Factor omitted P104 endpoint effects through eight queries

research/p104-omitted-completion/PROOF.md

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+# The omitted endpoint operator has rank eight
+
+All statements below concern chart-normalized payloads. The frame changes in
+section 5 are separate obligations.
+
+## 1. Exact object and conditional interpretation
+
+Use the two unchanged PR2586 inputs pinned in `SOURCES.json`. The word has
+5,398 events and 358 initial public banks. Original private banks start at zero.
+The native interpretation is conditional on the exact covariance
+`U(old,new) = J_new J_old^-1`, including its phase. This premise is inherited
+from the PR2586/native lineage and the PR3416 covariance result. This package
+binds every occurrence to its source frames and recipe rank. It does not
+independently prove the native matrix identity.
+
+Writing a physical bank in frame `f` as `J_f q` gives normalized payload `q`.
+A native transport leaves `q` unchanged. Every ADD has equal source and target
+frames, so it updates `q_target += c q_source`. Signs act by minus one.
+This gives rational linear maps on the 358 public normalized payloads after
+eliminating the eight zero-initialized private banks. The verifier proves all
+eight erased payloads are the zero linear form in both maps below.
+
+Let `F` be the complete word's normalized endpoint map. Define `S` by the
+PR3446 terminal-history projection, extended by the identity on omitted public
+banks. Its construction is fully specified here, without importing PR3446 code:
+
+1. Let `L` contain Y22 through Y32, Y63 through Y103, D27 through D31,
+   G27 through G31, and Q7.
+2. Observe each ADD at event at most 4516 whose source is in `L` and whose
+   target is outside `L` and is not the private C bank. There are 675 events.
+3. Retain their full syntactic histories and the final histories of `L` and
+   those ADD targets. Whenever a retained history reads an initial public bank,
+   also retain that bank's full terminal history. Whenever a private allocation
+   is retained, retain its erasure history. Repeat to a fixed point.
+
+Histories include each target's old value, source reads, signs and transports,
+even if their scalar coefficients later cancel. The forward bitset construction
+retains exactly 4,027 events and 265 initial banks. Its sorted event-list hash
+matches PR3446's published `66113bcd...c676dc8`. This is the projection being
+completed, rather than an algebraically minimized alternative slice.
+
+## 2. The full residual
+
+Set `Delta = F-S`. Exact rational execution establishes the following identities
+entry by entry. A separate reverse-covector calculation checks all 358 rows of
+both `F` and `S`, for 716 complete row comparisons.
+
+- `Delta` vanishes on all 265 retained output banks.
+- Among the 93 omitted output banks, 52 X rows are nonzero and 41 rows vanish.
+  The latter are 38 N banks and G2, G3, G9.
+- Every nonzero residual column is a Y input. Write `R` for the nonzero
+  52 by 104 block of `Delta`. It has 2,500
+  nonzero entries. Its exact sparse representation is authenticated by the
+  matrix digest in `CERTIFICATE.json`.
+- Both maps preserve every Y payload. Every full non-X row is its own identity
+  row. Every full X row is its own identity row plus a Y-only linear form.
+
+The 696 omitted ADDs whose source is retained and target is omitted are not the
+entire omitted program. In total the slice omits 794 ADDs, 104 signs and 473
+transports. The complete terminal difference includes all of these operations.
+In particular, adding 696 isolated increments at the endpoint is not the
+calculation used here.
+
+Let `K` denote the 358 by 358 matrix obtained by embedding `R` in its X output
+rows and Y input columns. Since `S` preserves Y, `K S = K`. Thus
+
+```
+F = S + K = (I + K) S.
+```
+
+Also `K^2=0`, because X and Y banks are disjoint. The normalized completion is
+an invertible shear with inverse `I-K`. It requires no reads of omitted N/G
+payloads and leaves those payloads unchanged. This is an endpoint identity,
+not permission to omit their physical frame changes.
+
+## 3. Exact factorization and sharp query dimension
+
+Use row order
+
+```
+X0, X33, X101, X102, X11, X12, X15, X24
+```
+
+and column order
+
+```
+Y0, Y10, Y100, Y101, Y102, Y103, Y11, Y12.
+```
+
+The corresponding minor of `R` has determinant `-1/128`. Equivalently, that
+minor of `2R` has determinant `-2`. This proves rank at least eight over Q,
+Q(i), and every field of characteristic other than two.
+
+Let row `i` of `A` be twice the complete Y row of `R` at the named basis X bank.
+For each X row, solve for its coefficients in these eight basis rows using the
+specified minor. The resulting 52 by 8 matrix `C` has integer entries. The
+verifier checks the reconstruction on every one of the 5,408 entries, not just
+the minor columns. Consequently
+
+```
+R = C A / 2.
+```
+
+Both matrices, with every integer coefficient, are in `CERTIFICATE.json`.
+`A` has 385 nonzeros and entries in {-2,-1,1,2}. `C` has 209 nonzeros and
+entries in {-3,-2,-1,1,2,3,4}. This proves rank at most eight and completes the
+rank proof. These identities have coefficients in Z[1/2], so they also prove
+the characteristic-other-than-two assertion.
+
+Any factorization of this fixed endpoint matrix through `q` linear query values
+has rank at most `q`, hence `q >= 8`. This is a bound on linear summary dimension
+in such a factorization. It is not a lower bound of eight simultaneous helper
+banks, a native-rank tariff, or a restriction on arbitrary interactive programs.
+No sparsity optimality is asserted for the displayed `A,C`.
+
+## 4. One arbitrary dirty helper suffices algebraically
+
+Let `M` be a Z[i] module embedded in a Q(i) vector space. Suppose the X
+targets start in `M` and the Y inputs belong to `2M`. Every `A_i Y/2` belongs to `M`, since `A` is integral. The prior
+projection's sufficient Y entrance is `4M`, which satisfies this condition.
+This does not independently prove that projection's native or scalar entrance
+contract. The present `2M` statement concerns the endpoint completion alone.
+
+Borrow one helper with arbitrary value `h` in `M`. For each `i=0,...,7`, perform
+these four normalized operations, in this order:
+
+```
+X <- X - C[:,i] h
+h <- h + A[i,:] Y / 2
+X <- X + C[:,i] h
+h <- h - A[i,:] Y / 2.
+```
+
+The two X changes sum to `C[:,i] A[i,:]Y/2`. The helper returns exactly to its
+initial dirty value after each iteration. No X value is read by a gather, so
+previous X updates do not affect subsequent queries. Summing the eight updates
+gives `RY`, proving the claimed completion and arbitrary dirty restoration.
+All intermediate helper and target payloads remain in `M` under the stated
+domain. Gathering the eight queries concurrently into eight helpers gives the
+same identity, but is unnecessary for peak algebraic workspace.
+
+Expanding each matrix entry into one weighted transvection gives exactly
+`2*(385+209) = 1188` nonzero weighted updates. These include halves and integer
+coefficients up to four. This count is not a bill in the source instruction
+alphabet and does not price frame adapters, scalar expansion or native calls.
+The standard-library replay checks every one of the 104 Y unit columns at twice
+a module element, with three arbitrary dirty values, in addition to six complete
+word/projection fixtures. Rational identities establish the general statement;
+these finite fixtures check the actual schedule and failure paths.
+
+An arbitrary `M` entrance cannot be inferred uniformly. For `M=Z[i]`, the
+certificate identifies a coefficient `1/2` or `-1/2` in `R`, so a unit input
+in that Y coordinate produces a nonintegral scalar residual. We claim sufficiency of `2M`, not a general
+classification of correlated entrance domains.
+
+## 5. Physical frame and cost boundary
+
+All 93 omitted public banks remain in their initial frames in the projection,
+and all 93 full-word final frames differ. The authenticated gap list records
+both endpoint frames for each bank. The dimension pairs are:
+
+| Omitted banks | Number | Projection frame dimension | Full final dimension |
+|---|---:|---:|---:|
+| X | 52 | 1 | 9 |
+| N | 38 | 0 | 9 |
+| G | 3 | 0 | 9 |
+
+The 41 zero normalized residuals therefore still need physical frame changes.
+Moreover, normalized Y readouts and X updates are not automatically legal
+same-frame physical ADDs. The actual retained Y endpoints have dimension eight.
+A common chart, legal adapters, source restoration and a complete paid native
+implementation of the summary program remain unknown.
+
+The new proved obligation is the exact omitted normalized endpoint operator and
+its one-dirty-helper factorization. Native covariance is inherited. Physical
+access/routing, ownership, adapter recipes/costs, numerical heights and precision,
+recursive realization, idle/cleanup charges and a complete supplier invoice are
+not supplied. The selected frontier is unchanged. The unchanged-profile stop
+remains in force, and this package does not begin another construction round.
+A follow-up must first bind a concrete physical caller and all adapter costs to
+the displayed coefficient program, then apply a fully charged budget screen.

research/p104-omitted-completion/README.md

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+# Rank-eight completion of the omitted P104 endpoint effects
+
+The 4,027-event PR3446 projection omits 696 outgoing ADD effects on 93 banks.
+This package computes its difference from the full 5,398-event PR2586 word on
+all 358 public normalized payloads. The final difference is a Y-to-X shear
+with 52 nonzero rows and exact rank eight. The other 41 omitted banks restore
+identically in normalized coordinates.
+
+The explicit integer factorization is `R=C A/2`. Its eight queries can run
+sequentially through one arbitrary dirty helper, restoring that helper after
+every query. The completion alone preserves a Gaussian module `M` when Y is in
+`2M`; the predecessor's sufficient `4M` Y entrance meets that condition.
+A fixed 8 by 8 minor has determinant `-1/128`. The factorization and this minor
+prove rank eight, including over fields of characteristic other than two.
+The bound concerns linear query dimension, not simultaneous workspace or native
+cost. The written argument is in [PROOF.md](PROOF.md).
+
+All 93 omitted banks still have unfilled physical frame changes, including the
+41 zero normalized rows. This is an algebraic endpoint completion, conditional
+on native covariance for its interpretation in the physical word. It is not a
+paid native implementation, supplier admission or improved kappa. The selected
+frontier and unchanged-profile stopping rule remain unchanged.
+
+## Evidence and replay
+
+Python 3.12, standard library only. From the repository root:
+
+```sh
+python3.12 -I -Werror -B research/p104-omitted-completion/verify.py
+python3.12 -I -O -Werror -B research/p104-omitted-completion/verify.py
+python3.12 -I -O -Werror -B research/p104-omitted-completion/test_verify.py
+python3.12 -B scripts/research_package.py --package research/p104-omitted-completion
+python3.12 -B scripts/check_research_commit.py --package research/p104-omitted-completion --check-source-history
+```
+
+The verifier uses forward provenance bitsets to reconstruct the projection and
+exact rational state rows to compute both endpoint maps. Reverse covectors
+separately check all 716 public output rows. It checks all 5,408 factorization
+entries, eight original private erasures, native class/occurrence frame bindings,
+93 physical frame gaps, six full-map fixtures and six rejecting controls.
+The tests also execute all 104 source columns with three dirty helper values and
+exercise the real optimized CLI with Git unavailable, altered/missing source
+bytes and altered certificate data. No predecessor executable is imported.
+
+`CERTIFICATE.json` contains the complete `A,C` matrices, basis minor, residual
+matrix digest, omitted outgoing event IDs, all omitted frame gaps and exact
+counts. Default verification compares every receipt field. `--out NEW_PATH`
+writes a fresh receipt and refuses overwrite. It does not regenerate frozen
+inputs. These are exact structural counts, not runtime measurements.
+
+## Source and overlap
+
+Canonical main is `6f100d4bb5083f8c35d96fd95a539811c62f245c`, priority revision 6.
+The input bytes already on main are PR2586's word and compressed native archive,
+retained by the rounded-collector package. `SOURCES.json` pins their hashes,
+blobs and the original card. PR3446 at
+`886222c1d3e2c2d878b263dc8d165ef03d5bac7f` supplies the projection definition,
+its published event-list hash and the released omitted-effects question.
+Runtime reconstructs that projection directly from the two canonical inputs;
+it needs neither PR3446 files nor an unmerged commit in the object store.
+
+The advance claim is [in the existing P104 investigation](https://github.com/CrocSwap/integer-mult-bounds/discussions/2750#discussioncomment-19009903).
+Entry overlap covered 368 open PRs and 2,285 Discussions with complete pagination.
+Targeted omitted-effects/residual/PR3446 search found the credited predecessor
+and its handoff. Unpublished work, every historical diff and omitted sparse
+local notes are outside that coverage. No match is a novelty guarantee.
+Submission refresh covered 374 open PRs and 2,299 Discussions with complete
+pagination and no warnings; the relevant matches were this claim and PR3446.
+Canonical main, guidance and the PR3446 head were unchanged. Concurrent
+correlated-entrance work changes a different obligation.
+
+## Open next questions
+
+- Can an actual physical caller realize `A,C` with legal frame access and
+  restore the one borrowed dirty helper? Start by binding the certificate's
+  93 target frame gaps and retained dimension-eight Y outputs to exact native
+  adapter recipes. Adapter availability, source restoration and payment block
+  a physical completion; a normalized zero row does not remove its adapter.
+- Can a complete changed profile pay for those adapters and the summary circuit?
+  First specify a full invoice, including scalar expansion, physical owners,
+  source/target movement, idle and cleanup costs, precision and recursive calls.
+  Unknowns are not zero. Stop before further construction if a fully charged
+  favorable screen fails. This package does not license an unchanged-profile
+  optimization campaign.
+
+## Credit
+
+Nick Petalas. Apache-2.0. Hiroki Kumazaki/PR3446 supplies the recipient-history
+closure and omitted-effects handoff; Clear1nsight/PR3423 and PR3382 supply the
+open-port/first-collector predecessors. ywz65/PR2586 and PR2497 supply the
+physical dual and logical word. Arham Tahir's rounded-collector package retains
+the canonical source bytes. Retain the contributions of Tekalign/PR3025,
+Ross Richardson/PR3068, Jose Manuel, Vasek, Philippe Van Velzen, Chafik Boukhalfa,
+Rohan Arun, Adam Gall, Josie Elliston, Ali Romman, icekylinx and the credited
+geometry/native/compiler lineage. PR3406, PR3416, PR3426 and PR3448 retain their
+separate source-domain, native-covariance and tensor-domain credit. No frozen
+predecessor bytes or notices were changed.

research/p104-omitted-completion/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "6f100d4bb5083f8c35d96fd95a539811c62f245c",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    },
+    {
+      "path": "research/p104-rounded-collector/inputs/NATIVE.json.gz",
+      "usage": "input",
+      "git_blob": "04f2877aa5444948b7ae37efd267745af82cb34d",
+      "sha256": "ab91c78835c3e10cf3b2cb8a5d4e0b7cfc3df1ae5a811298d27842f2193e1e3d"
+    },
+    {
+      "path": "research/p104-rounded-collector/inputs/WORD.json",
+      "usage": "input",
+      "git_blob": "93acaabc0a523789ef2f1b06f6eeeefed5551397",
+      "sha256": "cec1fb3fbd0067b500ad5119a62ba60685be718e769374d7ec8abdd2beaa7e50"
+    }
+  ]
+}