Prove a physical 1/sqrt(19) error floor for the partial-cache observer

research/partial-cache-minimax-floor/INPUTS.json

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+[
+  {
+    "local": "source/provider.json",
+    "path": "research/native-rank8-helper-provider/PROVIDER.json",
+    "repository": "CrocSwap/integer-mult-bounds",
+    "commit": "32890bc5de034ab8fc93ac88ead7279a6a416d04",
+    "sha256": "676de7551d3364598f4a7e29ae97cae0e4364348786bbe70435b84294ba7dd65",
+    "git_blob": "0afd30ed01202eb6ce62462dcf8bc9164f3b0455"
+  },
+  {
+    "local": "source/provider-proof.md",
+    "path": "research/native-rank8-helper-provider/PROOF.md",
+    "repository": "CrocSwap/integer-mult-bounds",
+    "commit": "32890bc5de034ab8fc93ac88ead7279a6a416d04",
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+    "git_blob": "5cbfc4fd02dce926c3d235f1549bdb2035dea088"
+  },
+  {
+    "local": "source/normalizers.json",
+    "path": "research/coded-native-caller/NORMALIZERS.json",
+    "repository": "CrocSwap/integer-mult-bounds",
+    "commit": "32890bc5de034ab8fc93ac88ead7279a6a416d04",
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+    "git_blob": "bca92c222c395d3d75c047fbb678f3cd0947c60f"
+  },
+  {
+    "local": "source/observer-proof.md",
+    "path": "research/partial-cache-history-observer/PROOF.md",
+    "repository": "JoyboyBrian/integer-mult-bounds",
+    "commit": "da7f5a0c005130873f45c8e5073cd5c989cd5feb",
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+  },
+  {
+    "local": "source/kernel.json",
+    "path": "research/partial-cache-history-observer/KERNEL.json",
+    "repository": "JoyboyBrian/integer-mult-bounds",
+    "commit": "da7f5a0c005130873f45c8e5073cd5c989cd5feb",
+    "sha256": "def77e3bcd50a4c992d9fecc660bbbc9dcdad37da22151351e346f62c9c4593f",
+    "git_blob": "92ca0f83330a8e6ef265ca05df2b5f6b472af9a0"
+  }
+]

research/partial-cache-minimax-floor/README.md

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+# A physical error floor for the partial-cache observer
+
+For every positive tensor width, the exact observation channel in PR #5498
+cannot approximate its coupled-query target uniformly on the physical input
+unit ball to error less than `1/sqrt(19)`. The decoder can be nonlinear and
+can use the complete supplied history. A randomized decoder has worst-case
+expected squared error at least `1/19`.
+
+This is a new quantitative application of an existing exact kernel. It uses
+the actual PR #5402 entrance basis. It is sharp on the specified kernel
+subspace, not a claim about the optimum on the full input ball. No new kappa,
+paid-call bound or complete multiplication theorem follows.
+
+## Source and observation contract
+
+Use canonical main `32890bc5de034ab8fc93ac88ead7279a6a416d04`, priority revision 7.
+JoyboyBrian's [PR #5498](https://github.com/CrocSwap/integer-mult-bounds/pull/5498),
+head `da7f5a0c005130873f45c8e5073cd5c989cd5feb`, supplies the
+[exact kernel and proof](https://github.com/JoyboyBrian/integer-mult-bounds/blob/da7f5a0c005130873f45c8e5073cd5c989cd5feb/research/partial-cache-history-observer/PROOF.md).
+Dillon Amburgey's [PR #5402](https://github.com/CrocSwap/integer-mult-bounds/pull/5402)
+supplies the [physical provider](https://github.com/CrocSwap/integer-mult-bounds/blob/32890bc5de034ab8fc93ac88ead7279a6a416d04/research/native-rank8-helper-provider/PROOF.md).
+The five exact source copies and their Git/SHA256 pins are in `source/` and
+`INPUTS.json`. Source copies retain their original bytes and attribution.
+
+The observer sees every full-nine-axis-frame value during the raw backward
+word and all eight partial-frame endpoint caches. It sees 5,897 full-frame
+events and the caches at roles 633 through 640. It does not see other main
+values, other partial histories, original helper retractions or additional
+paid responses. All 632 original inputs are independent. The nine owned
+inputs can be zero for the lower-bound instances.
+
+The predecessor proves that its scalar vector k annihilates this entire
+channel in the free-word algebra. Thus k times any field v has zero
+observation at every positive width. It also proves that the actual physical
+target b changes only in mode one, by
+
+    delta b_1 = -2 Km_1^-1 R v.                         (1)
+
+This complete history identity is a credited proof premise. We do not rerun
+its 21,473-record history interpreter. The native map R is the full Gaussian
+on nine axes. It and the monomial Km are unitary in the Euclidean field norm.
+Their all-width physical realization remains a source assumption.
+
+## Pull the witness back through the physical entrance
+
+Let V be the complex field space at one fixed positive width, with its usual
+Euclidean norm. Give the physical input bank the direct-sum Euclidean norm.
+Let L be the provider's scalar helper entrance basis. The checker obtains
+
+    sum_X k_j^2 = 60,  sum_Y k_j^2 = 12,
+    sum_H k_j^2 = 68,  L^-1 k_H = -4 e_225.             (2)
+
+It checks (2) both by the supplied exit basis and by reversing and inverting
+every entrance operation. A forward application of L to -4 e_225 gives
+exactly the 44 nonzero helper entries in the source kernel. Both full inverse
+products are also checked on all 122 basis owners.
+
+Physical main inputs use unitary monomial entrance wrappers B_X,j and B_Y,j.
+The raw X role has a unitary native birth frame F_X,j; its raw logical value
+is F_X,j^-1 B_X,j x_phys,j. The raw Y roles have ground birth frames. Set
+the physical X displacement to k_j B_X,j^-1 F_X,j v, and the physical Y
+displacement to k_j B_Y,j^-1 v. The backward conjugation convention is kept
+in F_X,j. Each native frame is a product of unitary Gaussian factors, so
+this extra X frame preserves the norm. It is not omitted from J.
+The physical helper entrance applies 2 Km_1 on role 225 before L. Therefore
+set the sole nonzero physical helper displacement to -2 Km_1^-1 v. All other
+physical helpers and all owned inputs have zero displacement. Call this
+physical input map J:V -> input bank. Its raw image is precisely k v, and
+
+    O J = 0,   ||J v||^2 = (60+12+4)||v||^2 = 76||v||^2,
+    T J v = (0,-2 Km_1^-1 R v,0,...,0),
+    ||T J v||^2 = 4||v||^2.                            (3)
+
+Here O is the stated observation channel and T is the eight-mode physical
+target. The 61 nonzero physical roles comprise 57 X, three Y and one helper.
+Using the raw norm 140 would give the weaker squared ratio 1/35. Treating the
+helper basis as an isometry would be incorrect. Formula (3) accounts for its
+actual nonunitary scalar basis and its factor two.
+
+The scalar coefficients in (2) act once. They are not raised to the tensor
+width. The unitary one-column address maps remain unitary under tensor powers.
+Thus 76 and 4 in (3) hold at every positive width.
+
+## Minimax proof
+
+Choose any unit v in V and put u=Jv/sqrt(76), t=TJv/sqrt(76). Both u and -u
+are in the physical input unit ball. Their complete observations are zero,
+and their targets are t and -t, with ||t||=1/sqrt(19).
+
+A deterministic decoder must give the same output z on both inputs. The
+triangle inequality gives
+
+    max(||z-t||, ||z+t||) >= ||t|| = 1/sqrt(19).         (4)
+
+For a randomized decoder the output distribution is also identical for the
+two inputs. The parallelogram identity gives, pointwise in its randomness,
+
+    (||z-t||^2 + ||z+t||^2)/2 = ||z||^2 + ||t||^2.
+
+Taking expectations proves that at least one input has expected squared
+error at least 1/19. Infinite second moment already satisfies the bound.
+No assumption of linear postprocessing or a bounded instruction count enters
+this argument.
+
+On the restricted domain `{Jv: ||Jv||<=1}`, the constant-zero decoder has
+maximum error exactly 1/sqrt(19), by (3). It also attains squared risk 1/19.
+This proves sharpness on that subspace. It supplies only a lower bound on
+the full physical unit ball. It does not show a sharp global optimum.
+
+## Verification and limitations
+
+Run `python3.12 -B scripts/research_package.py --package research/partial-cache-minimax-floor`.
+Python 3.12 and its standard library suffice. Both registered commands use
+isolated, warning-strict Python, normally and under `-O`. Each checks five
+source copies, 244 basis inverse products, 115,200 literal monomial address
+actions, the exact native one-axis unitarity identity, 117 rational complex
+square-loss identities and 14 rejection controls. The output is
+`PASS_PARTIAL_CACHE_MINIMAX_FLOOR`.
+
+The all-width statement is the written proof, not an extrapolation from
+finite address tests. The finite tests bind the consumed basis, unitary
+wrappers and target coefficients. The complete predecessor history proof and
+native semantics are explicit premises; their full suites, a cold selected
+replay, human review and a proof-kernel check were not run. This bound uses
+continuous complex inputs and Euclidean error. It does not establish a bound
+for a restricted integer lattice, a particular rounding algorithm, paid work,
+whole-machine observations or the changed selected completion.
+
+## Open next questions
+
+- Can one additional retained observation separate the subspace J(V)? This
+  would change the observer contract. Start with a versioned partial-history
+  row and compute its action on J; physical availability and retention are
+  still required.
+- What is the full norm of T restricted to ker O? This determines the sharp
+  linear-information error on the full ball. Start with a fixed-width exact
+  observation matrix. Its size and an all-width spectral argument remain open.
+- Can admissible discrete inputs realize a comparable pair? Start with the
+  actual input lattice and all entrance denominators. The continuous bound
+  alone does not answer that precision question.
+
+Credit JoyboyBrian (@JoyboyBrian) for the exact partial-cache kernel and target
+binding; Dillon Amburgey (@dillona) for the native coupled provider and basis;
+and the retained Mark/Matarener, THU New Media Research Center, zxv, Cao Yifan,
+Cinna Davis, Dugongue, Chris R, native and compiler lineage in those sources.
+Ali Romman (@aliromman) supplies this norm calculation and minimax application,
+with substantial OpenAI Codex assistance. Apache-2.0. No independent human or
+formal-kernel acceptance is claimed.

research/partial-cache-minimax-floor/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "32890bc5de034ab8fc93ac88ead7279a6a416d04",
+  "files": [
+    {
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+      "usage": "provenance",
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+      "path": "research/partial-cache-history-observer/PROOF.md",
+      "repository": "JoyboyBrian/integer-mult-bounds",
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+      "git_blob": "4cfe97e3a4fbd2a6a8c5f2260b68550e0d794e70"
+    },
+    {
+      "local": "source/kernel.json",
+      "path": "research/partial-cache-history-observer/KERNEL.json",
+      "repository": "JoyboyBrian/integer-mult-bounds",
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research/partial-cache-minimax-floor/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 7,
+  "base_commit": "32890bc5de034ab8fc93ac88ead7279a6a416d04",
+  "original_card": {
+    "id": "uniform-transfer-audit",
+    "question": "Does one quantified step of the proposed #4100 all-size construction hold at its actual caller?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/unconditional-batched-crt/README.md"
+    ],
+    "methods": [
+      "Prove one missing implication with explicit quantified hypotheses.",
+      "Find an exact counterexample at the claimed interface boundary."
+    ],
+    "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read PROOF.md and AUDIT.md in #4100. Select one distinct obligation: CRT total repair, native/free-algebra lifting, precision/conditioning, or complete fixed-tape cost. This is an author-submitted proof, not an accepted unconditional theorem.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Check live handoffs; name one proposition, its caller, quantified domain and dependency closure. Independently prove the implication or exhibit a precise gap/counterexample. Prioritize CRT.md total repair or COMPLEX.md free-algebra lift only if unclaimed. Stop at the chosen obligation; do not substitute finite samples or root arithmetic for an all-size proof, or launch the complete package by default.",
+    "bottleneck_link": "Finite supplier improvements are useful only under the retained transfer obligations.",
+    "allowed_variations": [
+      "One CRT repair, native lift, precision/conditioning or tape-cost proposition, with its exact caller."
+    ],
+    "scope_change_triggers": [
+      "Replacing a quantified obligation with another finite smoke test.",
+      "Treating archive intake, an author PASS receipt or a finite composed fixture as discharge of the universal hypotheses."
+    ],
+    "overlap_terms": [
+      "uniform transfer",
+      "all-size",
+      "precision",
+      "4100",
+      "unconditional-batched-crt"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4100: proposed unconditional construction and finite checks, without independent acceptance.",
+        "evidence": "research/unconditional-batched-crt/README.md",
+        "remaining": "Independent proof audit of one caller-bound all-size implication; all other premises remain explicit.",
+        "status": "author-checked",
+        "source_commit": "337d04939571a637aeab54a0f58f54fde811663a",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Quantify the physical unit-ball approximation error for the exact partial-cache observer of PR #5498.",
+  "scope_change_reason": "A completed live observer result leaves its approximation boundary open. Bind that boundary to the actual native provider entrance, as required by priority revision 7.",
+  "bottleneck_link": "This excludes an approximate passive decoder under a concrete physical observation contract.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after the physical norm, minimax proof and exact controls; do not extend the observer or infer a paid-call bound.",
+  "outcome": "completed",
+  "evidence": "README.md and registered normal/optimized verify.py checks.",
+  "author": "Ali Romman",
+  "ai_assistance": "Substantial OpenAI Codex assistance.",
+  "open_questions": [
+    {
+      "question": "Which additional retained observation separates J(V)?",
+      "why_it_matters": "A changed observer can evade this bound.",
+      "first_step": "Evaluate a versioned partial-history row on J.",
+      "blocked_by": "Physical availability and retention costs remain open."
+    }
+  ]
+}

research/partial-cache-minimax-floor/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Check source bytes, physical basis, norm, unitary maps and rejecting controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "-O",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Check source bytes, physical basis, norm, unitary maps and rejecting controls."
+    }
+  ]
+}