Research: bind canonical raw prefixes to a selected coefficient site

research/selected-prefix-coefficient-site/NOTICE.md

@@ -0,0 +1,39 @@
+# Attribution and scope
+
+Jose Manuel (`jmbarrancoml`) prepared the raw-prefix refinement, canonical
+decoder, serialized evaluator and joined finite site checks with substantial
+OpenAI Codex assistance. New work uses Apache-2.0. We claim no independent
+human review, new formalization or multiplication-exponent improvement.
+
+We adapt the local-ring routines and selected request reconstruction from
+Jose Manuel's PR2234, including its source527, commdata2338 PR2160/2188,
+Ross Richardson PR2159 and preceding stream-adapter lineage. We import the
+pinned main PR1627 rational frame reader and retain credit for alephnan,
+DreamingOfClouds, eumemic and the selected hosting/owner contributors. We
+reconstruct two selected sites here, not another independent full census.
+
+PR2310 supplies the iterative grammar, binary coefficient-table pages and
+conservative uniform-control envelope. commdata2338's PR2332 supplies the
+actual selected table-binding theorem and exact global-slot/source contract.
+We reimplement their value grammar with serialized program and continuation
+records; we do not claim the first selected producer/table binding or the
+first binary coefficient-table encoding. PR2364 supplies the completed
+generic payload evidence and embedded-child interpretation.
+
+The marked relative-origin stack discipline comes from commdata2338's
+PR1379 and rfu08's retained push/pop machines. We do not claim a new stack
+algorithm. The new finite-alphabet record implementation serves the local
+continuation and prefix interface; it is not a new Lean proof of those tapes.
+
+In concurrent [PR2386](https://github.com/CrocSwap/integer-mult-bounds/pull/2386),
+commdata2338 refines the complete-vector request count and conditional
+preparation envelope. In [PR2399](https://github.com/CrocSwap/integer-mult-bounds/pull/2399),
+the same author substitutes a completed ordinary binary callee at leaf/header
+sites. We retain PR2310's conservative allowance and the existing leaf/header
+contracts, leaving those results their separate credit and scope.
+
+SOURCES.json identifies exact main inputs and pending proof/code references.
+We keep pending sources as attributed adaptations and proof dependencies;
+we import no pending predecessor executable at runtime. The source fixtures
+are author-checked evidence under the stated scope, not independent approval
+of the selected supplier or its full fixed-tape compiler.

research/selected-prefix-coefficient-site/PROOF.md

@@ -0,0 +1,208 @@
+# Raw-prefix refinement at the selected coefficient boundary
+
+We connect three existing interfaces: PR2234's selected raw-header producer,
+PR2332's generated-weight table binding, and PR2364's completed stream
+operations. We add a canonical prefix decoder and serialize the generated
+program and its continuation stacks. We retain the earlier normal-form
+identities, control envelope and marked parking construction.
+
+## 1. Quantified raw-header interpretation
+
+Fix an odd prime q, w>=1, B=q^w, Q=q^(w-1), and an actual selected request
+rho of dimension m=64 and rank r. Let P_rho be its source projector and
+let B_rho be the retained image/kernel basis. Thus
+
+    P_rho = B_rho E_r B_rho^-1.
+
+Let sigma_rho be the request's owner chart, and put S=sigma_rho B_rho.
+For a COPY request the payload already has the common header d, and
+sigma_rho=I. Its source chart belongs to the preceding source alignment.
+These are the source conventions in PR2234's provider; a same-rank label
+cannot replace rho. All matrices in this section act over Z/B.
+
+Keep the physical raw field g=g0+qG, for a fixed g0 in GL_m(F_q) and the
+complete high cube G in [Q]^(m*m). We compute h=gS in control arithmetic.
+For fixed representatives of S, define entrywise
+
+    h0 = (g0 S) mod q,
+    c = (g0 S-h0)/q mod Q,
+    psi(G) = G(S mod Q)+c mod Q.
+
+Then h=h0+q psi(G) modulo B. For w>1, S remains invertible modulo Q,
+so psi is an affine bijection of the complete high cube, with inverse
+
+    psi^-1(Y) = (Y-c)(S mod Q)^-1.
+
+For w=1 the two high cubes are singletons. The identity also gives
+
+    h E_r h^-1 = g sigma_rho P_rho sigma_rho^-1 g^-1.
+
+We therefore prepare the intended selected projector while keeping the
+payload field g fixed. We need no payload route through B_rho or S.
+We do not assume that psi preserves lexicographic order.
+
+## 2. Canonical prefix coordinates
+
+The input specification lists frozen raw matrix fields in chronological
+global slots. A slot records q,w, the exact request and its raw low class.
+The specification also records the complete spectator factors between these
+fields. Unit resets discard weight dependence, not physical coordinates or
+slot numbers. The current fresh field occupies the next slot and is outside
+the incoming-weight domain.
+
+For each generated field, keep its m*m high entries as a lexicographic
+mixed-radix counter of radix Q. A finite odometer increments the last entry,
+resets it on overflow and carries left. Induction on the number of increments
+gives the canonical lexicographic high matrix. Nested field loops give the
+canonical chronological prefix. At w=1 each field has one state.
+
+Keep these generated counters in a compact band. Keep unused-row, collapsed
+body and arbitrary spectator counters on separate bands. A coefficient
+request scans only the generated band and the local program. It must not
+derive a field by division of a flattened address that contains arbitrary
+spectators. The decoder copies the requested high digits, joins them to the
+specified raw low class, and restores its local head.
+
+This is an input representation contract, not a test on payload values.
+We never infer a header from opaque payload bits. Nor may an external list
+choose a different header order while retaining only the same prefix length.
+For example, over Z/9, the two matrices
+
+    [[1,1],[1,2]],  [[4,1],[1,2]]
+
+have the same low matrix and rank-one label but first good pivots 2 and 5.
+Interchanging their external header entries changes the weighted operator.
+The canonical decoder fixes that association by construction.
+
+## 3. Serialized local evaluation
+
+Encode each generated step with its kind, source slot, parent/child extent,
+rank, diagonal field and orientation. Encode natural numbers as finite binary
+words with delimiters. A fixed alphabet suffices for words, records and local
+origins. Read a program record by a sequential scan from its own delimiter;
+restore the head after the read. No program scan includes a spectator band.
+
+For fixed q, the selected requests and raw low classes form a finite source
+vocabulary. A code denotes one exact pair, with q and w supplied by the common
+context. We use equality of those codes after fixing this vocabulary, rather
+than probabilistic equality of hashes. The decoder's schema records each
+global slot's code and dimension on a separate delimited binary tape.
+
+Evaluate the retained Good/Pair/unit grammar with separate marked pending
+and value stacks. Pending records contain a finite operation tag and binary
+operands. A root or unit step returns its constant. A Good step pushes a
+multiply continuation and its parent request. A Pair pushes its quotient
+continuation and two ordered bank requests. The bank index includes the
+physical reversal when the selected pair lies in the tail bank.
+
+Before reading a table, validate the original global slot against the exact
+field specification. Decode that raw high field, prepare gS, and construct
+the same full-ring pivot or terminal-diagonal/permutation page as PR2332.
+Read the needed page, retain its bounded local operands, and erase the
+numerical page. The incoming evaluator cannot request the current fresh
+field or an unused-row coordinate. A unit returns without requesting dead
+numerical preparation, while later steps retain their original slot numbers.
+To reach a later live slot, the field head can cross earlier encoded fields;
+the decoder ignores their symbols and charges those head movements.
+
+Induction over the pending expression forest proves equality with the
+PR2310 evaluator and PR2332's bound expression. Each push appends above the
+current marker. Each pop scans back to that marker, restores the preceding
+top, and erases its own record. Neither operation reads an absolute parking
+address. Induction over evaluation steps proves empty pending/value stacks
+and restored program/field heads on return. The marked-stack argument is the
+existing PR1379 construction; we apply it to serialized continuation data.
+
+Local empty work means marked-empty tapes. The caller supplies the local
+origin markers and includes their fixed installation/removal cost when it
+opens or closes those work roles. The finite tests preload that interface;
+they count the subsequent symbol operations. After erasure, successive
+evaluations can reuse the same finite list of tape roles. Constructor calls
+in the Python simulator do not imply a depth-dependent tape inventory.
+
+## 4. A complete selected coefficient site
+
+Consider the selected good head slot, with i>k and column j in [f):
+
+    H_(if+j) <- H_(if+j) + c H_(kf+j),
+    c = ell_ik(gS) A_(if+j) / A_(kf+j).
+
+The current preparation may read raw g. The two incoming A evaluations may
+read only older frozen fields. For the first elimination column,
+ell_i0=-P_i0/P_00; later coefficients use the retained chronological
+elimination state. The denominator is a unit on the admitted good class.
+
+Regroup the complete fields around these two atoms as
+
+    frozen prefix x before x X x middle x Y x suffix.
+
+For each canonical prefix state, prepare c, then traverse the complete
+before/source/middle loops. For a source x, set t=c*x mod B. Copy the first
+B-t target blocks to one sequential scratch stream and the remaining t
+blocks to another. During the merge pass regenerate the same canonical
+prefix and c, then copy the second part before the first. Preserve the
+records within each suffix block. This maps Y to Y+cX and fixes the prefix
+and spectators, for arbitrary correlated payloads. Erase split scratch and
+numerical preparation, and restore the local heads.
+
+For the inverse site, regenerate the same raw prefix and use -c. The two
+passes compose to identity. No stored array indexed by all high matrices is
+needed. In particular, a nontrivial psi does not authorize permuting raw
+prefix rows: we compute c(gS) while visiting raw G in its own order.
+
+The same substitution applies to the other good slots and to each fallback
+shear with its own original-row permutation snapshot. Its validity follows
+from pointwise coefficient equality. Induction over the fixed-low trace
+therefore refines the complete selected residual, assuming its completed
+child contracts. At a child, append the raw current field and its exact
+FieldSpecification; keep the whole high packet. PR2364's embedded-field
+interpretation then supplies the child view without moving records.
+For f=1 the completed leaf implements each selected child of width at most
+42. For general f retain the proper-child contract and its separate cost.
+An all-leaf replacement would not establish the selected saving.
+
+## 5. Local work and the uniform charge
+
+Keep the original root coupling w=ceil(e^(17/10000)) and the retained global
+depth bound h<=2 ceil(log2(max(2,e))). Fix q and the selected finite catalogue.
+The generated band has O_q(hw) bits. Program records, continuations and local
+operands have O_q(h(w+log(e+1))) bits. Request identities and low matrices
+have fixed q-dependent sizes. The raw-to-prepared matrix multiplication and
+bounded-dimensional modular preparation have the existing polynomial cost.
+
+The sequential program and stack scans add polynomial local work already
+within PR2310's conservative O_q(w^1800) allowance. We do not sharpen its
+request count. Schoolbook arithmetic supplies the scalar lowering; finite
+Python arithmetic timings are not transition counts. For w>=1,
+
+    w^1800 / q^(w-1) <= 2^1800 * 1800!.
+
+Each body target fibre contains B=q^w complete blocks. Recomputing the
+coefficient on split and merge therefore contributes O_q(V), as do payload
+copies and erasure. Generated counters use their own ripple-carry scans;
+arbitrary spectator counters use separate complete traversals. Their resets
+are charged to the blocks they precede. We do not scan an L-bit spectator
+band on each coefficient request, which would introduce an uncontrolled
+factor proportional to L/B.
+
+The site has one bound C_q V, independent of high values, spectator sizes,
+row multiplicity and parking origin. A selected residual contains O(n) such
+slots, with O(Vw) fixed header work under the retained route contracts.
+The resulting local toll is the existing C_q V(n+w), provided the caller
+implements the stated canonical schema and the remaining selected macrosteps.
+
+## 6. Scope of the result
+
+We prove a raw-prefix and local-control refinement under the named producer,
+stream and child contracts. We implement the canonical decoder, serialized
+continuation operations and a joined selected head-coefficient site. We
+separate source-bound selected fixtures from complete small stream domains.
+We do not enumerate the GL64 high cube or execute the selected global
+family/owner/XOR/COPY/ERASE interpreter. That caller must still instantiate
+this interface at its actual physical fields and close its other work.
+
+We retain credit for source527, commdata2338's normalizers, descriptor and
+table binding, Ross Richardson's fallback algebra, rfu08's marked tapes,
+and the selected supplier contributors. Jose Manuel (`jmbarrancoml`) prepared
+this work with substantial OpenAI Codex assistance. Independent review and
+formalization remain open; we claim no new multiplication exponent.

research/selected-prefix-coefficient-site/README.md

@@ -0,0 +1,79 @@
+# Canonical raw prefixes at a selected coefficient site
+
+We prove a refinement from selected raw-header preparation to a complete
+controlled-ADD site. We keep the physical header g in place and compute
+g sigma B0 in control. We decode generated fields from their canonical
+mixed-radix coordinates and encode the program, field schema and local
+continuation stacks as delimited binary words.
+
+For each fixed low class, the prepared high field is an affine bijection
+of the raw high cube. We use that fact to preserve complete packets while
+pulling each coefficient back to raw-header order. Induction over the
+fixed-low instruction trace extends the substitution to the selected
+residual, under its completed-child contracts. [PROOF.md](PROOF.md) gives
+the quantified statement and the retained uniform local charge.
+
+## Evidence
+
+We check:
+
+- All 81 high matrices in a two-dimensional Z/9 cube, with a nonzero carry
+  and nontrivial high-coordinate map. We execute the complete finite site
+  on 6,561 opaque 16-bit records and its inverse.
+- Actual selected COPY510 and nonidentity-owner MOVE requests from the pinned
+  word and hosting inputs. Exact raw fixtures from PR2234 prevent matching
+  chart errors in preparation and fixture generation from canceling.
+- A selected head coefficient with older generated weights. The coefficients
+  4,7,1 feed the actual split/merge preparation callback. We check all 1,944
+  records in the reduced finite site rectangle and inverse restoration.
+- Four serialized Good/Pair/reset chains with 36 coordinate comparisons
+  against the attributed recursive grammar. We retain global field slot 2
+  after a reset and check that dead high fields do not affect the value.
+- Complete odometers with 139 states across four schemas. We check identical
+  local symbol counts after translating origins as far as 10^40, replacing
+  buried data and growing a separate spectator-metadata band to 4,096 bits.
+
+Controls expose wrong raw/prepared interpretation, a second COPY chart,
+reordered coefficient prefixes, current-field capture, wrong source codes,
+compacted reset slots, wrong pair orientation, an inadmissible branch and a
+wrong selected rank. We retain exact output hashes and counters in
+[RESULT.json](RESULT.json).
+
+## Reproduction and measurement boundary
+
+Use Python 3.12 and the standard library:
+
+```sh
+python3.12 scripts/research_package.py --package research/selected-prefix-coefficient-site
+```
+
+The registered hook authenticates source bytes before importing the retained
+reader and runs normal and optimized replays. Use `verify.py --emit` to print
+a fresh receipt without changing files.
+
+We count single-symbol payload and local-control tape reads, writes and moves.
+We retain host integer arithmetic, decoded local operands, finite source
+dispatch and outer loop bounds as macros. Their fixed-tape lowering and
+polynomial cost use the written schoolbook/counter argument and PR2310's
+existing allowance. We claim no measured transition count for the whole
+compiled machine. We also retain the exact request/low vocabulary as an
+admitted source context; a caller must supply its correct physical schema.
+
+The selected64 execution uses three raw-prefix snapshots and a reduced
+complete two-atom site rectangle. We do not enumerate its full high cube or
+128-atom body. The complete 81-lift cube has dimension two. The written
+refinement covers the stated general domain; the finite tests do not prove
+that theorem or the selected global caller. We retain the selected frontier.
+
+## Open next questions
+
+Connect this interface to the actual family/owner and global XOR/COPY/ERASE
+interpreter, preserving its physical prefix schema at routes and recursive
+boundaries. Separately, replace the remaining host arithmetic and loop macros
+with a checked operational lowering. [TASK.json](TASK.json) records concrete
+first steps and blockers.
+
+Jose Manuel ([jmbarrancoml](https://github.com/jmbarrancoml)) prepared this
+contribution with substantial OpenAI Codex assistance. [NOTICE.md](NOTICE.md)
+retains source and concurrent credit. Coordination:
+[Discussion2319 follow-up](https://github.com/CrocSwap/integer-mult-bounds/discussions/2319#discussioncomment-18942932).