Expose 222 early BIT helper queries with exact port routes

research/bit-early-helper-queries/PROOF.md

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+# Forward BIT helper queries at the source entrance
+
+This file gives a source interface. It does not give a new kappa value.
+
+Use the retained PR589/PR555 word at repository commit
+`1ec1119972c9aefe42525172d07b332fcff81cb4`. The source is
+[WORD.json.gz](https://github.com/CrocSwap/integer-mult-bounds/blob/1ec1119972c9aefe42525172d07b332fcff81cb4/research/kappa-rebuild/bit/sources/pr589-regenerated/WORD.json.gz).
+Its compressed SHA256 is
+`e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f`.
+The source has 111 X fields, 111 Y fields, and 439 independent helper fields.
+Its nine owned center queries remain paid.
+
+The rescheduling uses the entrance prologue from
+[PR5388](https://github.com/CrocSwap/integer-mult-bounds/pull/5388), with its
+[proof at commit 568717635d8d11175a80024c9c596e16f54b1ea4](https://github.com/CrocSwap/integer-mult-bounds/blob/568717635d8d11175a80024c9c596e16f54b1ea4/research/native-helper-query-bank/PROOF.md).
+This BIT source needs no scalar first-departure basis change or helper
+completion child.
+
+## Exact source cuts
+
+Every helper starts in frame zero. Before its first MOVE, it is never the
+destination of an ADD. Earlier source instructions can read it. Thus every
+first-departure row is exactly its original helper input. The scalar matrix
+of all 439 such rows is the identity.
+
+Of these first MOVEs, 286 have rank one. Each of the 111 port lines has between
+two and five of them. The receipt gives all owners, event indices, frames,
+and prior read counts. All helpers finish at the same full rank-eight frame.
+
+Let P be the projector on one port line. Write Q=D_P, D=D_(I-P), and R=D_I.
+The two earliest first-query helpers on that line give two independent Q
+observations. Their final job is R=DQ. Select these two helpers at every port.
+This gives 222 local observations with no input, output, or observation
+normalizer. All other helpers keep their original jobs. A later global stage
+requires the separate port-route extension below.
+
+## Optional common-axis maps
+
+The larger 286-query interface can use one common rank-one axis. For a source
+root r, put u=r/2, so u has squared norm one. Let e be the first coordinate
+vector. If u=e, set K=I. Otherwise set v=u-e and
+
+    K = I - 2 v v^T / (v^T v).
+
+Then K is symmetric, K squared is I, and K e=u. For the head and tail address
+coordinates, use the block map diag(K,K). This is in coordinates where the
+fixed tail reversal has already been removed. Conjugation by that fixed
+reversal gives the native convention.
+
+For a projector A, the address matrix in these coordinates is
+
+    D_A = [[I-A, A], [A, I-A]].
+
+Direct rational matrix products give
+
+    D_(u u^T) diag(K,K) = diag(K,K) D_(e e^T),
+    D_I diag(K,K) = diag(K,K) D_I.
+
+Thus input map K, observation map K, and output map K give the common Q
+observation and preserve the full R target. The checker verifies these exact
+16-by-16 identities for all 111 source lines. It also compares every line with
+the actual source first-frame basis. The maps are fixed address maps in the
+retained common-cover contract. This check does not claim direct execution of
+a dense address map on a smaller cover.
+
+The direct two-per-port interface does not need these maps.
+
+## Matching the actual global stage chart
+
+The local identity R=Q D_(I-P) does not identify R with the fresh-stage helper
+job Q D_B automatically. The retained global controller moves main data through
+port-specific routes. A helper must also have an explicit route.
+
+Use the actual 15-by-15 matrix rho from
+[BIT-P3-NATIVE-BINDING.json](https://github.com/CrocSwap/integer-mult-bounds/blob/1ec1119972c9aefe42525172d07b332fcff81cb4/research/kappa-rebuild/bit/BIT-P3-NATIVE-BINDING.json).
+For each port, its old local projector L has rank eight and the fresh plane B
+has rank seven. The source matrix satisfies
+
+    rho^2 = I,
+    rho P rho = P,
+    rho L rho = P+B.
+
+Give a selected helper the input rho xi. Apply rho to its held observation
+and to its final output. The resulting observation is Q xi, since rho fixes
+P. The output is D_(P+B) xi = Q D_B xi. Thus it has the same Q and fresh D as
+the main controller port. The source accepts these separately routed inputs
+because its helper-to-main block is zero on all source columns.
+
+The checker loads and hashes the actual binding. It checks all 111 matrices,
+the three head identities, and the corresponding 30-by-30 native address
+identities. The same matrices embed at every stage by placing B on that
+stage's seven axes and fixing all other axes. This uses the retained fixed
+right-route contract on one common future cover. No smaller cover is assumed.
+
+For 222 selected helpers, a fresh-stage call adds 666 fixed cover-route maps:
+one at each input, observation, and output. It adds no native child. The
+222-cut source replay and all helper target ranks remain unchanged. The local
+stage can omit these routes. A complete caller must pay ordinary movement for
+the added maps.
+
+## Rescheduling the existing child
+
+For each selected helper, copy its entrance value into a fresh owned raw field.
+Run its actual first rank-one MOVE on that raw field. Copy the result into a
+fresh owned observation field. Apply the observation map only in the optional
+common-axis variant. Keep the raw field for the source cut.
+
+Then run the original source. Omit only the selected first MOVEs. At each
+omitted event, the original helper still has its unchanged entrance value.
+The raw field has exactly the value and frame that the child would produce.
+Swap these whole fields and their frame tags. Clear the owned raw field. The
+persistent source state is now exactly the original state after that child.
+The source can continue with every later instruction unchanged.
+
+Induction over the replacement cuts proves the whole source identity. In
+particular, all reads in the initial Y prefix receive the correct original
+helper values. No dirty persistent field is cleared. The old value left in
+an owned raw field is a duplicate. The transformed persistent field retains
+all its input information. Clear each owned observation after its last use.
+
+For 222 selected queries, the extra work is 444 owned COPYs, 888 owned CLEARs,
+222 whole-field SWAPs, and at most 444 extra owned fields. There is no new
+paid child. The 286-query common-axis version uses 572 COPYs, 1144 CLEARs,
+286 SWAPs, at most 572 owned fields, and 843 nonidentity head chart maps.
+Ordinary field movement and storage still need a complete caller tariff.
+
+## Forward source for both shear directions
+
+At each port the original main entrance frames are P and zero. The exit
+frames are I and I-P. Its scalar main map is (x,y) to (x,y+x). Therefore its
+physical main block is
+
+    F = [[D, 0], [DQ, D]].
+
+Let S swap the two external main fields. Then
+
+    S F S = [[D, DQ], [0, D]].
+
+Since Q squared is I for BIT, this is the opposite shear direction with the
+same Q. The swaps act only on the main fields. The helper prologue and every
+helper output remain unchanged. This uses the forward physical source twice
+with different main assignments. It does not use a reverse first-departure
+claim.
+
+The reverse-source census is different: it has 253 first rank-one helper
+MOVEs, but every one of its 439 first-departure rows contains a main input.
+The forward raw-helper argument does not establish a reverse helper bank.
+
+## Changes to helper arguments before the source
+
+At the end of the prologue, let a helper pair hold normalized inputs xi_h and
+xi_k, observations Q xi_h and Q xi_k, and raw fields with the corresponding
+native frames. To make the invertible change
+
+    xi_k += Q xi_h,
+
+update its stored input, its observation, and its raw field. The observation
+update is
+
+    Q xi_k += xi_h,
+
+because Q squared is I. For two helpers on the same port and stage, the raw-field update adds the
+stored input rho xi_h. No new native child is required. In the common-axis variant the raw update uses K_k xi_h.
+All source inputs must be fixed before the original source begins.
+
+The checker verifies this update on all 439 arbitrary helper columns for 111
+disjoint pairs. It also runs the complete source and all 222 replacement cuts
+on those changed inputs, keeping both I and Q coefficient columns. The source
+then returns R times the changed helper vector.
+This is a coupled output block. It is not the original independent helper
+job list until a complete controller proves the required decoding. No helper
+target may be credited twice or removed from the invoice.
+
+## Checks and limits
+
+Run from the repository root:
+
+    python3 -B research/bit-early-helper-queries/verify.py
+    python3 -O -B research/bit-early-helper-queries/verify.py
+
+Both runs give `PASS_BIT_EARLY_HELPER_QUERIES` and the same exact report. They check all 661 persistent input columns, all owned input columns,
+the complete source frames, all actual global port routes, both the 222- and
+286-cut variants, the nine
+original COPY lifetimes, and the full original paid profile:
+
+    1323 H1 + 1342 H2 + 79 H3 + 44 H4 + 102 H5 + 11 H6 + 19 H7.
+
+The helper target profile stays 439 H8 in the interface without feedback.
+The verifier rejects changed bytes in each of the four declared source inputs.
+It also rejects a wrong held input, a wrong frame-tag swap, a dirty owned input,
+an uncleared raw field, and an uncleared observation.
+
+The exact rational address identities extend to each column multiplicity by
+using K tensor I_f and P tensor I_f. The scalar owner program is unchanged.
+Its sums are not tensor powered. The all-column source identity permits
+arbitrary operator-valued helper inputs, so the stated pre-source feedback is
+a valid substitution when every stored copy is updated.
+
+No profitable caller, outer multiplication assembly, precision allocation,
+full GL cover materialization, or native payload execution was run here. This
+interface alone does not improve kappa.
+
+The retained source and its
+[attribution](https://github.com/CrocSwap/integer-mult-bounds/blob/1ec1119972c9aefe42525172d07b332fcff81cb4/research/kappa-rebuild/bit/README.md)
+remain unchanged. The source credits eumemic, Anthropic Claude assistance, and
+its cited predecessors. This BIT adaptation and checker have substantial
+OpenAI Codex assistance.

research/bit-early-helper-queries/README.md

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+# Early helper queries for the retained forward BIT source
+
+This package proves a finite source interface. It exposes two original helper
+queries at each of 111 ports before the retained forward word starts. It adds
+no native child and binds the responses and outputs to all 111 actual port
+routes. The complete kappa calculation does not improve: no profitable caller
+or full ordinary-work tariff is supplied.
+
+Read [the proof](PROOF.md) and [the exact result](RESULT.json). The source pins
+are in [SOURCES.json](SOURCES.json). The publication base is
+`32890bc5de034ab8fc93ac88ead7279a6a416d04`. The source was studied at
+`1ec1119972c9aefe42525172d07b332fcff81cb4`; all four live inputs have the same
+bytes at both commits. The selected record in the package metadata is only
+historical provenance. No selected record or frozen predecessor is changed.
+
+## What the interface gives
+
+All 439 helper first-query rows equal their independent entrance inputs.
+Of these first queries, 286 have rank one. Each port has at least two. Select
+the two earliest at every port to obtain the direct 222-query interface.
+
+For each selected helper, keep an owned copy of the raw first query and an
+owned observation. Run the original source. At its original first-query cut,
+swap the whole helper and raw fields, including their frame tags. Clear the
+owned duplicate. Every original source column and subsequent scalar event
+then has the original value and frame.
+
+The actual port route rho fixes the port projector P and sends the local
+rank-eight projector to P plus the fresh seven-plane B. Input, observation,
+and output routes therefore give the response Q xi and final job Q D_B xi.
+The proof checks the 15-by-15 head matrices and their 30-by-30 native address
+matrices. It uses the retained fixed common-cover route contract.
+
+| Interface | Early queries | Owned COPY | Owned CLEAR | Whole-field SWAP | Extra owned fields | Additional maps |
+| --- | ---: | ---: | ---: | ---: | ---: | --- |
+| Direct local ports | 222 | 444 | 888 | 222 | 444 | None |
+| Direct ports at a fresh global stage | 222 | 444 | 888 | 222 | 444 | 666 fixed cover routes |
+| Optional common-axis local variant | 286 | 572 | 1,144 | 286 | 572 | 843 nonidentity head chart maps |
+
+These are different variants. The 286-query counts must not replace the
+222-query counts. The optional common-axis row is a local interface; its 843
+maps do not pay the 222-query fresh-stage routes.
+
+The unchanged paid profile is
+
+    1323 H1 + 1342 H2 + 79 H3 + 44 H4 + 102 H5 + 11 H6 + 19 H7.
+
+Without feedback, all 439 helper targets remain H8. The package also checks
+111 disjoint pre-source feedback pairs. In BIT, Q squared is I, so the held
+input, response, and raw field can all receive the exact paired update. The
+source then returns a coupled helper block. A caller must decode that full
+block before it can claim the original independent target list.
+
+The reverse source has main-input contamination in all 439 first helper rows.
+This package does not expose a reverse cold-helper bank. Swapping the two main
+roles gives the opposite main shear with the same forward source.
+
+## Reproduce
+
+Use Python 3.12. The verifier uses only the standard library.
+
+```sh
+python3 -B research/bit-early-helper-queries/verify.py
+python3 -O -B research/bit-early-helper-queries/verify.py
+python3 -B scripts/research_package.py --package research/bit-early-helper-queries --check-only
+python3 -B scripts/research_package.py --package research/bit-early-helper-queries
+```
+
+Expected verifier output:
+
+```text
+PASS_BIT_EARLY_HELPER_QUERIES queries=222 routes=111 source_controls=4 owned_controls=5 new_kappa=false
+```
+
+The complete replay checks 661 persistent input columns and 445 owned input
+columns for the direct interface. It also checks the 286-cut variant, all
+nine original COPY lifetimes, all 111 native routes, and the changed helper
+arguments. Normal and optimized runs must give the same exact result.
+
+The verifier rejects a byte mutation in each declared source, a wrong held
+input, a wrong frame-tag swap, a dirty owned input, an uncleared raw field,
+and an uncleared observation. It does not rerun a discovery search.
+
+The direct verifier also works in the committed package export without Git.
+The registered metadata checks use Git to resolve provenance. No native
+payload, full GL cover, uniform compiler, ordinary precision or outer
+multiplication assembly is executed by this package.
+
+## Open next questions
+
+[TASK.json](TASK.json) records the next questions. First give a literal caller
+that uses the observations and closes every coupled helper output. This is
+needed before any complete paid-versus-target claim. Then bind its common
+cover and charge all route, movement, storage, and precision costs. A
+profitable complete caller is the present blocker.
+
+## Credit and related work
+
+The retained forward word comes from
+[PR #589](https://github.com/CrocSwap/integer-mult-bounds/pull/589) and
+[PR #555](https://github.com/CrocSwap/integer-mult-bounds/pull/555). Their
+[pinned source notice](https://github.com/CrocSwap/integer-mult-bounds/blob/32890bc5de034ab8fc93ac88ead7279a6a416d04/research/e8-bit-p111-tuned/NOTICE.md)
+credits eumemic with substantial Anthropic Claude assistance and preserves
+its earlier contributors. The
+[tuned-source credit](https://github.com/CrocSwap/integer-mult-bounds/blob/32890bc5de034ab8fc93ac88ead7279a6a416d04/research/e8-bit-p111-tuned/tune/README.md)
+records the transport changes. The
+[retained BIT reconstruction](https://github.com/CrocSwap/integer-mult-bounds/blob/32890bc5de034ab8fc93ac88ead7279a6a416d04/research/kappa-rebuild/bit/README.md)
+supplies the regenerated word and actual route binding used here.
+
+The entrance prologue follows
+[PR #5388](https://github.com/CrocSwap/integer-mult-bounds/pull/5388) and its
+[pinned proof](https://github.com/CrocSwap/integer-mult-bounds/blob/568717635d8d11175a80024c9c596e16f54b1ea4/research/native-helper-query-bank/PROOF.md).
+That proof is conceptual credit, not an executable dependency.
+[PR #5421](https://github.com/CrocSwap/integer-mult-bounds/pull/5421) instead
+uses nine owned BIT relations in the reverse word. This package preserves
+arbitrary forward helper inputs and does not delete their target credits.
+
+This BIT adaptation, route proof, checker, and package have substantial
+OpenAI Codex assistance. New files are Apache-2.0. No predecessor file is
+copied or changed by this package.