Consolidate research results and choose the next bounded depth test

docs/OPEN_PROBLEM.md

@@ -13,6 +13,86 @@ The [publication scope](../manuscript/PUBLICATION_SCOPE.md) retains the
 established frame results. The [verification map](VERIFICATION.md)
 distinguishes analytic proofs from implemented and finite evidence.
 
+## Revision checkpoint and selected next test
+
+This checkpoint consolidates the proved frontier and selects one bounded
+analytic test. It introduces no new theorem. Write $`N=2^n`$. The uniform-precision and
+low-precision bounds below are established; their remaining questions
+should not be conflated with the completed Hopf-QBP contract.
+
+| Remaining gap | Current boundary | Selected treatment |
+|---|---|---|
+| Large-width T-depth | At fixed accuracy, two clean flags and sufficiently large $`b=\Theta(\sqrt N)`$, $`\Omega(1)\le D_T^\star\le O(n)`$, with $`T=O(\sqrt N)`$ and $`G=O(N)`$ | Audit the linear-depth contributions, beginning with the exact dirty-indicator target below |
+| High-precision complete-frame count endpoint | At $`a=2,L=N,b=N+n+7,n\ge3`$, $`\Omega(N)\le T^\star\le O(N\ell_*(n))`$, where $`\ell_*(n)=1+\log_2^*(n+2)`$ | Park until a new complete native identity supplies its symbolic precision, workspace, and work-return ledger; another local source fixture is not selected |
+
+At fixed accuracy the current schedule has
+$`O(n/\log(n+2))`$ early groups. Five separately charged contributions
+can retain order n in its upper ledger:
+
+| Contribution | Current depth allowance | What a faster indicator would change |
+|---|---|---|
+| Early logical stages and incremental selectors | $`O(g)`$ per height-g group, summed over at most n heights | Unchanged |
+| Early phase-source preparation, unary conversion, and actual inverse | $`O(\log(n+2))`$ per group | Unchanged |
+| Early outer activity predicate and its inverse | $`O(\log(n+2))`$ per group | Unchanged |
+| Early program prefetch and unload | $`O(\log(n+2))`$ per group | The proposed indicator still has this order for an address of length $`O(n)`$ |
+| Tail dirty indicators | The sum of $`O(n(k+1)2^{-k/12})`$ over remaining heights k is $`O(n)`$ | This is the contribution the next test targets |
+
+These are costs of the present schedules, not five independent lower
+bounds. The tail's other source, predicate, and unweighted query terms
+already fit $`o(n)`$ under the unary cutoff. Improving only its indicators
+would not prove a complete-frame $`o(n)`$ bound: the four early
+contributions would remain separately charged.
+
+### Candidate: nonuniform chunks in an exact dirty indicator
+
+For an s-bit address and $`S=2^s`$ arbitrary output bits Y, the target
+is an exact native circuit
+
+```math
+|x,Y,W\rangle\longmapsto|x,Y\oplus e_x,W\rangle,
+\qquad T,G,w=O(S),\qquad D_T=O(\log_2(s+2)),
+```
+
+with absolute constants, no initialized work, literal phase, and return
+of all arbitrary dirty work W, including references. Here w is additional
+dirty width, and G counts elementary Clifford gates. The case s equal
+to zero is the single-output Clifford X.
+
+The candidate uses the existing
+[dirty tree and root/leaf echoes](CHUNKED_DIRTY_INDICATOR.md#1-a-reversible-tree-on-arbitrary-dirty-inputs)
+with unequal positive chunk lengths. Start with r equal to s remaining
+address bits. While $`r\gt64`$, propose
+
+```math
+r'=\lceil4\log_2(r+2)\rceil,\qquad b=r-r',
+```
+
+consume that b-bit chunk, and continue with r equal to r prime. Finish
+the remaining at most 64 bits in one chunk. This is a candidate schedule,
+not a proved resource bound. Its intended stage budget is
+$`O(2^{s-r'}(b+3)^3)`$: there is one private conjunction per edge,
+controlled by its dirty parent and that chunk's literals.
+
+Before implementation or a new frontier claim, require all five checks:
+
+- Prove that arbitrary unequal chunks preserve the existing full-input
+  tree identity, literal conjugation order, and both actual-inverse echoes.
+- Check every rounded chunk is positive; prove linear total T and
+  elementary Clifford count with no residual $`\log^*s`$ factor.
+- Count all retained tree nodes and the maximum simultaneous private
+  conjunction pool; reuse a pool only after its exact return.
+- Prove the depth sum $`\sum_i\log_2(b_i+3)=O(\log_2(s+2))`$
+  while preserving the read-only shared-control interface.
+- Reinsert a successful lemma into the complete ledger, keeping every
+  early contribution and every return/error charge visible.
+
+Stop this candidate if any check requires clean helpers, overlapping live
+work, an uncharged phase, or a nonconstant count overhead. Record the
+first failed obligation without converting that failure into a general
+lower bound. A successful local test would strengthen the query interface;
+it would not by itself close the large-width frame-depth gap or change
+the parked high-precision endpoint.
+
 ## Established frontier
 
 Write $`N=2^n`$, $`q=n+a+b`$,
@@ -553,6 +633,10 @@ supplies the missing native synthesis by itself.
 
 ## Revision decision and next bounded pass
 
+The current selection is the
+[revision checkpoint above](#revision-checkpoint-and-selected-next-test).
+This section retains the earlier source-carry decision and its boundaries.
+
 The coupled-merge passes have completed their structural task. The
 [commutator repair](RESIDUAL_ASSEMBLY.md#9-a-repair-word-without-an-ill-conditioned-transported-basis)
 works on every signal port without normalizing transported differences.

docs/QBP_COST_COMPARISON.md

@@ -597,6 +597,17 @@ This is a frame-compiler theorem, not a lower bound for gradient estimation
 or a change to the state-based schedules. The A/B comparison below remains
 a comparison of its stated expressions; all eligible schedules may be used.
 
+The subsequent [uniform real-frame theorem](UNIFORM_PRECISION_DEPTH.md)
+is also eligible at $`b\ge17(K+n+7)`$. It retains the last
+same-circuit T/Clifford bounds and improves depth to
+$`O(NK/b^2+nK)`$. For $`6\le K\le\log_2(n+2)/16`$, it gives
+$`T=O(\sqrt{NK}+NK/b)`$ and $`D_T=O(NK/b^2+n)`$.
+The fixed-accuracy predecessor already gives $`O_K(n)`$ depth at
+sufficient square-root dirty width. Thus the historical original-B
+expression compared below is not the best current fixed-accuracy frame
+bound. These improvements still use K, charge both frame calls and all
+shots, and assert no optimal gradient-estimation or complex-frame depth.
+
 The common A pool satisfies the old A threshold. For a common B comparison,
 require literally
 

docs/RELATED_WORK.md

@@ -1016,3 +1016,24 @@ only under a sufficient condition such as $`L\le N/n^2`$.
 All preparation, queries, actual inverses and work return remain charged.
 The selected high-precision endpoint and unrestricted large-width depth
 optimality remain open. No priority claim follows from this composition.
+
+## 22. Scope of recent depth lower bounds (3 October 2026)
+
+These comparisons guide further research; neither is an imported compiler
+premise. [Parham, arXiv:2504.19966v1](https://arxiv.org/html/2504.19966v1),
+Proposition 1.8, relates T-depth to alternations of unrestricted Clifford
+and shallow circuits. Theorems 1.14–1.15 connect sufficiently strong
+explicit-state and Boolean-function lower bounds to classical threshold
+circuit lower bounds, with polynomial clean workspace in the model.
+Section 6 explicitly says that no analogous reduction is known for
+general prescribed-unitary implementation. This is therefore not a
+blanket complexity barrier to complete-frame operator lower bounds.
+
+[Al-Ghattas–Gamarnik–Kiani, arXiv:2610.02166v1](https://arxiv.org/html/2610.02166v1),
+submitted 1 October 2026, treats arbitrary Clifford blocks. Corollary
+1.7(iii) and Section 4.2 cover every fixed number of shallow blocks at
+total width $`M=O(n)`$; arbitrary-width extensions concern specific
+one-round classes. Lemma 4.4 retains an $`O(kM^2)`$ entropy term, so
+this theorem does not cover the current $`b\asymp\sqrt N`$ allocation.
+No applicable growing depth bound was identified in this comparison;
+that audit outcome does not prove that such a bound is impossible.

docs/SOURCE_MAP.md

@@ -153,6 +153,8 @@ from the exact clean-workspace size–depth theorem.
 | F36 | [Jones et al., arXiv:1204.0567](https://arxiv.org/pdf/1204.0567), Section 2.1, Eqs. (2)–(4); Section 4.1, Fig. 15 and Eqs. (19)–(20) | Fourier-state phase kickback, programmed shifts, and reusable phase references | the unary source changes the encoding and implements coherent one-hot-selected shifts in conditional Hopf work; neither phase kickback nor reference reuse is new |
 | F37 | [Iggy van Hoof, arXiv:1910.02849v2](https://arxiv.org/html/1910.02849v2), Section 4.2; Section 4.3, Theorem 4.1 | standard three-product Karatsuba recursion and subquadratic binary-polynomial multiplication | the unary-source proof uses the bilinear rank recursion followed by linear cyclic reduction; it does not import constant depth from the space-efficient reversible multiplication schedule |
 | F38 | [Wu et al., arXiv:2609.36574v1](https://arxiv.org/html/2609.36574v1), Sections II.C, III.B and IV.C–E; Theorem 1 and Proposition 1 | shared phase arithmetic, reversible encoding, and amortized reference preparation | comparison checked 3 October 2026; its binary ripple-adder construction permits measurement/feedforward and has linear-width depth; no coherent unary-program, two-clean Hopf, or constant-depth convolution interface is imported |
+| F39 | [Parham, arXiv:2504.19966v1](https://arxiv.org/html/2504.19966v1), Proposition 1.8, Theorems 1.14–1.15 and Section 6 | magic-hierarchy and classical-complexity comparisons | comparison only: the state/Boolean reductions do not establish a blanket barrier for prescribed-unitary lower bounds; no compiler premise or growing frame-depth lower bound is imported |
+| F40 | [Al-Ghattas–Gamarnik–Kiani, arXiv:2610.02166v1](https://arxiv.org/html/2610.02166v1), Corollary 1.7, Lemma 4.4 and Proposition 4.5 | state-complexity comparisons with unrestricted Clifford blocks | comparison only: the multiple-round theorem requires linear total width; arbitrary-width extensions concern specific one-round classes and do not supply a growing bound at $`b\asymp\sqrt N`$ |
 
 Standard Pauli linear combinations, reversible arithmetic, and oblivious
 amplitude amplification are used with their actual preparations and adjoints.

docs/VERIFICATION.md

@@ -67,7 +67,7 @@ identifying the missing interface. The prescribed complete-frame
 endpoint and general T-depth frontier remain
 [active research questions](OPEN_PROBLEM.md#next-bounded-task-and-stopping-rule);
 count and depth now match in explicit accuracy/workspace ranges by the
-[amortized construction](AMORTIZED_DIRTY_LOOKUP.md). The existing
+[uniform-precision construction](UNIFORM_PRECISION_DEPTH.md). The existing
 state-based result does not require the remaining questions' resolution.
 Application-level advantage is outside the current research scope; the
 existing resource comparisons and classical baselines remain documented.