Prove O(n) fixed-accuracy Hopf T-depth with charged unary phase sources

README.md

@@ -144,18 +144,18 @@ For $`L=\Theta(n)`$, sufficient
 $`b=\Theta(n)`$ gives optimal worst-case $`T=\Theta(N)`$ and
 $`D_T=\Theta(N/n)`$ in one complete real-frame circuit.
 
-At fixed accuracy, [grouped programs](docs/GROUPED_PROGRAM_PREFETCH.md#8-complete-frame-theorem-at-fixed-accuracy)
+At fixed accuracy, [unary phase-source groups](docs/UNARY_PHASE_GRADIENT.md#7-complete-frame-theorem-at-fixed-accuracy)
 and chunked queries give, with two clean flags and sufficient
 $`b=\Theta(\sqrt N)`$, one complete real-frame circuit with
 
 ```math
 T=O(\sqrt N),\qquad G=O(N),\qquad
-D_T=O\!\left(n\log\log(n+2)\right).
+D_T=O(n).
 ```
 
-T-count is optimal in worst-case order. Depth optimality and the
-high-precision endpoint remain open; the
-[variable-precision theorem](docs/OPEN_PROBLEM.md) is unchanged.
+Phase-source preparation, reuse, and return are charged. T-count is
+optimal in order; depth optimality and the high-precision endpoint
+remain open. The [variable-precision theorem](docs/OPEN_PROBLEM.md) is unchanged.
 
 Beyond Hopf frames, **literal diagonals and general one-target U(2)
 multiplexors** attain $`\Theta(\sqrt{NL}+L+NL/b)`$ with one clean

WORKSPACE.md

@@ -3,9 +3,9 @@
 This is the entry point when a previous conversation or execution workspace
 is unavailable. Proofs and decisions live in the repository.
 
-The 2026-10-02 group-selector and source-reuse pass starts from verified main
-`9f7906f57ba379bd2cd934d8c625ba9bb06dcca4`, after the grouped-program
-fixed-accuracy depth theorem. Check later commits before
+The 2026-10-03 unary phase-source pass starts from verified main
+`005debe2ccf7a0b2e7cbc032a0f1ae285b83d2ab`, after the incremental
+group-selector and common-source audit. Check later commits before
 continuing.
 The selected state-based Hopf QBP construction and its bounded-input audit are complete; the
 [consolidated theorem](docs/STATE_BASED_QBP_THEOREM.md) is their entry point.
@@ -25,7 +25,8 @@ and finite checks, without large simulations, QRAM, resets inside a
 compiler execution, supplied catalysts, or hidden initialized work.
 
 1. Begin active depth work with the
-   [grouped complete-frame theorem](docs/GROUPED_PROGRAM_PREFETCH.md#8-complete-frame-theorem-at-fixed-accuracy),
+   [unary phase-source theorem](docs/UNARY_PHASE_GRADIENT.md#7-complete-frame-theorem-at-fixed-accuracy),
+   then the [grouped complete-frame theorem](docs/GROUPED_PROGRAM_PREFETCH.md#8-complete-frame-theorem-at-fixed-accuracy),
    [chunked dirty indicator](docs/CHUNKED_DIRTY_INDICATOR.md),
    [conditional geometric source](docs/CONDITIONAL_GEOMETRIC_SOURCE.md),
    [two-layer obstruction](docs/SHALLOW_SOURCE_OBSTRUCTION.md),
@@ -78,7 +79,7 @@ compiler execution, supplied catalysts, or hidden initialized work.
 | Additional dirty banks | Improve the state preparation T bound while charging the exact coarse circuit; [banked proof](docs/COMPLEX_COARSE_COMPILER.md#8-additional-dirty-banks-improve-fine-state-preparation) |
 | State-based T-depth | Two complete schedules, one retaining the sharper count at a stronger dirty reservation; [depth proof](docs/STATE_QBP_DEPTH.md) and [fair comparison](docs/QBP_COST_COMPARISON.md#7-state-based-t-depth-comparison) |
 | Complete-frame T-count and T-depth | Same-circuit bounds at every accuracy; matching in an explicit workspace range, including inverse-polynomial error; [amortized tradeoff](docs/AMORTIZED_DIRTY_LOOKUP.md) |
-| Fixed-accuracy large-width depth | Two external flags, sufficient square-root-scale dirty width, optimal-order square-root T-count, and O(n log log n) depth; [grouped program theorem](docs/GROUPED_PROGRAM_PREFETCH.md#8-complete-frame-theorem-at-fixed-accuracy) |
+| Fixed-accuracy large-width depth | Two external flags, sufficient square-root-scale dirty width, optimal-order square-root T-count, and O(n) depth; [unary phase-source theorem](docs/UNARY_PHASE_GRADIENT.md#7-complete-frame-theorem-at-fixed-accuracy) |
 | Complete-frame error accumulation | Sharp ideal-angle stability and finite relative spectra; coherent linear leakage in actual shared-flag source layers; [scoped error audit](docs/HOPF_ERROR_ACCUMULATION.md) |
 | Filtered complete-frame source | Quadratic radial error on the same two flags, charged native selective phases, and a smaller source-precision cap; [filter proof](docs/HOPF_RADIAL_FILTER.md) |
 | Conditional precision depth | Logarithmic source/reflection depth using an active zero suffix and two external flags; [conditional source](docs/CONDITIONAL_GEOMETRIC_SOURCE.md). Query/predicate depth remains charged |
@@ -152,18 +153,21 @@ The original routed schedule retains an additive $`n^2`$ term. The
 [dirty-counter hybrid](docs/PARALLEL_DIRTY_LOOKUP.md#6-a-polylogarithmic-depth-indicator-using-dirty-counters)
 improves fixed-accuracy large-workspace depth to
 $`O(n\chi(n))`$ while retaining optimal-order T-count.
-The [grouped-program theorem](docs/GROUPED_PROGRAM_PREFETCH.md#8-complete-frame-theorem-at-fixed-accuracy)
+The [unary phase-source theorem](docs/UNARY_PHASE_GRADIENT.md#7-complete-frame-theorem-at-fixed-accuracy)
 now improves this fixed-accuracy, sufficient-square-root-width bound to
 
 ```math
 T=O_\eta(\sqrt N),\qquad G=O_\eta(N),\qquad
-D_T=O_\eta\!\left(n\log\log(n+2)\right),
+D_T=O_\eta(n),
 \qquad b\ge C_\eta\sqrt N.
 ```
 
 These are simultaneous bounds on one complete real-frame circuit using
-two external clean flags. Early groups store their programs in conditional
-logical zeros; chunked dirty indicators control the late-query depth.
+two external clean flags. Early groups store one-hot programs and a charged
+unary phase source in conditional logical zeros. Bilinear cyclic shifts
+have constant T-depth; source preparation and its actual inverse occur
+once per group. Chunked dirty indicators and capped source precision
+control the entire remaining tail.
 The worst-case count is optimal in order, but the available depth lower
 bound remains only Omega(1). Fixed accuracy is essential to this theorem;
 the general-precision matching interval and endpoint remain unchanged.
@@ -277,10 +281,10 @@ claim and its proof obligation. A general compiler is not required to
 close this selected Hopf-QBP task.
 The guarded-batch logarithm is now amortized, and the variable-accuracy
 composition is complete; do not repeat either task.
-The next bounded depth target is fixed accuracy at sufficiently large
-$`b=\Theta(\sqrt N)`$, retaining optimal-order $`T=\Theta(\sqrt N)`$.
-Its available depth bounds are now $`\Omega(1)`$ and
-$`O(n\log\log(n+2))`$ by the grouped-program schedule.
+The selected fixed-accuracy depth milestone at sufficiently large
+$`b=\Theta(\sqrt N)`$ now has $`D_T=O(n)`$, retaining
+optimal-order $`T=\Theta(\sqrt N)`$. Its available depth bounds
+are $`\Omega(1)`$ and $`O(n)`$; depth optimality remains open.
 The capped precision allocation has removed the accumulated source widths
 as a quadratic contribution: sources and suffix predicates now cost
 $`O(n\log(n+1))`$ depth at fixed L. The retained per-layer routing
@@ -448,7 +452,7 @@ leaves, use ell=min(address-half length,2^floor(k/12)). Its count is
 summable within O(sqrt N), while all final O(log n) queries have O(n)
 total depth. No unknown dirty program is treated as an initialized cache.
 
-The global theorem uses the unchanged unfiltered additive precision:
+The earlier geometric grouped theorem uses the unfiltered additive precision:
 m=L+4+ceil log2(8n), cutoff R=min(n,256m), and early height
 g=floor log2(k/(64m)). These groups fit the literal suffix reservation,
 have O(n/log n) prefetches, and cost O(n log log n) depth internally.
@@ -460,7 +464,8 @@ and erase the retained prefix tree in reverse. It has O(g) group depth,
 fits the existing 16m2^g suffix reservation, and returns its work exactly
 through source leakage and on arbitrary inactive inputs. Across early
 groups this contribution is O(n). Source/reflection depth still retains
-the log log n factor, so the complete-frame frontier is unchanged.
+the log log n factor in that geometric construction. The unary phase
+source below gives the current complete-frame frontier.
 
 The [common-source audit](docs/GROUPED_PROGRAM_PREFETCH.md#11-a-common-source-identity-and-the-remaining-reflection)
 gives an exact two-layer conjugation identity, but two conjugated success
@@ -470,22 +475,30 @@ by one-use fresh banks also leaves constant rejection unless a later
 operation mixes that bank's success and failure sectors. These are scoped
 failures of specified substitutions, not general depth lower bounds.
 
-The next bounded target is a literal implementation of the conjugated
-success reflection, or a maintained encoding that updates its monitor
-through the programmed word. Aim for O(g+log(m+2)) group depth at
-O(m2^g) native count and the stated suffix reservation; a larger charged
-reservation needs a revised global allocation. Start with two unequal
-legal rows and carry the same rule through a third stage, preserving
-literal phases, all rejected action, and exact inactive identity.
-An unpriced conjugated reflection or a fresh bank per Q is not a solution.
-If no rule meets these obligations, record that there is no selected
-construction rather than starting a larger fixture. Keep the high-precision
-endpoint separate. Do not repeat the completed selector schedule,
-common-source audit, carry pipeline, short echoes, conditional preparation,
-grouped program identity, or chunked-query proof; do not apply ideal-angle
-stability to unfiltered source leakage.
-A matching unrestricted large-width frame-depth lower bound remains
-separate.
+The [unary phase source](docs/UNARY_PHASE_GRADIENT.md) supplies a different
+route. A Karatsuba rank decomposition implements a coherent one-hot cyclic
+shift with private conditional work and literal full inactive identity.
+The ideal Fourier source supplies signed target phases; its charged native
+preparation and actual inverse cost at most twice the preparation error
+for the entire group. No geometric success reflection is retained.
+The natural suffix cutoff reserves the convolution pool. Early groups
+have O(log n) depth and number O(n/log n). The entire remaining tail uses
+chunked queries at the original capped source precision, giving O(n)
+total depth while preserving O(sqrt N) T-count and O(N) Clifford count.
+The six bounded tests cover the rank identity, native guard, arbitrary
+source shift, preparation/inversion, unequal rows, and full source-return
+error. They do not constitute a scalable native group compiler.
+
+The next bounded research question is whether this fixed-accuracy route
+extends to a width-dependent same-circuit bound of O(N/b²+n) depth with
+O(sqrt N+N/b) T-count. That extension is not established here. Start with
+a complete shared-work reservation and a summed prefetch schedule; do not
+assume the large-width parallel query bound survives reduced dirty width.
+A stronger unrestricted depth lower bound and the high-precision endpoint
+remain separate. Do not repeat the completed selector, common-source,
+unary phase-source, or chunked-query proofs. Any alternative must retain
+literal phases, actual inverses, full work return, and charged source
+preparation; ideal-angle stability does not bound unfiltered source leakage.
 The completed modest-width
 matching theorem does not require solving the high-precision endpoint.
 For any new component, keep literal phases and actual inverses, declare

docs/GROUPED_PROGRAM_PREFETCH.md

@@ -18,6 +18,12 @@ Both ingredients are needed: retaining logarithmically many old late
 queries would retain the previous $`O(n\log(n+2))`$ depth allowance.
 Depth optimality and the high-precision endpoint remain open.
 
+The later [unary phase-source construction](UNARY_PHASE_GRADIENT.md#7-complete-frame-theorem-at-fixed-accuracy)
+changes the source interface and improves the fixed-accuracy depth to
+$`O(n)`$, with the same asymptotic count and external-workspace orders.
+The geometric-source schedule proved here remains a valid construction;
+its individual conjugated reflections are not resynthesized by that result.
+
 ## 1. Group, program, and local statement
 
 Fix a group of heights $`d,\ldots,d+g-1`$, with $`g\ge1`$.

docs/OPEN_PROBLEM.md

@@ -41,6 +41,7 @@ hybrid depth bounds, write
 | Simultaneous T-count and T-depth | $`T=O(\sqrt{NL}+L\ell_*(n))`$, $`D_T=O(\min\{nL+n^2,L\ell_*(n)+n^3\})`$, $`G=O(NL)`$, at $`a=2`$, $`b\ge C(L+n+7+\sqrt{NL})`$ | Same real-frame circuit, for sufficiently large fixed C; [parallel dirty lookup](PARALLEL_DIRTY_LOOKUP.md); T-depth optimality remains open |
 | Fixed-accuracy count and depth at modest width | $`T=O(\sqrt N+N/b)`$, $`D_T=O(N/b^2+n\chi(n))`$, $`G=O(N)`$, at $`a=2`$, fixed L, $`b\ge17(L+n+7)`$ | Same complete real-frame circuit; count is optimal in order, and depth is matching for $`b\le\sqrt N/n`$; [amortized dirty lookup](AMORTIZED_DIRTY_LOOKUP.md) |
 | Variable-accuracy count and depth | $`T=O(\sqrt{NL}+NL/b+nL)`$, $`D_T=O(NL/b^2+nL+n\chi(n))`$, $`G=O(NL)`$, at $`a=2`$, $`L\ge6`$, $`b\ge17(L+n+7)`$ | Same complete real-frame circuit; both are matching when $`b\le\sqrt{NL/(nL+n\chi(n))}`$; [hybrid composition](PARALLEL_DIRTY_LOOKUP.md#every-eligible-width-and-precision) |
+| Fixed-accuracy large-width depth | $`T=O_\eta(\sqrt N)`$, $`G=O_\eta(N)`$, $`D_T=O_\eta(n)`$, at $`a=2`$, $`b\ge C_\eta\sqrt N`$ | Same complete real-frame circuit with charged unary source preparation/return; [unary theorem](UNARY_PHASE_GRADIENT.md#7-complete-frame-theorem-at-fixed-accuracy); T-count is optimal in order, depth lower bound remains $`\Omega(1)`$ |
 
 Take the best applicable construction. For fixed L, $`a=2`$ and
 $`b=L+n+7=\Theta(n)`$, the arbitrary-budget matching splice gives
@@ -84,8 +85,8 @@ uses $`a=2`$ throughout and respects each sufficient allocation threshold.
 | Variable L, $`17B_0\le b\le\sqrt{NL/(nL+n\chi(n))}`$ | $`\Omega(NL/b^2)`$ | $`O(NL/b^2)`$ with $`T=\Theta(NL/b)`$ | Matching throughout this interval when nonempty |
 | $`L=\Theta(n)`$, sufficient $`b=\Theta(n)`$ | $`\Omega(N/n)`$ | $`O(N/n)`$ with $`T=\Theta(N)`$ | Matching at inverse-polynomial error in N for sufficiently large n |
 | Fixed L, $`2B_0\le b\lt17B_0`$ | $`\Omega(N/n^2)`$ | $`O(N/n)`$ | Earlier schedule remains the proved fallback at this literal reservation |
-| Fixed L, sufficiently large $`b=\Theta(\sqrt N)`$ | $`\Omega(1)`$ | $`O(n\log\log(n+2))`$ with $`T=O(\sqrt N)`$ | [Grouped program reuse](GROUPED_PROGRAM_PREFETCH.md#8-complete-frame-theorem-at-fixed-accuracy); depth lower bound remains unmatched |
-| Fixed L, $`b=\Theta(N)`$ | $`\Omega(1)`$ | $`O(n\log\log(n+2))`$ with $`T=O(\sqrt N)`$ | Extra width is not needed by this schedule; depth optimality remains open |
+| Fixed L, sufficiently large $`b=\Theta(\sqrt N)`$ | $`\Omega(1)`$ | $`O(n)`$ with $`T=O(\sqrt N)`$ | [Unary phase-source groups](UNARY_PHASE_GRADIENT.md#7-complete-frame-theorem-at-fixed-accuracy); depth lower bound remains unmatched |
+| Fixed L, $`b=\Theta(N)`$ | $`\Omega(1)`$ | $`O(n)`$ with $`T=O(\sqrt N)`$ | Extra width is not needed by this schedule; depth optimality remains open |
 | $`L=N`$, $`b=\Theta(N)`$ | $`\Omega(1)`$ | $`O(N\ell_*(n))`$ | Serial precision cost remains |
 | Selected endpoint $`L=N,b=B_0`$ | $`\Omega(1)`$ | $`O(N\ell_*(n))`$ from $`D_T\le T`$ | The larger-bank depth theorem does not apply |
 
@@ -101,9 +102,13 @@ The exact dirty-counter indicator, [masked-sum refinement](DIRTY_SUM_COMPRESSION
 and bilinear-query hybrid now improve
 the square-root-width upper bound to $`O(n\chi(n))`$. The
 [grouped-program refinement](GROUPED_PROGRAM_PREFETCH.md#8-complete-frame-theorem-at-fixed-accuracy)
-now gives $`O(n\log\log(n+2))`$ at fixed accuracy, with returned
+gives $`O(n\log\log(n+2))`$ at fixed accuracy, with returned
 work and the same optimal-order T-count. It combines cached conditional
 programs with a summable chunked-indicator budget for the late layers.
+The [unary phase-source refinement](UNARY_PHASE_GRADIENT.md#7-complete-frame-theorem-at-fixed-accuracy)
+now gives $`O(n)`$ depth under the same external-flag and sufficient
+dirty-width orders. It charges source preparation/inversion and the
+coherent one-hot program interface.
 The lower bound is still constant in this regime; depth optimality is open.
 The lower bound does not assume count optimality; the upper circuit also
 retains optimal-order T-count. No high-precision endpoint improvement follows.
@@ -164,15 +169,21 @@ $`16m2^g`$ suffix reservation. They erase each suffix enable before its
 controls change and retain prefix nodes until the final reverse traversal.
 Their work returns exactly through source leakage and on arbitrary
 inactive inputs. The resulting selector contribution is $`O(n)`$;
-the source/reflection contribution still determines the displayed frontier.
+the source/reflection contribution retains the old geometric bound.
 The [common-source identity](GROUPED_PROGRAM_PREFETCH.md#11-a-common-source-identity-and-the-remaining-reflection)
 moves a shared preparation to the group boundaries but retains two
 conjugated success reflections per stage. A legal exact row rules out a
 stale success monitor and a one-use source-bank substitution, with constant
 rejected norm. These are interface restrictions, not depth lower bounds.
-The next bounded task is a charged shallow reflection or monitor-update
-identity that closes through two unequal rows and a third stage, including
-all rejected action. No such improved source construction is yet selected.
+The [unary construction](UNARY_PHASE_GRADIENT.md) instead uses a Fourier
+eigenstate and constant-T-depth programmed cyclic shifts. Conditional
+bilinear work returns exactly on all source inputs, the inactive sector
+is literal identity, and source preparation plus actual inversion costs
+at most twice its preparation error for the entire group. Its global
+allocation proves the displayed $`O(n)`$ upper bound. The next bounded
+question is a width-dependent $`O(N/b^2+n)`$ depth extension retaining
+$`O(\sqrt N+N/b)`$ T-count; this remains unproved. Its first obligation
+is a complete reduced-width prefetch and workspace schedule.
 
 The [two-layer obstruction](SHALLOW_SOURCE_OBSTRUCTION.md) separately
 allows unrestricted Clifford interlayers: the original source at width

docs/README.md

@@ -26,6 +26,7 @@ topic has one primary chapter below.
 | [Two-layer source obstruction](SHALLOW_SOURCE_OBSTRUCTION.md) | Width-independent approximation gaps for full-input sources with arbitrary Clifford interlayers and returned dirty helpers |
 | [Conditional geometric source](CONDITIONAL_GEOMETRIC_SOURCE.md) | Logarithmic precision depth using the active logical suffix as temporary clean work; lookup and suffix-predicate costs remain separate |
 | [Grouped program reuse](GROUPED_PROGRAM_PREFETCH.md) | Complete real-frame depth O(n log log n) at fixed accuracy and sufficient square-root dirty width; O(n) selector maintenance and scoped source-reuse audit |
+| [Unary phase-source groups](UNARY_PHASE_GRADIENT.md) | Complete real-frame T-depth O(n) at fixed accuracy and sufficient square-root dirty width, retaining optimal-order T-count; charged preparation and exact guarded cyclic shifts |
 | [Chunked dirty indicator](CHUNKED_DIRTY_INDICATOR.md) | A tunable exact dirty-tree indicator and a summable late-query budget that removes the late routing bottleneck |
 | [Hopf error accumulation](HOPF_ERROR_ACCUMULATION.md) | Sharp ideal-angle stability, finite relative spectra, and coherent leakage in the actual shared-flag sources; scoped precision boundaries |
 | [Flag-echo audit](HOPF_FLAG_ECHO.md) | Exact errors of four diagonal Pauli echoes, their generic linear leakage, and an exact equal-mask exception |