Close the fixed nine-query tariff at 486 using prior geometry (no κ)

research/nine-query-fixed-tariff/CASES.json

@@ -0,0 +1,34 @@
+{
+  "schema_version": 1,
+  "queries": 9,
+  "sources": 5,
+  "closed": [
+    0,
+    1,
+    3,
+    2,
+    6,
+    14,
+    10,
+    26,
+    16,
+    0
+  ],
+  "prices": {
+    "b": 38,
+    "a": 2,
+    "x": 2
+  },
+  "limits": {
+    "records": 4000,
+    "oracle_candidates": 400000,
+    "artifact_bytes": 8388608,
+    "command_seconds": 180
+  },
+  "credited_premises": {
+    "unrestricted_R_lower_bound": 68,
+    "X2_minimum_R": 70,
+    "fixed_price_optimum": 486
+  },
+  "scope": "Fixed-price path-lowered even-subset9query invoice only. Geometry6059Lemma2/1340table,6696X2floor/labeledcount/attainingpath remain prior. No pointcensus/newgeometry/newwitness/allpricefrontier/physicalbackend. Repeated prefixvalues allowed withoutfilter."
+}

research/nine-query-fixed-tariff/NOTICE

@@ -0,0 +1,24 @@
+Nine-query fixed-tariff closure and independent premise/ABI verification
+Prepared for ToikY with substantial OpenAI Codex assistance; Apache-2.0.
+Separate AI primary/oracle/static review is not independent human or proof-kernel
+acceptance. Original source notices and licences remain exact in inputs/.
+
+apoursh PR6059: unrestricted nine-point difference bound, with its independent
+geometry proof; Arham Tahir PR1340: prior normalized table and mutable-source
+interface. The R68 consequence is prior geometry, not a new theorem here.
+Nick Petalas PR6696: released nine-query question, X2minimumR70, labeledcount453600
+and attaining path; PR6695/5935 retain adjacent support/XOR context. No new floor,
+forest counting law, point normalization/census or witness is claimed.
+Wentao Li PR2677/2686/2643: path-lowered model/invoice and related questions;
+Arham Tahir PR1898: abstract alphabet and six-query word;
+ywz65 PR2455: component inequality; Ross Richardson PR943 and retained PR1370
+lineage: connector and ten-query context.
+
+Classical incidence rank, forests, subset dynamic programming, parity-state
+shortest paths and reversible linear updates retain their usual credit.
+Repository task/replay/export tooling retains Douglas Colkitt and other
+contributors' attribution. Gzip payload equality follows Maxime Fleury's
+reproducibility convention after Antoine Olivier's independent replay findings.
+No predecessor program is imported or executed; model.py is inert text only.
+This is a modest fixed-price corollary plus independent checks, not a new physical
+backend, supplier admission, kappa improvement or selected-frontier update.

research/nine-query-fixed-tariff/ORACLE.json

@@ -0,0 +1,72 @@
+{
+  "case_manifest_sha256": "fc2c9be60d410124b45da970af462acc749275a35e20a16969527dca5b56f4ea",
+  "floor": {
+    "classes": 3780,
+    "coordinate_labelings": 120,
+    "histogram": [
+      [
+        70,
+        234
+      ],
+      [
+        74,
+        432
+      ],
+      [
+        78,
+        594
+      ],
+      [
+        80,
+        513
+      ],
+      [
+        82,
+        135
+      ],
+      [
+        84,
+        864
+      ],
+      [
+        88,
+        756
+      ],
+      [
+        96,
+        252
+      ]
+    ],
+    "labeled_walks": 453600,
+    "minimum_R": 70,
+    "records_sha256": "652136f7b89793acfe5c437a939953bb06ddfbca2cf04f142b529e331aa849f2"
+  },
+  "schema_version": 1,
+  "source_manifest_sha256": "0d1cbe7daa2cd1092c1a64a3615521b651faf8f83102b7db03896c17a0a09a1a",
+  "witness": {
+    "Q": 9,
+    "R": 70,
+    "X": 2,
+    "closed": [
+      0,
+      1,
+      3,
+      2,
+      6,
+      14,
+      10,
+      26,
+      16,
+      0
+    ],
+    "invoice": {
+      "borrowed_dirty_rows": 4,
+      "echo_nodes": 81,
+      "map_leaves": 322,
+      "root_dispatches": 81,
+      "source_XORs": 2,
+      "total": 486
+    },
+    "tariff": 486
+  }
+}

research/nine-query-fixed-tariff/PROOF.md

@@ -0,0 +1,150 @@
+# The fixed nine-query tariff is486
+
+This is a modest fixed-price corollary closing Nick Petalas's PR6696 README
+question1, with independent premise/ABI verification. The geometry, unrestricted
+readout consequence, X2minimum, forest count and attaining path retain predecessor
+credit. No new point census, geometry law, witness or all-price frontier is claimed.
+
+## Exact inherited interface
+
+Use the five-source common-B, path-lowered even-subset model of Wentao Li2677,
+built on Arham Tahir1898/1340. Five source rows,31output rows, connectorz and
+private scratch are distinct and arbitrarily dirty. B and the31mapsA_m are
+read-only exact linear maps; they need not be invertible or commute. The output
+adds A_m B(sum_{j in m} x_j), restoring sources, connector and private rows.
+Queries Q_j add Bx_j toz. Readouts R_m add A_m z toy_m. Source shears X_{p,j}
+addx_j tox_p, and must be undone as prescribed by each path-local query lowering.
+These are credited complete calls, not newly supplied physical implementations.
+
+Fix exactlyQ=9 queries and a restored closed nonzero-edge prefix path
+
+    0=w_0,w_1,...,w_9=0 inF2^5.
+
+Time slots arew_0,...,w_8; the final zero is identified with the initial slot.
+Labels and zero revisits may repeat. The minimum readout for a nonzero maskm
+is the least even subset of these nine slots whose XOR ism. All31masks must
+be available. WithS=sum_t weight(w_t+w_(t+1)), the path-local source-shear count
+isX=2(S-9). The fixed inherited tariff is
+
+    T=38Q+2R+2X.
+
+We prove its exact minimum atQ=9 is486. The six-query tariff436 remains better;
+this is completion of one existing invoice row, not a new supplier improvement.
+
+## Credited unrestricted readout bound
+
+Every nonzero mask needs at least two readouts. It has cost two exactly when
+it is a difference of two distinct support values. IfD(A) denotes the nonzero
+pair differences of the at-most-nine-point slot support, apoursh6059 Lemma2
+proves |D(A)|<=28 without restricting the point set to a particular walk.
+Although that package's main theorem concerns partitioned independent walks,
+this lemma is an unrestricted finite-geometry statement and applies here.
+The package's partitioned Q=9 readout row must not be transferred to this model.
+
+Thus at least three of the31masks cost at least four and
+
+    R>=28*2+3*4=68.                                         (1)
+
+Smaller supports and repeated slot labels are included; duplicates create no
+new nonzero pair difference. Arham Tahir1340's prior normalized table also has
+maximum coverage29 INCLUDING zero, hence28nonzero differences. This immediate
+R68consequence is prior geometry, not a new floor here. No2600-support census
+is executed by this package. Other masks might cost more than four, so(1)is
+only a lower bound, not an attainment assertion.
+
+## The source-XOR gap and fixed price
+
+Coverage requires the query increments to spanF2^5. Each coordinate therefore
+appears in at least one increment; closure makes its occurrence count positive
+even, hence at least two. ConsequentlyS is even andS>=10, giving
+
+    X=2(S-9) in{2,6,10,...}.                               (2)
+
+IfX>=6, (1) gives
+
+    T>=38*9+2*68+2*6=490.                                 (3)
+
+IfX=2, Nick Petalas6696 already establishes minimumR=70 in this floor layer,
+and supplies the attaining closed path
+
+    (0,1,3,2,6,14,10,26,16,0).                              (4)
+
+So that layer hasT>=342+140+4=486, attained by(4). Combining it with(3)
+proves the exact fixed-price optimum486. Neither the witness nor the X2floor
+is new here. The useful added result is this explicit priced closure plus an
+independent full-data check of the premise and the credited word's actual ABI.
+No claim is made about all positive prices or arbitrary non-path source shears.
+
+## Independent finite control of the X2 premise
+
+AtX=2, S=10, so every source coordinate occurs in exactly two of the nine
+query increments. Represent each coordinate by an edge joining its two query
+positions. Full rank forbids parallel edges and cycles, because either would
+make these incidence columns linearly dependent overF2. Nonzero increments
+forbid isolated query vertices. Thus there is a simple forest with nine labeled
+vertices and five edges, no isolates, and four components. Each component has
+at least two vertices. The only component sizes are3+2+2+2, giving one three-
+vertex path and three separate edges.
+
+Choose the three-vertex center in9ways, its two leaves inC(8,2)=28ways, and a
+perfect matching of the remaining six vertices in15ways. This gives3780graphs.
+Coordinate labels may be assigned to their five edges in5!=120ways, giving
+453600labeled walks, the count already recorded by6696. Coordinate permutations
+preserve the sum over ALL31readout demands and the uniform source-XOR/query
+prices. They do not quotient query-position rotations or reversals. Assigning
+bits0..4to lexicographically sorted edges gives one exact representative per
+coordinate-label class. Repeated prefix values remain allowed; none are filtered.
+
+The primary enumerates these center/leaves/matching graphs, reconstructs all
+nine slots and enumerates all512subsets for each even-subset minimum. The
+independent engine instead scans allC(36,5)five-edge subsets and uses a64-state
+(mask,parity) BFS. They retain and compare every actual graph-code/slot/minimum
+row, not merely a common minimum or histogram. This corroborates the credited
+X2premise without importing its checker. The forest classification/count is
+not presented as a new counting theorem.
+
+## Actual word and complete abstract invoice
+
+For the credited path(4), the primary chooses one minimum even subset per output
+mask. At each query edge, it shears all other active sources into the least-bit
+pivot, callsQ on that pivot, and undoes shears in reverse order. Readouts occur
+at the selected prefix slots. Own tie choices are not a new witness claim.
+The submitted data must bind the literal query prefixes to(4), each readout's
+minimum even subset/XOR, and the full dirty endpoint.
+
+The literal interpreter retains distinct noncommuting operator words. A readout
+of a query contribution isA_m B x_j in that order; no commutation is used.
+Every old dirtyy_m is preserved, dirtyz cancels, and every source/private row
+returns. Reversing the complete literal instruction list is its inverse because
+each elementary update is self-inverse overF2. This proves the formal ABI rather
+than just testing zero inputs or commuting scalar operators.
+
+The inherited complete-call fixture decomposes the actual countsQ9,R70,X2 as
+
+    map leaves=28Q+R=322,
+    echo nodes=9Q=81,
+    source XORs=2,
+    root dispatches=Q+R+X=81,
+    borrowed dirty rows=4,
+    total=322+81+2+81=486.
+
+These are the source2677abstract prices and resource accounting, not a new
+physical backend, recursive implementation or primitive tariff. Actual serialized
+word data and all floor records are retained inFLOOR.json.gz and independently
+checked after blind first seals. The source model itself stays inert text.
+
+## Limits and predecessor credit
+
+The exact scope is path-lowered restored single-walk nine-query programs with
+all31even-subset readouts and uniform fixed prices(38,2,2). It does not price
+physical source access, map delivery, routing, admission, stock, precision,
+restoration implementations or a complete supplier/outer caller. No newkappa,
+selected-frontier change, human acceptance or formal proof-kernel result follows.
+
+Credit apoursh6059 for unrestricted nine-point geometry; Arham Tahir1340 for
+its prior table and mutable-source contract; Nick Petalas6696 for the released
+question, X2minimum/count and path,6695/5935 for adjacent support/XOR context;
+Wentao Li2677/2686/2643, Arham Tahir1898, ywz652455, Ross Richardson943 and the
+retained1370lineage. The new work is the modest fixed-price closure and separate
+premise/data/ABI verification. Classical incidence rank, forest counting,
+subset dynamic programming and shortest paths retain their usual credit.

research/nine-query-fixed-tariff/README.md

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+# Exact fixed-price nine-query tariff486
+
+Close Nick Petalas's [PR6696 question1](https://github.com/CrocSwap/integer-mult-bounds/pull/6696):
+in the five-source path-lowered even-subset class at prices(b,a,x)=(38,2,2),
+the minimum at exactly nine queries is486. This is a modest corollary with
+independent premise/ABI verification, not a new geometry or counting law.
+
+The credited unrestricted nine-point geometry of apoursh6059 Lemma2 (also
+Arham Tahir1340's prior table) givesR>=68. Path-local closure forcesX in
+{2,6,10,...}; thereforeX>=6 costs at least490. PR6696's knownX2minimumR70
+and credited path(0,1,3,2,6,14,10,26,16,0) attain486. No new witness,
+2600-point normalization/census, all-price frontier or physical supplier is claimed.
+The six-query tariff436 remains better. See [PROOF.md](PROOF.md).
+
+## Evidence and reproduction
+
+The primary independently enumerates3780coordinate-unlabeled X2forest graphs,
+using one three-vertex path plus three paired edges on the nine query positions.
+Every graph's actual code, nine prefix slots and31even-subset minima are retained
+in [FLOOR.json.gz](FLOOR.json.gz). Repeated prefix values are allowed, not filtered.
+All120coordinate labelings preserveR/X, recovering the credited453600walks.
+The primary uses512subset states. The independent engine scans five-edge subsets
+and uses64(mask,parity) BFS states; actual complete rows are compared after blind
+first seals. MinimumR70/classification/count remain credited premises checked
+independently, not newly claimed floor or enumeration results.
+
+The submitted actual witness word has9queries,70readouts,2sourceXORs,81root
+dispatches and fixture invoice322map leaves+81echo nodes+2XORs+81dispatches=486,
+with4borrowed dirty rows. A free noncommuting operator-word interpreter checks
+its actual query prefixes, minimum even readout slots, every dirty output/source/
+connector/private row, full map and literal inverse. The old model remains inert
+text. This checks the abstract ABI and inherited complete-call price fixture;
+no physical backend, native multiplier, stock or precision claim follows.
+
+From the repository root, CPython3.12, standard library only:
+
+```sh
+python3.12 -I -W error -B research/nine-query-fixed-tariff/verify.py
+python3.12 -I -W error -B research/nine-query-fixed-tariff/oracle.py --check-primary research/nine-query-fixed-tariff/RESULT.json
+python3.12 -W error -B scripts/research_package.py --package research/nine-query-fixed-tariff
+```
+
+[VERIFY.json](VERIFY.json) adds both optimized modes and declaresFLOOR.json.gz as
+a runtime input. Direct commands are read-only and need noGit/network/compiler/
+Lean/Docker. Source/cases/schema authenticate before computation. Exact typed
+retained values, duplicate/nonfinite/float/boolean-alias guards and ten primary
+parser/ABI/omission negative controls remain active under-O. Proof artifacts are
+bounded at8MiB,4000records and180s per command. New functions stay below50physical
+lines. CPython dynamic allocation, arbitrary-precision integers and bounded
+recursive JSON comparison are declared managed-runtime deviations. Regenerated
+gzip equality compares complete canonical payloads, not container bytes.
+
+[RUN-RECORDS.json](RUN-RECORDS.json) and [VALIDATION.md](VALIDATION.md) retain exact
+first timings, subsequent checks, corrections and pending transport stages.
+Formal-kernel and independent human acceptance are unclaimed.
+
+## Open next questions
+
+Structured copies are in [TASK.json](TASK.json).
+
+1. What is the full nine-query frontier for arbitrary positive prices? This
+   fixed-price closure does not classify the higher-X tradeoffs. First define
+   one source-bound price region and its relevant layers before a wider census.
+2. Does the fixed nine-query486 tariff extend to non-path restored source-XOR
+   programs? Prior projection/geometry already gives the genericR68bound. First
+   classify their actualX2source cost; the path-localX=2(S-9)/forest reduction
+   does not establish a non-pathX2minimum or fixed486tariff.
+3. Can one physical caller implement the inherited price fixture profitably?
+   Specify complete calls, access, dirty restoration, routing, coefficient/
+   precision/stock and all extra caller charges. None is supplied by this invoice.
+
+## Credit
+
+apoursh6059 supplies the unrestricted geometry; Arham Tahir1340 its prior table
+and mutable-source contract. Nick Petalas6696 supplies the question, floor count,
+X2minimum and attaining path;6695/5935 retain adjacent support/XOR context.
+Wentao Li2677/2686/2643, Arham Tahir1898, ywz652455, Ross Richardson943 and the
+retained1370lineage keep predecessor credit. Exact original notices remain in
+inputs/ with [SOURCE-PINS.json](SOURCE-PINS.json). Classical incidence rank,
+forest counting, subset DP and shortest paths are used. Prepared for ToikY with
+substantial OpenAI Codex assistance and separate AI primary/oracle/static review,
+Apache-2.0. No predecessor source, κ or selected frontier changes.

research/nine-query-fixed-tariff/RESULT.json

@@ -0,0 +1 @@
+{"case_manifest_sha256":"fc2c9be60d410124b45da970af462acc749275a35e20a16969527dca5b56f4ea","floor":{"classes":3780,"coordinate_labelings":120,"histogram":[[70,234],[74,432],[78,594],[80,513],[82,135],[84,864],[88,756],[96,252]],"labeled_walks":453600,"minimum_R":70,"records_sha256":"652136f7b89793acfe5c437a939953bb06ddfbca2cf04f142b529e331aa849f2"},"schema_version":1,"source_manifest_sha256":"0d1cbe7daa2cd1092c1a64a3615521b651faf8f83102b7db03896c17a0a09a1a","witness":{"Q":9,"R":70,"X":2,"closed":[0,1,3,2,6,14,10,26,16,0],"invoice":{"borrowed_dirty_rows":4,"echo_nodes":81,"map_leaves":322,"root_dispatches":81,"source_XORs":2,"total":486},"tariff":486}}