Research: Nonempty production windows

research/core-production-nonempty/CHECKPOINT.md

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+# Wave 2: exact production nonempty-interior admission
+
+Status: **verified local package, ready for lead publication review**.
+Canonical main was rechecked and remains
+`7c49fe32c55de02c2c873655c804b16e4c3fef0a`.
+Branch: `research/core-production-nonempty-20261109`.
+Worker: analytic_join. Package: `research/core-production-nonempty`.
+
+## Established and independently reviewed
+
+The exact target-coordinate inverse image and longest phase-pure selected run
+M give `I nonempty iff M>6W` for the unchanged production capped carve. The
+uniform guard `S>=e(6W+1), L>=12W+2` admits every ordinary original start,
+including short endpoint half-cells. Both one-unit adverse families retain
+`N-L>=W`; sharpness is not asserted for dyadic aligned windows.
+
+The lead independently reviewed the inverse cuts, middle-cell maximum,
+cap criterion, all-window proof and fixed-tape setup cost. The lead independently
+checked the source-ratio adverse family; the worker separately checked both
+families against the original source bodies. Common per-axis descriptors
+permit O(g d^2) setup; heterogeneous or local tests retain up to O(K g d^2).
+No payload pass, numerical recovery theorem or kappa gain is claimed.
+
+## Reproduction and executed checks
+
+`python3 -B scripts/research_package.py --package research/core-production-nonempty --check-only`
+returned `PASS_PACKAGE_METADATA_AND_PINS`.
+
+`python3 -B scripts/research_package.py --package research/core-production-nonempty`
+returned `PASS_REGISTERED_RESEARCH_PACKAGE` after both ordinary and optimized
+runs passed the full frozen receipt. Coverage: 772,850 exact literal-coordinate
+windows; 77,793 authenticated source partitions; 855 uniform-guard admissions;
+17 negative controls; both sharp families W=1 through 16. No numerical or
+native replay is represented by these geometry checks.
+
+The complete-Core source family is a cited conditional application. All three
+executable dependencies are exact canonical-main files listed in SOURCES.json
+and VERIFY.json. The phase-cell facts of #6259 and the capped density lineage
+are prior work, explicitly credited.
+
+## Outstanding obligations and next action
+
+Source row gap, numerical accuracy/norms, uniform preparation, native lifecycle
+and whole recovery remain. Empty short sections outside the uniform guard
+still need a complete numerical/native solver. Next action: commit the checked
+package, run the committed-byte/mode gate and give the exact head to the lead.
+The worker has created no remote branch, external post or upstream PR.
+
+Wave 1 is preserved separately at public bridge commit
+`73d5adaba2b7dc662e86eb901e81369a3f73be9f`; this package has no executable
+dependency on that branch or another unmerged package.

research/core-production-nonempty/NOTICE.md

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+# Credit and scope
+
+Prepared for **@CubeJerry** with substantial OpenAI assistance in mathematical
+research, implementation, independent exact checking and packaging. Apache-2.0.
+Independent AI review is not human acceptance or proof-assistant verification.
+
+Jose Manuel / jmbarrancoml authored the common rational Reference and Core
+source. Vlado Vojdanovski, Jeremy Kothe and Cinna Davis supply its retained
+geometry and capped-carve lineage; Mason Hale supplies interval arithmetic
+used by the full numerical source, which this geometry projection does not run.
+All three executable source inputs remain unchanged on canonical main.
+
+Nick Petalas's CrocSwap/integer-mult-bounds#6259 supplies the exact source-phase
+cut formulas, endpoint half-cell lengths and boundary counts for the different
+4w physical-source carve. Its credited Beatty-run/Gaussian-word predecessors,
+including Dugongue and Jose Manuel's lineage, retain their attribution. The
+capped phase-fusion source already supplies the 6W loop, separators and
+asymptotic boundary density.
+
+The new work is the target-window inverse-image decision with identity holes,
+the exact longest source-phase run, the sharp finite guard without long
+endpoint half-cells, and its binding to literal current production bodies.
+The root reviewer independently verified the mathematical implication and
+supplied the complementary-gap refinement of the source-ratio sharp family;
+the worker independently executed both refined sharp families on the original
+partition bodies. This does not claim a new density order or Gaussian theorem.

research/core-production-nonempty/PROOF.md

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+# Exact nonempty-interior admission at every original target window
+
+Prepared for **@CubeJerry** with substantial OpenAI assistance. Apache-2.0.
+**Source-bound geometry theorem; no new kappa.** Base:
+`7c49fe32c55de02c2c873655c804b16e4c3fef0a`.
+
+## 1. Obligation, source and predecessor boundary
+
+The actual common rational Reference defines selected coordinates by
+
+`phi(j)=floor(e*j/S+1/2)`, `q(j)=j+phi(j)`, `0<=j<S`,
+
+where S is the original source period, e its positive excess, `0<e<S`, and
+`N=S+e` the target period. The other target coordinates are identity holes.
+These are the literal `Reference.__init__`, `Coordinates` and
+`SelectedAddresses` rules in `research/rational-unitary-fft/gaussian_reference.py`.
+No phase is recomputed after translating a target window.
+
+The literal `Core.partition` walks the original target window. Every hole
+becomes a boundary singleton. It divides selected coordinates into maximal
+consecutive source runs of one original phase, then invokes the authenticated
+capped `carve(m,W,cap)` on each run. That body is
+`research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_capped_gauges.py`.
+It returns only boundary when `m<=6W`; otherwise its loop immediately creates
+an interior. The original complete Core's source and numerical contracts
+remain in force; this theorem supplies one geometric admission.
+
+Nick Petalas's CrocSwap/integer-mult-bounds#6259 already proves the source-phase
+cut formulas, endpoint half-cell lengths and a constant-size boundary count
+for Jose Manuel's **different 4w physical-source carve**. The capped-interior
+lineage already proves the dyadic cap, W-separation and asymptotic boundary
+density. We cite those facts and do not claim a new density order, phase-floor
+identity, Gaussian inverse, or Schur-bandwidth bound.
+
+The present result is the exact decision at the actual **target-coordinate
+window with identity holes** and the current **6W capped carve**, followed by a
+sharp finite uniform admission of every original start. Its hypothesis is
+weaker than requiring both endpoint half-cells to be long. Earlier density
+estimates already suggest eventual nonemptiness; the contribution is an exact
+source-specific finite guard and its literal production-body check.
+
+## 2. Exact largest selected phase run, without enumerating cells
+
+Consider an ordinary window `[a,b)` with integer `0<=a<b<=N`. Let `L=b-a`.
+Because j is integral, `q(j)=floor(N*j/S+1/2)`. Thus `a<=q(j)<b` is equivalent to
+
+`S(2a-1)/(2N) <= j < S(2b-1)/(2N)`.
+
+The selected source indices in the target window are exactly `[j0,j1)`, where
+
+`j0=ceil(S(2a-1)/(2N))`, `j1=ceil(S(2b-1)/(2N))`.           (1)
+
+The ordinary-window domain gives `0<=j0<=j1<=S`, including a=0, b=N and a
+window consisting only of a hole. This inverse image is not `[a,b)` in source
+coordinates; deleting holes without changing the endpoints is invalid.
+
+Use the credited exact source-phase cuts
+
+`A_0=0`, `A_(e+1)=S`,
+`A_c=ceil(S(2c-1)/(2e))` for `1<=c<=e`.                  (2)
+
+Phase c occupies `[A_c,A_(c+1))`. Its first and last phase lengths are
+`ceil(S/(2e))` and `floor(S/(2e))`. An internal phase length is either
+`h=floor(S/e)` or `h+1`; the latter disappears if S/e is integral. Empty
+endpoint phases are harmless in the general formula.
+
+If j0=j1, the largest selected phase run has length `M=0`. Otherwise set
+
+`c0=floor(e*j0/S+1/2)`, `c1=floor(e*(j1-1)/S+1/2)`.
+
+When c0=c1, `M=j1-j0`. When c0<c1, the two clipped endpoint runs have lengths
+
+`u=A_(c0+1)-j0`, `v=j1-A_c1`.                            (3)
+
+There are `m=c1-c0-1` complete middle cells, all internal source phases. If
+m=0 then `M=max(u,v)`. If m>0, put `D=A_c1-A_(c0+1)`. Since these m lengths
+are h or h+1, their maximum is exactly `h+[D>m*h]`. Consequently
+
+`M=max(u,v,h+[D>m*h])` when m>0.                         (4)
+
+Equations (1)--(4) use a fixed number of integer operations independent of S,
+e, N, L and the number of cells. Within a source phase, q(j+1)-q(j)=1; a phase
+increase creates one hole and breaks the target run. Hence M is exactly the
+maximum piece length passed to the original capped carve by `Core.partition`.
+The exact total D detects whether any h+1 middle cell occurs; inspecting an
+arbitrary representative middle cell would be insufficient.
+
+## 3. Exact admission for the unchanged capped carve
+
+Assume integer W>=1 and a positive dyadic cap with `2 cap<=S`, as required by
+the source's dyadic interior interface. For a selected run of length m:
+
+* If `m<=6W`, the literal carve returns one boundary piece.
+* If `m>6W`, its initial remaining length is `R=m-2W>4W`. Its first interior
+  length is `min(cap,2^floor(log2(R-W)))`, a positive dyadic integer at most
+  cap. The loop therefore produces at least one interior.
+
+For every ordinary target window in Section 2,
+
+`I is nonempty iff M>6W`.                                (5)
+
+There is no substitute interior or change to the source partition. The source
+`Reference.interior` is invoked at the actual retained start and checks the
+same phase and contiguous selected coordinates. The bound `2cap<=S` supplies
+its source-size admission for every subblock. The first and last W selected
+vertices of a long run remain boundary, so (5) also gives `|J|>=2W>0` and
+`0<|I|<L`. Existing separation and matrix-support arguments are inherited;
+(5) does not prove their numerical consequences.
+
+## 4. A uniform guard for every original start
+
+**Theorem.** Suppose the domain above holds, and
+
+`S >= e(6W+1)`, `L >= 12W+2`, `L<=N`.                    (6)
+
+Then every ordinary original target window of length L has a nonempty interior
+and a nonempty boundary under the literal production carve.
+
+**Proof.** Every complete **internal** source phase cell has length at least
+`floor(S/e)>=6W+1`. The two endpoint half-cells may be short. Also
+`e/S<=1/(6W+1)<1/2`, so the first selected coordinate is 0, the last is N-1,
+and all holes are singletons between successive nonempty phase cells.
+
+Suppose a window had no selected run longer than 6W. It cannot meet three
+selected phase cells: the middle one would be a complete internal cell, whose
+length is at least 6W+1. If it meets two cells and starts or ends on an external
+hole, the adjacent selected cell is complete and internal, again a
+contradiction. Otherwise those two clipped runs contribute at most 6W vertices
+each, with one hole between them, giving length at most `12W+1`.
+
+If it meets only one selected cell, it cannot include both neighboring holes,
+because that would make the cell a complete internal cell. Its length is at
+most `6W+1`. A window with no selected coordinate contains at most one hole.
+These alternatives all contradict `L>=12W+2`. Thus M>6W; Section 3 proves the
+assertion. ∎
+
+Requiring `floor(S/(2e))>=6W+1` would also suffice, but unnecessarily requires
+both original half-cells to be long. Equation (6) avoids that factor of two
+while preserving clipped endpoint phases and identity holes.
+
+### Both integer thresholds have exact adverse families
+
+For each W>=1 choose `S=18W+3`, e=3, `a=3W+3`, `L=12W+1`. The two internal
+phase cells both have length 6W+1, with one target hole between them. The window
+clips one selected coordinate from each end of their union, leaving lengths
+6W and 6W and that hole. The source-ratio guard holds with equality but I is
+empty. Moreover `N-L=6W+5>=W`. Thus the ordinary-window length threshold cannot
+be lowered by one even with the complementary-gap guard retained.
+
+Conversely choose `S=18W+2`, e=3, a=0 and `L=12W+2`. Then
+`S=e(6W+1)-1`, N=18W+5, and the first two phase lengths are 3W+1 and 6W.
+The window includes a prefix of 3W-1 selected coordinates from the next phase,
+with two holes in total. Its largest selected run is 6W, so I is empty.
+Here `N-L=6W+3>=W`. Thus the source-ratio threshold cannot be lowered by one
+while retaining the length and complementary-gap guards.
+
+These are sharpness statements in the **ordinary-window/complementary-gap domain**. Their L
+need not be dyadic. No separately optimal threshold for the narrower dyadic
+aligned Core family is claimed; that family may use (6) or exact decision (5).
+
+## 5. Conservative complete-Core family and surviving admissions
+
+The [complete Core's source family](https://github.com/CubeJerry/integer-mult-bounds/blob/6fb9ab776b7d0956b8bcc2d3cfde0f3034421630/research/complete-core-physical-join/NUMERICS.md)
+binds the unchanged `Core.partition` source to ordinary
+original windows. Its additional admissions include dyadic side and cap,
+original alignment, `2cap<=S`, the complementary gap `N-L>=W`, exact source
+accuracy and norm certificates. These are preserved. Suppose its retained
+conservative regime has uniform constants with
+
+`theta=e/S <= p^-2`, `W=O(sqrt(p) polylog p)`,
+`L=2^ceil(400 log2 p)`.
+
+Then `theta(6W+1)->0` and `(12W+2)/L->0`, so both inequalities (6) hold for
+all sufficiently large p. Every original Core start on every active axis
+therefore has a nonempty interior and boundary. There is no separate
+nonemptiness assumption for its K contexts. The conclusion is uniform in K
+and the original positions, including the first and last complete sections.
+
+This removes only the empty-interior case in that stated asymptotic source
+family. It neither supplies the theta or W bounds nor proves row gap,
+conditioning, matrix/Schur formation, uniform precision, compiler, native
+stock, owner lifecycle, whole halo delivery or final recovery. Short finite
+families outside (6) retain exact classifier (5); an empty classification is
+not an implementation of a missing zero-dimensional solver.
+
+The parameter family is a cited conditional source contract, supplied by the
+complete-Core and global-cohort predecessors. The geometric statement does
+not depend on their unpublished data, modified code or acceptance status:
+its three executable inputs are already pinned on canonical main.
+
+## 6. Full added cost and model boundary
+
+The guard reads input-independent source descriptors S,e,W,L,cap. Let d be the
+maximum of their binary lengths and the original endpoint lengths. The uniform
+guard uses a fixed number of comparisons, shifts, additions and products,
+including dyadic-cap validation. Exact single-window decision (1)--(4) adds a
+fixed number of integer divisions. Ordinary binary arithmetic supplies
+O(d^2) bit work and O(d) scratch for either calculation. This is not constant
+bit time. Input access, produced descriptors and disposal have O(d) size and
+are included in the bound.
+
+When each axis has the same S,e,W,L,cap descriptors across all K contexts,
+the uniform guard is checked once per actual axis, costing O(g d^2) without
+scanning or materializing contexts. Every distinct descriptor must otherwise
+be checked and charged; heterogeneous families may require K g checks. If
+a caller instead uses the exact local test at every context and axis, its full
+extra setup is O(K g d^2). All existing coefficient, catalogue, route, backing,
+payload and native bills are retained. This is source setup: there is no new
+payload pass, floating-point computation, numerical error or supplier gain.
+
+These checks fit the cited complete-Core preparation envelope
+`K 2^(O(ell_original)) poly(p)` with its full K factor retained. No new saving
+in the dominant online bill follows. The finite-domain geometry theorem is
+unconditional; the whole multiplication construction remains conditional on
+every other source and physical premise.
+
+The verifier compiles only the authenticated Reference **geometry prefix**,
+its original `interior` method, and unchanged `Core.partition` and capped
+`carve` bodies. It excludes the constructor continuation that makes exponential
+intervals and Gaussian values. This is a transparent source projection for
+the geometric theorem, not an actual numerical Gaussian or native-tape run.
+The decision formula is independent of that source projection.
+
+## 7. Evidence and remaining work
+
+The independent coordinate oracle covers all 772,850 ordinary windows for
+every `2<=S<=40`, `1<=e<S`, original start and positive window length. It
+constructs rational phases by independent nearest-half-up comparison and walks
+target coordinates literally, including holes. All closed-form answers agree,
+including 10,660 hole-only windows and 63,180 physical-endpoint windows.
+
+The original source geometry/partition/carve bodies check 77,793 finite
+configurations, including 855 admitted by the uniform guard. In the complete
+N=128, L=64 two-window example S=127,e=1,W=4,cap=4, both original windows have
+five interior blocks, 20 interior rows and 44 boundary rows. These are source
+geometry outputs, not numerical Core outputs.
+
+Sixteen W values corroborate each sharp adverse family. Controls detect
+target/source-coordinate confusion, origin rebasing, substitution of a smaller
+width for the actual carve width, wrong tie-to-even rounding, invalid
+windows/caps/widths and corruption of each actual source input. Descriptors of
+64, 257 and 1024 bits exercise the fixed-size formula without enumerating
+their exponentially large periods. Written Sections 2--6, not finite counts,
+establish the universal claims.
+
+Open questions are threading this finite guard into the actual uniform source
+constructor with its remaining matrix/precision certificates, optimizing the
+narrower dyadic-aligned admission, and implementing genuinely empty short
+windows at the full native interface. None is answered by replacing a missing
+solver with an identity or treating finite geometry checks as analytic recovery.