Prove the exact reader-497 single-merge cache floor (no κ)

research/p104-single-merge-cache-floor/PROOF.md

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+# Exact tariff for a single merged operand cache
+
+This scoped proof is produced by Nick Petalas. Apache-2.0.
+
+## Model
+
+Let the frame spaces be over any field. Write S,U,H,L,R for subspaces with
+
+    dim S=s, dim U=u, dim H=dim R=u+1, dim L=1,
+    1 <= s < u, H intersection R=U, S,L <= U, L not contained in S.
+
+The field quantifier concerns frame geometry. Every nested MOVE is granted
+as an available operation in the relaxation; native availability over an
+arbitrary field is not asserted. The separate scalar payload check uses rational numbers.
+
+The reader ascends from zero to R. The root descends from H to L and is read
+once at F. Two independent operands V and B are captured at S and U. Each
+owned cache descends to one common merge frame M; one cache forms V+B there.
+The sum cache then descends to A and is delivered once to the reader. The
+reader also receives -4Q from the root at F. The required logical endpoint is
+A_in+V+B-4Q, with all public sources unchanged. Coefficients are scalar and
+fixed. The argument does not require independence in the actual source word.
+Independence defines this local replacement contract.
+
+Every nested native MOVE of rank r costs phi(r). Here phi(0)=0 and phi is
+nonnegative, increasing and concave on nonnegative integer ranks. All native
+cache transport is charged. Waiting, clean owned storage, capture, the merge,
+disposal and ordinary control are free in this deliberately favorable model.
+Disposal can discard arbitrary cache contents. This is a tariff relaxation,
+not a physical erasure or native implementation claim.
+
+There is exactly one merge and one subsequent sum delivery. There are no
+additional reads, split deliveries, public ascending caches, cloned roots,
+changed productive targets or coordinated removal of other work. Intermediate
+stops on each monotone path are unrestricted. Scalar logical payload checks
+do not establish the native meaning of those paths.
+
+Jerry Chen / CubeJerry's PR6248 supplies the question of mixed operands and
+the inherited reader-497 geometry. Its separate-cache theorem is not replayed
+or reproved here. The new model merges the sources before delivery.
+
+## Reduction to three dimensions
+
+Both pre-merge caches descend, so M<=S<=U. The sum descends to A<=M. Since
+L<=F<=H and F lies on the reader path to R, L<=F<=U. Reader contacts A,F
+are comparable. F cannot be below or equal to A: that would give L<=S.
+Thus A<F. Put a=dim A, m=dim M, f=dim F. Exactly the following domain remains:
+
+    0 <= a <= m <= s, a < f <= u.
+
+Every triple is possible. Choose an a-space A in S, extend it to an m-space
+M in S, and extend A+L to an f-space F in U. Only A needs to lie in F; M
+need not. Longer paths cannot be cheaper than the compressed ones, since
+concavity with phi(0)=0 implies subadditivity.
+
+The compressed paid ranks, including zero ranks of zero cost, are
+
+    reader:     a, f-a, u+1-f
+    root:       u+1-f, f-1
+    V cache:    s-m
+    B cache:    u-m
+    sum cache:  m-a.
+
+Write C(a,m,f) for their total tariff. The unchanged schedule has bill
+
+    C0 = 2phi(1)+phi(s)+phi(u-s)+phi(u-1).
+
+It receives V at S, receives B and the root at U, and preserves the same
+reader and root endpoints.
+
+This is a fixed comparison profile, not an inference from the predecessor's
+optimality claim.
+
+## The exact minimum
+
+Define
+
+    C_low  = phi(1)+2phi(u)+phi(s)+phi(u-s),
+    C_high = 2phi(1)+phi(s)+2phi(u-s)+phi(u-1).
+
+Then the minimum over all single-merge itineraries is min(C_low,C_high).
+
+If f=1, then a=0. Subadditivity and monotonicity give
+
+    phi(s-m)+phi(m) >= phi(s),
+    phi(u-m) >= phi(u-s).
+
+The remaining terms are phi(1)+2phi(u). Therefore C>=C_low. Equality is
+attained at (a,m,f)=(0,s,1), with M=S, A=0 and F=L.
+
+If f>=2, both root segment ranks lie between 1 and u-1 and sum to u.
+Concavity minimizes their sum at the two extreme ranks:
+
+    phi(u+1-f)+phi(f-1) >= phi(1)+phi(u-1).
+
+Group the other terms as follows:
+
+    phi(a)+phi(s-m)+phi(m-a) >= phi(s),
+    phi(f-a)+phi(u+1-f) >= phi(1)+phi(u-a),
+    phi(u-a) >= phi(u-s),
+    phi(u-m) >= phi(u-s).
+
+For the second inequality, its two arguments are at least one and sum to
+u+1-a. Concavity again places the minimum at the extremes 1,u-a. Adding
+the inequalities gives C>=C_high. Equality is attained at
+(a,m,f)=(s,s,u), with A=M=S and F=U. This proves the claimed exact minimum
+without enumerating fields, subspaces or path lengths.
+
+The excess over the unchanged schedule is exactly
+
+    min(2phi(u)-phi(1)-phi(u-1), phi(u-s)).
+
+Both terms are nonnegative for increasing phi. For phi(r)=r^t, 0<t<1,
+both are strictly positive: u>1, and u-s>=1. Thus no itinerary in this
+single-merge class improves the complete signed moment when other work and
+productive targets stay fixed. Any additional nonnegative physical charge
+only increases this obstruction.
+
+## Reader 497 and finite evidence
+
+PR6248 identifies s=3,u=6 for frames S=519,U=520,H=529,L=595,R=914 in
+Ali Romman's PR6167. These source identities and inclusions are inherited
+premises, not a new audit of the word. The new exact minimum is
+
+    min(H1+2H3+2H6, 2H1+3H3+H5),
+
+whereas C0=2H1+2H3+H5. The excess is
+
+    min(2H6-H1-H5, H3) > 0, for H_r=r^t and 0<t<1.
+
+There are 50 dimension recipes. The verifier constructs a binary flag witness
+for each, independently extracts all native rank differences, and checks each
+profile's capped sums against its f=1 or f>=2 endpoint. Since every increasing
+concave tariff is a nonnegative combination of min(r,k) on a bounded integer
+domain, these integer certificates corroborate the complete tariff comparison.
+The same finite check covers 2<=u<=14 and 1<=s<u. These finite checks support
+the written universal proof; they do not replace it.
+
+The normalized scalar payload is checked on every public basis column and on
+rational dirty examples, including nonzero incoming reader values. Those
+checks illustrate the favorable merge/disposal contract only. They are not
+an operator, phase, ownership, caller, compiler or rounded multiplier proof.

research/p104-single-merge-cache-floor/README.md

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+# A single merged cache cannot improve reader 497
+
+Scoped written proof with exact finite corroboration. No new kappa.
+
+Mixing the two operands does not evade the reader-497 obstruction in the
+single-merge model defined below. For H_r=r^t, 0<t<1, its exact optimistic
+minimum is
+
+    min(H1+2H3+2H6, 2H1+3H3+H5).
+
+The unchanged bill is 2H1+2H3+H5. The exact excess is
+
+    min(2H6-H1-H5, H3) > 0.
+
+This closes one mixed-cache route explicitly left outside Jerry Chen /
+CubeJerry's [PR6248](https://github.com/CrocSwap/integer-mult-bounds/pull/6248).
+That result keeps the two caches separate. Here they descend to a common
+frame, form V+B once, and deliver that sum to the ascending reader. Every
+native cache MOVE is charged. This is a new representation and tariff proof,
+not a replay of the predecessor's numbers.
+
+## General result and scope
+
+Let S,U,H,L,R have dimensions s,u,u+1,1,u+1, with 1<=s<u,
+H intersection R=U, S,L<=U and L not contained in S. Operand caches start at
+S and U, descend to M<=S and merge once. Their sum descends to A<=M and is
+received once. The reader ascends from zero to R. The uncloned root descends
+from H to L and meets the reader once at F. The public logical endpoint is
+A_in+V+B-4Q, with arbitrary dirty public inputs preserved as specified.
+
+For any nonnegative increasing concave tariff phi with phi(0)=0, the exact
+minimum in this relaxation is
+
+    min(phi(1)+2phi(u)+phi(s)+phi(u-s),
+        2phi(1)+phi(s)+2phi(u-s)+phi(u-1)).
+
+The proof covers every field, permitted subspace and finite monotone path.
+Its two attaining flag recipes are (a,m,f)=(0,s,1) and (s,s,u), where a,m,f
+are the dimensions of A,M,F. [PROOF.md](PROOF.md) gives the argument.
+
+Waiting, capture, clean storage, merging, disposal and ordinary control are
+optimistically free. Disposal may discard arbitrary cache values. Therefore
+attainment is an attainment of this tariff relaxation, not a physical erasure
+program. Extra nonnegative implementation costs cannot repair its loss.
+
+No extra source reads, repeated merging/splitting, public ascending caches,
+root clones, changed targets or coordinated deletions are allowed. More
+general linear recoding is not excluded. The source geometry in PR6248 is
+inherited, not independently audited; this package does not replay PR6167.
+Native phases, physical delivery, owned erasure, caller, compiler, precision,
+tape and analytic recovery remain unproved here. The selected frontier and
+its five framework assumptions are unchanged.
+
+## Evidence and reproduction
+
+The standard-library verifier constructs all 50 dimension recipes for s=3,u=6.
+It extracts MOVE ranks from explicit binary flag witnesses independently of
+the symbolic invoice formula. All six capped sums for each recipe dominate
+the appropriate attaining endpoint. This proves a finite concave-order
+certificate for those recipes. Broader corroboration checks 20,839 recipes
+over 91 dimension pairs with 2<=u<=14 and 1<=s<u. The all-dimension result
+rests on the written proof, not extrapolation from those checks.
+
+Four public basis columns and 81 rational dirty examples check the relaxed
+scalar payload. Thirteen controls test illegal incidence, unpaid transport,
+missing operands/root and lost incoming dirt or cache cleanup. These scalar
+checks do not verify native operators.
+
+From the repository root, using CPython 3.12:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/p104-single-merge-cache-floor/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/p104-single-merge-cache-floor/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-single-merge-cache-floor
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-single-merge-cache-floor --check-source-history
+```
+
+Both modes report `PASS_SINGLE_MERGE_CACHE_FLOOR`. An optional
+`--receipt /absolute/path/outside/checkout/receipt.json` writes the generated
+50-recipe certificate, interpreter version and hashes of all seven package
+files. Receipts, exports and run logs are not committed.
+[VALIDATION.md](VALIDATION.md) records actual checks and their limits.
+
+## Open next questions
+
+Can a cache representation with splitting and more than one merge change the
+complete signed bill? It would escape the single-delivery reduction and could
+change the useful next construction. Start with one chronological diagram
+whose values, source captures, all cache MOVEs and erasure are explicit.
+The blocker is a physically legal, fully charged delivery contract. No such
+construction is supplied or reserved here.
+
+## Sources, credit and coordination
+
+This contribution was produced by Nick Petalas. Apache-2.0. Independent human
+and formal proof-kernel review are unclaimed.
+
+Jerry Chen / CubeJerry's PR6248 at
+`9aaf76f43493cd2392eaae248274c651f29bc28e` supplies the mixed-source question
+and source geometry. Its proof is available at the
+[pinned source](https://github.com/CubeJerry/integer-mult-bounds/blob/9aaf76f43493cd2392eaae248274c651f29bc28e/research/p104-reader497-cache-floor/PROOF.md).
+Ali Romman retains [PR6167](https://github.com/CrocSwap/integer-mult-bounds/pull/6167)
+and the retained-operand construction at
+`564a076b113621a93edc4e39968ee4e4a8c9caf0`. DreamingOfClouds retains
+PR6144/PR6030/PR5907 and toolkit PR6170 credit. Matarener retains PR6214's
+independent audit and handoff. The native, caller, compiler, BIT and earlier
+P104 contributors retain their existing attribution. No predecessor source
+files or credit lines are changed.
+
+The scientific base is canonical main
+`2485805a6d47f36ea83ddde6327d453ac82d44dc`, priority revision 8. The selected
+binding-only orientation passes at conditional kappa .003125804. The
+[advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19291954)
+defines this scope. The 316-open-PR and 5,998-Discussion-record cache, main
+README and merged searches, recent account work and local reservations found
+no competing single-merge result. The sparse overlap helper searches 99 of
+8,722 main note records; separate Git README searches cover the omitted main
+files for the distinctive terms. This is bounded overlap coverage, not a
+proof of novelty. Frozen provenance is in [SOURCES.json](SOURCES.json).

research/p104-single-merge-cache-floor/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "2485805a6d47f36ea83ddde6327d453ac82d44dc",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/p104-retained-operand-descent/README.md",
+      "usage": "provenance",
+      "git_blob": "cbe390b9e7167b7aac019f52ed2c0c0707f8e7cc",
+      "sha256": "42704b4bf59b6f83ddc3a37c1e26d4c13d7e5f5da64fd6c1874e2391e6b7c4ff"
+    }
+  ]
+}

research/p104-single-merge-cache-floor/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 8,
+  "base_commit": "2485805a6d47f36ea83ddde6327d453ac82d44dc",
+  "original_card": {
+    "id": "complex-rank-shape-decoder",
+    "question": "Can a legal decoder or coframe change improve the full signed moment, including changed helper targets?",
+    "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",
+      "docs/research/recursive-invoice-preflight.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 source-bound arbitrary-input identity and legal producer/consumer boundary with the complete paid and productive target invoice, including restoration, ownership, native alignment and extra charges. Price under the actual current model; historical ceiling.py assumptions must be established before use. A favorable optimistic relaxation keeps a route open but is not a completed supplier. A scoped local result remains useful; expansion requires the family-level budget gate.",
+    "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": "Price one complete changed itinerary with both paid and productive histograms. Check #5693/#5698 for overlap and their exact hypotheses before another coframe search. Use the signed invoice preflight where its boundary contract agrees; an inconclusive concave comparison is not a power crossing. Stop if an optimistic complete budget fails at the chosen target or native delivery remains unspecified.",
+    "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": [
+      "coframe",
+      "signed moment",
+      "5693",
+      "5698"
+    ],
+    "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": "Exact increasing-concave rank tariff for two descending operand caches merged once before a single delivery to reader 497, and the general 1<=s<u geometry. Scoped proof with finite corroboration; no native implementation or new kappa.",
+  "scope_change_reason": "Take the mixed-source question explicitly outside PR6248. The single-merge representation changes that predecessor's separate-cache premise. This is a scoped cost obstruction, not a complete supplier or decoder construction.",
+  "bottleneck_link": "Eliminates one changed data-path proposal for the binding complex supplier: mixing two owned caches cannot pay for deleting reader 497's root meeting in this model. Other productive targets and paid work stay fixed.",
+  "budget_minutes": 60,
+  "stop_rule": "Package the first exact single-merge cost decision. Do not extend to repeated merging, split deliveries, root clones or a new supplier.",
+  "outcome": "completed",
+  "evidence": "PROOF.md proves the exact two-profile tariff for every field and 1<=s<u. verify.py checks 50 reader-497 recipes, 20,839 recipes over 91 bounded dimension pairs, 85 relaxed scalar payload cases and 13 controls. Source geometry and predecessor theorems are inherited, not independently audited.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Can splitting and repeated merging of caches improve the fully charged reader-497 signed bill?",
+      "why_it_matters": "That changes the single-delivery reduction and may permit data paths outside this obstruction.",
+      "first_step": "Specify one chronological source/capture/merge/split/delivery diagram, including every native cache MOVE and physical erasure.",
+      "blocked_by": "A physically legal complete consumer and cost contract are missing."
+    }
+  ],
+  "source_question": {
+    "pr": 6248,
+    "author": "CubeJerry",
+    "head": "9aaf76f43493cd2392eaae248274c651f29bc28e",
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/6248",
+    "proof_sha256": "7c011dd406a678d149b3632b22096175e75d1b329df92f788767bc8fbbc38e01",
+    "usage": "Question and geometry premise only; not a runtime input or cold-available Git pin."
+  },
+  "coordination": {
+    "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19291954",
+    "open_pr_cache_records": 316,
+    "discussion_cache_records": 5998,
+    "local_materialized_note_records": 99,
+    "main_note_records": 8722,
+    "selected_kappa": "0.003125804"
+  }
+}