Unit 3, supplement: closure of main projections and localization

This supplement develops D §6.1, pp.42–44, equations (6.1)–(6.5). In the source order it precedes variation of localized outputs. It defines the groups, masks, and main projections used later, and is where depth-varying functions first enter H's localization argument.

H's graph-mask construction, different-label separation, and same-label detection remain explicit inputs. We develop finite projection closure, Bessel synthesis on whole endpoint intervals, and the binary search that places the four functions on an original cell in one current group.

1. Labels, groups, and local orthogonal projections

1.1 Graph vertices and new columns

An actual occurrence of a basic column corresponds to a graph vertex. Each vertex carries H's full normalized curvature label, including horizontal-coordinate data. In position $l$ , the quadratic phase is the label phase multiplied by

\kappa_l,\qquad (\kappa_0,\kappa_1,\kappa_2,\kappa_3)=(-1,3,-3,1).

Linear modulation is not part of the curvature label.

Graph distance depends on the labels and absolute depth from the original $J$ , and is nonincreasing with depth. Later main or auxiliary columns are attached to existing vertices with the same labels. Adding columns creates no new vertices.

1.2 Groups and masks

A group $G$ has a fixed color and a collection of actual center uses. Its mask with enlargement $h$ ,

\mathcal M_{G,h}(I)

is the set of vertices permitted on interval $I$ . It records earlier center uses, so it may continue to be inherited after the group ceases to be current on a branch.

For rank $r$ , define

P_{r,\mathcal M,I}

as the orthogonal projection in $L^2(I;\mathcal H)$ onto the span of existing columns with rank at most $r$ and labels in $\mathcal M$ , using coefficients constant on the whole of $I$ . Coefficients may depend on $I$ but cannot vary arbitrarily pointwise within one $I$ .

For fixed $I$ , enlarging a mask or adding columns increases the projection range. Each individual mask projection is also inherited along complete depth partitions. Required continuations must be defined even on intervals without actual use.

A positive averaging filter averages $P$ expanded projections:

Q=\frac1P\sum_{p=1}^P P_p.

The operator $Q$ is a positive contraction, generally not an orthogonal projection. A history operator is a prescribed ordered product of these filters and their complementary operators $1-Q$ . Fixing individual projection choices in each average and expanding complements gives finitely many ordered orthogonal-projection products. Projections in different positions need not commute.

1.3 Usable endpoints

An endpoint interval $U$ is usable for a group mask when the mask is nonempty and $U$ or a descendant has an actual use of that group. The endpoint need not itself be a current center use.

H Lemma 6.4 gives three geometric properties:

If the actual center is $v$ , every existing vertex whose distance from $v$ is at most $h$ lies in the level- $h$ mask.
Graph neighbors of vertices in the level- $h$ mask lie in the level- $h+1$ mask.
For different groups of the same color on overlapping usable endpoints, their mask vertices have distance greater than one at the finer endpoint depth.

These are the geometric inputs for localization and synthesis across groups. We do not reconstruct the colors, center competition, and historical masks needed to prove them.

2. Closure tests fix the function on the current interval

Fix an original cell $I$ at depth $s$ and an actual basic function

b=z_{l,s},

of type $B$ or $U$ . Throughout closure on this cell, $b$ remains fixed and may contain the original $F_{l,s}$ .

For functions $v,w$ in two specified positions, set

\mathfrak d_I(v,w) = \sup_{\|g_a\|_{2,I},\,\|g_b\|_{2,I}\le1} |\mathcal H_I(\ldots,v,\ldots,w,\ldots,g_a,\ldots,g_b)|.

The other two positions on the right contain normalized $L^2(I)$ functions.

Closure at rank $r$ requires that, for every prescribed history $H$ , every prescribed individual projection choice, and every lower-rank pure column function $v$ whose label lies in the target mask,

\mathfrak d_I\big(v,(1-P_{r,\mathcal M,I})Hb\big) \le \frac{\tau_3}{D_{\rm base}\sqrt{n_{r-1}}} \|b\|_{2,I}, \qquad \tau_3=D_{\rm base}^{-C_\tau}. \tag{Closure}

The quantity $n_{r-1}$ bounds the pathwise number of columns below the current rank. Once individual projection tests hold, averaging gives the corresponding tests for positive averaging filters.

Only $n_{r-1}$ is under the square root in the denominator. This is equation (6.1) on D p.43, and it cancels the source-column coefficient bound below.

3. A failed test produces an orthogonal increment

The same-label detection input in H Lemma 6.6 is as follows. If a pure column with label $v$ in one position, a physical residual in another, and two normalized $L^2$ functions give a local form of size greater than $t_0$ , then the residual has normalized moment at least

t_0/H_0,\qquad H_0\le \exp(C_qk^C).

against a template with the same label. Templates permit arbitrary linear modulation and have uniformly bounded physical components, with no growth in insertion rank or modulation. Both original and later inserted templates may serve as source columns.

If (Closure) fails, insert a copy of a detected template at the target rank with a fresh private orthogonal coordinate of weight one: $\Phi=(\phi,e_{\rm new})$ . Write the target projections before and after insertion as $P_{\rm old}\le P_{\rm new}$ , the old test state as $X=Hb$ , and the old residual as

r=(1-P_{\rm old})X.

The new column $\Phi$ has an absolute normalized norm bound. Its private coordinate vanishes in $X$ and in the old projection, so the physical moment equals the full-vector inner product:

|\langle r,\Phi\rangle_I| \ge c\,\frac{\tau_3}{D_{\rm base}\sqrt{n_{r-1}}H_0} \|b\|_{2,I}.

Set $\psi=(1-P_{\rm old})\Phi$ . Since $r$ is orthogonal to the old range,

\langle r,\psi\rangle_I=\langle r,\Phi\rangle_I, \qquad \|\psi\|_{2,I}\le\|\Phi\|_{2,I}\le C.

The new range contains at least the direction $\psi$ , hence

\|(P_{\rm new}-P_{\rm old})X\|_{2,I} \ge \frac{|\langle r,\psi\rangle_I|}{\|\psi\|_{2,I}} \ge c\,\frac{\tau_3}{D_{\rm base}\sqrt{n_{r-1}}H_0} \|b\|_{2,I}. \tag{Fresh-gain}

If $\|b\|_{2,I}=0$ , every state is zero and no test fails.

4. Bounding insertions when test states vary

4.1 A fixed vector with changing histories

Other projections may enlarge during insertion, so at step $i$ the state

X_i=H_i b

can change. The sum $\sum_i\|(P_{i,\rm new}-P_{i,\rm old})X_i\|^2$ cannot simply be treated as an orthogonal decomposition of one vector.

Fix a named history and target mask. During at most $T$ insertions, every positive projection family increases in the insertion index. The target increments

B_i=P_{i,\rm new}-P_{i,\rm old}

are mutually orthogonal rows, so

\sum_i\|B_i x\|_{2,I}^2\le\|x\|_{2,I}^2

for every fixed $x$ .

Expand the complementary operators in the history and apply the proved dyadic-prefix row bound termwise. The history has $O(\log L)$ factors, giving

\sum_i\|B_iH_i b\|_{2,I}^2 \le (C\log(2T))^{O(\log L)}\|b\|_{2,I}^2. \tag{Changing-history}

The original vector on the right is always $b$ . Only prefixes of nested projections vary, exactly as permitted by the prefix-row estimate.

4.2 Failed tests are counted separately from source labels

Each failure gives the squared lower bound from (Fresh-gain). Comparing with (Changing-history), the number of failures for one target history and mask is at most

C\,D_{\rm base}^2n_{r-1}H_0^2\tau_3^{-2} (C\log(2T))^{O(\log L)}.

The number of possible target masks, named histories, positions, and auxiliary basic functions is $\exp(k^{O(1)})$ . Combining these counts and prefactors bounds insertions at rank $r$ on one cell by

n_{r-1}\exp(k^{C_1}) (C\log(2T))^{O(\log L)}. \tag{Insertion-count}

There is no additional factor for every source column that could trigger failure. Different sources may trigger the same target residual, but each failure is counted through that target's orthogonal increment in the same row sequence. Source identity introduces no extra sum.

4.3 Choosing rank bounds and ruling out the cutoff

Absorb the pathwise active-cell count and fixed rank and position costs into $\exp(k^{C_1})$ as well. Choose in advance

\frac{n_r}{n_{r-1}}=\exp(k^{C_2}),

with $C_2>C_1$ sufficiently large. There are $L+5=k^{O(1)}$ total ranks, so

\log n_{\max}=k^{O(1)}.

Set a trial cutoff $T=n_{\max}$ for the total insertions allowed on each path, leaving a fixed margin. This also bounds the insertion sequence on a fixed current cell. It does not bound all insertions on disjoint branches of the entire tree by $n_{\max}$ . Each path has only the prescribed number of active cells, already absorbed into $\exp(k^{C_1})$ . Now

\log\!\left((C\log(2T))^{O(\log L)}\right) =O((\log k)^2),

which, for large $k$ , is much smaller than $k^{C_2}$ . Thus (Insertion-count) is strictly below the prescribed allowance at each rank, and no trial can reach the cutoff.

This proves finite closure. Auxiliary closure is performed after complete side functions are fixed. Its sources come only from main columns, and its new private directions differ from all old ones. The same argument applies.

4.4 The order of parameter choices is not circular

The count uses the number of named contexts, template complexity, and history length. It does not use the numerical graph-edge height $H_{\rm ed}$ or the later mode-approximation precision. Thus first determine

\log n_{\max}\le C_qk^C,

then choose a sufficiently small different-label error $\tau_2$ . Finally choose $H_{\rm ed}$ using H's different-label separation theorem and construct compatible masks.

Fixing counts before errors is essential here. If a count depended on the edge height selected afterward for that count, the argument would not provide a finite bound in advance.

5. Bessel synthesis across groups on whole endpoint intervals

We develop the use of H Lemma 6.5 in D equation (6.3). Fix a position and color, and let

h_G=\sum_{v,U}a_{v,G,U}\mathbf1_U\Phi_v,

where $U$ is a usable endpoint interval, coefficients are constant on the whole of $U$ , and the mask permitting that term contains label $v$ . One label may occur at arbitrarily many endpoints.

5.1 First combine every use of one label

Define

a_{v,G}(y)=\sum_Ua_{v,G,U}\mathbf1_U(y).

If column $\Phi_v$ has private coordinate $\sqrt{\epsilon_v}e_v$ and $\epsilon_v^{-1/2}\le W_0$ , then

\sum_v|a_{v,G}(y)|^2\le W_0^2\|h_G(y)\|^2. \tag{Private-coefficients}

Combining uses first preserves cancellation between repeated occurrences of one label.

5.2 The last separated depth and coefficient measurability

Fix labels $v,w$ in different groups. Graph distance is nonincreasing with depth, so the depths where it exceeds one form an initial interval. Let its last depth be $s_*$ , capped at the end of the current finite tree if needed.

If there is no separated depth in the current root, mask separation makes the label supports disjoint. The cross term vanishes and no $s_*$ need be defined. We consider only the case with a separated depth below.

For overlapping supporting endpoints $U,V$ , their intersection is the finer interval. Mask separation bounds its depth by $s_*$ . Therefore

a_{v,G}\overline{a_{w,G'}} = \sum_{U,V}a_{v,G,U}\overline{a_{w,G',V}} \mathbf1_{U\cap V}

is measurable for the depth- $s_*$ partition. A finer endpoint failing this condition cannot overlap an endpoint of the other label, so its product vanishes.

Apply H's different-label correlation estimate on each cell of this partition and sum to obtain

|\langle h_G,h_{G'}\rangle| \le \tau_2\sum_{v,w}\int|a_{v,G}a_{w,G'}|. \tag{Cross-group}

Different groups cannot use one private label on an overlap: its distance from itself would be zero, contradicting separation.

5.3 Summing over all groups and labels

If the total pathwise number of groups and labels is at most $N_0$ , pointwise Cauchy–Schwarz and (Private-coefficients) give

\sum_{G\ne G'}\sum_{v,w} |a_{v,G}a_{w,G'}| \le C N_0^2W_0^2\sum_G\|h_G\|^2.

Choose $\tau_2 C N_0^2W_0^2\le1$ . Then

\left\|\sum_Gh_G\right\|_2^2 \le2\sum_G\|h_G\|_2^2. \tag{Endpoint-Bessel}

The constant contains neither endpoint repetition counts nor the depth of future use establishing usability. The crucial points are measurability of coefficient products and combining identical labels first.

5.4 Restrictions must occur on the correct side of the adjoint

If outputs are used only on a union of finer cells $E$ , the operator is $\mathbf1_EP_U$ , with adjoint

(\mathbf1_EP_U)^*=P_U\mathbf1_E.

The adjoint range still consists of constant-coefficient column combinations on the whole of $U$ . Synthesize these ranges using (Endpoint-Bessel), then apply duality to obtain

\sum_G\|\mathbf1_{E_G}L_{G,t}h\|_2^2\le C\|h\|_2^2

for a common input function.

Arbitrarily restricting the original range to $E$ would violate the whole-endpoint hypothesis. The adjoint order explains why fine-cell output restrictions are allowed.

6. Binary search over colors

First specify the cumulative masks and ranks used in the source. Define

\mathcal U_{m,h}(I) = \{v:d_s(v,\{w:\operatorname{color}(w)\le m\})\le h\}.

These masks are inherited with depth, and one extra enlargement contains the graph neighbors of the old mask. At a color split $m$ , the cumulative filter uses

\text{rank}=L-m+1,\qquad \text{base enlargement}=10P(L-m+1).

The four successive ball-filter insertions use

\text{rank}=L+j,\qquad \text{base enlargement}=20P(L+1)+10(j-1)P, \quad j=1,2,3,4.

Each positive average adds $p=1,\ldots,P$ to the relevant base enlargement and averages. These values and the required one-neighbor enlargements lie within the source's permitted mask range.

Following the proof of H Proposition 6.8, we now explain D equation (6.5). An active pattern has at least three basic coarse functions; choose one as the initial upper guard position.

Maintain a color range $[\ell,h]$ , an upper guard position, and, when $\ell>1$ , a lower residual guard position. Start from $[1,L]$ with no lower guard.

If $\ell<h$ , choose midpoint $m=\lfloor(\ell+h)/2\rfloor$ and a third position distinct from the existing guards. Decompose its current function $z$ by the identity

z=Q_mz+(1-Q_m)z.

Quadrilinearity splits the original term into two exact terms.

In the positive-projection term, the new position becomes the upper guard, the old upper guard is released, and the color range becomes $[\ell,m]$ .
In the residual term, the new position becomes the lower guard, the old lower guard is released, and the range becomes $[m+1,h]$ .

No further operator is applied to a guard position before it is released. Position choices are fixed for each branch; the same rule applies even when some colors are absent on a cell. Every branch therefore has length $O(\log L)$ , at most $L$ leaves occur, and each leaf's inner history remains fixed across the tree.

7. Retaining only current groups at a single-color leaf

7.1 Excluding sources of earlier colors

Let the leaf color be $c$ . The upper guard is either the original coarse function or has as its outermost operator a cumulative positive projection of rank

L-c+1

Expand only this outermost range, retaining the complete inner history.

If $c>1$ , the lower guard has a residual projection of rank $L-c+2$ whose base enlargement exceeds the upper one's by at least $10P$ . After individual choices $1\le p\le P$ , the required containment margin remains. Thus every upper source label close to a vertex of color at most $c-1$ lies in all required lower masks and has lower rank. Equation (Closure) controls these old-source terms. For $c=1$ , there are no earlier colors.

7.2 A unique matching current group

A remaining source label lies near a vertex of current color $c$ and therefore matches that vertex's current group $G$ . Mask containment puts the source in the mask needed for the group's first ball projection.

It cannot match two groups of one color, by same-color mask separation. It is also separated from pure sources in the ball projections of other current groups.

Choose a third position distinct from the guards and insert the sum of first ball projections over current groups. For the unique matching group, (Closure) controls projection minus identity. For other groups, expand the new ball projection's pure sources and use different-label separation. The unchanged lower guard residual continues to control earlier-color terms.

7.3 Bringing the other positions into the same group

The third position now has the first-ball mask support of one group $G$ . Keep it as a lower-rank source while inserting ball projections of successively higher rank and larger enlargement in the other three positions.

Each new target mask contains the source mask and has higher rank, so (Closure) controls the inserted residual. Leave the third position unchanged. All four positions now have outermost ball filters for the same group, while their inner histories depend only on the original search branch.

This gives

\mathcal H_I(z_{\cdot,s}) = \sum_\lambda\sum_{G\text{ current on }I\text{ }} \mathcal H_I(Z^\lambda_{0,G,s},\ldots,Z^\lambda_{3,G,s}) +e_I,

Z^\lambda_{l,G,s} =L^\lambda_{l,G,s}B^\lambda_{l,s}z_{l,s}, \qquad \sum_{I\in\mathcal A_R^{\rm main}}|I|\,|e_I| \le Ce^{-k}|R|. \tag{Localization}

Here $L$ is a positive averaged ball filter. The operator $B^\lambda_{l,s}$ is an ordered history of cumulative filters and complements of length $O(\log L)$ , independent of $G$ . Current groups must have actual center use; otherwise the inner group sum is empty.

8. Summing the numerical errors

A test at rank $r$ expands only one outermost source of lower rank. Its total absolute coefficient mass satisfies

\sum_v|a_v| \le D_{\rm base}\sqrt{n_{r-1}}\, \|\text{original input of the source projection}\|_{2,I}. \tag{Source-coefficients}

This follows from private-coordinate lower bounds, Cauchy–Schwarz for coefficients, and projection contraction. Cancellation with the denominator in (Closure) leaves $\tau_3$ times the local function sizes.

The remaining numbers of active uses, search leaves, groups, and insertion steps are fixed powers of the basic parameters. Choose $C_\tau$ sufficiently large to make their length-weighted error sum $O(e^{-k}|R|)$ . Different-label terms may cost an additional fixed power of the main-column count, but $\tau_2$ is chosen after that count and can make them equally small.

The main term of (Localization) thus carries no cost from a complete column-coefficient sum. Coefficient expansion is used only to bound small errors.

9. From cellwise identities to varying-depth estimates

On each cell $I$ , localization uses only the four actual functions $z_{l,s(I)}$ fixed there. Throughout closure, every projection update acts on the same basic cell function. The number of updates is independent of the deterministic depth block supplying that function.

Equation (Localization) alone gives no cross-depth energy bound. The later D Lemma 6.1 must treat

Z_{G,s}=L_{G,s}B_s z_s

by separating variation of the outer projection, the common inner history, and $z_s$ itself. Only after these estimates can one proceed to frozen mode rows and side-vector summation.

For the minimum, first retain the group sum on the original cell, then use across fixed search leaves

\min\!\left(\eta,\sum_\lambda a_\lambda\right) \le\sum_\lambda\min(\eta,a_\lambda), \qquad a_\lambda\ge0.

At this localization stage, separate minima for current groups are not part of the original target. Later matching and counting still require the full group sum on each original cell.

Source references and conditional scope

D pp.42–43: original depths, actual uses, graph labels, local orthogonal projections, and the normalization of the test in (6.1).
D p.43: fixed cell functions, orthogonal-row counting with changing histories, and equation (6.2).
H Lemma 6.6, pp.46–47: same-label template detection. Its precise statement is used here; the template Fourier decomposition is not reproved.
H Lemma 6.7, pp.48–49: named-context counts, finite closure, and counts before error tolerances.
H Lemmas 6.1 and 6.4: different-label correlations and mask geometry, retained as inputs.
H Lemma 6.5, pp.45–46, and D p.44: coefficient aggregation and Bessel synthesis on whole endpoints.
H Proposition 6.8, pp.49–50, and D equation (6.5): binary guard positions, successive ball-projection insertion, and error summation.

Under these geometric and detection assumptions, the same localization can be completed on each fixed cell despite the depth variation of D's current functions. This supplies the specific connection between the surrounding parts without treating H's full geometric proof as verified.