Unit 3, continued: why two and three non-core positions require different arguments

This part treats the beginning of D §6.5, pp.52–54, especially equation (6.18). Three functions with pathwise square-sum bounds can be summed directly by Hölder's inequality. With exactly two, the projections in the complementary positions must first be removed so that the small correlations of the full $Y$ become available. A matched difference sum with coefficient $P^{-1}$ remains.

Far differences in the projection transfer are controlled by the closure and label-separation conclusions of H Lemma 6.9. We prove the finite algebraic identity, identify where the source assumptions enter, and derive the freezing errors from the established variation bounds. The fixed short-block comparison and final $P^{-1/2}$ gain are deferred to the next part.

1. Three pathwise square sums control the $p$ -power sum

Let the local sizes in three non-core positions be $a_{s,G},b_{s,G},c_{s,G}$ . Set

a_s=\left(\sum_Ga_{s,G}^2\right)^{1/2}, \quad b_s=\left(\sum_Gb_{s,G}^2\right)^{1/2}, \quad c_s=\left(\sum_Gc_{s,G}^2\right)^{1/2}.

Each of these quantities has pathwise square sum bounded by $P^C$ on retained paths. The fourth position needs only the existing local bound $P^C$ .

The local absolute estimate and Cauchy–Schwarz in the group index give

\left|\sum_G\mathcal H_I(\text{these four positions})\right| \le P^C a_sb_sc_s.

For example, first use the local bound for the fourth position, then

\sum_Ga_{s,G}b_{s,G}c_{s,G} \le\left(\sum_Ga_{s,G}^2\right)^{1/2} \left(\sum_Gb_{s,G}^2c_{s,G}^2\right)^{1/2} \le a_sb_sc_s.

A shared size in the source transfer is counted once as a shared quantity, rather than repeated for every group. The two corresponding applications of Cauchy–Schwarz in group indices respect this convention.

Fix $2/3<p<1$ . Apply three-factor Hölder in depth, then use $3p>2$ to obtain

\begin{aligned} \sum_s(a_sb_sc_s)^p &\le \left(\sum_sa_s^{3p}\right)^{1/3} \left(\sum_sb_s^{3p}\right)^{1/3} \left(\sum_sc_s^{3p}\right)^{1/3}\\ &\le \left(\sum_sa_s^2\right)^{p/2} \left(\sum_sb_s^2\right)^{p/2} \left(\sum_sc_s^2\right)^{p/2} \le P^C. \end{aligned} \tag{Three-side-power}

The second step is the fact that a finite sequence's $\ell^{3p}$ norm is at most its $\ell^2$ norm.

Integrate over path points, converting the constant value on each original interval into a length-weighted sum. This gives total $p$ -power cost $P^C|R|$ . If $x$ is the absolute value of the entire prescribed group sum on one original interval, then

\min\{\eta,x\}\le\eta^{1-p}x^p \quad(x\ge0),

which gains the positive power $\eta^{1-p}$ . The prescribed group sum stays intact within each original interval before taking its $p$ -th power. We have not summed the $p$ -th powers of all individual group terms.

2. With exactly two non-core positions, first restore the complete complementary functions

Let the non-core positions be $a,b$ and the complementary positions be $c,d$ . Using the physical decomposition

Z=A+S+E,

replace the core part $A$ in each complementary position by the full current output $Z$ . Every correction has a third non-core position or a previously controlled physical error. The former is handled by (Three-side-power); the latter has total absolute error $O(\mathrm e^{-k}|R|)$ .

It remains to treat the two fixed complete non-core vectors $S_a,S_b$ with two full current complementary outputs. A basic $U$ position has no core part and is always non-core, so the complementary positions here are both basic $B$ positions.

H Lemma 6.9 removes the complementary filters in the prescribed order, starting with the larger outer positive filters, and transfers the corresponding auxiliary projection averages to position $a$ . Auxiliary closure uses the already fixed complete $S_a$ as its test basis. Expand each complementary operator as identity minus a positive average, keeping every resulting product in its original order.

3. The full finite algebra of one transfer step

This is D equation (6.18), p.53, and H equation (15), p.51.

In a step transferring from position $c$ to position $a$ , let $Q_{l,p}$ , for $1\le p\le P$ , be nested orthogonal projections in position $l=a,c$ . Set

Q_{l,0}=0,\quad Q_{l,P+1}=1, \quad A_l^p=Q_{l,p}-Q_{l,p-1}, \quad Q_l=\frac1P\sum_{p=1}^{P}Q_{l,p}.

For one position, the $A_l^p$ are pairwise orthogonal difference projections and

\sum_{p=1}^{P+1}A_l^p=1.

Since $Q_{l,u}=\sum_{p\le u}A_l^p$ , the average can be written as

Q_l=\sum_{p=1}^{P+1}\frac{P-p+1}{P}A_l^p. \tag{Average-weights}

The terminal weight at $p=P+1$ is zero, but its difference must be retained so that the sum of all differences is the identity.

Let $x_c$ be the full inner ordered product left after removing the current outer filter, applied to $B_c$ . Let $x_d$ be the entire remaining function in the other complementary position. Let $x_a$ be the transferred auxiliary projection product applied to $S_a$ , and set $x_b=S_b$ .

Insert (Average-weights) and the two identity decompositions into the quadrilinear form. This gives

\begin{aligned} &\mathcal H_I(x_a,x_b,Q_cx_c,x_d) -\mathcal H_I(Q_ax_a,x_b,x_c,x_d)\\ &\quad=\sum_{p,q'=1}^{P+1} \frac{p-q'}P \mathcal H_I(A_a^px_a,x_b,A_c^{q'}x_c,x_d). \end{aligned} \tag{Transfer}

Place the functions in their original positions $a,b,c,d$ . The coefficient is the difference of the averaging weights:

\frac{P-q'+1}{P}-\frac{P-p+1}{P} =\frac{p-q'}P.

This identity includes all terminal differences and requires no commutation between projections in different positions.

4. Discarding far differences while retaining the whole near sum

The source closure and label-separation results bound the total absolute contribution of all far differences with $|p-q'|>2$ , within each interval and group, by

D_{\mathrm{base}}^{-C_3} \|S_a\|_{2,I}\|S_b\|_{2,I}

where $C_3$ may be prescribed first and the source numerical precision chosen afterward. All main and auxiliary counts must precede the corresponding mismatch tolerances.

The two directions in H Lemma 6.9 use different hypotheses:

For $p>q'+2$ , the lower complementary difference consists of source columns from a positive projection. Their labels lie in both sets of use required by the higher auxiliary residual and have lower rank. Apply auxiliary closure with the entire $S_a$ as its basis.
For $q'>p+2$ , start from lower-rank source columns in the other complementary position $x_d$ . Nearby labels enter the higher complementary residual's set of use through closure under neighboring sets. Distant labels are separated from the auxiliary positive-projection source. Use main closure for the former and the source mismatch estimate for the latter.

The order of filter removal makes the other complementary source strictly lower in rank. A gap greater than two supplies the required one-step adjacency margin. Denominators in the source tests cancel the coefficient bounds for the expanded lower-rank columns. Retained complete functions are not fully expanded in coefficients. These are structural hypotheses of H's transfer result, beyond the algebra of (Transfer).

For the remaining terms, fix

\ell=q'-p\in\{-2,-1,0,1,2\}.

The coefficient is $-\ell/P$ , and we retain the complete matched sum

\sum_{\substack{1\le p\le P+1\\1\le p+\ell\le P+1}} \mathcal H_I(A_a^px_a,x_b,A_c^{p+\ell}x_c,x_d) \tag{Matched-bands}

The zero-offset coefficient vanishes. Separating absolute values for every $p$ in (Matched-bands) would introduce a cost for the number of differences.

5. The transferred main term uses the small correlations of the full $Y$

After the finitely many prescribed transfer steps, the main term is

\mathcal H_I(D_{a,s}S_a,S_b,B_c,B_d),

where $D_{a,s}$ is a product of auxiliary positive projection averages of length $O(\log L)$ , hence is contractive on the current interval.

Write

B_c=Y_c-U_c,\qquad B_d=Y_d-U_d.

Corrections containing $U_c$ or $U_d$ have a third basic non-core position with an established pathwise square-sum bound. The remaining $Y_c,Y_d$ is the full complementary pair from Proposition 5.6, with form norm at most $\eta$ . Thus the main term has absolute value at most

\eta\,\|S_a\|_{2,I}\|S_b\|_{2,I}

with the costs of finite ordered products and search branches absorbed into the indicated polynomials. Cauchy–Schwarz over groups and depths, together with the pathwise square sums, gives total absolute cost

\eta P^C|R|.

Small correlation is used only after restoring the full $Y$ . It is not assigned to the two original individual core parts.

The summation of numerical transfer errors and the choice of their precision belong to the cited source transfer result. After fixing the main and auxiliary counts, the source chooses precision so that these errors total $O(\mathrm e^{-k}|R|)$ .

6. Freezing the complementary positions of the near matched sum at the block start

For a fixed transfer step, search branch, and offset $\ell$ , write

C_{G,s}^{q'}=A_c^{q'}x_{c,s}, \qquad D_{G,s}=x_{d,s}.

For the following size and error estimates, combine all valid matched coordinates into one Hilbert-space vector

\boldsymbol C_{G,s} =(C_{G,s}^{p+\ell})_{p:\,1\le p,p+\ell\le P+1}, \qquad \|\boldsymbol C_{G,s}\|_{2,I}^2 =\sum_{p:\,\text{valid}}\|C_{G,s}^{p+\ell}\|_{2,I}^2.

For fixed $\ell$ , the map $p\mapsto p+\ell$ has no repetitions. Orthogonality of the difference projections at one endpoint bounds the current vector's norm by that of $x_{c,s}$ .

Freeze these vectors and the other complementary function at the block start $b$ , obtaining $\boldsymbol C_{G,b}$ and $D_{G,b}$ . This freezing time is $b$ , distinct from the mode-expansion time $u_b=b-4K$ .

D Corollary 6.2 already gives depth-variation bounds for these exact source ordered products. In a ball-filter transfer, nonterminal differences are differences of two permitted group endpoints, while the terminal difference is a shared inner function minus a positive group endpoint. In a cumulative-filter transfer, the outer ball has been removed and every endpoint is shared. The other complementary position may likewise retain an outer ball or already be shared.

Count a shared term once on the union of its actual intervals of use. The intervals $[b,s]$ have edge overlap $O(K)$ ; summing over $P+1$ differences incurs only the fixed-power cost in $P$ permitted here. All required endpoints are evaluated on complete ancestor intervals, with actual descendant use.

7. Freezing errors supply a third pathwise square sum

In this section, relabel $C_{G,s}^{p+\ell}$ as $C_s^p$ , still retaining only valid indices $1\le p,p+\ell\le P+1$ . The vector $\boldsymbol C_s$ consists of these matched coordinates. Replacing the current complementary pair by its frozen pair coordinatewise gives the bilinear difference

[C_s^p,D_s]-[C_b^p,D_b] =[C_s^p-C_b^p,D_s]+[C_b^p,D_s-D_b].

For the first term's complete matched sum, the local absolute estimate for four $L^2$ functions and Cauchy–Schwarz in $p$ give

\begin{aligned} &\left|\sum_p \mathcal H_I(A_a^px_a,S_b,C_s^p-C_b^p,D_s)\right|\\ &\quad\le C\|S_b\|_{2,I}\|D_s\|_{2,I} \left(\sum_p\|A_a^px_a\|_{2,I}^2\right)^{1/2} \|\boldsymbol C_s-\boldsymbol C_b\|_{2,I}\\ &\quad\le C\|S_a\|_{2,I}\|S_b\|_{2,I}\|D_s\|_{2,I} \|\boldsymbol C_s-\boldsymbol C_b\|_{2,I}. \end{aligned}

The last step uses orthogonality of auxiliary differences and contraction of the transferred ordered product. It controls the complete matched sum, including the count of differences. First make this calculation with common projection choices fixed, then restore their normalized average.

The extra pathwise squared size in the first term is the norm of the full vector difference. For the second term, again apply Cauchy–Schwarz in $p$ . Since $\boldsymbol C_b$ may be large, first write

\boldsymbol C_b=\boldsymbol C_s+(\boldsymbol C_b-\boldsymbol C_s).

This produces one term with three pathwise square sums and another with four. The other current complementary functions have local bounds from their orthogonal difference sums and contraction.

Section 6 and the preceding first-crossing construction already control these square sums. Equation (Three-side-power) therefore bounds the total $p$ -power cost of freezing errors by $P^C|R|$ .

The source also excludes frozen complementary sizes too large for the short-block comparison. On the difference side, size always means the norm of the entire $\boldsymbol C$ vector; on the other side it means the norm of $D$ . If the current size is at most $C_{\mathrm{fr}}\ge1$ while the frozen size exceeds $2C_{\mathrm{fr}}$ , their difference exceeds $C_{\mathrm{fr}}$ . Hence

\mathbf1_{\{\text{frozen size}>2C_{\mathrm{fr}}\}} \le C_{\mathrm{fr}}^{-2} \|\text{frozen value}-\text{current value}\|^2.

The square root of this indicator is another pathwise squared size controlled by variation. Near-difference contributions excluded by the large-value indicator enter this power-error sum; the original summation interval does not stop immediately. Only when the accumulated nonnegative costs first exceed their threshold are the corresponding original interval and descendants transferred to a new local construction.

The source freezing-and-restoration construction includes the required whole block-ancestor intervals and boundary leaves in its tests, with an additional propagation cost of at most $K$ . On near-difference uses not excluded by the large-value indicator, both frozen complementary functions have uniform local bounds on all whole ancestor intervals and boundary leaves required for short-block comparison. Bounds only at the finest current uses do not supply this whole-interval condition.

8. The remaining near-difference estimate

The argument has now handled:

The $p$ -power sums with three or four non-core positions;
The transferred main term with exactly two non-core positions;
Far differences, physical errors, and errors from freezing, excessive complementary sizes, and restoration.

What remains is the complete matched sum (Matched-bands) for fixed $\ell$ , with coefficient $O(P^{-1})$ . This small coefficient does not finish the argument by itself: the sum contains $P+1$ differences and ranges over many groups and blocks.

The next part constructs fixed Hilbert-space-valued function families for the two non-core positions, with respective total square energies

P G_0|R|,\qquad G_0|R|.

Whole-ancestor maximal estimates and a sparse decomposition then bound the sum of size products over short-block roots by $P^{1/2}G_0|R|$ . Only after this step does $P^{-1}$ leave the gain $P^{-1/2}$ .

Source coverage

D p.52: the $p$ -power estimate from three pathwise square sums.
D p.53, equation (6.18): projection-average weights, the full transfer identity, and matched sums at fixed near offsets.
D pp.53–54: restoration of the full $Y$ , freezing complementary endpoints at $b$ , and the third square-sum error.
H Lemma 6.9, pp.50–52: rank order, closure, and label separation retained as explicit assumptions.