Research note: exact incident-safe parity frame dimensions (no new kappa)

research/parity-incident-completions/CLAIM.md

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+# Sprint claim
+
+- Track: structural decoder research; OS assignment seed 47 (unbiased UInt32 rejection sampling, 2026-10-13 08:43 PDT).
+- Base: 407d52ff15b2217b6e22bce44728b4c3907b6515.
+- Object: the 692 pointwise-completable degenerate nodes in the deterministic historical ParitySide(10) DAG; all minimum-dimension nondegenerate sandwiches, with every actual source-line and target-line residual required to be zero or nonalternating.
+- Method: exhaustive exact enumeration of one-dimensional extensions of radical-one source spans inside target-perp; independent explicit-vector residual checks. Seed 47 selects h=10 and this refinement of the odd-label interface.
+- Deliverable: exact census/certificates, proof of coverage, meaningful negative controls, source pins and one reproducible offline command. No supplier or new kappa claim.
+- Expected finish: 2026-10-13 09:25 PDT, including final verification/handoff.
+- Alternatives considered: simultaneous nested frames for the same DAG (too broad); several added E8 decoder totals with fixed readers (crowded and larger search). This is distinct from PR972's pointwise sandwich census and PR920's two-source/two-target rectangles.
+- Coordination: public issues page and a repo-scoped sprint-issue query checked; no pinned coordination issue found. Claim retained locally as requested. Relevant open PRs and review ledger inspected; PR972 fetched for exact source comparison. Delegated work stays within this residual-completion census.
+
+## Result update
+
+Completed the fixed-domain census and exact local dimension refinement:
+560 nodes admit a safe minimum, 12 require one extra dimension, 100 require
+two extra dimensions, and 20 have no incident-safe frame at any dimension.
+The producer and independent vector-set checker agree on all 3,096 minimum
+frames. The 22-test suite passes normally and with assertions disabled;
+fresh regeneration is byte-identical. No new kappa or legal supplier is claimed.
+
+The open-PR overlap search was refreshed before submission. PR #920 and
+PR #972 remain open at the exact heads recorded in SOURCES.json; this note
+adds full-DAG line residuals, larger-frame minima and the 20 absolute local
+obstructions beyond those predecessors. The current main moved to
+`8bddbee377e2f3ce1f1247196c4b290d55767d20`; the contribution retains its
+declared base and source pins. An isolated worktree was used after concurrent
+work was detected in the initial checkout.
+
+Result package: [woklerx research branch](https://github.com/woklerx/integer-mult-bounds/tree/research/seed47-incident-completions/research/parity-incident-completions).
+The final chat handoff links the focused PR. No coordination issue was available
+to receive a public claim update.

research/parity-incident-completions/PROOF.md

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+# Incident residuals and minimum local ranks
+
+This note concerns the 692 completable radical-one source spans in the fixed
+`ParitySide(10)` circuit. It is a local geometric result, not a compatible
+assignment of frames to that circuit, a transfer primitive, an operation
+invoice, or a multiplication-bound improvement. All dimensions and counts
+below are exact over the binary field.
+
+## Definitions and the finite domain
+
+Let `V = F2^10` have its standard dot product, and let `j = 1023` be its
+all-ones characteristic vector. Integer masks encode vectors. At an active
+node, `S` is the set of actual ancestor source labels and `T` is the set of
+actual descendant target labels. Set `A = span(S)` and `B = span(T)`.
+Every label in `S` and `T` is a weight-three, hence norm-one, vector.
+
+A separating frame is a nondegenerate subspace `U` with
+`A subset U subset B-perp`. Its incident source residual at `s in S` is
+`U intersect s-perp`. Its incident target residual at `t in T` is
+`U-perp intersect t-perp`. These are the residuals from the source line to
+`U`, and from the target line to the reverse-complement frame `U-perp`.
+Call `U` incident-safe if each of these residuals is zero or nonalternating.
+Nondegeneracy of the residuals follows from the norm-one lines and the
+nondegeneracy of the containing frames.
+
+The domain requires `A perpendicular B`, `A intersect B = {0}`, and
+`dim rad(A) = 1`. It consists of 692 nodes, rather than 692 distinct source
+subspaces: different nodes can have the same source span or different actual
+incidences. The predecessor's two impossible sandwiches and 1,577
+nondegenerate-source nodes are outside this domain.
+
+The sandwich minimum theorem used below is established in ToikY's
+[PR #972](https://github.com/CrocSwap/integer-mult-bounds/pull/972), pinned at
+`f39f92a085fc95fe4245b06d4a695739d79763bc`. It gives the minimum separating
+dimension `dim A + dim rad(A)`. The present note adds complete incident
+residual tests and minimum incident-safe ranks on the stated radical-one
+domain; it does not repeat that contribution's all-node census.
+
+## Exhausting all minimum separating frames
+
+Write `a = dim A`, `b = dim B`, and `rad(A) = span(r)` with `r != 0`.
+Every minimum frame has the form
+
+    U_z = A + span(z),  z in B-perp,  r dot z = 1.
+
+To prove sufficiency, let `x = x_A + c*z` be in the radical of `U_z`.
+Pairing with `r` forces `c = 0`. Pairing with `A` then places `x_A` in
+`span(r)`, and pairing with `z` forces `x_A = 0`. Thus `U_z` is
+nondegenerate and has dimension `a+1`.
+
+Conversely, a nondegenerate `(a+1)`-dimensional frame containing `A` has
+the displayed form, and its additional vector must pair nontrivially with
+`r`; otherwise `r` would remain in its radical. This proves coverage, not
+just a construction of some frames.
+
+The functional `z -> r dot z` is nonzero on `B-perp`, since otherwise
+`r` would lie in `(B-perp)-perp = B`, contrary to `A intersect B = {0}`.
+There are therefore `2^(10-b-1)` permitted vectors `z`. Two such vectors
+give the same frame exactly when they differ by an element of `A`:
+the quotient `U_z/A` is a one-dimensional binary space, with a unique
+nonzero coset. Each frame has exactly `2^a` representatives. Consequently
+the exact number of distinct minimum frames at the node is
+
+    2^(10-a-b-1).
+
+Signs of scalar coefficients do not change binary source lines. Actual
+label sets are essential; replacing them by all vectors of their spans
+would impose additional, unrequested incidence conditions.
+
+## Characteristic vectors give exact incident tests
+
+For a nondegenerate space `E`, write `w_E` for its unique characteristic
+vector, characterized by `w_E dot x = x dot x` for every `x in E`.
+For a nondegenerate subspace `L` of `E`, the orthogonal decomposition
+`E = L direct-sum (E intersect L-perp)` shows that the latter summand
+is alternating exactly when `w_E in L`.
+
+For a norm-one line `L = span(s)`, `w_E` cannot be zero, since it pairs
+to one with `s`. Thus a **nonzero** residual `E intersect s-perp` is
+alternating exactly when `w_E = s`. If `dim E = 1`, the residual is zero
+and is accepted even though this equality holds. The zero-residual
+exception must not be dropped.
+
+The orthogonal decomposition `V = U direct-sum U-perp` gives
+
+    w_(U-perp) = j + w_U.
+
+Hence the complete incident test is:
+
+* Reject at source `s` exactly when `dim U > 1` and `w_U = s`.
+* Reject at target `t` exactly when `dim U-perp > 1` and `j + w_U = t`.
+
+The characteristic vector of a minimum frame has an especially small
+search domain. The vectors `w in A` satisfying
+
+    w dot x = x dot x for every x in A
+
+form a pair `{w0, w0+r}`. Existence follows because the linear functional
+`x -> x dot x` vanishes on the radical, and the image of the bilinear map
+`A -> A*` is precisely the annihilator of that radical. Its solution set
+is a coset of the one-dimensional radical.
+
+In any `U_z`, its characteristic vector belongs to `A`: write it as
+`x_A + c*z` and pair with `r`; the required value is `r dot r = 0`, so
+`c = 0`. Exactly one of the two candidates also satisfies
+`w dot z = z dot z`, because the candidates differ by `r` and
+`r dot z = 1`. Explicitly,
+
+    w_(U_z) = w0          if (j+w0) dot z = 0,
+              w0 + r      otherwise.
+
+This selector is constant on each coset `z+A`, so it is a well-defined
+test of the frame. The two possible characteristics need not both occur.
+On the permitted affine set of `z`, the selector is constant exactly
+when `j+w0` belongs to `B+span(r)`; otherwise both values occur equally
+often. This follows by restricting the functional to the translation
+space `B-perp intersect r-perp`, whose annihilator is `B+span(r)`.
+
+## Exact minimum-frame census
+
+Exhausting the described cosets gives 3,096 distinct minimum frames at
+the 692 nodes. Exactly 2,538 frames pass every actual source-line and
+target-line test. There are 158 source-only failures, 400 target-only
+failures, and no frame failing both kinds of incidence.
+
+| dim A | dim B | Minimum frames per node | Safe minimum frames per node | Nodes |
+|---:|---:|---:|---:|---:|
+| 2 | 4 | 8 | 0 | 10 |
+| 2 | 5 | 4 | 0 | 2 |
+| 3 | 1 | 32 | 32 | 16 |
+| 3 | 5 | 2 | 2 | 2 |
+| 4 | 4 | 2 | 1 | 50 |
+| 4 | 4 | 2 | 2 | 80 |
+| 4 | 5 | 1 | 0 | 20 |
+| 4 | 5 | 1 | 1 | 2 |
+| 5 | 1 | 8 | 8 | 170 |
+| 5 | 2 | 4 | 4 | 20 |
+| 5 | 3 | 2 | 2 | 40 |
+| 6 | 1 | 4 | 0 | 100 |
+| 6 | 2 | 2 | 2 | 10 |
+| 7 | 1 | 2 | 2 | 100 |
+| 8 | 1 | 1 | 1 | 70 |
+
+Thus 560 nodes admit a safe minimum frame: 510 accept every minimum
+frame and 50 accept some but not all. The other 132 accept none. These
+counts were also obtained by an independent in-memory enumeration using
+explicit XOR-closed sets and all 1,024 ambient masks. That cross-check
+shared the scalar DAG generator, but did not use the producer's frame
+enumerator. It is not independent human peer review.
+
+## Minimum incident-safe dimensions, including larger frames
+
+An even-dimensional nondegenerate `U` is automatically incident-safe
+here. Indeed, both `U` and `U-perp` have even dimension because the
+ambient dimension is ten; removing any contained norm-one line leaves
+an odd-dimensional nondegenerate residual. An alternating nondegenerate
+space has even dimension, so each such residual is nonalternating.
+This observation does not assert that a frame of the next even dimension
+exists.
+
+For the 12 rejected nodes with `dim A = 2`, the minimum separating
+dimension is three. Complete enumeration of dimension-four separating
+frames gives 16 frames at each of the ten `dim B = 4` nodes and four
+frames at each of the two `dim B = 5` nodes. Every one is incident-safe
+by parity. Their minimum incident-safe dimension is therefore four.
+There are 168 such dimension-four frames in total.
+
+One complete way to enumerate these frames is to take every minimum
+frame `U` and every norm-one vector
+`q in U-perp intersect B-perp`, and form `U + span(q)`, deduplicating
+subspaces. This is exhaustive: any nondegenerate frame `E` of dimension
+`a+2` containing `A` contains a minimum frame `U`. Choose in `E` a
+vector pairing to one with `r` to construct it. Then
+`E intersect U-perp` is a nondegenerate line, and its unique nonzero
+vector is one of the enumerated `q`.
+
+The 100 rejected nodes with `dim A = 6` all have `B = span(t)` for one
+actual norm-one target `t`. Every one of their four minimum
+dimension-seven frames has characteristic `j+t`; equivalently, every
+one fails at that target. No dimension-eight separating frame exists.
+To see this, put `W = t-perp`. This is a nondegenerate
+dimension-nine space with characteristic `j+t`. For any minimum `U`,
+the plane `W intersect U-perp` is alternating, since `w_W = w_U`
+belongs to `U`. A dimension-eight nondegenerate frame `E` containing
+this `U` would require a nondegenerate line in that alternating plane,
+which is impossible. Any proposed dimension-eight `E` containing `A`
+would contain some minimum `U` by the partner argument above, so it
+cannot avoid this obstruction by choosing different intermediate frames.
+
+The frame `W` itself is incident-safe: its target residual is zero, and
+its characteristic `j+t` has weight seven, hence cannot equal any
+actual weight-three source label. The minimum incident-safe dimension
+at these 100 nodes is therefore nine, two above the separating minimum.
+
+At each of the 20 rejected nodes with `dim A = 4` and `dim B = 5`,
+the only minimum frame already equals `B-perp`, of dimension five.
+It fails a source incidence. Any nondegenerate frame containing `A`
+has dimension at least five, while any separating frame has dimension
+at most five. Thus **no incident-safe separating frame of any dimension
+exists** at these 20 nodes.
+
+Together with the 560 nodes already safe at the separating minimum,
+this exhausts the original 692-node domain:
+
+| Local conclusion | Nodes |
+|---|---:|
+| Safe at minimum separating dimension | 560 |
+| First safe one dimension above minimum | 12 |
+| First safe two dimensions above minimum | 100 |
+| No incident-safe separating frame of any dimension | 20 |
+
+Among the 672 locally feasible nodes, the minimum incident-safe
+dimension histogram is `4:30, 5:132, 6:230, 7:10, 8:100, 9:170`.
+These are geometric dimensions and node counts, not paid helper counts
+or a recursive resource bound.
+
+## Concrete witnesses and obstructions
+
+At node **229**, `S = {38,49}`,
+`A = span(49,23)`, and `B = span(896,448,192,64)`.
+Its actual targets are `{448,704,832,896}`. Its radical generator is
+23. Each of its eight dimension-three frames has characteristic 38 or
+49, four frames of each type. Both characteristics are actual sources,
+so every minimum frame fails. For example, partners `z=1` and `z=2`
+produce characteristics 49 and 38 respectively. Its sixteen
+dimension-four separating frames all pass.
+
+At node **1217**,
+`A = span(56,27,15,3)` and `B = span(515,259,131,67)`.
+Its actual sources are `{35,44,52,56}`, and actual targets are
+`{67,131,259,515}`. There are exactly two dimension-five minimum
+frames. Partner `z=961` produces characteristic 35 and fails at the
+actual source 35. Partner `z=965` produces characteristic 32 and
+passes every actual incidence. This is a small example of why choosing
+one arbitrary minimum completion does not decide existence of a safe one.
+
+At node **818**,
+`A = span(518,262,134,20,12,6)` and `B = span(97)`.
+The radical generator is 30. The two characteristic solutions inside
+`A` are 926 and 896, but only 926 occurs among the four minimum frames.
+Because `926 = 1023+97` over the binary field, all four fail at target
+97. No dimension-eight separating frame exists. The dimension-nine
+frame `97-perp` is incident-safe, as proved above.
+
+For an example of absolute incident impossibility, node **302** has
+`A = span(896,352,224,96)` and `B = span(112,17,9,5,3)`.
+Its sole minimum frame is `B-perp` and has characteristic 896, an
+actual source. Its source residual has positive dimension and is
+alternating. There is no larger separating frame.
+
+## Why the tests are necessary for nested transfers
+
+For a nested chain of nondegenerate spaces
+`L = U0 subset U1 subset ... subset Uk = U`, there is an orthogonal
+direct-sum decomposition
+
+    U intersect L-perp
+      = direct-sum over i=1,...,k of (Ui intersect Ui-1-perp).
+
+It follows successively from `Ui = Ui-1 direct-sum
+(Ui intersect Ui-1-perp)`. If every successive residual is zero or
+nonalternating, a nonzero endpoint residual must be nonalternating:
+some nonzero summand supplies a vector of norm one, which keeps that
+norm in the full orthogonal sum. Therefore a nonzero alternating
+endpoint residual cannot be repaired merely by subdividing that same
+nested transfer into more intermediate frames.
+
+Apply this argument along a source-to-node path, and along a
+target-to-node path in the reversed complement frames. Every actual
+source or target incidence must pass in a legal assignment under this
+nested orthonormal-residual interface. The 20 absolute incident
+obstructions therefore reject such assignments at these otherwise
+completable source spans. They are additional local obstructions;
+the predecessor's two impossible sandwiches already reject the
+unchanged full circuit.
+
+The converse is not proved and is not implied: safe choices at
+individual nodes need not nest along an edge, and the intervening
+edge residuals need not pass. Alternative topologies, copied carriers,
+paid non-nested transport, and other phase interfaces are outside this
+result. There is no claim that the locally feasible nodes form a legal
+supplier, nor that local rank increments equal operation costs.
+
+## Attribution
+
+The historical signed parity circuit and its obstruction/repair analysis
+are due to Douglas Colkitt. DreamingOfClouds' format-limits work supplied
+the sandwich restriction and radical rank floor. ToikY's pinned PR #972
+proved the constrained minimum-completion theorem and identified the
+692-node radical-one domain. ArnavSwiggy's
+[PR #920](https://github.com/CrocSwap/integer-mult-bounds/pull/920)
+already emphasized the stronger incident-residual condition and supplied
+an explicit alternating-residual counterexample. These predecessors
+are the basis for this narrowly scoped extension; no priority is claimed
+for the underlying bilinear-form or characteristic-vector facts.
+
+Prepared with substantial OpenAI Codex assistance in the mathematical
+derivations, enumeration, and independent computational cross-check.