Selected BIT: generate a mixed good/fallback child-weight descriptor

research/selected-bit-mixed-descriptor/PROOF.md

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+# One mixed good/fallback weight descriptor
+
+The contribution is an explicit descriptor and its executable continuation,
+not a new normal-form or fallback theorem. Diagonal covariance is retained
+from source527 and Ross Richardson's PR #2159. Jose Manuel's PR #2171
+implements good-only ancestral weights and asks for a mixed path. This note
+gives the row and atom indices for its diagonal-pair subcase. The selected
+good-child binding is from commdata2338's PR #2160. Exact heads and proof
+hashes are in [SOURCES.json](SOURCES.json).
+
+## 1. Conditional interface
+
+Let `R = Z/q^w`, with q an odd prime and w >= 1. An ADDRESS residual on
+`N = 2m` coarse fields acts independently on f atom columns. Its full
+unweighted matrix M is invertible over R, with determinant +1 or -1.
+At column j its weighted matrix is
+
+```
+M_j = B_j M B_j^-1,
+B_j = diag(b_0(j), ..., b_(N-1)(j)),
+b_i(j) = A[i*f+j] if i < m, and 1 otherwise.
+```
+
+Every A entry is a unit. M and A depend only on exposed parent/ancestor
+fields and fixed labels, never active ADDRESS values, VALUE payloads, or
+fresh descendant stock. These are explicit premises. The native selected
+to stopped-GL domain bridge is not proved here.
+
+For n incoming head atoms set `f=floor(n/m)>0`, use the first mf head
+atoms and the corresponding reversed tail block, and retain the peeled
+remainder as spectators under the inherited peeling contract.
+
+The first field of a diagonal pair has index i < N-1; its partner is N-1.
+Physical atom t represents logical column `sigma_i(t)=t` or `f-1-t`.
+The selected layout uses the first orientation for heads and the second
+for tails. This map matters even when both fields are reversed.
+
+## 2. The row-permutation quotient
+
+Run unnormalized unit-pivot Gaussian elimination on M, clearing all other
+rows at each stage and retaining zero shear slots. Choose the first unit
+in the current pivot column. Let D be the final diagonal, with entries
+d_i. Track original row labels through swaps: `pi(i)` is the original row
+occupying final row i, not the list of pivot choices.
+
+For one column abbreviate B_j to B. During elimination the weighted and
+unweighted matrices satisfy
+
+```
+T_weighted = L T_unweighted B^-1,
+```
+
+where L is diagonal and initially equals B. A row swap S changes L to
+`S L S^-1`. A shear clearing row i using pivot row k has weighted
+coefficient `(L_i/L_k)` times the unweighted coefficient: the right
+diagonal entry cancels from the quotient of entries in the same column.
+This shear preserves the displayed relation.
+
+Unit scaling preserves the low-residue zero pattern, so both runs choose
+the same first unit row at every step. At termination `L_i=b_pi(i)(j)`:
+
+```
+d_i(j) = d_i * b_pi(i)(j) / b_i(j).                          (1)
+```
+
+All divisors are units. The row itinerary is determined by M modulo q,
+but d_i must be computed over the full ring. Replacing M by its low
+representatives can change d_i with no change of itinerary. Incoming A
+values likewise need their full ancestors.
+
+The physical descriptor for the weighted reversed child is
+
+```
+a_child[t] = d_i * b_pi(i)(sigma_i(t)) / b_i(sigma_i(t)).      (2)
+```
+
+`FallbackPair.value` implements (2). Its possible incoming reads are
+`pi(i)*f+sigma_i(t)` and `i*f+sigma_i(t)`; a tail-bank read becomes the
+constant 1. Both head indices lie in the parent's first mf atoms. The
+fallback generally has no same-flat-index product rule.
+
+## 3. Composition with good ancestors
+
+For the retained northeast good normal form whose active head coordinates
+are `0,...,r-1`, source527 and PR #2160 give
+
+```
+A_good[k*f+j] = A_parent[k*f+j] * delta_k.                    (3)
+```
+
+Delta is the unweighted full-ring pivot vector. `Good.value` implements
+(3); no different `i_k` map or unbound local-frame label is assumed.
+Starting with a unit root weight, (2) and (3) inductively give only units.
+Every node reads older permitted controls and at most two entries of the
+previous descriptor. A requested weight can therefore be generated
+without a supplied ancestor-weight oracle or access to unused child stock.
+This also covers a good node after a fallback node: its incoming descriptor
+is the fallback's, not the original root array.
+
+This closure statement covers the named good normal forms and fallback
+diagonal-pair children. It does not compile all fallback shear coefficients,
+arbitrary role labels, or complete selected invocations.
+
+## 4. Source-bound rank-42 example
+
+The pinned selected template has idle child 6, rank 42, from level 2 to
+level 8 on pair 0. Copy 213 hosts no part of it. Its retained projector is
+`P0=diag(0_22,I_42)`; the selected receipt contains 288 rank-42 children.
+The verifier authenticates the template, its decoded hash and selected-word
+binding, and the PR1094 assembly's Git-blob citation of PR589's receipt.
+
+For the GL-domain examples, S sends P0's last 42 axes to the first 42.
+Each chart is explicitly `g=h S`. This is a chart in the assumed GL orbit,
+not an uncharged physical route or an emitted orthogonal-cover invocation.
+
+Modulo 25, a good ancestor has disjoint blocks `[[1,1],[c,1+c]]` on axes
+`(0,43),(1,42),(25,44)`, with c=5,10,15. All determinants are one. Exact
+northeast elimination gives `delta_0=6`, `delta_1=11`, `delta_25=16`, all
+others one. At root column width 5 its child has 210 atoms and
+`A[x]=delta_floor(x/5)`.
+
+The next parent uses f=3 and peels 18 atoms. Its chart has blocks
+`[[5,4],[1,1]]` on `(0,43)` and `[[1,0],[1,1]]` on `(1,42)`, both of
+determinant one. Its projector has `P[0,0]=5`, a nonzero nonunit, so the
+generic good branch is unavailable. Its full 128-coordinate residual is
+
+```
+M = [[I-P, P C_64], [C_64 P, I-C_64 P C_64]].
+```
+
+This is an involution of determinant one. Its exact fallback plan gives
+`pi(1)=42, d_1=-1` and `pi(126)=1, d_126=1`.
+For the first pair, (2) reads two different ancestor locations:
+
+```
+numerator:   A[126], A[127], A[128] = 16,16,16
+denominator: A[3],   A[4],   A[5]   =  6, 6,11
+child:       -16/6,  -16/6,  -16/11 = 14,14,19 mod 25.
+```
+
+Field 126 has denominator one and reversed first-field orientation, so
+its descriptor is `A[5],A[4],A[3] = 11,6,6`. Omitting orientation gives
+`6,6,11`; using only the good ancestor's low residues gives `1,1,1`.
+
+A further fixture starts with 6,400 atoms. Its good child has 4,200; the
+bad parent uses f=65 and peels 40; the diagonal-pair child has 65. The next
+good parent uses f=1, peels one atom and produces 42 weights. Entries
+`0,1,29,30,41` are `16,21,11,6,6`. A separate forward-elimination oracle
+computes all 65 incoming fallback weights before the final comparison.
+This exercises a good/fallback/good descriptor chain, with callees still
+conditional.
+
+## 5. Chronology, reversals and charges
+
+The completed weighted reversed-child premise is
+
+```
+F_a(H,D) = (diag(a) C_f D, C_f diag(a)^-1 H),
+```
+
+preserving exposed controls and complete spectators and restoring its row.
+A logical diagonal pair is `J_a J_1`: execute J_1 first, then J_a.
+For equal field orientations, each ordinary swap is `R_tail F_a R_tail`.
+Each reversal is one unweighted reversed child on the first/last
+`floor(f/2)` atoms, with a possible middle atom fixed. The tested tail126
+pair at f=3 thus pays six children of widths `1,3,1,1,3,1`. For opposite
+orientations each swap is already a reversed child: field1's pair uses two
+width-three children. In both cases the physical weights are (2).
+
+No reversal or variable scaling is free. The inherited fallback allowance
+remains
+
+```
+6N(N-1) + 3N + 6(N-1) = 98,682 < 32*64^2 = 131,072,
+```
+
+with every child at most f atoms. This is the prior conservative allowance,
+not a selected cost improvement. The preceding good call still pays its
+one 42f child and adapters. Later-target shifts, the fixed determinant
+sign, preparation and required routing remain charged.
+
+For fixed q and m=64, an unweighted plan needs a fixed number of ring operations
+on O(w)-digit values and is prepared from the full exposed chart once.
+A good descriptor follows one parent index; (2) follows at most two. With
+D ancestors a direct depth-first evaluator uses at most
+`O(2^D poly(w,log n_root,D))` control work per value, and
+`O(D(w+log n_root))` storage for fixed-dimensional prepared data and its
+stack. The executable realizes this finite recursion; no memoization or
+unit-cost arbitrary-prefix oracle is assumed.
+
+As in source527 and PR2171, impose `n_root<=E0`,
+`D<=c(1+log2 E0)`, and `w=ceil(E0^s)` for fixed `0<s<1`. Then
+`2^D<=C E0^c` and `E0<=w^(1/s)`, making preparation polynomial in w.
+It must remain in the retained ordered-affine tape estimate:
+
+```
+T_adapters <= N_slots V (C + 2 E_max / q^w).
+```
+
+Only under that stopping relation does the complete fiber absorb the
+polynomial preparation. For unrelated w and D retain E_max explicitly.
+This supplies a descriptor to the interface, not a new execution of its
+physical tapes or proof of the weighted callee.
+
+## 6. Evidence and limits
+
+The replay checks 46,848 cases: all 1,344 determinant +/-1 matrices of
+size two modulo 3 and 9, each under every unit diagonal conjugator. The
+oracle uses normalized forward elimination; the compiler uses unnormalized
+all-row elimination. Four selected-size fixtures check all 127 diagonal
+pairs in both orientations over 11 weighted columns, with full operator
+reconstruction. The additional mixed chain checks 65 fallback and 42 good
+weights. The same-orientation test covers 5,625 complete coordinate-pair
+and stock cases; the opposite-orientation test covers six basis columns.
+Fifteen semantic controls reject.
+
+Small primes are arithmetic fixtures, not admitted selected primes. This
+is a written conditional proof with author-run exact evidence, not formal
+kernel verification, external independent review, a complete high-fiber
+census, native admission, or a new exponent. The stopped-GL/orthogonal-cover
+bridge, other local-frame bindings, all fallback instruction descriptors,
+completed weighted callees and internal borrowing, physical scheduling,
+restoration, full invocation hosting, precision and outer transfer remain
+outside this result.

research/selected-bit-mixed-descriptor/README.md

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+# Selected BIT: a mixed good/fallback child-weight descriptor
+
+This package implements the ancestor-weight descriptor for one fallback
+diagonal-pair child after a good rank-42 ancestor, and passes that generated
+descriptor into a further good child. It takes Open next question 1 of
+[PR #2171](https://github.com/CrocSwap/integer-mult-bounds/pull/2171), whose
+evaluator covers good ancestors. The claim is in
+[discussion #2165](https://github.com/CrocSwap/integer-mult-bounds/discussions/2165).
+This is a conditional interface result, with no supplier or exponent change.
+
+If unweighted elimination ends at diagonal d and original-row permutation
+pi, the weighted fallback diagonal at logical atom column j is
+
+```
+d_i(j) = d_i * b_pi(i)(j) / b_i(j),
+b_i(j) = A[i*f+j] for a head field, and 1 for a tail field.
+```
+
+The child must also convert physical atom t to j according to its field's
+orientation. This follows inherited diagonal covariance in source527 and
+Ross Richardson's PR2159. We add explicit recursive indices, an executable
+evaluator and mixed-path checks, not a new covariance or fallback theorem.
+
+On a source-bound rank-42 GL-orbit example modulo 25, a head child needs
+weights **[14,14,19]**, from two differently indexed ancestor weights. A
+reversed tail child needs **[11,6,6]**. Low-only ancestors, a lost row
+permutation and the good-only flat-index rule fail. The
+[proof](PROOF.md) states the assumptions and preparation charge.
+
+## Reproduction and evidence
+
+From the repository root, using Python 3.12 and no third-party packages:
+
+```sh
+python3 -I -B scripts/research_package.py --package research/selected-bit-mixed-descriptor
+```
+
+Both normal and optimized replay must print:
+
+```
+PASS_SELECTED_BIT_MIXED_DESCRIPTOR covariance=46848 mixed_columns=11 controls=15 no_new_kappa
+```
+
+Replay compares [RESULT.json](RESULT.json), without regenerating a missing
+or changed receipt. `--write-result NEW_PATH` writes a candidate exclusively
+when explicitly requested. [SOURCES.json](SOURCES.json) binds the selected
+template, receipt, assembly citation and source527 proof. No predecessor
+executable is imported.
+
+- 46,848 covariance cases cover every 2 by 2 determinant +/-1 matrix
+  modulo 3 and 9 and every unit diagonal conjugator.
+- All 127 diagonal pairs and both orientations are checked over 11 weighted
+  columns in four 128-coordinate fixtures. Full operators reconstruct
+  exactly; a separate forward-elimination oracle agrees.
+- A good/fallback/good path has widths 6400 → 4200 → 65 → 42, with 40-atom
+  and one-atom peeled remainders explicit. All 65 intermediate weights are
+  checked by direct weighted elimination.
+- 5,625 complete coordinate-pair/stock cases check the six-child
+  same-orientation pair; six basis columns check the two-child opposite
+  pair; 15 semantic controls reject.
+
+The prior fallback allowance remains **98,682 children**, below the charged
+131,072. The tested same-orientation pair pays six children, including four
+reversals. No bill reduction is claimed. Preparation uses the retained
+root-width stopping relation before complete fibers absorb its cost.
+
+## Scope and credit
+
+The selected object is PR589 idle child 6, unhosted copy 213, with axes
+22 through 63. Constructed charts lie in its assumed GL64 implementation
+domain, not asserted emitted orthogonal-cover invocations. Small primes
+are arithmetic fixtures. The full domain bridge, other local-frame labels,
+all fallback shear descriptors, completed weighted callees, physical tape
+execution and hosting remain open. Address-map tests are not a complete
+payload/tape implementation. Evidence is author-checked, without external
+independent or formal review.
+
+Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.
+
+This account's [PR #2160](https://github.com/CrocSwap/integer-mult-bounds/pull/2160)
+supplies the selected good residual and normal-form binding. Ross Richardson's
+[PR #2159](https://github.com/CrocSwap/integer-mult-bounds/pull/2159) supplies
+the fallback schedule and covariance proof. Jose Manuel (@jmbarrancoml)
+supplies PR2171's good-only evaluator, physical controlled ADDs and handoff.
+The generic weighted/fallback and ordered-affine methods retain the
+source527/eumemic and original compiler lineage. PR589's tuning is
+DreamingOfClouds' work on eumemic's PR555, with the retained geometry and
+hosting lineage. Adam Gall and Nick Petalas supply fresh-field, stock and
+carry context. Rohan Arun and Jeremy Kothe supply the selected assembly
+lineage. Apache-2.0.
+
+## Open next questions
+
+1. Can all fallback shear slots use equally explicit mixed descriptors?
+   Begin with `(b_pi(i)/b_pi(k))*c_ik` at one clearing step, tracking the
+   intermediate row permutation and both orientations. This is needed for
+   the whole fallback. Paid ordered-shift and child-routing integration
+   remain blockers; the final permutation here cannot replace an
+   intermediate one.
+2. Can a generated mixed descriptor drive the physical evaluator through a
+   complete selected invocation? Substitute `FallbackPair` for one incoming
+   weight source in PR2171 and account for preparation. The weighted callee,
+   additional local frames, domain bridge and correlated hosting remain
+   separate blockers. Active discussion #2179 addresses whole invocations.
+
+The structured task and limits are in [TASK.json](TASK.json).