Prove sharp two-for-one terminal rank capacity (no κ)

research/bit-terminal-two-for-one/PROOF.md

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+# Sharp terminal capacity with changed unmarked arrivals
+
+This theorem concerns a binary rank relaxation. It constructs synthetic source
+libraries. It neither implements a native operation nor changes the selected frontier.
+
+## Model
+
+Work over F_2. An old terminal basis of W consists of n unmarked vectors
+u_1,...,u_n, b base vectors a_1,...,a_b, and s singleton vectors p_1,...,p_s.
+Write U=span(u), B=span(a), and m=b+s. A marked payload changes from v to v+d_v.
+The fixed allowed difference libraries obey
+
+* d_a belongs to U+B for each of the b base positions;
+* d_pi belongs to U+span(p_i) for each singleton position.
+
+These are containment hypotheses. Libraries may be proper subspaces. At most h
+of the n unmarked arrivals may change arbitrarily within W. All n+m final
+arrivals must form a basis of W. Let U' span the n final unmarked arrivals.
+Return stock is the dimension of the span of the marked differences in W/U'.
+Reconstruction from the final basis is free. The model promises no chronological
+program, restored named output or paid realization.
+
+## Upper bound, credited context
+
+ToikY's [PR #7258 NEXT.md](https://github.com/ToikY/integer-mult-bounds/blob/90a63fbefcf79c3e0c0453ef0741dff8517b6167/research/bit-fixed-payload-quotient-bound/NEXT.md)
+gives the b=12, s=10 instance of this dimension bound. Its literal library census
+and prices are cited results, not verified here. The short argument below fixes
+the exact abstraction for the new sharpness theorem.
+
+Let k count singleton differences whose image in W/U is p_i. Their new marked
+arrivals vanish in W/U, since p_i+p_i=0. Other marked arrivals have joint quotient
+rank at most b+s-k. The h changed unmarked arrivals add at most h quotient
+dimensions. Full terminal rank implies k<=h.
+
+All differences have rank at most b+k modulo U. Both U and U' have dimension n
+and share the n-h unchanged unmarked basis vectors. Thus dim(U/(U intersect U'))<=h.
+For every subspace X of W,
+
+    dim((X+U')/U') <= dim((X+U)/U) + dim(U/(U intersect U')).
+
+Indeed, project (X+U+U')/U' onto (X+U+U')/(U+U'). Its kernel has dimension
+dim(U/(U intersect U')); its image is a quotient of (X+U)/U. Apply this to the
+span of the differences. Return stock is at most b+k+h, hence min(m,b+2h).
+
+F_2 matters. Over F_3 even a lone marked position with d=p has a nonzero new
+arrival 2p, without any changed unmarked arrival.
+
+## New attaining construction
+
+For every even b>=0, every n,s>=0 and 0<=h<=n, the upper bound
+
+    maximum return stock = min(b+s, b+2h)
+
+is attained with the fixed maximal libraries U+B and U+span(p_i). The same
+libraries serve every budget h. Only the chosen payloads and unmarked arrivals
+vary. There is no stronger universal bound from these dimension and singleton
+containment hypotheses alone.
+
+For one unmarked vector u and two singleton vectors p,q, choose
+
+    old unmarked u -> new unmarked p+q
+    marked p      -> new marked u       with d_p=p+u
+    marked q      -> new marked q+u     with d_q=u.
+
+Both differences belong to their fixed singleton libraries. The new arrivals
+(p+q,u,q+u) are a basis. From their values (x,y,z), recover
+
+    u=y, q=z+y, p=x+z+y.
+
+Modulo U'=span(p+q), the differences p+u and u are independent. In an equality
+alpha(p+u)+beta u=gamma(p+q), the q coefficient forces gamma=0, then the p
+coefficient forces alpha=0, and finally beta=0. One changed unmarked arrival
+therefore earns two independent difference directions.
+
+For one remaining singleton p, use u->p and p->u, with d_p=p+u. This earns one
+direction modulo the new unmarked span. Leave other singletons and unmarked
+arrivals unchanged. Use min(h,floor(s/2)) disjoint three-vector gadgets, followed
+by the two-vector gadget only when s is odd and budget remains. They earn exactly
+min(s,2h) directions. Disjoint supports make both the endpoint bases and quotient
+directions combine by direct sum.
+
+For every pair of base vectors a,c, choose d_a=c and d_c=a+c. New arrivals are
+a+c and a. Both difference and arrival pairs are bases. These disjoint blocks earn
+all b base directions with no unmarked changes and no intersection with the
+singleton gadgets. This proves the formula.
+
+## The 523+22 shape and its boundary
+
+For n=523, b=12 and s=10, the synthetic construction has terminal rank 545 and
+return stocks 12,14,16,18,20,22 at budgets h=0,1,2,3,4,5. All 523 new unmarked
+arrivals remain independent. No extra arrival slot is used.
+
+It can retain the base support partition 3+3+2+4 reported by #7258. A three-vector
+base block uses differences (c,d,a+c) on (a,c,d). New arrivals are
+(a+c,c+d,a+c+d); both triples have determinant one. The two-vector block above
+and its direct sum handle sizes two and four. The partition itself therefore
+cannot exclude the synthetic attainment.
+
+Five changed unmarked arrivals suffice for stock 21 or 22 in this abstraction,
+and four cannot suffice. This proves sharpness of the source's dimension-only
+necessary condition. It does not assert that five actual BIT arrivals have these
+payloads, that actual singleton libraries contain the required u components, or
+that free basis reconstruction has an affordable legal caller. Those are the next
+discriminating constraints. This package does not enumerate the source's 523 ports
+or recheck its stock values.
+
+## Finite corroboration and implementation bounds
+
+The verifier checks every constructed endpoint and quotient stock by binary
+elimination. Small instances also use an independent span-cardinality oracle.
+It enumerates every unmarked replacement and allowed difference in ten complete
+small maximal-singleton models. These are complete finite certificates for their
+declared domains; the written direct-sum argument supplies the all-size scoped proof.
+
+Every loop has a finite domain. Ambient dimension is at most 545; rank calls have
+at most 1,090 rows. Span sets have at most 64 elements in the construction checks
+and 32 in the complete models. The largest complete model has 16,384 assignments. The construction
+sweep fixes n<=4, b in {0,2,4,6,12}, s<=10, h<=n, then checks six 545-dimensional
+witnesses. Functions stay below 60 lines. Explicit exceptions keep checks active
+under optimized Python. Python integer, container, JSON and hashing allocation is
+a managed-runtime deviation from fixed steady-state allocation; the finite
+dimensions above bound live mathematical state. No recursion, dynamic code,
+external process or third-party dependency is used. Warning-strict execution
+supplies interpreted diagnostics.

research/bit-terminal-two-for-one/README.md

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+# Sharp two-for-one terminal exchange
+
+Scoped written proof with exact finite corroboration, produced by Nick Petalas.
+
+In the binary terminal-rank relaxation with n unmarked arrivals, b base marked
+directions, s singleton marked directions and at most h changed unmarked
+arrivals, the universal stock bound min(b+s,b+2h) is sharp for every even b.
+One three-vector exchange earns two independent marked differences while
+changing one unmarked arrival. Fixed libraries suffice for every budget.
+
+For the synthetic 523+22 shape with b=12 and s=10, five changed unmarked
+arrivals attain stock 22 and preserve terminal rank 545. The construction also
+respects the base support partition 3+3+2+4. Consequently the dimension-only
+argument in [ToikY #7258](https://github.com/CrocSwap/integer-mult-bounds/pull/7258)
+cannot raise its five-arrival necessary condition. An obstruction beyond it must
+use actual source libraries, donor projections, scheduling or costs.
+
+These are synthetic libraries. No claim of stock 22 on the literal BIT word,
+physical delivery, native admission, complete multiplication theorem or new
+kappa follows. The selected frontier and its five assumptions are unchanged.
+
+The [proof](PROOF.md) defines the relaxation, credits the predecessor upper bound,
+and proves attainment. The standalone verifier constructs endpoints, checks full
+rank and quotient rank, and exhausts every assignment in ten small singleton
+models with an independent span oracle. It generates a receipt with source
+hashes. No predecessor scientific verifier runs.
+
+Local CPython 3.12.15 verification covers 825 small construction cases, six
+545-dimensional witnesses and all 26,286 assignments in the ten complete models.
+Four mathematical controls and three invalid-parameter controls pass.
+
+## Reproduce
+
+From the repository root, using CPython 3.12:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error scripts/research_package.py --package research/bit-terminal-two-for-one
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/bit-terminal-two-for-one/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/bit-terminal-two-for-one --check-modes --check-source-history
+```
+
+Registered normal and optimized checks have 120-second bounds. Direct verification
+prints its generated hash-bound receipt. An optional `--receipt` path must be
+outside the checkout. Runtime needs only the standard library and package files;
+the source-history check additionally needs full Git history.
+
+## Open next questions
+
+Determine whether funded changed paths or expanded marked libraries can realize
+even one three-vector gadget with the required shared U component and quotient
+projection. Concurrent [#7260](https://github.com/CrocSwap/integer-mult-bounds/pull/7260)
+studies actual old-source donor constraints, unlike this maximal-library relaxation.
+Use its source-pinned donor map as a cited starting point rather than repeating
+its census. A legal paid schedule, joint endpoint, ownership, restoration and
+the complete invoice remain separate obligations.
+The structured question is in [TASK.json](TASK.json).
+
+## Credit and source context
+
+The new exchange, sharpness theorem and verifier were produced by Nick Petalas.
+Credit ToikY's #7258, exact head `90a63fbefcf79c3e0c0453ef0741dff8517b6167`,
+for the 12+2h upper bound, singleton/base partition and altered-unmarked-arrival
+route. Its census and prices are cited inputs, not deliverables rechecked here.
+Credit Jerry Chen / CubeJerry #7001/#6726 and their retained contributors for the
+BIT word motivating that route, and Clear1nsight #7244 for its complete comparison.
+Binary elimination, rank-nullity and direct sums are classical methods.
+
+[SOURCES.json](SOURCES.json) records immutable main provenance and the exact public
+predecessor path and hash. There are no external runtime inputs. Independent human
+and formal proof-kernel review are unclaimed. New work is Apache-2.0.

research/bit-terminal-two-for-one/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "75d77b328ea02ded535e7f23aee5b1e55d0bb1a8",
+      "sha256": "154c3d30455d778fd3343b6b4851f0fa4665f0f7625b0adb3675e2bafbbf6bf1"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    }
+  ],
+  "external_provenance": [
+    {
+      "repository": "ToikY/integer-mult-bounds",
+      "commit": "90a63fbefcf79c3e0c0453ef0741dff8517b6167",
+      "path": "research/bit-fixed-payload-quotient-bound/NEXT.md",
+      "url": "https://github.com/ToikY/integer-mult-bounds/blob/90a63fbefcf79c3e0c0453ef0741dff8517b6167/research/bit-fixed-payload-quotient-bound/NEXT.md",
+      "sha256": "67e11381abef156fb3d24da94c53a4ee74fbbc2140638be98278a49c67b8a364",
+      "usage": "credited mathematical context; no runtime import or predecessor replay"
+    }
+  ]
+}

research/bit-terminal-two-for-one/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "bit-helper-contract",
+  "priority_revision": 8,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "original_card": {
+    "id": "bit-helper-contract",
+    "question": "Can a changed G20 helper path or independent-target split improve the complete BIT moment without losing its physical endpoint?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+      "research/g20-helper-split/README.md",
+      "research/g20-conjugation-admission/README.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "A full boundary contract and paid profile; unchanged scalar counts alone are insufficient. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "First read #5679/#5708 and their audits as pending component proposals. Choose one uncovered helper group/cut on an exact word; compare arbitrary-input native action, targets, owned storage, both orientations and full signed bill. Stop if the gain is only a scalar count or repaid by lost targets. Do not rerun the frozen #5609 common-frame domain without changing its hypothesis.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "G20",
+      "helper split",
+      "5679",
+      "5708",
+      "independent target"
+    ],
+    "existing_checks": [
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Prove sharpness of the binary dimension-only terminal stock bound min(b+s,b+2h) by a fixed-library exchange construction.",
+  "scope_change_reason": "The predecessor leaves altered unmarked arrivals open. This isolates the rank relaxation before any actual donor census, rather than auditing the predecessor or attempting an unpriced native construction.",
+  "bottleneck_link": "A stronger BIT terminal obstruction must use actual source libraries, donor projections or costs if the dimension-only relaxation is sharp. This is a local interface result, not a complete supplier.",
+  "budget_minutes": 60,
+  "stop_rule": "Ship the first verified exchange and sharpness theorem; do not extend to the actual port census.",
+  "outcome": "completed",
+  "evidence": "PROOF.md and the standalone exhaustive verifier. Generated hash-bound receipts print to stdout or an external path.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Can funded changed paths or expanded marked libraries realize a three-vector exchange with the required shared U component and distinct terminal ports?",
+      "why_it_matters": "The abstract dimension bound cannot distinguish this route from a feasible rank endpoint.",
+      "first_step": "Use the exact source-constrained donor map in concurrent PR7260 as a cited input, then test funded new paths or changed marked libraries against the explicit gadget and full endpoint. Do not repeat its census.",
+      "blocked_by": "Actual fixed marked libraries, donor projections, eligible distinct funded slots, legal chronology, ownership, restoration and complete costs."
+    }
+  ]
+}

research/bit-terminal-two-for-one/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "-W",
+        "error",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Construct full-rank terminal bases and sharp quotient stock; exhaust ten small singleton models with independent elimination/span ranks; reject invalid parameters and lost endpoint rank."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "-O",
+        "-W",
+        "error",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Repeat every mathematical check with optimized Python; explicit exceptions must retain all checks."
+    }
+  ]
+}