Unit 3, continued: three sources of depth variation in local outputs

This part develops D Lemma 6.1 and Corollary 6.2, pp.44–47. A local output contains an outer group projection, an inner ordered product of projections, and an original function that varies with depth. Comparing two depths requires treating these three changes separately.

The frequency-group construction from H enters through a separation inequality on specified endpoint spaces, together with its Compression operator statement. We state these assumptions first and then verify D's varying-function argument. The construction of the groups and endpoint spaces from full curvature labels in H is not reproved here.

1. The localization problem and parameter sizes

Fix a late precision, a root, and a $B/U$ pattern from D Proposition 5.6. Write

The quantity to be controlled on the original active intervals is

The small pairwise bounds for the full $Y$ will be used later when the complementary $B$ functions are restored to $Y$. They do not automatically hold for the current pattern.

Fix a root $R$ for one localization. All depths keep their indices relative to the original root. These localization roots belong to a different stage from the earlier projection-approximation roots. If an original interval is later cut out, only that interval and its original descendant summands pass to a new root. Previously fixed ancestor terms and operators remain unchanged.

The main column-family constant is $D_{\mathrm m}$, and the number of group colors is $L$. The source uses

Every ordered projection product in this part has length $O(\log L)$. The constant $G_0$ may be enlarged within the displayed form.

2. The endpoint-space assumption from H

Fix one position and one color, and sum only over groups $G$ of that color. Group indices remain fixed. Even after a group ceases to occur at a particular depth, its projection family continues by inheriting the original columns and locally reprojecting.

Each individual positive projection family is defined on complete depth partitions. Its range consists of constant-coefficient combinations of main columns, with sets of use inherited as depth increases. For a fixed group and projection choice, the outer projections $L_{G,s}$ therefore form an orthogonal projection family nested in $s$.

An endpoint interval $U$ is permitted for group $G$ when the corresponding projection's set of use is nonempty and either $U$ or a descendant has an actual use of $G$. The source calls this a usable endpoint. Actual use on a finer descendant also establishes usability of its ancestor endpoint.

The endpoint-space separation result from H, used as D equation (6.3), is the following. Suppose each $h_G$ is a finite combination of main columns, each coefficient is constant on a whole permitted endpoint interval, and the column belongs to the corresponding group projection's set of use. Then

By choosing the mismatch tolerance after the counting bounds, as in H, one can bound $C_{\mathrm{sep}}$ by an absolute constant, independent of $k$ and $P$. The same label or interval may recur at different endpoint times. The constant counts neither these repetitions nor the depth of a descendant establishing usability. Column counts, group counts, and private weights must satisfy (Main-sizes) and the separation tolerance chosen in H.

This is an operator statement about combinations of columns on whole intervals. Separation of only the current output functions, or of their restrictions to fine intervals, cannot replace (Endpoint-separation).

2.1 A projection square sum at one endpoint

Fix a right-endpoint depth $t$. Let $E_G$ be the union of depth-$t$ intervals satisfying the actual-use condition above. Set

Its adjoint is $A_G^*=L_{G,t}\mathbf1_{E_G}$, with range in the permitted full depth-$t$ endpoint space, and $\|A_G^*c_G\|_2\le\|c_G\|_2$.

Apply (Endpoint-separation) to $\sum_GA_G^*c_G$ and use Hilbert-space duality. This gives D equation (6.4):

All groups act on the same $h$. This must remain true in subsequent applications.

3. Ordered projection products in the outputs

An original basic $B$ function is a signed sum of at most two current underlying projections. Localization applies further positive projection averages and their complementary operators outside it. Fix a search branch, expand the complementary operators, choose one underlying component, and fix one projection choice in each positive average. This gives

Each $P_s^j$ is an inherited orthogonal projection family, with main column constant bounded by $D_{\mathrm m}$. The order is fixed and the length is $m=O(\log L)$. These inner families are independent of $G$; projections with distinct $j$ need not commute.

A group-independent shared output may have no outer $L_{G,s}$. The case $m=0$ is also allowed, with the inner operator product equal to the identity. The source includes the identity output itself among the shared types.

Every required endpoint is evaluated using the complete continued families, even if the group has no current actual use there. When a required finite endpoint lies outside the original function-index range, extend the sequence constantly from the nearest endpoint. Repeating endpoint values does not increase within-block variation. Outputs before the current localization root are still defined to be zero.

4. The full variation statement

Let $\mathcal J$ be a finite family of integer-depth intervals $[r,t]$ with edge overlap at most $a$. For each $G,[r,t]$, let $E_{G,r,t}$ be the union of depth-$t$ intervals with actual descendant use of that group. Lemma 6.1 asserts that

For one specified shared output $Z_s^{\mathrm{sh}}$, let $E_{r,t}$ be the union of the right-endpoint intervals required for all its uses. Then

There is no sum over groups in the shared-output statement. Use by several groups does not cause the same shared output to be counted repeatedly.

5. The exact three-term difference

Partition the depth intervals into at most $a$ subfamilies with disjoint interiors. In one subfamily, remove zero-length intervals and order the rest as

For now, assume all endpoints lie within the current localization root.

Equation (One-term) gives the exact identity

The first two terms change only the operators. First prove an ordinary joint Bessel bound for an arbitrary fixed function, then substitute the function varying with $r$, namely $F_r$. The last term changes only the common function, so the square-difference bound for the original sequence applies directly.

6. The first term: changing the outer group projection

6.1 First omit the inner ordered product

For each interval at the left-endpoint depth $r$, denoted by $U$, define a row

Both endpoint projections commute with $\mathbf1_U$. For a fixed $[r,t]$, these row outputs have disjoint spatial supports, so the split preserves the exact sum of squared norms.

For fixed $G$, the nested-projection increments on intervals with disjoint interiors have pairwise orthogonal ranges. Restricting outputs to $E_{G,r,t}$ only decreases the square sum. Hence

6.2 The ordinary joint bound across groups follows from adjoints

For arbitrary finitely supported row test vectors $c_{G,r,t,U}$, set

The dual form of (One-group) gives

Each row adjoint takes values in a full group-projection range at the left or right endpoint. If a small depth-$t$ interval has actual descendant use, its ancestor $U$ has the required use as well. An empty set of use gives the zero projection. These adjoint sums therefore satisfy the hypotheses of (Endpoint-separation), yielding

Duality then proves a joint Bessel bound for all $C_{G,r,t,U}$ acting on any common function.

Even though outputs are restricted to fine intervals in $E_{G,r,t}\cap U$, the left-endpoint projection in the adjoint may extend over all of $U$. The proof uses the column space on this whole ancestor interval; it does not assume the adjoint remains supported on the fine output set.

6.3 Restoring $\mathcal B_r$ and checking the varying-function row hypotheses

On the decision interval $U$, use H Compression to compose the row on the right with the common inner projection product. This gives an ordinary joint operator bound with cost

For $m\ge1$, the adjoint of the composed row has the order

The last operator applied is $P_U^1$, so the range consists of constant-coefficient combinations of main columns on the whole of $U$. The support and observation depths are both $r$, and the function index is also $r$.

If the inner operator product is empty, both endpoint ranges can be represented by main columns on depth-$t$ cells, restricting the earlier range to these fine cells by inheritance. The support depth is then $r$, the observation depth is $t$, and the function index remains $r$.

In both cases, the next left endpoint satisfies $r_{\mathrm{next}}\ge t$, giving the required refinements. D Lemma 4.4 can now be applied to the varying $F_r$ and bounds the square sum of the first term by $G_0|R|$.

7. The second term: changing a noncommutative inner projection product

First apply (Endpoint-Bessel) to the group projections at the common right endpoint. The inner difference

is the same for every $G$. This removes the group sum, leaving the square sum of these inner differences to be controlled.

Keeping the original order gives the noncommutative telescoping identity

In each term, the left-hand product of $t$-projections is contractive and does not increase the norm.

Fix $j$. The middle increments $P_t^j-P_r^j$ have pairwise orthogonal ranges on the ordered disjoint intervals. Splitting them over complete depth-$r$ intervals preserves ordinary Bessel constant one by the exact square-sum identity in Corollary 4.5. Add the right-hand product using H Compression.

If the right-hand product is nonempty, its adjoint applies $P_r^1$ last, so the observation depth is $r$. If $j=1$, that product is empty; inheritance represents both increment ranges on depth-$t$ cells, giving observation depth $t$. The support depth and function index are always $r$, and $t\le r_{\mathrm{next}}$ still ensures the required successive refinement.

Thus an ordinary operator bound is established for each $j$ before D Lemma 4.4 is applied to varying $F_r$. Sum over $j$ in the finite row direct sum. The cost is controlled by a fixed power of $m$ and (Compression-cost), and is still absorbed by $G_0$. This handles the second term.

For a shared output without an outer group projection, the same proof directly uses the full $L^2(R)$ bound for the inner difference. The identity shared output has zero operator difference.

8. The third term: changing only the original function

The inner projection product is common to all groups, and every projection is contractive, so

The blockwise variation lemma for the original sequence, with edge overlap $a$, gives

For shared outputs, likewise bound the full norm first, then restrict once to $E_{r,t}$. For the identity shared output, the required estimate is exactly this difference bound for the original functions.

If the inner projection product varied arbitrarily with the group $G$, (Endpoint-Bessel) would not apply directly. This explains the source requirement that the inner ordered product be independent of $G$.

9. Zero outputs before the root and restoration of the averages

If both endpoints precede the current root, their difference is zero. At most $a$ depth intervals cross the starting depth. On these intervals, the earlier output is zero. The right-endpoint projection square sum, inner contraction, and

together bound the cost by $Ca|R|$.

Now restore the original positive projection averages. When comparing depths, use the same projection choice in each average so that its projection family stays nested. The choices for shared inner averages must also be common across all groups, keeping the ordered product independent of $G$. Each group's outer projection choices are coupled across the compared depths as well. Regard all choices as one normalized finite probability average and apply Jensen's inequality to the squared norm in the finite row direct sum. This restores the averages without a factor for their number of terms.

Choosing different projections independently at the two depths would make the first term cease to be an increment of one nested family, so the orthogonality argument would no longer apply directly.

Expand complementary operators in the original order into at most $2^{O(\log L)}$ ordered projection products. The basic $B$ has at most two fixed signed components. Restore these finitely many terms using the triangle inequality in the row direct sum, then sum over at most $a$ interval colors. This proves (Variation-local) and (Variation-shared).

10. Why all losses remain within $G_0$

By (Main-sizes),

The logarithm of (Compression-cost) is at most

This is bounded by a fixed power of $(1+\log(2k))$. The logarithms of the additional row-estimate factor $\log(2D_{\mathrm m})$, the finite expansion costs, and the squared costs have the same type of bound. The total can therefore be written as

No factor counts the terms in a positive average, and the larger mode-family constant $D_*$ has not been used here. If later $P=\lceil\eta^{-\alpha}\rceil$ with fixed $\alpha>0$, then $\log P$ grows like $k^\theta$, whereas $\log G_0$ is only a fixed power of a logarithm. Thus $G_0=P^{o(1)}$. This is the size needed to preserve a negative-power gain in $P$ later.

11. Application to later adjacent projection differences

This is D Corollary 6.2, pp.46–47.

In a one-step output-filter transfer specified by H, removing the outermost positive filter leaves an inner ordered product in the source order, of length $O(\log L)$ and independent of the group. Its endpoint function is a shared output of the type above. If a permitted group projection remains outside it, the endpoint is of group type.

Let the individual projections in that step be $Q_{p,s}$ for $1\le p\le P$, and set

The source distinguishes two kinds of transfer. In a ball-projection transfer, $Q_{p,s}x_s$ for $1\le p\le P$ are permitted group endpoints, while $x_s$ is shared. In a cumulative-filter transfer, the outer ball projection has already been removed: $x_s$ and all $Q_{p,s}x_s$ are shared endpoints.

The complete remaining endpoint $x_{d,s}$ in the other complementary position also has only two permitted forms. Either it retains an outermost positive ball-projection average with a group-independent inner projection product, or that outer layer has been removed and the output is shared. Apply (Variation-local) in the first case and (Variation-shared) in the second.

For a nonterminal difference

the two terms are group endpoints or shared endpoints according to the transfer type above. Their difference across depths is controlled by two corresponding proved variation bounds, costing only a fixed factor.

The terminal difference is

In a ball-projection transfer, count the first term $x_s$ once on the union of all required right-endpoint cells for that shared output; count the second as a permitted group endpoint. In a cumulative-filter transfer, both endpoints are shared, and each is counted on its corresponding union of uses. Repeating the same shared term for every group would not be justified by (Variation-shared).

The total cost for $P+1$ differences may grow as a fixed polynomial in $P$. Subsequent summation must still preserve the matched combinations specified in the source. The single-endpoint variation bound alone does not permit arbitrary separation of all group and difference terms.

Source coverage and remaining work

• D pp.43–44, equations (6.3)–(6.4): the separation of full usable endpoint spaces assumed from H and its projection square-sum consequence.
• D pp.44–46, Lemma 6.1 and equations (6.6)–(6.8): the exact three-term difference, ordinary operator bounds before varying-function substitution, the zero-output convention, and coupled projection averages.
• D pp.46–47, Corollary 6.2: group and shared endpoint types in the source filter transfers.

This part does not prove the curvature charts or group-mask construction in H, or the complete localization decomposition in D equation (6.5). It proves the depth-variation bounds required by D assuming the stated endpoint separation and ordered-projection conditions. The next part, on frozen mode rows, must also identify the final range of the composed adjoint so that a loss involving $D_{\mathrm m}$ is not replaced by one involving the larger $D_*$.