Price address negations in uniform signed SWAP words

research/integer-address-signed-swap/PREDECESSORS.json

@@ -0,0 +1,14 @@
+{
+  "schema_version": 1,
+  "files": [
+    {
+      "path": "research/integer-address-signed-swap/predecessor-2544.md",
+      "usage": "input",
+      "git_blob": "e4647abcd7056dd205252f8d1e4d20368df66c9d",
+      "sha256": "bd9e1a3346adbb8b99a30eaa2819006760adff5258f629b2ffc628cc397e93c4",
+      "source_commit": "9fb1552f12ca4777ca660461a8b602e709285f99",
+      "source_path": "research/gl2-swap-weighted-tariff/README.md",
+      "source_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2544"
+    }
+  ]
+}

research/integer-address-signed-swap/PROOF.md

@@ -0,0 +1,166 @@
+# Integer address SWAPs need a separately charged sign
+
+## Exact model
+
+Work in `M_R(F2[GL2(Z)])`. Each of the R scalar roles is an arbitrary dirty
+binary stream. Address matrices act on its two integer coordinates; a NEG
+below negates one **address coordinate**, not a binary payload value.
+The paid labels are
+
+```
+a+ = [1 1;0 1],  a- = [1 -1;0 1],
+b+ = [1 0;1 1],  b- = [1 0;-1 1],
+nx = [-1 0;0 1], ny = [1 0;0 -1].
+```
+
+For any binary rank-one idempotent `N=uv^T`, with `v^T u=1`, a call is
+`D_N(g)=I_R+([g]+1)N`. The prices of a+/a- are A, those of b+/b- are B,
+and those of nx/ny are C. A,B,C are fixed nonnegative real numbers,
+independent of role count, projector, state and word position. Arbitrary
+scalar GL_R(F2) operations may be interleaved for free. No other address
+map, clean input, reset, erasure, higher-rank paid unit or free restoration
+is provided. Every actual inverse call is charged.
+
+Let r roles have any mixture of the four signed anti-diagonal targets
+
+```
+ S=[0 1;1 0],  -S=[0 -1;-1 0],
+ J=[0 1;-1 0], -J=[0 -1;1 0].
+```
+
+Let t of them be S or -S (determinant -1). The other r-t targets are J or
+-J (determinant +1). The complete endpoint is their diagonal direct sum,
+with identity on all R-r spectators and every off-diagonal coupling zero.
+Assume finite R>=r>=0. Equality is coefficientwise in the full formal group
+algebra, not equality only after applying a nonfaithful representation.
+
+**Theorem.** The exact minimum price is
+
+```
+                  r*(2*min(A,B)+max(A,B)) + t*C.
+```
+
+At unit prices, this is 4t+3(r-t). The zero-target case is the empty word.
+This is a sharp tariff in this specified relaxation, not a physical tape
+invoice, primitive elapsed time, recursive supplier or new exponent.
+
+## The credited ADD bounds survive the integer extension
+
+Reduction modulo2 is a group homomorphism `GL2(Z) -> GL2(F2)`. Both signs of
+a shear have the same image; nx and ny have identity image. All four targets
+map to coordinate SWAP. The induced group-algebra map preserves a formal
+identity and the scalar projectors, while deleting every address NEG call.
+
+Thus PR2544's credited individual crossing bounds and PR2415/PR1989's total
+bound apply to the remaining word, with the same arbitrary scalar mixing
+and restored dirty spectators:
+
+```
+                  n_a >= r, n_b >= r, n_a+n_b >= 3r.       (1)
+```
+
+For clarity, each individual bound uses the upper/lower crossing block of
+the natural two-dimensional representation. In address-major order, a paid
+a has image `[I N;0 I]`, a paid b has `[I 0;N I]`, scalar mixing is block
+diagonal, and each target has both crossing ranks one per target role.
+
+The total bound acts on the three nonzero binary vectors, ordered
+`(1,0),(1,1),(0,1)`. a and b are the two adjacent transpositions. Their
+action on ordered distinct pairs splits increasing and decreasing pairs:
+each paid shear has crossing rank one; SWAP has crossing rank three.
+This is the predecessor's ordered-pair argument, not a new proof principle.
+
+All these crossing ranks are subadditive because for a fixed decomposition
+`(UV)_12=U_11 V_12+U_12 V_22`, hence `rank((UV)_12)` is at most the sum of
+the two crossing ranks. No intermediate triangularity or commuting
+projectors are assumed.
+
+## A disjoint determinant-character charge
+
+Take the homomorphism `det:GL2(Z)->{+1,-1}`, then its two-point permutation
+representation over F2. The +1 element acts as I2, while -1 swaps the two
+sheets. Extend it linearly to the full group algebra and tensor with roles.
+
+Every ADD has identity image, hence crossing rank zero. Every paid address
+NEG has image `[I+N N;N I+N]`, whose crossing block N has rank one. Free
+scalar mixing is `[Q 0;0 Q]` and crosses zero. At the endpoint, exactly the
+t negative-determinant targets exchange sheets, with crossing rank t.
+The same subadditivity therefore proves
+
+```
+                              n_NEG >= t.                (2)
+```
+
+This is a separate necessary inequality. The ADD cuts charge no NEGs, and
+the determinant cut charges no ADDs. Summing their weighted consequences
+does not charge one instruction twice. In particular, even when an even
+number of target swaps has overall determinant +1, its block crossing rank
+is t rather than merely the parity of t. Scalar role mixing cannot hide it.
+
+For A<=B, (1) gives
+`A*n_a+B*n_b=A*(n_a+n_b)+(B-A)*n_b >= r*(2A+B)`.
+The other ordering is symmetric. Adding C times (2) proves the lower bound,
+including zero prices. Negative prices are excluded substantively: for R>=1,
+a paid label followed by its charged inverse is a formal identity loop;
+if its price is negative, arbitrary repetitions make the cost unbounded below.
+
+## Literal integer attaining words
+
+Use left-to-right matrix multiplication in the displayed words. Direct
+integer multiplication gives
+
+```
+J  = a+ b- a+ = b- a+ b-,
+-J = a- b+ a- = b+ a- b+,
+S  = J nx,             -S = J ny.
+```
+
+Choose the cheaper three-shear orientation for each target and append the
+one indicated NEG exactly for negative-determinant targets. On role j use
+N=E_jj. Each factor changes only that role, so the full formal product has
+exactly the prescribed monomial there and identity elsewhere. Concatenating
+over r roles attains (1) and (2) jointly. It needs no extra role, scalar
+preparation, inter-role copying or uncharged restoration. This verifies the
+entire dirty endpoint, not only a natural-representation image.
+
+## Uniform address realization and its boundary
+
+Each attaining integer matrix word reduces to the same signed permutation
+on `(Z/MZ)^2` for every M>=2. Every label is a unit coordinate ADD or address
+NEG; neither whole-SWAP nor free coordinate relabeling is an extra paid label.
+This supplies literal uniform upper identities, not their physical compiler,
+prefix/suffix layout, word representation or recursively weighted cost.
+
+The lower bound is for formal GL2(Z) identities, equivalently for a fixed
+finite word that realizes the complete stream endpoint on integer addresses.
+To see faithfulness of the latter implication, for any finite set of distinct
+integer matrices choose v=(1,z) outside the finitely many collision values
+of z. Each nonzero matrix difference kills at most one such v. The distinct
+images then separate all coefficients of every finite group-algebra entry.
+The same separation survives reduction modulo any sufficiently large M,
+since a fixed finite list of nonzero integer differences cannot all vanish
+modulo arbitrarily large M. Thus a fixed finite word valid at every modulus
+in an unbounded family also satisfies the formal identity.
+
+This is **not** a lower bound for every separately optimized word at one
+finite modulus. At M=2, both address NEG labels become identity and J=S,
+so three shears already swap coordinates. Nor does the theorem establish
+admission for width-dependent alphabets, varying coefficient labels, changing
+role counts, restricted input subspaces or incomplete spectator restoration.
+
+The quotient images are lower-bound witnesses only. For example a+ squared
+is a nonidentity integer matrix with identity image in both the mod2 and
+determinant quotients. Adding `[I]+[(a+)^2]` to an endpoint entry is therefore
+invisible to all four cuts but changes the full formal map. The verifier
+uses exact formal coefficients for endpoints and checks this control.
+
+## Credit
+
+PR2544 at `9fb1552f12ca4777ca660461a8b602e709285f99` supplies the asymmetric
+GL2(F2) tariff and its individual label counts, crediting Aaron Wang's
+PR2444 handoff and Ross Richardson's PR1989/PR2415 total crossing bound.
+Those bounds and standard rank subadditivity are reused, not claimed new.
+The contribution is their source-bound integer-address extension with a
+separately paid determinant-sign requirement, mixed signed endpoints and
+the explicit finite-modulus boundary. Prepared for ywz65 with substantial
+OpenAI Codex assistance, Apache-2.0. No human or formal-kernel review claimed.

research/integer-address-signed-swap/README.md

@@ -0,0 +1,87 @@
+# Price address signs in uniform integer SWAP words
+
+For any mixture of `r` signed anti-diagonal targets in formal
+`F2[GL2(Z)]`, with `t` of determinant -1, the exact minimum price is
+
+```
+r*(2*min(A,B)+max(A,B)) + t*C.
+```
+
+Here the two unit shear orientations, with either sign, cost A and B;
+negating either **address coordinate** costs C. Prices are nonnegative.
+All R>=r roles begin arbitrarily, scalar `GL_R(F2)` mixing is free, and all
+spectators and off-diagonal couplings must be restored. A paid unit acts on
+one binary rank-one scalar idempotent, not an arbitrary-rank block.
+At unit prices, SWAP and minus-SWAP cost four each; the two determinant-positive
+quarter turns cost three each. A binary payload sign is not an address NEG.
+
+[PROOF.md](PROOF.md) gives the all-R, all-target, all-nonnegative-price theorem.
+The inherited mod2 ADD cuts give `n_a>=r`, `n_b>=r`, `n_a+n_b>=3r`.
+A separate two-sheet determinant representation gives `n_NEG>=t`: even when
+t is even, the crossing rank sees t, not just its parity. The charges are
+disjoint. Literal signed three-shear words, with one NEG where needed, attain
+all inequalities together on complete formal endpoints.
+
+This extends [PR2544](https://github.com/CrocSwap/integer-mult-bounds/pull/2544)
+at `9fb1552f12ca4777ca660461a8b602e709285f99`; its asymmetric mod2 tariff is
+credited, not rediscovered. That predecessor credits Aaron Wang's PR2444
+handoff and Ross Richardson's PR1989 general ordered-pair cut and PR2415
+coordinate-SWAP bridge. Its unchanged README is retained as
+[predecessor-2544.md](predecessor-2544.md) for source-bound review; no source
+implementation is imported. [PREDECESSORS.json](PREDECESSORS.json) pins it;
+[SOURCES.json](SOURCES.json) pins the unchanged main ordered-pair proof,
+with SHA-256 and Git blob IDs.
+
+## Domain boundary
+
+Formal equality is also forced by a **fixed finite word** valid on full integer
+streams, or at every modulus in an unbounded family; the proof supplies the
+finite-support separation argument. The attaining identities work modulo
+every M>=2. The lower bound does **not** apply to every word optimized
+separately at a fixed finite M. At M=2 address NEG is identity and three shears
+already implement SWAP. Restricted clean inputs, width-dependent alphabets,
+higher-rank units, free address maps and incomplete restoration are outside
+the model. No physical tape compiler, completed recursive supplier, selected
+frontier change or integer-multiplication theorem follows.
+
+## Reproduce and evidence
+
+Python 3.12+, standard library, from the repository root:
+
+```sh
+python3 -I -O -B research/integer-address-signed-swap/verify.py
+python3 -I -O -B research/integer-address-signed-swap/test_boundary.py
+python3 scripts/research_package.py --package research/integer-address-signed-swap
+```
+
+The fixed [RESULT.json](RESULT.json) records 930 rank-one paid-factor cut checks
+(all binary projectors for R=1..4), 20 free scalar factors, 1,364 complete formal
+endpoints (every mixture of up to four targets, two orientations and zero/one
+spectator), 42,625 exact rational price checks and 720 deterministic mixed
+paid/free prefixes. These bounded checks support the written proof; they are
+not an exhaustive word search or a proof assistant formalization.
+
+Controls detect an omitted NEG invisible to mod2 cuts, a nonzero formal
+perturbation invisible to **all** quotient representations, fourteen deleted
+attaining calls and a nonidempotent projector. Separate controls exhibit the
+M=2 exception, negative-price identity loops and the changed higher-rank-unit
+premise. Six subprocess tests check unchanged Git-free exports, source drift,
+missing sources/receipt, typed receipt drift and exclusive regeneration.
+
+Verification is read-only. `--write-result NEW_PATH` regenerates a complete
+receipt and refuses an existing path. Explicit failures remain active under
+`-O`. Runtime provenance and independent AI review are recorded with delivery;
+hosted CI and independent human review are not asserted.
+
+## Open next questions
+
+1. Establish a lower bound for one named physical finite-modulus family at its
+   actual widths. Start by pinning the native alphabet and proving its action
+   faithful on the allowed word supports; M=2 blocks an unconditional transfer.
+2. Price the literal signed words in a legal tape representation. Start with
+   all ADD/NEG calls, temporary roles and restoration; native compilation,
+   all-size layout and recursive eligibility remain blockers.
+
+The structured obligations and original picker scope change are in
+[TASK.json](TASK.json). Contributor: ywz65 with substantial OpenAI Codex
+assistance. Apache-2.0; predecessor attribution is retained.

research/integer-address-signed-swap/RESULT.json

@@ -0,0 +1,22 @@
+{
+  "all_register_proof": "PROOF.md",
+  "claim_kind": "finite_checks_for_scoped_proof",
+  "controls": {
+    "all_quotients_miss_nonzero_formal_perturbation": true,
+    "finite_modulus_two_breaks_formal_lower_bound": true,
+    "fourteen_deleted_attainer_calls_rejected": true,
+    "higher_rank_unit_would_change_model": true,
+    "nonidempotent_projector_rejected": true,
+    "omitted_neg_detected_only_by_sign_cut": true,
+    "six_negative_price_identity_loops": true
+  },
+  "exact_rational_price_cases": 42625,
+  "formula": "r*(2*min(A,B)+max(A,B))+t*C",
+  "free_scalar_factor_cases": 20,
+  "full_formal_endpoint_cases": 1364,
+  "mixed_word_prefix_cases": 720,
+  "new_selected_exponent": false,
+  "rank_one_paid_factor_cases": 930,
+  "schema_version": 1,
+  "source_files": 3
+}

research/integer-address-signed-swap/SOURCES.json

@@ -0,0 +1,31 @@
+{
+  "schema_version": 1,
+  "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "e8870e5917057d59b730eb14721a778639eca050",
+      "sha256": "d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7"
+    },
+    {
+      "path": "research/symmetric-adjacent-cut-bound/README.md",
+      "usage": "input",
+      "git_blob": "cdb3f431acc07b4a75abdd3998093fd68d9a498f",
+      "sha256": "4fb453c83d7cd15a93fc7c65b82f2f67cb609a31285ac9ebd1f44b675777d154"
+    },
+    {
+      "path": "research/symmetric-adjacent-cut-bound/PROOF.md",
+      "usage": "input",
+      "git_blob": "30dad3856246c9cef3b8014ee967dd545f3a092d",
+      "sha256": "008be68867a8e85f4e18455d206e03f20aebfc28e9f08efba576180e7e1edc50"
+    }
+  ],
+  "retained_source_note": "Only the cited predecessor README is retained for Git-free review, unchanged with its credit and Apache-2.0 notice; no predecessor implementation is imported."
+}