research: bind joint-child batching to the staged544 native rank interface

research/staged544-joint-child-rank-interface/PROOF.md

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+# A source-bound rank interface discriminator
+
+Chris R / Clear1nsight with substantial OpenAI Codex assistance. Apache-2.0.
+
+## Actual group and the proposed transfer
+
+The two PR3945 providers at `91f81d131252db704f945891da1ea54e29d46b8b`
+contain the same four frame climbs on helpers600,601,602,604 at zero-based
+B936,B937,B938,B941. Each entrance is span(45), each destination is
+span(409,80,40,5,2), and the charged dimension increase is4.
+The source window includes native maps on564 and405 at936 and on565 at939.
+They remain at their literal positions. The checker traces the complete A
+prefix and the B prefix through941, including every intervening native frame
+boundary. It also verifies that the future helper inputs are untouched in
+this window. It does not emit a retimed provider or claim a full dirty replay.
+The intervening cut is checked to write only target families; its centre
+descents are temporary readouts under the retained BORROW-WORD contract and
+do not change the persistent helper frames used by this trace.
+
+Gonzalo Mier's PR4031 at `87899f26ccceb36d32985eab59dbeb467d90825b`
+checks an old selected gauge44 rank4 grouping and the valid matrix identity
+diag(A1,...,A4)diag(B1,...,B4)=diag(A1B1,...,A4B4) for4-by4 matrices.
+It explicitly waives physical layout, native interfaces, hosting and dirty
+restoration. Its optimistic profile is a screen, not a physical supplier.
+We test the additional implication needed to transfer that rank16 invoice
+to the actual PR3945 group. This is not a rerun of its80-call census.
+
+## The inherited rank parameter counts Fourier axes
+
+The retained exact compiler `bw_adapters.py:S_of,wrapped_ok` identifies a
+rank-r residual with L (C_[r] tensor I) R, for monomial L and R. Here
+C_[r]=C tensor ... tensor C on r binary axes, where
+C=((1+i)I+(1-i)X)/2. `clifford_exact.py:fourier,lower_child` implements this
+precise convention. A rank4 child therefore has16 nonzero entries per row
+and per column; it is not a4-by4 matrix-multiplication oracle.
+
+The actual residual S=T_U T_E^-1 for the named source frames acts on512
+complex coordinates. Using the credited PR3987 scalar Gaussian arithmetic,
+the checker evaluates all512 basis columns exactly. Each row and column has
+16 nonzero entries, all with squared magnitude1/16. This fresh local check
+does not import or execute the original compiler. Its arithmetic engine is
+reused and credited; the proof below is independent of an observed hash.
+
+## Monomial and arbitrary-dirty padding obstruction
+
+Let A be the direct sum of the four actual residuals. It acts on2048
+coordinates and every row has support16. The desired all-input contract,
+with any additional borrowed coordinates restored, is A direct-sum I_D.
+It has row-support multiset {16 repeated2048 times,1 repeated D times}.
+An arbitrary invertible monomial matrix permutes coordinates and multiplies
+them by nonzero scalars. Left and right multiplication by such matrices
+preserves the multiset of row supports. This includes source native L/R,
+arbitrary monomial routing and arbitrary placement of the dirty auxiliaries.
+It does not matter whether those monomials are free or charged.
+
+A single inherited rank16 child C_[16] tensor I has65536 nonzero entries in
+every active row. Even after enough dirty padding to match dimensions, its
+row-support multiset cannot equal that of A direct-sum I_D. Thus no one-call
+rank16 implementation of this group exists using only monomial pre/post
+maps and auxiliary coordinates returned identically for every initial value.
+The same argument excludes any rank r>4 in this one-call contract, including
+rank12. Identity spectator coordinates within a child are different from
+an identity direct-sum bank: C_[16] tensor I still has support65536 per row.
+
+The dimension obstruction already appears at the literal base instance:
+four512-coordinate banks give2048 coordinates, whereas a rank16 child needs
+at least65536. Adding63488 arbitrary dirty coordinates resolves only that
+dimension mismatch; the support mismatch remains. Clean initialized
+coordinates, projections, nonmonomial encoders, multiple child queries or
+changed output contracts are outside this proof and need their own invoices.
+
+For a tensor-power lift f>=1 of the source convention, each bank has2^(9f)
+coordinates and support2^(4f), while a purported rank16 child has support
+2^(16f). Four banks add two selector bits, not twelve Fourier axes. Here f
+means the number of tensor copies of each source axis, not linear spectator
+multiplicity used in the separate finite-lattice implementation PR4017.
+The executable verifies the source base instance f=1; the all-f statement
+follows from tensor-product support multiplication, not sampled larger f.
+
+There is a valid free algebraic batching: I_4 tensor C_[4] tensor I_32,
+with the inherited block-diagonal monomial wrappers. It processes four
+banks with two spectator selector axes. Its active Fourier rank remains4.
+Consequently rank16 pricing does not follow from that identity.
+
+## Exact optimistic allowance and the complete obligation ledger
+
+At the pinned PR3945 tariff endpoint kappa=1730663236534549309146/10^24,
+write H_r=r^(1-kappa). Replacing four H4 by one H16 would save
+
+    0.038249447579079975271991 < 4 H4-H16
+      < 0.038249447579079975271992.
+
+The executable encloses log2 by an atanh series with a rational remainder
+and exp(-x) by odd/even Taylor truncations, using exact fractions throughout.
+This is an allowance before every additional interface cost. With every
+other bill and inventory fixed, one added positive-rank recursive child costs
+at least H1=1, exceeding the allowance. This conditional budget does not
+exclude compensating savings elsewhere, changed hosting, or a new model.
+
+| Obligation | Status in this result |
+|---|---|
+| Four source frame transitions and literal native ordering | Freshly checked on both immutable providers |
+| Exact local native all-input maps |512 basis columns checked; source arithmetic credited |
+| One rank16 child with monomial wrappers and arbitrary-dirty identity return |Proved impossible under the stated contract |
+| Valid spectator batching |Algebraic identity; active rank remains4 |
+| Optimistic4H4-H16 allowance |Exact rational enclosure, before extra charges |
+| Physical gathering/routing and larger hosting |Unknown; not zero |
+| Changed nonlinear/linear nonmonomial encoders and dirty compensation |Unknown; excluded from the obstruction |
+| Supplier stock, precision, tape, complete owner realization and root |Inherited/unrun; no changed provider or kappa claimed |
+
+## Open next questions
+
+The useful changed premise is a genuine multi-query or nonmonomial encoding
+that preserves the four arbitrary inputs, permits their actual consumers,
+and restores every borrowed coordinate. Its first test should exhibit one
+explicit map outside the row-support argument and explain how its complete
+cost fits the0.03825 allowance or which separately credited cost disappears.
+A larger zero-charge grouping or a matrix direct-sum identity alone cannot
+answer that question. This route stops here; no larger census is justified.

research/staged544-joint-child-rank-interface/README.md

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+# Joint-child rank interface on the staged544 source
+
+**Scoped obstruction and exact allowance; no new kappa.**
+
+Four actual PR3945 rank4 frame climbs share their entrance and destination.
+Their direct sum has16 nonzero entries per row. A single inherited rank16
+child has65536, so arbitrary monomial gathering/wrappers and identically
+returned arbitrary-dirty padding cannot turn one into the other. Rank counts
+Fourier axes in this supplier; ordinary block-matrix multiplication uses a
+different interface. Valid selector batching retains active rank4.
+
+This resolves one missing transfer boundary from Gonzalo Mier's explicitly
+zero-charge PR4031 screen, without disputing its matrix identity or claiming
+a general batching obstruction. At the pinned PR3945 endpoint the hypothetical
+4H4-to-H16 saving is only0.038249447579079975271991–992 H1 units, before
+all omitted costs. [PROOF.md](PROOF.md) states the contract, proof, cost-status
+ledger and open next questions. [RESULT.json](RESULT.json) retains the source
+group, native events, exact support and rational allowance.
+
+Run the read-only registered normal/optimized replay:
+
+```sh
+python3 scripts/research_package.py --package research/staged544-joint-child-rank-interface
+```
+
+Direct `python3 -B research/staged544-joint-child-rank-interface/verify.py`
+also works without Git. `--record` intentionally regenerates RESULT.json;
+ordinary verification checks its equality without writes. Frozen input hashes
+are authenticated before the scientific checks. Five positive/negative
+mathematical controls distinguish spectator identity from active Fourier
+axes and reject the rank12/rank16 padded substitutions. This is not a full
+provider dirty-state or owner replay.
+
+## Open next questions
+
+Can a nonmonomial or multi-query encoding preserve all four actual inputs,
+their consumers and arbitrary borrowed state within the complete allowance?
+First exhibit the changed map and price each new operation and any offsetting
+credit. Physical encoding and restoration remain unresolved. A larger
+zero-charge census does not remove the support obstruction in this contract.
+
+Credit: Gonzalo Mier/PR4031 provides the motivating rank4 retimed screen;
+Dugongue/PR3945 provides both staged544 words; PR3969 binds their owner-cost
+interpretation; PR3987 supplies the retained exact Gaussian phase arithmetic
+and native compiler snapshots. The native code credits DreamingOfClouds,
+chafreaky/PR458, PR2226 and its retained source lineage. Preserve the source
+notices in frozen files. New proof/checker: Chris R / Clear1nsight with
+substantial OpenAI Codex assistance. Apache-2.0. No independent human or
+proof-assistant verification is claimed.