Audit the physical cap-two swap price against the direct target (no new κ)

research/three-swap-cap2-physical-ceiling/.gitattributes

@@ -0,0 +1,2 @@
+# Preserve immutable scientific evidence bytes, including CRLF.
+** -text

research/three-swap-cap2-physical-ceiling/EVIDENCE.json

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+}

research/three-swap-cap2-physical-ceiling/README.md

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+# Three-SWAP cap-two physical interface and direct-comparator ceiling
+
+**Source-pinned research package.** The physical interpretation of Nick Petalas's
+[#5083](https://github.com/CrocSwap/integer-mult-bounds/pull/5083) needs a stronger
+price gate than its sound restricted formal-word comparison. With reversed BIT
+generators the target is an ordinary swap, and the retained source supplies a
+direct implementation. Beating that explicit recipe requires
+
+```
+B < (2F + 6Q - Delta)/4,       Delta = wNew - wDirect.
+```
+
+This is a scoped necessary comparison against a constructive upper recipe.
+The recipe is not asserted optimal. No actual cap-two supplier, bound eligible
+caller, changed P/T inventory, universal physical obstruction or new κ follows.
+The selected record remains unchanged. The retained independent review checks
+the exact arithmetic and orientation; it does not certify a physical supplier.
+
+## Result and physical boundary
+
+The formal source has owner-rank profile `(2,2,1,2,2)`, price `c1+4c2`, and
+comparator `3c1+3c2`. Its formal saving `2c1-c2` is credited to Nick Petalas.
+That theorem and its formal-word checker are not duplicated or rerun here.
+Ross Richardson's GL2(F2)/S3 bridge identifies `a=(12), b=(23), s=(13)`;
+David Arias's rank-priced theorem and the retained group/cut lineage remain
+predecessor work.
+
+The missing attaining ABI is
+
+```
+Cap2(g,{i,j}): (u0,u1,u2,dirty...)
+  -> (rho_f(g)u_i, rho_f(g)u_j, all other original fields unchanged), g=a or b.
+```
+
+Each owner is an arbitrary independent payload function on the same complete
+address range. Two original inputs require two independent transformed outputs;
+the unselected active owner, arbitrary dirty owners, untouched rows and every
+spectator must each be restored. A sum of owners, a zero input or a discarded
+owner does not meet this ABI. Entry/exit charts, atom order, orientation,
+weights, width lifting, buffers, copies, reset and all outputs must be explicit.
+Scalar owner rank, address width, logical volume, column multiplicity and native
+recursive-child rank are separate dimensions. The existing BIT supplier's
+width-`rf` call acts on one role at volume `V/W`; putting two owners in ROW
+values increases volume to `2V/W`. It does not establish the cap-two discount.
+
+For unweighted reversed generators, write `C=C_f`, `C^2=I`. Source convention
+reverses atom positions, without reversing digits inside an atom. Then
+
+```
+(x,y,z) --F12--> (Cy,Cx,z) --F23--> (Cy,Cz,x) --F12--> (z,y,x).
+```
+
+Thus `aba` produces ordinary `J13` and restores the middle chunk. The source's
+`J_A=R_tail F_A R_tail` recipe gives one width-`f` reversed call and two
+width-`floor(f/2)` calls, all on the same complete logical volume. Put `F,Q`
+for their complete prices, `B` for the complete width-`f` cap-two transaction,
+and `wNew,wDirect` for costs outside these complete calls only. The candidate
+cost is `F+4B+wNew`; the available direct upper recipe costs `3F+6Q+wDirect`.
+Their strict comparison gives the displayed threshold. In-call work must not
+be charged again in `Delta`; unknown outside work is unknown.
+
+Uniform prices must hold for both labels and every selected owner pair.
+Otherwise use the actual sum, initially `F_a+2B_a+2B_b`, and any further
+pair-dependent terms. The unweighted identity does not identify different
+ancestor-dependent weight arrays. Weighted jobs require matched actual weights,
+reversed pairing, orientation, precision and complete returned inventory.
+For raw-F targets the word has the wrong endpoint: the particular wrapper
+adds six half-width calls and loses against three direct F calls when `B>=F`.
+A proved caller layout absorbing reversals would be a changed premise.
+For ordinary-J generators the direct three-target competitor is already `3J`.
+
+## Exact conditional tariff screen
+
+Under the separately declared hypothetical same-volume model
+
+```
+F=K*f^(1-C), Q=K*floor(f/2)^(1-C), B=2^(1-C)*F,
+K>0, H_0=0, Delta=0, 0<=C<=17/5000,
+```
+
+the nominal cap-two price loses. The exact integer inequality
+`399^5000 < 2^4983 * 200^5000` proves `2^(1-C)>399/200`.
+The necessary price satisfies `B/F<333/266`, while the hypothetical nominal
+price is `B/F>399/200`; the normalized recursive disadvantage exceeds
+`19767/6650`. Even widths attain the underlying bound; odd floors strengthen
+it, and width one is covered with `Q=0`. The checker includes sign reversal
+outside the screened regime. These conclusions reject this tariff/comparator
+combination, not every cap-two implementation. A sufficiently favorable actual
+outside-cost difference or different complete B could change the full decision.
+
+## Evidence, provenance and reproduction
+
+The scientific packets under `evidence/` are exact retained bytes:
+
+- [Physical ABI and feasibility report](evidence/physical-interface/REPORT.md),
+  its complete source snapshots and original manifests.
+- [Direct-comparator derivation](evidence/outputs/new-approach-ceilings-20261023/DIRECT-COMPARATOR-PROOF.md),
+  exact checker/receipts, candidate interface and baseline obligation ledger,
+  with all 16 scientific snapshots.
+- [Independent direct-comparator review](evidence/outputs/new-approach-verification-20261023/DIRECT-COMPARATOR-REVIEW.md),
+  unchanged direct-only checker and both frozen direct receipts.
+
+`EVIDENCE.json` binds all 51 retained files. Original reports and proposed drafts
+retain historical local status and worker names; they are evidence of that
+checkpoint, not current publication text or instructions. The full original
+`FREEZE.json` is retained with SHA256
+`22deada00ef988bdc99b7fd01480fc86136111a8483513d00ce2ae0eea711975`.
+Its unrelated centre-p2 entries are not vendored or replayed in this package.
+The direct-only checker authenticates only this package's direct source packet.
+
+The originating catalogue card `bit-new-register-model` and selected record
+remain pinned to canonical main `81e9bd69e4a5b00ba9c713a1620e36d576d5522e`.
+The scope is explicitly **changed** to a physical ABI/orientation/price audit.
+Root supplied the bounded overlap checkpoint and owns the final live/publication
+check; this package does not claim a complete current board survey or novelty.
+It complements #5083 instead of claiming its formal construction again.
+
+The original #5083 source head
+`d2b3f35275cab6fbf18bee40e3fb411496732e54` was not an ancestor of that canonical
+base. Provenance-only merge `ea40685a28ebf1489f74e91e39b143930dbfee53` uses
+`git merge -s ours --no-ff` to retain that exact history, with an unchanged tree.
+The separate arithmetic/source head
+`20bb93cd990d2ff683e4d0fc56b87a42cd5d3296` was already ancestral.
+`SOURCES.json` keeps original paths and original head pins as provenance, and
+pins every retained live input at its same path in ancestor commit
+`a84a22d3f6e0ce75a9e433c6438f01cf25ed384c`. No silent source repin or scientific
+rewrite is used. Package `.gitattributes` disables text conversion to preserve
+scientific bytes, including CRLF.
+
+Standard-library Python 3.12; run either from the repository root:
+
+```sh
+python3 -I -W error -B research/three-swap-cap2-physical-ceiling/verify.py
+python3 -I -O -W error -B research/three-swap-cap2-physical-ceiling/verify.py
+python3 scripts/research_package.py --package research/three-swap-cap2-physical-ceiling
+```
+
+Both direct commands print `PASS_CAP2_PHYSICAL_CEILING_PACKAGE` in about two
+seconds each locally, with a 180-second registered cap each. The adapter copies
+unchanged scientific code and packets into a private temporary layout matching
+their original relative paths. It captures stdout, compares the author result
+and independent generated direct receipt byte for byte with frozen receipts,
+checks all original manifests and retained freeze entries, and rejects a real
+corruption of a frozen source in both unchanged checkers. It verifies package
+bytes before and after. Scratch results are removed. No Git, network, compiler,
+Lean, native execution or third-party library is needed for direct replay.
+
+The retained author controls cover 40 exact rational factor/sign cases,
+64 floor cases, five complete-tariff scenarios, six semantic rejections and
+three reserve controls. The independent review adds exact orientation/dirty
+spectator cases and independent rational economics. Normal/optimized results
+must equal their corresponding frozen receipts. This bounded check does not
+execute the source author's formal-word verifier or admit all-width physical
+implementations.
+
+For a committed package, also use the repository's byte/source-history preflight
+and official exporter. Transport integrity and ancestry availability are separate
+from mathematical validation. Root retains local validation receipts and the
+checked Git-free export; publication and hosted CI remain separate actions.
+
+## Remaining obligations and stopping rule
+
+No eligible chronological occurrence or autonomous output inventory is bound.
+No actual complete `F,Q,B,wNew,wDirect` bill, strict child contraction, finite
+work, preparation/routing/storage/cleanup, precision/rounding amplification,
+ordinary BIT stock, outer rows, both paid tails or funded compiler reserve is
+supplied. In the complex ABI, Fourier rank and exact Clifford phases differ
+from scalar owner rank and binary permutation labels; a BIT-only improvement
+does not establish a changed complex supplier. The retained #4610 obligations
+remain partial/conditional. No root search, changed histogram, emitter, physical
+fixture or new P/T/κ is reported.
+
+Stop implementation expansion until the complete bill for one actual occurrence
+passes the strict direct-recipe comparison. A positive relaxed moment must also
+retain `T<=U`, `P>=L`, hence `M<=U-L`, and the strict reserve screen
+`L<(1-e)U`; passing an optimistic bound is not admission.
+
+## Open next questions
+
+The structured questions in `TASK.json` give relevance, first steps and blockers.
+First name one eligible caller, actual cap-two ABI and complete prices, and test
+the strict inequality. A failure stops that candidate; unknown prices remain a
+blocker. Only after a pass should a separately scoped physical fixture
+with width two or larger distinguish ordinary/reversed orientation using
+independent owner impulses and dirty spectators. A finite fixture still needs
+a uniform exact operator/address argument. Then audit the changed full complex
+inventory, finite work, both tails, compiler reserve and outer transfer.
+
+## Credit and license
+
+Prepared by **Chris R / Clear1nsight**, with substantial **OpenAI Codex**
+assistance in research, independent review and packaging. Predecessor authors
+are credited for their source work, not described as commissioning clients.
+Nick Petalas owns #5083's formal word; David Arias, Ross Richardson, Adam Gall
+and the retained group/cut lineage retain their mathematical credit. The source
+BIT, native, row-cover, Clifford, precision/compiler and analytic contributors
+include Cao Yifan, Cinna Davis, zxv, Dugongue, Jacob Alan Sussman, eumemic and
+their predecessors. The frozen reports retain fuller attribution. Source
+extraction, economic derivation and the independent review are retained with
+their original attribution and scope. Apache-2.0, with all source notices kept.