a positive ball, or the ball has been removed and its entire history is group-independent. These range conventions are used later for the near-band defects and do not change the main-term transfer in (28).

Finally restore the complete arguments $Y_e=B_e+U_e$. Subtracting slot by slot yields the exact identity

Each error term has at least one additional basic side argument $U$. Only the complete $Y_c,Y_d$ restore the smallness for pairs of full arguments from Proposition 5.6. The additional side terms and coupled near-band sums require separate estimates in Section 7 of H. Equation (31) does not assert that they have all been summed here.

18. Parameter order and the scope of the conclusion

H, p. 48 · H, p. 49 · H, p. 51 · H, p. 52 · Catalog bounds and the graph threshold · Construction of the basic bound · Later theorems still required

First fix the basic bounds, the search rule, $P,L$, the template complexity, and an a priori bound for the number of complete side bases required. Choose $C_\tau$ for the prescribed error power. Then use the name count in Section 6 and Lemma 6.7 to obtain a priori column bounds for the main and helper catalogs. These bounds depend only on the number of context names, not on numerical graph distances or $H_{\mathrm{ed}}$. Next choose $\tau_2$ to control all mismatch coefficient sums, and finally choose $H_{\mathrm{ed}}$ by Lemma 6.1. Only after the actual graph, successful colors, and masks are available do we run the column-insertion constructions. Bounds are fixed before the objects are constructed.

The main catalog is completed first. Complete side states are then constructed and fixed independently of helper insertion, after which the helper catalog is run. An a priori bound for the number of side bases neither specifies their concrete vectors in advance nor permits them to change at every helper insertion.

Under these explicit inputs, the actual search histories belong to a finite list of controlled size. Proposition 6.8 holds on the original active cells, retaining the complete group sum for each leaf and the original minimum. The exact band identity in Lemma 6.9 holds, both far-band directions satisfy (27), and the final main term is (28). Construction of complete side states, linear side energy, coupled near-band estimates, and their sums over all cells and groups are not completed here.

The individual histories and error quantifiers are detailed in the history and precision appendix. The source links in each section point to the relevant pages of the fixed original version.