Sharp binary frame intervals through native warps and carried swaps (no new κ)

research/carried-warp-frame-intervals/PROOF.md

@@ -0,0 +1,135 @@
+# Sharp marginal frame intervals and the extremal price test
+
+This is a written proof about the binary frame contract defined below. The
+finite exact replays supplement it. No formal proof kernel, physical phase
+admission or multiplication theorem is claimed. Prepared by 1337isnot1337 with
+substantial OpenAI Codex assistance; Apache-2.0.
+
+## The graph and the quantified statement
+
+Let V be F₂^h for any finite positive h. Let D be a finite directed acyclic
+multigraph whose vertices are topologically numbered. Each edge e:a→b carries
+an invertible linear map G_e:V→V. A specified subset of vertices has prescribed
+subspaces F_v. A feasible assignment consists of subspaces U_v≤V such that
+
+    U_v = F_v at each prescribed vertex;
+    G_e(U_a) ≤ U_b for every edge e:a→b.
+
+Assume at least one feasible assignment exists. Define L in topological order:
+
+    L_v = F_v                              at prescribed vertices;
+    L_v = sum_{e:a→v} G_e(L_a)              otherwise.
+
+The empty sum is the zero subspace. Define H in reverse topological order:
+
+    H_v = F_v                              at prescribed vertices;
+    H_v = intersection_{e:v→b} G_e^(-1)(H_b) otherwise.
+
+The empty intersection is V. Then:
+
+1. L and H are feasible, and every feasible U satisfies L_v≤U_v≤H_v.
+2. For each single vertex v and each subspace S with L_v≤S≤H_v, there
+   exists a feasible assignment U with U_v=S. It is constructed by adding
+   the prescription U_v=S and recomputing the lower propagation.
+
+This is a standard order propagation argument applied to this caller. Choices
+at different vertices remain coupled by the edges. The statement concerns
+one prescribed marginal at a time, over all subspaces, not a registry search.
+
+## Proof of the bounds
+
+Compare against any feasible U. Forward induction gives L_v≤U_v: this is
+immediate at prescribed vertices, and a sum of images of contained subspaces
+is contained in U_v at an unprescribed vertex. Backward induction gives U_v≤H_v,
+using inverse images and intersections. Thus L_v≤H_v.
+
+For an edge a→b, if b is unprescribed its recurrence contains G_e(L_a).
+If b is prescribed, the existence of a feasible U gives
+G_e(L_a)≤G_e(U_a)≤F_b=L_b. Hence L is feasible. For H, an unprescribed a
+satisfies G_e(H_a)≤H_b by definition. At a prescribed a, use feasibility and
+the already established backward bound to get G_e(F_a)≤H_b. Thus H is feasible.
+The implementation checks these prescribed-vertex compatibility conditions
+explicitly and rejects empty intervals.
+
+## Proof of complete marginal attainability
+
+Fix v and S between L_v and H_v. Keep L unchanged before v, prescribe S at v,
+and propagate sums of images forward. Incoming edges at v remain valid because
+their images are contained in L_v≤S.
+
+At each later vertex w, the extra contribution from S is the sum of G_path(S)
+over paths from v to w whose internal vertices are unprescribed. This follows
+inductively because linear isomorphisms preserve sums of subspaces. At the first
+prescribed endpoint w on any such path, the reverse recurrence gives
+H_v≤G_path^(-1)(F_w). Consequently G_path(S)≤F_w. The original lower contribution
+was also contained in F_w, so prescribing F_w causes no conflict. That vertex
+resets the downstream contribution to its fixed F_w. This proves compatibility
+at every prescribed vertex and constructs the required feasible assignment.
+
+## Binding the graph to the native word
+
+The input is the exact frame-carrying program constructed by Dugongue's PR4971
+at head 0130b196145a43a7d305de41f49e56e8e8556378. It retains Cao Yifan's native
+round-three source and the credited native/helper lineage.
+
+Each live field holds a pointer (v,A), meaning that its current frame, before
+any next paid climb, is A(U_v). Initially v is its prescribed entrance and A=I.
+A weighted whole-field swap exchanges the pointers. A native warp of signed
+operator k changes A to G_k A, using the inverse map for negative k.
+
+A scalar contact between live fields creates one vertex u and an edge
+(v,u,A) for each operand. Both pointers then become (u,I). A scalar SCALE
+commutes with the frame and creates no constraint. A scalar contact with the
+owned cell retains its supplied frame as a prescribed vertex. Every live
+terminal frame is prescribed. Owned-center events and all scalar coefficients
+are retained. The graph is acyclic because every new contact is later in the
+word than the pointers feeding it.
+
+Every feasible graph assignment has a sufficient chronological realization:
+before a contact u, climb each operand from A(U_v) to U_u. The edge constraint
+makes this an ascending nested climb. At a warp carry the actual frame by G_k;
+at a weighted swap carry it with the exchanged field. Restore each prescribed
+terminal frame by its final edge. This realizes exactly the binary contract.
+
+Conversely, any schedule in this contract with the same scalar contacts, swaps,
+warp maps and fixed boundaries defines such a graph assignment. Between two
+contacts, invertible maps preserve containment, so any ascending climbs can be
+deferred through the intervening warps to the next contact. The composed map
+on that edge therefore sends the earlier contact frame into the later one.
+This is the precise schedule class to which the sharp bounds apply.
+
+These statements concern subspaces and normalized linear actions. They do not
+prove that every new canonical frame transition has the required physical
+absolute phase or uniform implementation. The separate replay checks all
+632 independent-input normalized rows, actual G-image warp conditions, frame
+chronology, paid ranks and owned-cell clearing for each emitted candidate.
+
+## Paid rank and the two extremal profiles
+
+The sum of the dimensions of the 632 live frames changes only at paid climbs.
+Swaps permute its terms and the invertible warp maps preserve each dimension.
+Consequently every ascending realization with these fixed entrances and exits
+has the same total live paid rank. The eight retained rank-eight owned-center
+jobs add a fixed 64. The complete source and each realized schedule have total
+paid rank 5,327.
+
+For a paid profile n_r, define C_k(n)=sum_r n_r min(r,k). For any concave rank
+price f on {0,...,9} with f(0)=0, put d_j=f(j)-f(j-1). Direct summation gives
+
+    f(r) = r d_9 + sum_{k=1}^8 min(r,k) (d_k-d_(k+1)).
+
+Every curvature coefficient d_k-d_(k+1) is nonnegative. Equal total rank cancels
+the linear term when comparing profiles. Therefore C_k(source)≤C_k(candidate)
+for every k implies that the source price is no greater for every such f.
+A positive cut difference yields strict inequality for every strictly concave f,
+including f(r)=r^t for 0<t<1.
+
+The exact differences candidate minus source, for k=1,...,9, are
+
+    least:    [347, 375, 60, 30, 0, 0, 0, 0, 0];
+    greatest: [500, 427,115, 12, 0, 0, 0, 0, 0].
+
+Both extremal schedules are thus strictly more expensive than the source for
+every power price in the relevant range. This rejects these schedules as a
+paid-profile improvement with the other invoice terms held fixed. It does not
+optimize the intermediate assignments or include a new physical/compiler bill.

research/carried-warp-frame-intervals/README.md

@@ -0,0 +1,126 @@
+# Carried frame intervals for the PR4971 native word
+
+This package computes sharp per-contact binary frame bounds for the weighted
+field swaps of [PR4971](https://github.com/CrocSwap/integer-mult-bounds/pull/4971)
+while also carrying frames through the actual invertible native warp maps.
+It preserves the source's scalar word, helper boundaries and owned-cell contacts.
+No new κ, physical phase admission or selected-frontier change is claimed.
+
+The source is PR4971 head `0130b196145a43a7d305de41f49e56e8e8556378`, which
+retains Cao Yifan's round-three word beyond PR4610. It is a different native word from the selected PR4470 supplier. PR4971 was
+merged while this investigation was running; its retained source remains in
+this branch’s ancestry. No separate pending-PR fetch is needed for a cold replay.
+
+## What is established
+
+[PROOF.md](PROOF.md) defines the exact binary schedule class and proves sharp
+marginal bounds for every finite acyclic graph with invertible linear maps on
+its edges. The graph propagates subspace sums forward and inverse-image
+intersections backward. Every feasible contact frame lies between those bounds.
+For each single contact, every intermediate subspace extends to a feasible
+binary schedule by adding that prescription and propagating again. Different
+contacts' choices remain coupled by the graph edges.
+
+On the actual source, each orientation has:
+
+- 15,935 graph vertices and 29,790 mapped edges;
+- 14,671 scalar-contact vertices, of which 14,487 are variable;
+- 13,148 rigid variable contacts and 1,339 flexible contacts;
+- a maximum interval dimension difference of five.
+
+The checker independently executes the source, the reconstruction from its
+original contact labels, both extremal schedules and one strict intermediate
+pinned frame. Every execution checks the 632 independent-input normalized
+endpoints, all 408 arbitrary helper inputs, actual signed G-image warp frames,
+scalar contacts, paid ranks and owned C0 clearing. The emitted schedules retain
+534 weighted field swaps and 240 warp events per orientation.
+
+The least and greatest schedules give no paid improvement. Their profiles are:
+
+| Schedule | Rank 1 | Rank 2 | Rank 3 | Rank 4 | Rank 5 | Rank 8 |
+| --- | ---: | ---: | ---: | ---: | ---: | ---: |
+| Source / original contacts | 2201 | 412 | 510 | 127 | 40 | 8 |
+| Least frames | 2520 | 755 | 225 | 127 | 10 | 8 |
+| Greatest frames | 2774 | 651 | 301 | 36 | 28 | 8 |
+
+All have total paid rank 5,327. Exact min-rank cut differences show that the
+source costs no more for every concave price f with f(0)=0, and costs strictly
+less for every power price r^t with 0<t<1. This is an all-price comparison of
+these profiles, not a numerical power sample. The proof uses the standard
+discrete curvature decomposition of a concave sequence.
+
+`RESULT.json` contains the complete interval dimension census, profiles,
+integer cut differences and controls. It also records an exhaustive two-variable
+three-bit example: all 256 subspace pairs are checked, six are feasible, and
+all seven feasible marginal pins are extended constructively. Five toy controls
+and four actual-word controls per orientation reject wrong map direction,
+independently chosen incompatible marginals, an impossible pin, a changed fixed
+node, a singular map, a false paid rank, false warp frames and lost swap frame
+carry and an incorrect swap sign. The separate exact row replay imports no retained row primitives.
+
+## Reproduce
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3 -B scripts/research_package.py --package research/carried-warp-frame-intervals --check-only
+python3 -B scripts/research_package.py --package research/carried-warp-frame-intervals
+python3 -I -OO -B research/carried-warp-frame-intervals/verify.py
+```
+
+The registered replay authenticates the seven live source files. Its retained
+constructor runs in an isolated subprocess with an empty private bytecode-cache
+namespace. `verify.py --write` regenerates the deterministic receipt.
+
+`frame_dag.py` exposes `FrameDAG.bounds()` and `FrameDAG.extend(node, subspace)`.
+`NativeSchedule(canonical, program)` builds the actual mapped graph, and
+`realize(values)` emits the corresponding sufficient binary schedule with
+strictly nested paid climbs. The implementation accepts ambient dimensions one through nine; this caller
+uses nine. Frames are exact subspaces encoded as point bitsets.
+No subspace is selected using floating-point optimization.
+
+## Scope and provenance
+
+Only ascending live-field climbs, fixed scalar/weighted-swap order, fixed warp
+maps, fixed helper/main entrances and exits, and fixed owned-cell contacts are
+covered by the sharp bounds. The theorem includes every binary subspace in
+that contract. It establishes no global price optimum over their coupled
+assignments. The finite replays are normalized exact linear checks; physical
+Clifford absolute phases, new whole-job lowering, all-column implementation,
+movement, precision, the complete controller invoice and multiplication remain
+separate obligations. The two losing extrema do not justify those larger runs.
+
+Sources, Git blobs and SHA256 pins are in `SOURCES.json`. The catalogue and
+selected record are historical provenance. The task card is explicitly changed
+from historical four-provider admission to this carried-frame interface.
+Canonical main was refreshed at `bee95ce88f9f19802f0613f92838895cd0e14a1f`, priority
+revision 7. All seven scientific source files match that refreshed main byte
+for byte. The scientific source pin remains the original PR4971 head. Overlap checks found PR4971 as the direct predecessor and PR340 as
+older fixed-word free-form frame retiming. Earlier packages also enumerate local
+frame intervals. This package applies the standard graph recurrence to the new
+carried-swap/native-map contract; it claims no invention of lattice propagation
+or concave-price comparison. Remote coverage and sparse local limits are recorded
+in TASK.json, and no exhaustive novelty claim is made.
+
+## Open next questions
+
+Can a coupled intermediate assignment beat the source's complete paid profile?
+Start with the 1,339 flexible contacts and retain every mapped-edge constraint;
+the extrema and independent marginal choices are insufficient. A favorable
+profile is required before expanding into physical phase and controller/compiler
+validation. No improving assignment has been established. The concrete question,
+relevance, first step and blocker are also recorded in TASK.json.
+
+## Credit
+
+Dugongue's PR4971 supplies the weighted frame-carrying swap construction and its
+retained source. Cao Yifan supplies the round-three native word beyond PR4610.
+Cinna Davis supplies the scalar/helper and selected-result lineage, including
+PR4440/4470. Dugongue's PR4935/4948, zxv, Jacob Alan Sussman and the remaining
+native/compiler authors retain their original credit in the source NOTICE.
+The retained LICENSE and NOTICE are declared export inputs. James Reed's issue4571
+motivates checking actual G-image warp events in the separate replay.
+
+Prepared by 1337isnot1337 with substantial OpenAI Codex assistance, without
+subagents. New material is Apache-2.0. This is written mathematics and exact
+finite evidence, without formal-kernel or independent human acceptance.