Unit 3, continued: how bad-height counts control square sums of mode rows

This part treats the first half of D §6.4, pp.49–50, especially equation (6.12). The preceding frozen-row theorem assumes an ordinary joint Bessel bound for each group's mode projections. We show why the non-core mode rows actually selected in the source satisfy this hypothesis, then pass from orthogonal projections to exact partial sums of mode coefficients.

D uses the graphs, good/bad classification, and geometric conclusions of H §§8.4–8.6. Two source assumptions remain explicit here: the bounds on exceptional blocks and bad heights for each main column, and the small oscillatory inner products between nonadjacent modes. Their rational-height stability proofs and Fourier estimates are not all reproved here. The chain count, the deduction of Bessel bounds from small inner products, and the subsequent argument for exact partial sums are developed below.

1. Two graphs on the same frozen list

Fix a position, group, offset, and analysis block. Freeze the mode list at $u_b=b-4K$ and compare it at depth $v_b=b-2K$ on retained intervals $J$; the current block ends at $t_b=b+K$. Include every available mode, even with zero coefficient, and retain the list on the relevant descendants. A common constant $D_*\ge\mathrm e^k$ controls the permanent mode count, total absolute frozen coefficients, and reciprocal private weights along a path across all relevant groups. The horizontal parameter dimension satisfies $d_{\max}\le C_qk^C$; see D equation (6.9) and the discussion following it.

The frequency slope of a scalar quadratic mode is

H uses one horizontal frequency vector $\vartheta$ and height parameters

with $A_0\ge6$ sufficiently large. The quantity $L_h$ is the denominator of permitted shifts, and $E_h$ is the integer-shift range, not an orthogonal projection. Write the separation threshold as $\Xi$; the source chooses $\Xi=D_*^{1000}$.

At height $h$, if an integer vector $n$ satisfies $\|n\|_\infty\le E_h$ and

join $m,m'$ in the PRE graph. Replacing the right-hand side by $\Xi/r_{t_b}$ defines the CURRENT graph. Here $r_s=2^{-s}|J_{\mathrm{original}}|$.

Since $r_{t_b}<r_{v_b}$, every PRE edge is also a CURRENT edge. Denote the corresponding components by

Replacing an old witness $n$ by $6n$ shows that both graphs gain edges and lose none as $h$ increases. Smaller descendant intervals and larger comparison thresholds likewise preserve existing edges on the later lists specified by the source.

In this part, a minimal component means a component at height zero. For a set of modes $D$, the notation $\pi(D)$ denotes its set of main-column labels, rather than a probability-factor map.

2. Good heights, core modes, and non-core modes

The source distinguishes whether a main column is exceptional in the current block and, for a nonexceptional column, whether height stability holds at $h$. These tests come from H Lemmas 8.2–8.3 (pp.67–69). The quantitative conclusions used here are:

• A fixed main column is exceptional in at most $B_{\mathrm{bad}}\le C d_{\max}$ blocks along a path;
• In a fixed block, a nonexceptional main column fails stability at at most $A_{\mathrm{bad}}\le C d_{\max}R_{\mathrm{ht}}$ heights.

Here $d_{\max}$ controls the relevant horizontal parameter dimension, and $R_{\mathrm{ht}}$ is an auxiliary integer in the rational-height argument. It may depend on the larger mode constant. That dependence is absorbed first by choosing the total height $J_*$.

For $h\ge\Delta$, a mode $m$ is good at height $h$ if

and at least nine tenths of the main columns in $\pi(\mathsf F_h(m))$ are nonexceptional in the block and stable at that height. All heights below $\Delta$ are bad.

Fix a sufficiently small absolute constant $\delta_*>0$. A mode is core if it has at most $\delta_*(J_*+1)$ bad heights. Otherwise it is non-core, called a linear-side mode in the source.

Modes in the same minimal PRE component have the same component in every higher PRE and CURRENT graph. Their good/bad classifications are therefore identical: an entire minimal PRE component is either core or non-core. This elementary inheritance fact suffices for selecting the non-core events below.

3. Chains of non-core events

Fix a position and a group. An event is a triple

where $J$ is a retained depth-$v_b$ interval and $E$ is a non-core minimal PRE component in its frozen list.

Split analysis-block indices into ten residue classes modulo ten. Within one class, connect an earlier event to a later event when the latter lies on a descendant path and some modes in the two events coincide or are adjacent in the later event's minimal PRE graph.

Successive blocks in one class are separated by at least $10K$, so the later PRE comparison depth is no earlier than the previous block's end. Every CURRENT edge from the earlier block persists in the later PRE graph: the interval is smaller, the threshold is wider, and the modes and integer-shift witnesses are retained. Minimal PRE connections persist at every height.

Along an event chain, choose one mode from each event and denote its main-column sets at height $h$ by $P_{i,h},C_{i,h}$. Then

Both sequences also increase with $h$. Every set is nonempty and has size at most the main-column bound $D_{\mathrm m}$. The relevant count is the number of main columns, rather than the potentially much larger mode count $D_*$.

4. A finite summation lemma using order of entry

This is H Lemma 8.5, pp.69–70.

Suppose nonempty finite sets satisfy

and $X_t\subseteq S_t$, with each element belonging to at most $A$ of the sets $X_t$. Then

Rank elements by the time they first enter $S_t$, breaking ties arbitrarily. Any set containing the element of rank $r$ has at least $r$ elements. Each occurrence therefore contributes at most $1/r$, and the total contribution of that element is at most $A/r$. Sum over ranks.

The lemma does not require knowing when an element leaves $X_t$; only its total number of occurrences matters.

5. Counting chains with many bad heights

We now give the form of H Lemma 8.6 (p.70) used in this proof.

Suppose a chain has $N$ events and heights $0,\ldots,J$. The sets satisfy (Chain-nesting) and have size at most $M$. Each main column is exceptional in at most $B$ events. At a fixed event, each nonexceptional main column is unstable at at most $A$ heights.

Let $X_i$ be the exceptional set and $Y_{i,h}$ the height-unstable set. For $h\ge\Delta$, a height is bad if either the two cardinalities have different integer parts of their base-$1.1$ logarithms, or

Every event has more than $\delta(J+1)$ bad heights. Define

5.1 Bad heights caused by changes in cardinality

Write

These values lie in $0,\ldots,L_0$. When the integer parts differ, set inclusion makes their difference at least one. The indicator is therefore bounded by

For fixed $i$, summing the first term over $h$ gives at most $\Delta L_0$, since each one-step increase occurs in at most $\Delta$ windows.

For fixed $h$, the inclusion $C_{i,h}\subseteq P_{i+1,h}$ bounds the sum of the second term over $i$ by $L_0$. Including the first $\Delta$ bad heights of each event, the number of bad event–height pairs of this type is at most

5.2 Contributions from exceptionality and height instability

The indicator of the bad-proportion condition is at most $\beta^{-1}$ times the sum of the exceptional and unstable proportions.

For a fixed event $i$, the sets $C_{i,h}$ increase with $h$, and each nonexceptional main column occurs in at most $A$ sets $Y_{i,h}$. Equation (Entry-rank) bounds the total unstable proportion by $AH_M$.

For a fixed height $h$, the sets $C_{i,h}$ increase with $i$. Each main column appears in at most $B$ exceptional sets, so the total exceptional proportion is at most $BH_M$.

Combining the two types, the total number of bad heights is at most

It is also strictly greater than $\delta N(J+1)$. If

then $NQ$ is absorbed by half the left-hand side, giving

5.3 Where the larger height parameter disappears

Substitute

H first chooses $J_*$ sufficiently large to satisfy this absorption condition. The final chain-length bound is therefore

The parameter $R_{\mathrm{ht}}$ affects only how large $J_*$ must be. It no longer appears in the final chain length. Thus the crude total count of mode heights is not the row-count cost.

6. An ordinary Bessel bound at each chain level

Assign each event the length of the longest chain ending there. Connected events have different levels, so overlapping events at the same level have neither coincident nor adjacent modes. Distinct events on the same interval in the same block are already distinct minimal PRE components, with no edges between them.

Along any path, mode-label sets of same-level events are therefore disjoint and have total size at most $D_*$. The oscillatory part of H Lemma 8.7 further states that any mode pair from two overlapping events has normalized inner product on the smaller interval of absolute value at most

This analytic smallness follows in the source from PRE nonadjacency, permitted integer shifts, and cutoff Fourier tails. It is the second explicit geometric assumption used here.

6.1 A joint projection bound directly from small inner products

Let $P_e$ be the local orthogonal projection onto the mode span of event $e$. By duality, it suffices to bound sums of arbitrary functions $f_e$ in these ranges. Write

The lower bound on private weights gives

Disjoint intervals give zero inner product. For intersecting intervals, (Mode-pair) and the bound of $D_*$ modes per group give

There are at most $D_*$ nonempty events at one level along a path. Hence

Adding the diagonal terms gives an absolute synthesis bound. By duality, these local mode projections have an absolute ordinary joint Bessel constant.

6.2 Restoring block residue classes and chain levels

There are ten block residue classes, each with the number of chain levels bounded by (Short-chain). Adding the square sums over levels gives all event rows of a fixed group squared Bessel constant

This is the joint row hypothesis required by D Lemma 6.3 in the preceding part. It follows from the selected event structure, beyond the individual orthogonal-projection property of each row.

7. Frozen rows handle functions varying between blocks

The mode-output interval of an event row is $J_e$, and its main endpoint is the containing interval at depth $u_b$. The modes are already available there, and the retained interval is used in the later steps of this block. The preceding frozen-row theorem therefore applies.

It gives

The sum across groups keeps the position, search branch, and color fixed. Each block's frozen function comes from its own $F_{u_b}$. The varying-function row theorem already handles this summation; mode coefficients need not agree between blocks. What remains fixed is the mode labeling, expansion data, and associated operator structure.

8. Orthogonal projections versus exact coefficient partial sums

Let

be the exact coefficient partial sum over $E$ in the frozen mode expansion. It generally differs from $P_{E,J}\widetilde Z_{G,b}$, since distinct modes are only approximately orthogonal.

Every mode outside the component satisfies (Mode-pair) with every mode in $E$. If $v$ lies in the span of $E$ and has normalized $L^2(J)$ norm one, the private directions bound the $\ell^1$ norm of its coefficients by $D_*^{3/2}$. The frozen function's external coefficients have total absolute mass at most $D_*$. Thus

The pairing $\langle\cdot,\cdot\rangle_{2,J}$ uses normalized measure. Therefore

Denote the chain-level bound by

Fix a residue class modulo ten, a level, and a path point $y$. Every event containing that point uses at least one permanent mode label. These labels are disjoint within a group at one level, and permanent labels from different groups are distinguished. The common bound includes all relevant groups. There are therefore at most $D_*$ such events, giving

The ordinary $L^2$ squared error in (Subtotal-error) is at most $|J_e|D_*^{-380}$. Summing all squared errors gives at most

Using $d_{\max}\le C_qk^C$, $\log D_{\mathrm m}\le C_qk^C$, and $D_*\ge\mathrm e^k$, and increasing the initial precision, this is at most $|R|$. Finally, apply termwise

and combine with (Projection-energy) to bound the sum of squared norms of all exact partial sums $T_{E,J}$.

9. Square energy of the non-core part

On one interval $J$, distinct minimal PRE components have disjoint labels and satisfy the same small cross-mode inner-product bounds. The coefficient calculation in Section 6.1 therefore also applies to their exact partial sums, bounding the squared norm of their sum by the sum of their squared norms.

Define $V_{l,G,b}$ as the exact sum of all non-core modes in $\widetilde Z_{l,G,b}$, including private coordinates, and set it to zero on branches with no required output or basis for use. Sum over disjoint intervals $J$ at one depth and over all $G,b$. This gives D equation (6.12):

The entire argument holds for a fixed but arbitrary candidate offset. Later, the physical approximation error may first be averaged and then a suitable offset selected; the uniform square-energy bound survives. Nor does the proof require one mode to have identical frozen coefficients in different blocks.

Source coverage and the next step

• D pp.49–50: frozen lists, PRE/CURRENT classification, event rows, exact non-core partial sums, and equation (6.12).
• H §§8.4–8.5, pp.67–70: the two graphs, good/bad heights, core classification, entry-order summation, and bad-height chains.
• H Lemma 8.7 and Proposition 8.9, pp.70–73: small oscillatory inner products, event levels, joint Bessel bounds, and exact coefficient partial sums.

The source bounds on exceptional blocks and heights, and the small oscillatory inner products, remain assumptions here; ordinary orthogonality does not supply them. The next part, D pp.50–52, must choose offsets, control pathwise square sums for both rolling variation and the non-core part, and establish count and size conditions for core representatives. Those steps require their own checks.