From local forms in H to the estimates used in D

H denotes An $L^3$ bound for the trilinear Hilbert transform, a 93-page preprint whose title page is dated October 5, 2026. D denotes Pointwise convergence of triple ergodic averages for mixing transformations, a 61-page preprint whose title page is dated October 4, 2026. Reference [22] on page 61 of D explicitly cites the October 5 version of H. All page references here use the printed page numbers of these two fixed versions.

This guide distinguishes the local proofs expanded here from the assumptions that still need separate study. Tracing each step makes clear which analytic estimates are used before the main theorem is reached.

1. D does not use the main theorem of H directly

Theorem 1.1 on page 2 of H states

\|T(f_1,f_2,f_3)\|_{L^1(\mathbb R)} \le C\prod_{j=1}^3\|f_j\|_{L^3(\mathbb R)},

where

T(f_1,f_2,f_3)(x)=\operatorname{p.v.}\int_{\mathbb R} f_1(x-t)f_2(x-2t)f_3(x-3t)\,\frac{\mathrm{d}t}{t}.

D uses estimates for local forms from within the proof, together with curvature localization, endpoint spaces, two-sided estimates, and core frequency tests. The finite local theorem in D cannot therefore be described as a direct consequence of the main theorem of H.

The main direction of the argument within H is

\text{Normalized absorption in Proposition 2.3} \Longrightarrow \text{Uniform local estimate in Theorem 2.2} \Longrightarrow \text{Theorem 1.1}.

D takes inputs from the first two steps and the specified arguments supporting them, while adding control of inputs that vary with depth.

2. What is uniform in the local estimate?

For an interval of length $r$ , denoted by $I$ , both papers use

H_I(z_0,z_1,z_2,z_3)=r^{-2}\iint_{\mathbb R^2} w_I(x,t)\prod_{j=0}^3z_j(x+jt)\,\mathrm{d}x\,\mathrm{d}t.

The kernel support keeps all four positions inside a fixed relative interior of $I$ . All four marginal integrals vanish, and the normalized derivatives satisfy fixed Gevrey bounds. Kernels may be chosen independently in different cells. For Hilbert-valued inputs, only the physical scalar coordinate enters the form.

Page 4 of H defines $C_N(q)$ by requiring, for every finite collection of cells in at most $N$ consecutive depths,

\sum_I|I|\,|H_I(f_0,f_1,f_2,f_3)| \le C_N(q)\int_{\mathbb R}\prod_{j=0}^3M_qf_j.

Theorem 2.2 states that some $q\in(2,3)$ satisfies

\sup_{N\ge1}C_N(q)<\infty.

The estimate is uniform in the number of depths, the cell collection, the grid, and the allowed kernel family. The input functions, however, are fixed across depths. The input schedules in D are an additional issue.

3. D also allows variation with depth

Theorem 3.2 on page 11 of D fixes a finite dyadic tree below a root interval $J$ , with maximum depth $d$ . Each role has a full depth schedule $f_{j,s}:J\to\mathbb C$ , $0\le s\le d$ , extended by zero outside $J$ . Partition all the depths into $W_j$ prescribed nonempty consecutive blocks. On each block $B=\{a,\ldots,b\}$ , require

V_B(f_j)(y)=\max_{s\in B}|f_{j,s}(y)| +\sum_{s=a}^{b-1}|f_{j,s+1}(y)-f_{j,s}(y)| \le W_j^{-1/2}

almost everywhere. The four roles may use different block partitions. The conclusion is

\sum_{I\in\mathcal I}|I|\, |H_I(f_{0,s(I)},f_{1,s(I)},f_{2,s(I)},f_{3,s(I)})| \le C_{\mathrm{loc}}|J|,

with a constant independent of the depths, block counts, and partitions.

Functions must be specified in advance even at depths with no cells in the sum, because the comparison uses the full schedule. Sections 4–7 of D establish these additional variation bounds. Replacing $f_j$ by $f_{j,s}$ in H2.2 is not by itself justified.

4. Four main analytic steps

The first is local detection. Section 4 of H uses quadratic charts and returns at finite scales to obtain rational fast splittings and Fourier expansions, together with a specified discrete inverse theorem and separation arguments. The compression in H Section 3.4 and the basic increment expansion in Section 5 place correlations into a finite catalog and groups of scales.

The second is curvature localization. Section 6.1 of H forms a graph on tags: a large correlation detects an edge, and a short path passes the edge splittings to the endpoints. The colours and historical ball masks in Sections 6.2–6.3 separate distinct groups of the same colour at usable endpoints. Small inner products on whole cells and the private coordinates then give a Bessel bound.

The third is the two-sided estimate. Section 7 of H uses short-block comparisons, fixed stack functions, and coupled near-band terms. Lemma 7.4 constructs the fixed stacks; Section 7.6 handles maximal functions on whole hulls and a common stopping rule. Stopping a summand does not redefine the support of a previously fixed function.

The fourth concerns three or four core roles. Section 8 of H supplies modes, components, representatives, and squared sizes. Section 9.1 selects a common successful height; Section 9.2 converts the result into frequency tests; Lemma 9.1 in Section 9.3 controls finite sparse counting; and Sections 9.4–9.5 sum over groups and amplitude classes. Section 10 then chooses the parameters and carries out the restarting and absorption arguments.

Three features must be preserved when combining these steps: range spaces on whole cells, minima on the original cells, and the choice of complexity degrees before the exponents.

5. Why H and D need the same $q$

Page 60 of D checks this explicitly. The local detection argument in H allows any $q>2$ . The degrees in the logarithmic complexity bounds are independent of $q$ , although the coefficients may depend on $q$ . The short-block comparison allows $2<q<3$ and retains an explicit loss proportional to $q-2$ . The other complexity degrees are fixed before $q$ .

Write the relevant degrees in H and D as $C_{19}^{H},C_5^{H}$ and $C_{19},C_5$ , respectively. D chooses a common exponent sufficiently close to $2$ , denoted by $q$ , such that

\max(C_{19}^{H}C_5^{H},C_{19}C_5)(q-2)<1/8.

Only after rerunning the parameter and absorption arguments of Sections 2 and 10 of H at this $q$ do we obtain

C_H(q)=\sup_NC_N(q)<\infty.

The common $q$ thus comes from the parameter order within the proof of H. The statement in H2.2 that there exists some $q$ does not itself assert the estimate for every exponent near $2$ , with $q$ chosen as needed.

6. Companion readings and source locations

Scalar Fourier analysis after the successful height starts from the finite model in (29). It develops the comparison with Haar square averages, the four vanishing lines, grouping of full cosets, the three-core Schur bound, and probability summation over failure histories. Mode construction and component almost orthogonality remain explicit inputs.

One shared coloring and small exceptional roots develops Lemma 6.3 in Section 6.2. With the original vertex lists, admission data, and entire monotone graph family fixed in advance, it proves a finite color-change bound for every delay realization and selects one global realization whose first required marking failures have small total length relative to the root. It retains independence of the primary and high arrays and the requirement that high arrays be shared across contexts.

Historical ball masks and endpoint Bessel bounds proves Lemmas 6.4–6.5 from a fixed successful realization of the colouring. It treats historical centres, currently available vertices, the future witness needed only for the finer group, combining repeated coefficients, and preservation of the whole endpoint cell by the adjoint.

Finite algebra for fast spaces along a short path expands Lemma 6.2, assuming that the edge splittings already exist. It proves the height bounds for two rational eliminations, two replacements justified by isotropy, cancellation of joint coordinate blocks, and the endpoint image-space construction.

Analytic detection of returns in two slots treats the first part of the distinct-slot argument in Lemma 6.1. It derives a frequency-increment bound for every small return from a large correlation, accounts explicitly for large real parameters, and proves the Bessel contradiction with a fixed kernel. The number of multiples is chosen before the return threshold.

From bounded returns to a one-parameter rational fast splitting continues from the return condition obtained by two-slot detection. With the stated input from Minkowski’s second theorem, it develops the antisymmetric identity, integer isotropy, the polar body and projection heights, and bounds the slow part and effective curvature. The general matrix-valued, multiparameter case is outside this chapter.

Physical detection of same-slot inner products uses interval smoothing and an energy contradiction for one fixed Fourier transform to produce the return condition, then applies the same rational fast splitting. Its conclusion concerns physical inner products on whole cells; orthogonality of private coordinates still requires the separate endpoint separation supplied by labels and masks.

Fixed testing templates with the same tag develops Lemma 6.6 in Section 6.4. It turns a large four-linear correlation between an original or previously inserted source column and a target physical residual into a same-tag template moment. Exact integer-carry selection, horizontal Fourier expansions after residue reduction, and two observer square-sum estimates keep the complexity bound uniform. The nonlinear amplitudes form one fixed finite family, while external linear modulations remain arbitrary real numbers. This chapter does not prove catalog termination in Lemma 6.7.

Column-catalog termination with changing histories first proves the first prefix-row estimate of Lemma 3.3 in full, then develops main- and helper-catalog termination in Lemma 6.7 under explicitly stated fixed inputs. Complete root-to-leaf paths reserve the original budget, while actual projections still use only currently available columns. Main histories exclude the helper rank; the whole helper base vectors are fixed after the main catalog is complete, and the helper stage counts its insertions separately. The active-cell path count remains an input to this chapter from Proposition 5.8 on p. 38 of H; the required argument under supplied basic data is now developed in H10. The construction of the original catalog budget is now developed in H11, Section 16. The actual history list is now supplied in H12, Section 6. Construction and counting of complete side bases still require the inputs from Sections 7–8 of H.

Fixed triples and the active path budget starts from the basic data supplied by Proposition 5.6 and develops the fixed-triple scale count, small-value path sum, active-cell path count, and minimum reduction needed for Lemma 5.7 and Proposition 5.8 on pp. 37–38 of H. The three-variable technical appendix to the same H10 module derives the rational fast splitting and finite expansion needed here. The original catalog and its path budget remain inputs within this chapter; their construction is now developed in H11. The actual later history list is now supplied in H12. Complete side bases must still be constructed separately. The inactive-cell and two-delayed-difference errors are kept separate.

Original basic columns and the full-path label budget develops the construction of old columns, fixed prediction columns, and additional columns in the two high slots needed for Proposition 5.6 on pp. 35–36 of H, then combines their original path budgets. Spatial compression, local detection, the all-scale prediction catalog, and parameter choices appear as four technical appendices within the same H11 module. Detection explicitly uses an external discrete quadratic inverse theorem. A verifiable quantitative choice supplies the step affected by the precision conventions on pp. 31 and 35, without deciding the authors’ notational intent. The actual later histories, whole helper bases, and global absorption remain outside this chapter.

Actual search histories, current-group localization, and far-band transfer develops the filter arrangement, Proposition 6.8, and Lemma 6.9 on pp. 47–52 of H. It first lists the finite histories actually used in search and transfer, then proves localization on the original active cells while retaining the complete group sum for each leaf and the original minimum. Both terminal bands enter the exact identity, and the two far-band directions are estimated with their respective strict rank orders. The history and precision appendix within the same H12 module specifies the projection words, pad choices, and error quantifiers. Construction and counting of complete side bases, linear side energy, and global near-band estimates still require later proofs.

Comparing a short depth block with the same finite-depth constant develops H Lemmas 7.1–7.2 on pp. 52–57. Two weighted extrapolations preserve the mixed-norm order. Leaf-depth rectangles, shared quadratic randomization, clipping in a single heavy slot, and two sampling steps then return to the original kernel. The short-block constant may depend on the fixed kernel class and a uniformly fixed eighth-moment bound. Absolute complexity powers are fixed before the exponent is chosen. The same finite-depth constant remains, with no premature absorption. The calculations and boundary appendix expands the sampling masses and concrete counterexamples. The source appendix identifies the 2006 function-pair extrapolation theorem actually used and the 2011 citation not directly verified. Fixed stacks for varying side inputs, complete side bases, linear side energy, and global near-band estimates still require later proofs.

Each chapter has specified assumptions and conclusions. Completing one chapter does not establish all of H or remove every analytic dependency of D.

7. The arguments still needed

These chapters connect both the distinct-slot and same-slot detection arguments of Lemma 6.1 to the one-parameter rational fast splitting, retaining its explicitly stated classical input. The fixed-parameter scaling form in at most three variables needed for fixed triples is now developed in the H10 technical appendix. This supplies the form of Lemma 4.3 needed here. The H11 prediction appendix also proves the all-scale tail bounds and finite catalog needed for fixed low columns. The detection appendix develops the passage to continuous functions and local residual detection under the named external discrete quadratic inverse theorem. This still does not establish every strengthened Fourier-tail assertion of Lemma 4.4 or all of Section 4.

The successful fixed coloring required by the historical-ball-mask and endpoint Bessel arguments is now supplied. The original vertex lists, admission data, and entire monotone graph family being fixed in advance remain inputs to this step. The coloring chapter controls only the first exceptional roots caused by marking failure. The same-tag template chapter also supplies the uniform test of Lemma 6.6. Catalog closure under changing projection histories in Lemma 6.7 is now developed under explicitly stated fixed inputs. The active-cell path count used here comes from Proposition 5.8 on p. 38 of H; its proof under supplied basic data is now developed in H10. The construction of the original catalog and its path budget is now developed in H11, without using the active-cell count or later catalog termination. H12 lists the actual localization and transfer histories and proves local far-band bounds given fixed complete side bases. Construction and counting of those bases, linear side energy, and global near-band estimates remain later inputs. The error and root-length bounds for other stopping rules also still require separate treatment.

H13 develops the short-block comparison for fixed stacks. It does not construct those stacks from varying side outputs or prove a uniform bound for the finite-depth constant. The later two-sided and common-stopping arguments, complete side bases, and global near-band sums still require separate verification.

The two-sided stack and hull estimates, the mode and component construction in Section 8, and the final restart and absorption with a common parameter have also not been developed in full within these thirteen local modules. Each chapter distinguishes its proved conclusions from its retained inputs.