Mister-Meeseeks · Oct 21, 2026 · Oct 21, 2026
diff --git a/research/centre-nonlinear-fresh-floor/README.md b/research/centre-nonlinear-fresh-floor/README.md
@@ -0,0 +1,106 @@
+# Nonlinear and adaptive centre sharing keeps the fresh-information floor
+
+**Scoped negative result; no new κ.** The selected record and frontier are unchanged.
+
+[centre-causal-tangent](../centre-causal-tangent/README.md) (#3873) proves its 1,152-query floor for uniform
+causal *linear* scalar-oracle sharing and lists "a lower bound against nonlinear or adaptive algorithms" as
+not established. [centre-history-innovation](../centre-history-innovation/README.md) proves that no decoder,
+even nonlinear, of the earlier centre history recovers the call-10 query, and that a fresh *linear*
+observation needs complex dimension `m >= 8d`; fresh *nonlinear* observations are left open there.
+
+This package closes that remaining case under the same premises:
+
+- **Theorem 1.** Any fresh observation that is continuous, or semialgebraic (finitely many branches built
+  from `+ - x`, division by nonzero values including the dyadic `/2, /4`, real constants, comparisons and
+  adaptive reads), still needs **at least 8d complex coordinates** per future owner, the same as the linear floor.
+- **Theorem 2.** On a finite b-bit grid the only escape is packing several values into wider words
+  (Kronecker substitution). It conserves bits: the widths of the fresh coordinates sum to at least `16db`.
+  In the rank-mass model a k-times wider coordinate is a k-times larger child, so the coverage mass
+  `72w + 64` and deficit `w - 64` are unchanged.
+- **Screen.** Regrouping the 64 centre mass into larger children, which bounds every packing, raises the
+  [complex-local-word ceiling](../complex-local-word-ceiling/README.md) by at most **+0.67%**, before guard bits,
+  carries, unpacking or hosting are charged. The card's two-call sharing budget (x1.90) needs the fresh
+  information halved, which Theorems 1–2 exclude.
+
+| Variant (centre mass 64 kept) | ceiling `a_hi` | x selected ceiling |
+|---|---:|---:|
+| selected: 8 x rank-8 | 0.00169005411 | 1.000000 |
+| pack 2 totals: 4 x rank-16 | 0.00169362555 | 1.002113 |
+| pack 4 totals: 2 x rank-32 | 0.00169720789 | 1.004233 |
+| pack 8 totals: 1 x rank-64 | 0.00170080115 | 1.006359 |
+| ideal regroup: fractional rank-72 | 0.00170141282 | 1.006721 |
+| control: halved centre information (4 copies, charge 32) | 0.00321297289 | 1.901107 |
+
+Exact rational brackets are in [RESULT.json](RESULT.json).
+
+## Setting (inherited, not re-proved)
+
+From centre-history-innovation, under its native common-frame and full-space cover-view premises:
+`H` (all centre fields of calls 0..9 at every owner) and `Q` (the eight pre-transport fields of call 10 at
+common owner `I`) are complex-linear on input differences, and `Q(ker H) = M^8`, `dim_C = 8d`.
+Fix a complex subspace `V ⊂ ker H` with `Q|_V : V -> M^8` bijective. `V` has real dimension `16d`.
+
+A sharing scheme for call 10 may use (1) the complete earlier history `H(x)` in any form, cached,
+recombined or nonlinearly processed; (2) a fresh observation `E(x) ∈ R^n` (a complex coordinate counts as
+two real ones), produced adaptively: which reads happen and what is computed may depend on `H(x)` and on
+earlier fresh reads; (3) any decoder `D` with `D(H(x), E(x)) = Q(x)` on legal inputs. The "mutable
+eight-slot cache with fresh reads" left open by the predecessor is an instance: the cache is a function
+of `H`, the reads are `E`.
+
+## Proofs
+
+**Theorem 1.** On `V`, `H(x) = 0`, so the history gives no information and any adaptivity on it is one fixed
+choice. `D(0, E(x)) = Q(x)` and `Q` is injective on `V`, hence `E` is injective on `V`.
+Continuous case: no continuous injection of an open subset of `R^{16d}` into `R^n` exists for `n < 16d`
+(invariance of domain / Borsuk–Ulam). Semialgebraic case: `V ∩ box` is the finite union of the semialgebraic
+pieces on which the branches are fixed; one piece `P` has dimension `16d`; an injective semialgebraic map
+preserves dimension, so `16d = dim E(P) <= n`. ∎
+
+**Theorem 2.** The dimension argument fails only for maps with unboundedly many pieces, i.e. bit extraction
+on an unbounded grid. On a grid `G ⊂ V` with `2^{16db}` points, `E` injective on `G` needs at least
+`2^{16db}` distinct values, so the bit widths of its coordinates sum to at least `16db`. Kronecker packing
+attains this, and one bit less collides (checked exhaustively on six small grids by `verify.py`). ∎
+That such a grid is a legal fixed-width input set is an inherited premise: the predecessor notes that scaling
+its constant witness gives a bounded dyadic counterexample but supplies no fixed-width right inverse.
+
+## Scope and limits
+
+- All premises of centre-history-innovation are inherited (native common-frame contract, full-space cover
+  view, one future owner, call 10). Its verifier is replayed unchanged here; its compiler contracts are not rebuilt.
+- This is an information floor. The step "8d fresh coordinates => 64 rank-mass units" is the separate
+  rank-8-per-channel carrier premise of #3918; nonlinearity does not change it, and the screen bounds what
+  any regrouping of that mass could gain.
+- The screen uses the fixed selected local word, 112 ports, the nine-stage template and the coverage law.
+  Changed words, templates or coverage laws are outside it.
+- A terminal-only consumer that never forms the internal call-10 query is a changed contract and is not covered.
+
+## Reproduce
+
+From the repository root (Python 3.12, standard library; about 35 s, mostly the predecessor replay):
+
+```sh
+python3.12 -I -B research/centre-nonlinear-fresh-floor/verify.py
+python3.12 scripts/research_package.py --package research/centre-nonlinear-fresh-floor
+```
+
+Expected: `PASS_CENTRE_NONLINEAR_FRESH_FLOOR premise=8edf9aec270e packing_best=x1.006721 controls=4 no_new_kappa`.
+Controls: the halved-information lever reproduces x1.90; a rank-73 child, a mass-losing pack and a corrupted
+predecessor digest are rejected. `--write` regenerates `RESULT.json`.
+
+## Open next questions
+
+Structured in [TASK.json](TASK.json):
+
+1. **Terminal-only consumer.** Can a changed consumer avoid forming the internal call-10 query? First step:
+   state the complete varying-source-and-target terminal map and test the 112 kernel witnesses of
+   centre-history-innovation. Blocked by: no priced changed terminal contract.
+2. **Carrier premise.** Does `8d` fresh coordinates force 64 rank-mass units for carriers other than rank-8
+   channels? First step: price one alternative carrier with its restoration and hosting bill. The screen
+   here caps its ideal gain at +0.67%.
+
+## Credit
+
+Builds directly on Jose Manuel (`jmbarrancoml`)'s centre-history-innovation and its credited lineage,
+Clear1nsight's centre-causal-tangent (#3873), Douglas Colkitt's complex-local-word ceiling, the nonlinear
+copy-floor line (#703, #785, #814, #888) and the centre-sharing budgets (#1791, #1888, #3075, #3548, #3948).
+Prepared by Mark (Matarener) with substantial Claude (Anthropic) assistance. Apache-2.0.
diff --git a/research/centre-nonlinear-fresh-floor/RESULT.json b/research/centre-nonlinear-fresh-floor/RESULT.json
@@ -0,0 +1,125 @@
+{
+ "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+ "best_packing_ratio": "1.006720916",
+ "controls": [
+  [
+   "halved_centre_information_x1.90",
+   "1606486447/500000000000",
+   "1.901107"
+  ],
+  [
+   "rank_73_child_rejected"
+  ],
+  [
+   "mass_losing_pack_rejected"
+  ],
+  [
+   "corrupted_predecessor_digest_rejected"
+  ]
+ ],
+ "packing_sharpness": [
+  [
+   2,
+   1,
+   4
+  ],
+  [
+   2,
+   2,
+   16
+  ],
+  [
+   2,
+   3,
+   64
+  ],
+  [
+   3,
+   1,
+   8
+  ],
+  [
+   3,
+   2,
+   64
+  ],
+  [
+   4,
+   1,
+   16
+  ]
+ ],
+ "predecessor_receipt": "8edf9aec270ec97d9800d097e6bd6feebab7b8d55d8b81b104a53fe8789f0cb4",
+ "screen": [
+  {
+   "a_hi": "169005411/100000000000",
+   "a_lo": "1690054109/1000000000000",
+   "local": {
+    "1": "2412",
+    "2": "418",
+    "3": "432",
+    "4": "142",
+    "5": "82",
+    "8": "8"
+   },
+   "name": "selected 8 x rank-8",
+   "ratio_hi": "1.000000000"
+  },
+  {
+   "a_hi": "423406387/250000000000",
+   "a_lo": "1693625547/1000000000000",
+   "local": {
+    "1": "2412",
+    "16": "4",
+    "2": "418",
+    "3": "432",
+    "4": "142",
+    "5": "82"
+   },
+   "name": "pack 2 totals: 4 x rank-16",
+   "ratio_hi": "1.002113209"
+  },
+  {
+   "a_hi": "1697207887/1000000000000",
+   "a_lo": "848603943/500000000000",
+   "local": {
+    "1": "2412",
+    "2": "418",
+    "3": "432",
+    "32": "2",
+    "4": "142",
+    "5": "82"
+   },
+   "name": "pack 4 totals: 2 x rank-32",
+   "ratio_hi": "1.004232869"
+  },
+  {
+   "a_hi": "1700801149/1000000000000",
+   "a_lo": "425200287/250000000000",
+   "local": {
+    "1": "2412",
+    "2": "418",
+    "3": "432",
+    "4": "142",
+    "5": "82",
+    "64": "1"
+   },
+   "name": "pack 8 totals: 1 x rank-64",
+   "ratio_hi": "1.006358991"
+  },
+  {
+   "a_hi": "1701412821/1000000000000",
+   "a_lo": "85070641/50000000000",
+   "local": {
+    "1": "2412",
+    "2": "418",
+    "3": "432",
+    "4": "142",
+    "5": "82",
+    "72": "8/9"
+   },
+   "name": "ideal regroup: fractional rank-72",
+   "ratio_hi": "1.006720916"
+  }
+ ]
+}
diff --git a/research/centre-nonlinear-fresh-floor/SOURCES.json b/research/centre-nonlinear-fresh-floor/SOURCES.json
@@ -0,0 +1,84 @@
+{
+  "schema_version": 1,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "64bfc5af6709ed9811171e0a2108f05e1219cb65",
+      "sha256": "352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96"
+    },
+    {
+      "path": "research/centre-history-innovation/RESULT.json",
+      "usage": "input",
+      "git_blob": "441ab95422bc6d970f92816e52cf05bc0b8775df",
+      "sha256": "575970c3cf0c1aa8dfe22c5edfa22290b3b918b24ea8ecef55f9316a8edca399"
+    },
+    {
+      "path": "research/centre-history-innovation/SOURCES.json",
+      "usage": "input",
+      "git_blob": "f9ab49e9a4f1c4671950821ee2f91242510405b1",
+      "sha256": "a5ed0e3437fb89a84f1f3141aa7278e208e5f22229b0f341a1a5ce12ecba4f4a"
+    },
+    {
+      "path": "research/centre-history-innovation/VERIFY.json",
+      "usage": "input",
+      "git_blob": "34f798359708edf7397413a2603c454b97c71956",
+      "sha256": "9e56a60f94beb385c31da9690e34b3ec3bccb4f4e0d7775cd2df81b3ac193537"
+    },
+    {
+      "path": "research/centre-history-innovation/verify.py",
+      "usage": "input",
+      "git_blob": "ae11fa6ccdba61db98bbac7b19bb5d2247d60097",
+      "sha256": "f0874ac671f3ac797a8834c6f4ee07a24800b19bda33c099aee31051fb1250b9"
+    },
+    {
+      "path": "research/centre-native-observation-dimension/PROOF.md",
+      "usage": "input",
+      "git_blob": "ffe657f7e8a691b3dc3db5910cab8594b8b49285",
+      "sha256": "5f81f76cbb61e5d66ded17423045d02f959f182ada6158fd6428b3a909943b29"
+    },
+    {
+      "path": "research/centre-native-terminal-factor/README.md",
+      "usage": "input",
+      "git_blob": "fa3eea91d01f0f67fc4122a9d21e1e7bf0ca48c7",
+      "sha256": "2fc637efd214ec00a57f23428d6a1decc11f144768baada9517b6f9d0a38b20a"
+    },
+    {
+      "path": "research/community-frontier1094-audit/selected-replay.tar.gz",
+      "usage": "input",
+      "git_blob": "c24daf067f9c543421d89ca2eeaa4eb78a57fb6f",
+      "sha256": "f1da80610bf33c58527f18ad63b4350d0b864e1d4486141ee17ac9bed4d529d6"
+    },
+    {
+      "path": "research/complex-local-word-ceiling/ceiling.py",
+      "usage": "input",
+      "git_blob": "6c0f54527d879396f689523bfde19928a9b6157c",
+      "sha256": "86a1432870b3b21d8454106ec9162fddc0471472d9214c590a1420f2426cedfc"
+    },
+    {
+      "path": "research/idle-axis-hosting/PROOF.md",
+      "usage": "input",
+      "git_blob": "4468a00349b4f68dce103e785a8392099435e1d6",
+      "sha256": "cfe34bf0b64259661312dae22c3a23adc5bf6f6326951eee5c1488ccfc347542"
+    },
+    {
+      "path": "research/selected-centre-native-lift/README.md",
+      "usage": "input",
+      "git_blob": "98721e3eb4b955129321078f09c4b1b3b2ef5ae0",
+      "sha256": "aa50f2984b8f0af8781e1b0295f907606b1031bc284a396b683f3e5506148595"
+    },
+    {
+      "path": "research/split-centres-global/proofs/INHERITED-COMPLEX-GEOMETRY.txt",
+      "usage": "input",
+      "git_blob": "c4150cb693414c098690eae79b3aea5f47c184ff",
+      "sha256": "c1ff55536594222a8d2a30bc0e0956959774e1be2d5420b7838ad553ee556df2"
+    }
+  ]
+}
diff --git a/research/centre-nonlinear-fresh-floor/TASK.json b/research/centre-nonlinear-fresh-floor/TASK.json
@@ -0,0 +1,78 @@
+{
+  "schema_version": 1,
+  "task_id": "centre-path-sharing",
+  "priority_revision": 7,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "original_card": {
+    "id": "centre-path-sharing",
+    "question": "Can the 64-per-call centre path (eight rank-8 COPY/ERASE totals through the private temporary) be shared across calls or shrunk?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/complex-local-word-ceiling/README.md",
+      "research/split-centres-global/proofs/PR386-BIT-GEOMETRY.txt",
+      "certificates/selected-result.json"
+    ],
+    "methods": [
+      "Derive a fully charged favorable relaxation for sharing one centre path across k calls, including restoration, hosting and precision.",
+      "Prove a lower bound on copied information per call for the selected decoder contract."
+    ],
+    "acceptance_gate": "Show the changed centre charge and copy profile per call with every added restoration or hosting cost charged, and its ceiling via ceiling.ceiling(centre=...). 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": "Write the per-call centre charge and copy children for sharing across k = 2 calls, charge the sharing overhead, and screen it; stop this route if even the ideal sharing gains under 10%. Ideal budgets: free 2.80x, shared by two 1.90x (levers.py).",
+    "bottleneck_link": "The centre path consumes 64 of about 99.6 unpaid coordinates per call, i.e. about 64% of the raw deficit.",
+    "allowed_variations": [
+      "One sharing pattern across a fixed number of calls.",
+      "One reduced copied-total contract for the existing decoder.",
+      "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": [
+      "Removing copies without a replacement contract for the decoder's copied totals.",
+      "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": [
+      "centre path",
+      "copied totals",
+      "private temporary",
+      "shared centre",
+      "batching"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Close the nonlinear/adaptive case left open by research/centre-causal-tangent (#3873) and the fresh nonlinear-observation case left open by research/centre-history-innovation: any continuous or semialgebraic (including dyadic division, branching and adaptive reads) fresh observation still needs 8d complex coordinates per future owner, and finite-precision packing conserves bits; price that packing in the complex-local-word ceiling.",
+  "scope_change_reason": "The card asks whether the centre can be shared or shrunk; existing floors are linear or cover only decoders of the earlier history. This package answers the card method 2 (copied-information lower bound) for the remaining nonlinear and adaptive fresh-data interfaces, and screens the only finite-precision escape (packing) with ceiling.ceiling, rather than constructing a sharing circuit.",
+  "bottleneck_link": "The centre path is 64 of about 99.6 unpaid coordinates per call; the x1.90 two-call sharing budget requires halving fresh centre information, which this package shows nonlinearity cannot do under the inherited premises.",
+  "budget_minutes": 240,
+  "stop_rule": "Stop at the information boundary: continuous/semialgebraic fresh observations need >=8d complex coordinates, packing conserves bits and its ideal ceiling gain is below the card 10% stop threshold.",
+  "outcome": "negative-result",
+  "evidence": "verify.py replays research/centre-history-innovation/verify.py unchanged (receipt 8edf9aec...), checks Kronecker-packing sharpness on six small grids, and computes exact ceiling brackets: best packing x1.006721 (fractional rank-72), 2/4/8-total packs x1.002113/x1.004233/x1.006359, versus the halved-information lever x1.901107. Passed on CPython 3.12 (Windows) with and without -O.",
+  "author": "Mark (Matarener)",
+  "ai_assistance": "Substantial Claude (Anthropic) assistance in proof, code and prose; no independent human or formal-kernel review.",
+  "open_questions": [
+    {
+      "question": "Can a terminal-only consumer avoid forming the internal call-10 centre query?",
+      "why_it_matters": "Both this floor and its predecessor bind the internal query; a changed consumer contract is the remaining route to fewer fresh centre coordinates.",
+      "first_step": "State the complete varying-source-and-target terminal output map and evaluate it on the 112 kernel witnesses of centre-history-innovation.",
+      "blocked_by": "No changed terminal contract with priced consumer, restoration and hosting obligations is specified."
+    },
+    {
+      "question": "Does the information floor (8d fresh complex coordinates) imply the 64-unit rank-mass charge for carriers other than rank-8 channels?",
+      "why_it_matters": "The step from coordinates to rank mass is the rank-8-per-channel carrier premise of #3918, independent of nonlinearity.",
+      "first_step": "Price one alternative carrier (e.g. one rank-16 child carrying two totals) with its restoration and hosting bill.",
+      "blocked_by": "No admitted alternative carrier contract; the screen here shows its ideal gain is at most +0.67%."
+    }
+  ],
+  "coordination": {
+    "canonical_main": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+    "overlap": "Inspected PRs 3075, 3120, 3147, 3151, 3291, 3548, 3873, 3918, 3948, 1888, 785, 888 and packages centre-history-innovation, centre-causal-tangent, *-nonlinear-copy-floor; no nonlinear fresh-observation centre bound found. No novelty guarantee.",
+    "preservation": "Only the new package directory is added."
+  }
+}