Mister-Meeseeks · Oct 23, 2026 · Oct 22, 2026
diff --git a/research/center-output-compression/README.md b/research/center-output-compression/README.md
@@ -0,0 +1,178 @@
+# Sharp information tradeoff for retained center outputs
+
+This scoped algebraic proof determines what remains recoverable when a caller
+compresses the output of Cinna Davis's retained-center interface. Given all 632
+old output fields and only j independent scalar linear observations of its
+eight center outputs, exactly **624+j independent linear functions of the
+desired old endpoint** are recoverable, for 0 <= j <= 8. Recovering all 632
+requires eight observations. Arbitrary nonlinear decoding cannot improve this
+threshold. These are dimensions of information, not native call prices.
+
+The contribution was produced by Nick Petalas. New work is Apache-2.0.
+
+## Interface and exact target-specific theorem
+
+Let F be any field. Let x in F^n and z in F^k be arbitrary independent inputs.
+A is n by n, B is n by k, and R is k by n. The record matrix L has k columns
+and the target matrix T has n columns; their row counts are unrestricted.
+Assume the block map
+
+    y = A x + B z,        w = R x + z
+
+is invertible. Write S=A-BR, which is then invertible: a nonzero vector in
+ker(S), together with z=-Rx, would give a nonzero kernel vector of the block
+map. A caller sees all of y
+and the linear record Lw. There is no other input-dependent state. L is fixed
+independently of x and z. A target Tx is recoverable on every input, by an
+arbitrary decoder, if and only if
+
+    ker(L) is contained in ker(T S^-1 B).
+
+Equivalently, every row of C=T S^-1 B lies in the row space of L. Therefore the
+least possible rank of a center record recovering Tx is exactly
+
+    rank(T S^-1 B).
+
+This includes singular targets T and rank-deficient B. A basis of the rows of C
+is an attaining record. The theorem does not assume that individual center
+coordinates, rather than their linear combinations, must be retained.
+
+### Proof
+
+Substitute z=w-Rx. Then y=Sx+Bw, and hence
+
+    Tx = T S^-1 y - Cw.
+
+If ker(L) is contained in ker(C), row-space duality supplies a matrix D with
+C=DL. The displayed expression gives a linear decoder using y and Lw.
+
+Conversely choose v in ker(L) with Cv nonzero. Compare the zero input with
+
+    delta x = -S^-1 Bv,       delta z = v-R delta x.
+
+Their y and Lw observations agree, whereas their targets differ by -Cv. No
+decoder, linear or nonlinear, can distinguish these inputs. This proves the
+criterion. It also forces rank(L) >= rank(C). Choosing a row basis of C attains
+that lower bound. This is a standard kernel argument applied to this boundary;
+no priority is claimed for rank-nullity or block elimination themselves.
+
+## Sharp full-endpoint tradeoff
+
+Assume additionally that A is invertible and B has column rank k. For a fixed
+record L of rank j, all possible differences in x between inputs with equal
+observations form exactly
+
+    U = S^-1 B ker(L).
+
+Since S^-1 B is injective, dim(U)=k-j. The recoverable linear functions of Ax
+are exactly the annihilator of AU. That space has dimension n-k+j. Each of
+its functions has the decoder just proved, so this is an equality for every
+rank-j record L, rather than only a lower bound or an optimized choice of L.
+It does not say that n-k+j particular coordinate entries can always be chosen.
+
+Over a finite field of order q, every observation fiber has exactly q^(k-j)
+possible old endpoints. Thus arbitrary decoding does not distinguish two old
+endpoints in the same fiber. The statement about independent linear functions
+over an infinite field does not assert a bound on arbitrary nonlinear encodings.
+
+## Application and predecessor credit
+
+Cinna Davis's [PR #4694](https://github.com/CrocSwap/integer-mult-bounds/pull/4694),
+head `2add42d1edf14a263abe90a3bcde67fc1ba30768`, supplies n=632, k=8,
+A(X,Y,H)=(X,Y+X,H), invertibility of the whole interface, and rank(B)=8.
+The merged [#4698 helper normalization](../center-helper-normal-form/README.md)
+retains these properties and constructs eight explicit three-step chains.
+Its [proof](../center-helper-normal-form/PROOF.md) states B's injectivity.
+Those facts are cited premises, not reverified results of this package.
+Their rational matrices give the specialization over Q and C. Their dyadic
+native interface is not being specialized to characteristic two or three.
+
+The theorem gives the following complete rank table:
+
+    retained center rank j:  0   1   2   3   4   5   6   7   8
+    recoverable old rank:  624 625 626 627 628 629 630 631 632
+
+The new result is this sharp observation-compression law and its
+target-specific minimum. The predecessor's helper basis and conditional
+two-call consumer are not new here. Chris R's
+[PR #4839](https://github.com/CrocSwap/integer-mult-bounds/pull/4839) constructs
+and prices a particular physical terminal. Our theorem concerns the amount of
+linear center information required when all old endpoint functions are
+demanded; it neither reprices nor excludes other terminal policies.
+
+Credit Cinna Davis #4694/#4698 and the #4440/#4441/#4447/#4470 lineage,
+Cao Yifan #4610, Chris R #4839, zxv, Jacob Alan Sussman and the retained
+native, scalar, E8, Clifford, precision and outer contributors. ywz65's
+earlier center-observation-floor work supplies the information-boundary
+perspective; its owner-routing model is a different theorem.
+Ross Richardson's [#2480](https://github.com/CrocSwap/integer-mult-bounds/pull/2480)
+proves a sharp retention threshold for another fixed QQ16 observation cut.
+Ali Romman's [#4720](https://github.com/CrocSwap/integer-mult-bounds/pull/4720)
+proves historical routed-query information and snapshot bounds. These are
+credited precedents for kernel-based observation lower bounds. Neither counts
+compression of the #4694 coupled endpoint studied here.
+
+## Evidence and reproduction
+
+The all-field claim is the written proof above. The self-contained verifier
+uses two independent three-step chains at n=4,k=2. It enumerates every
+zero-, one- and two-row record L over F2 and F3, including dependent rows.
+For every L it enumerates every dirty input, checks every observation fiber,
+constructs collisions, computes the recoverable row-space dimension and
+checks target-specific recovery against literal zero fibers. It tests every
+one-row target plus a full endpoint and a two-row target, and enumerates the
+minimum record rank for each target. These small examples corroborate the
+new theorem; they do not substitute for a native 640-column replay.
+
+From the worktree root:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/center-output-compression
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/center-output-compression/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/center-output-compression --check-source-history
+```
+
+On another machine substitute its CPython 3.12 path. There are no external
+runtime inputs or dependencies. SOURCES.json pins the cited merged proof as
+historical provenance; the scientific verifier does not import or replay it.
+Checks use explicit exceptions and remain active with -O. The verifier writes
+only stdout. No receipt is handwritten or accepted as evidence.
+
+The card's historical first example was attempted and raised
+`KeyError: 'complex_profile'` after the selected schema changed. The current
+`scripts/research_selected_baseline.py` source-binding control passed. Neither
+is an input to this algebraic theorem, and neither was repaired here.
+
+## Scope, obligations and expansion decision
+
+This is a scoped proof with finite corroboration, not a new supplier, kappa,
+search failure, proof-kernel certificate or complete multiplication theorem.
+The observation contract grants all old output fields and fixed scalar linear
+center records. Additional input-dependent caller state, changed demanded
+targets, nonlinear measurements, multiple calls or intermediate observations
+can change the problem. The decoder is algebraic; endpoint chart conversions,
+native reads, precision, routing, restoration, retained helper jobs and a
+complete paid moment remain unknown. No information dimension is converted
+into a recursive call count. The selected frontier is unchanged.
+
+Power-of-ten inspection: loops have fixed n<=6 and p in {2,3}; no recursion or
+child processes occur. The finite census intentionally allocates Python tuples,
+sets and dictionaries, bounded by 729 inputs and 729 fibers per record.
+This is a managed-runtime allocation deviation, not a fixed-memory claim.
+No additional tooling or native/formal replay is required for the written proof.
+Independent human and formal proof-kernel review are unclaimed.
+
+Decision: stop at this information result. It does not justify another
+construction round without a fully charged changed consumer.
+
+## Open next questions
+
+For one literal proposed consumer, compute C=T S^-1 B for its actual demanded
+target T and find whether rank(C)<8. This can reduce the observation demand,
+but the first step needs that consumer's explicit target matrix and costs.
+Those are not supplied here. If all old outputs remain demanded, the sharp
+threshold is settled under this observation contract.
+
+Separately, price one attaining row-basis record in the actual endpoint charts.
+Start with its literal read and restoration word. Scalar rank alone supplies
+no native cost bound; this is the blocker to applying the result to a moment.
diff --git a/research/center-output-compression/SOURCES.json b/research/center-output-compression/SOURCES.json
@@ -0,0 +1,30 @@
+{
+  "schema_version": 1,
+  "base_commit": "624de9dc66858af0945e5f8a0942488d62280d43",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395",
+      "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/center-helper-normal-form/README.md",
+      "usage": "provenance",
+      "git_blob": "a1ab98dd6c717243b60b78fbb60d5512f05d902c",
+      "sha256": "c02694140e99d68e4226ba3c334acf5326ac3977f53f7a123edc67c0a8545cd9"
+    },
+    {
+      "path": "research/center-helper-normal-form/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "f7af6ae6224a08df2ec581efc746e1c38fafb643",
+      "sha256": "c52ce9eb18ebeca54aa7bbd30560215f0ab87af547bf770da6c3f23e105579ec"
+    }
+  ]
+}
diff --git a/research/center-output-compression/TASK.json b/research/center-output-compression/TASK.json
@@ -0,0 +1,113 @@
+{
+  "schema_version": 1,
+  "task_id": "complex-new-helper-inventory",
+  "priority_revision": 7,
+  "base_commit": "624de9dc66858af0945e5f8a0942488d62280d43",
+  "original_card": {
+    "id": "complex-new-helper-inventory",
+    "question": "Can a new local cost interface change the deficit-to-recursive-cost ratio beyond current packing?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/complex-local-word-ceiling/README.md",
+      "research/heterogeneous-owner-assembly/README.md",
+      "certificates/selected-result.json",
+      "research/selected-k8-unrestricted-ends/README.md"
+    ],
+    "methods": [
+      "Propose a changed helper inventory and derive an optimistic exact moment.",
+      "Design a small completed instance and independently verify its endpoint and rank bill."
+    ],
+    "acceptance_gate": "A new physically realizable contract outside the frozen catalogue whose per-call profile, screened with ceiling.ceiling(), beats the selected word's packing ceiling; a lower cap does not establish identical circuitry or order attainable optima, and is insufficient to pass the stronger-budget target under this model. 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",
+        "research/complex-local-word-ceiling/verify.py"
+      ],
+      "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING",
+      "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.",
+      "scope": "Positive control and planning budget only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Before any synthesis, write the candidate per-call local profile (and centre charge / unpaid coordinates if changed) and screen it with ceiling.ceiling(); reject it if even the ideal ceiling does not beat the selected word's by your stated margin.",
+    "bottleneck_link": "Packing of the selected word is capped at +0.87% (ceiling model); only a lower local excess spend or a changed coverage law raises the cap.",
+    "allowed_variations": [
+      "Narrow to one changed boundary or helper type with explicit counts.",
+      "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."
+    ],
+    "scope_change_triggers": [
+      "Mixing only providers already inside the pinned catalogue.",
+      "Only changing helper terminal subspaces and monotone frames within the fixed scalar word, entrances and cleanup contract already covered by #1995/#2013.",
+      "Continuing a partial cost improvement while leaving the same unpriced consumer, restoration or central-readout obligation unresolved; refresh the family expansion decision first. Local product-retiming evidence has a narrow scope and is not a ban on changed graphs or contracts."
+    ],
+    "overlap_terms": [
+      "helper inventory",
+      "heterogeneous-owner",
+      "mutable leaver"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Sharp all-field observation-compression theorem for an invertible coupled-center interface, with the 624+j specialization under the cited PR4694/4698 premises.",
+  "scope_change_reason": "The donor asks for original research rather than an audit. Replace helper synthesis and kappa optimization with a new scoped consumer-information tradeoff answering the predecessor consumer obligation. No physical moment is claimed.",
+  "bottleneck_link": "A caller cannot discard center information while demanding the complete old endpoint. The target-specific minimum rank(T (A-BR)^-1 B) identifies exactly which changed demands could need less information, before native cost synthesis.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop at the first verified original sharp observation tradeoff. Do not extend into synthesis, predecessor replay, selected-frontier changes or kappa optimization.",
+  "outcome": "completed",
+  "evidence": "README.md contains the all-field proof and 624+j corollary; verify.py enumerates every 0..2-row center record and every dirty input of two three-step chains over F2 and F3, all one-row targets and two multirow targets, exact fibers and collision witnesses, and seven rejection controls.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Can a literal consumer demand a target T with rank(T (A-BR)^-1 B)<8?",
+      "why_it_matters": "A smaller target can require fewer center observations than the full old endpoint.",
+      "first_step": "Supply one explicit consumer target matrix and compute the theorem's rank.",
+      "blocked_by": "No such changed demanded target or complete paid consumer is supplied."
+    },
+    {
+      "question": "What is the actual native cost of an attaining row-basis record?",
+      "why_it_matters": "An information dimension is not a recursive call price.",
+      "first_step": "Emit a read/restoration word in the actual endpoint charts for one target.",
+      "blocked_by": "Physical chart conversions, precision, routing, restoration and the complete moment remain unpriced."
+    }
+  ],
+  "upstream_checkpoint": {
+    "commit": "624de9dc66858af0945e5f8a0942488d62280d43",
+    "priority_revision": 7,
+    "checked_at_utc": "2026-10-22T21:49:07Z"
+  },
+  "predecessors": [
+    {
+      "pr": 4694,
+      "head": "2add42d1edf14a263abe90a3bcde67fc1ba30768",
+      "use": "Cited endpoint and injective B, not replayed."
+    },
+    {
+      "pr": 4698,
+      "head": "9079ab66ba9a1717df478a08eb957c20d84d3e61",
+      "use": "Merged helper normalization and explicit chain model."
+    },
+    {
+      "pr": 4839,
+      "head": "cd6ce17acc1eedb9843c6c346b6aa4327673639b",
+      "use": "Related concrete terminal cost, not an input or recheck."
+    }
+  ],
+  "orientation": {
+    "selected_binding_control": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+    "historical_card_first_step": "attempted; KeyError: complex_profile; historical schema not repaired"
+  },
+  "overlap": {
+    "open_prs_cached": 365,
+    "discussion_records_cached": 3817,
+    "complete_remote_pagination": true,
+    "latest_live_pr_and_claim_check_utc": "2026-10-22T21:49:30Z",
+    "terms": [
+      "retained-center-extension",
+      "center output compression",
+      "624+j",
+      "n-k+j",
+      "T S^-1 B",
+      "center observation compression"
+    ],
+    "limitations": "Sparse local README/TASK coverage; all-main distinctive README grep separately performed. No arbitrary code-diff or unpublished-work coverage. Merged searches are term-scoped."
+  }
+}
diff --git a/research/center-output-compression/VERIFY.json b/research/center-output-compression/VERIFY.json
@@ -0,0 +1,34 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Complete F2/F3 record/input/target census, exact ambiguity fibers, sharp minimum ranks and seven rejection controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Same scientific checks with Python assertions disabled."
+    }
+  ]
+}