Prove exact mode preservation needs a full rounding unit (no κ)

research/mode-preserving-rounding/PROOF.md

@@ -0,0 +1,109 @@
+# An exact radius-one mode fiber
+
+This scoped proof was produced by Nick Petalas. Apache-2.0.
+
+## Contract and source
+
+Let M be the 112-by-8 integer mode matrix in Dillon Amburgey's
+[PR4933](https://github.com/CrocSwap/integer-mult-bounds/pull/4933), head
+`d969c458c24278e0fd37e5b4c0ff8a53a496b430`.
+Its columns are `mode_columns` in the source-pinned NORMALIZERS.json.
+The cited inverse-code identity is A^-1 = I - MM^T/52. That identity
+is a predecessor premise; no new check of its factorization is claimed.
+
+Consider rounding x in R^112 to z in Z^112 subject to
+
+    M^T z = M^T x.
+
+The rule may use all coordinates, arbitrary nonlinear computation and
+arbitrary correlated randomness. There is no coordinatewise rounding premise.
+Mode equality is required for each returned output. Equality only in
+expectation is a weaker contract and is outside this obstruction.
+We require the target mode tuple to have an integer lift. This avoids an
+obstruction caused merely by an empty fiber.
+
+## The witness and exact optimum
+
+Use zero-based owner indices. Write m_i for row i of M. The source rows are
+
+    m_0   = ( 1, 1, 1, 1, 0, 0, 0, 0),
+    m_7   = (-1,-1, 1, 1, 0, 0, 0, 0),
+    m_20  = ( 0, 0, 0, 0, 1,-1, 1, 1),
+    m_111 = ( 0, 0, 0, 0, 1,-1,-1,-1),
+    m_48  = ( 0, 0, 1, 1, 1,-1, 0, 0).
+
+They give the exact relation
+
+    m_0 + m_7 + m_20 + m_111 = 2 m_48.                 (1)
+
+Put x = (e_0 + e_7 + e_20 + e_111)/2. Then M^T x = m_48,
+so z = e_48 is an integer lift. Its infinity-norm distance from x is one.
+
+Suppose an integer lift z had ||z-x||_infinity < 1. Every coordinate of x
+outside S={0,7,20,111} is zero, so every corresponding integer z_i is zero.
+On S, each z_i must be zero or one. Mode 0 and mode 2 now require
+
+    z_0 - z_7 = 0,          z_0 + z_7 = 1.             (2)
+
+The first equality follows from the zero target in mode 0. The second
+follows from the unit target in mode 2. The rows m_20 and m_111 contribute
+zero to both equations. Equations (2) force z_0=z_7=1/2, contradicting
+integrality. Consequently
+
+    min { ||z-x||_infinity : z in Z^112, M^T z=M^T x } = 1.    (3)
+
+The same argument applies to each realized integer output of a randomized rule.
+Thus no rule can guarantee a smaller uniform radius on all inputs with
+nonempty integer mode fibers, even if it is allowed arbitrary correlations.
+This does not show that radius one suffices on every such fiber.
+
+For any h>0 and any n in Z^112, translation and scaling give
+
+    min { ||z-h(n+x)||_infinity : z in h Z^112,
+          M^T z=M^T h(n+x) } = h.                     (4)
+
+Indeed z/h-n is an integer lift of M^T x. The attaining output is
+h(n+e_48). Thus retaining exact modes can require twice the usual
+half-grid coordinate radius, at every output-grid scale.
+
+## Why this is a rounding obstruction
+
+There are exactly sixteen integer outputs inside the open radius-one box.
+They are the {0,1} choices on S, zero elsewhere. The verifier enumerates all
+sixteen, compares column-oriented and row-oriented mode calculations, and
+finds no lift. Equation (2), rather than extrapolation from that census,
+proves the exclusion of every integer output in the box.
+
+If the mode constraint is removed, z=0 has distance 1/2. Even preserving
+the ordinary sum does not cause this example's obstruction: e_0+e_20 has
+the same sum as x and distance 1/2, but different modes. Allowing radius
+one admits e_48. The obstruction therefore retains both its mode and
+grid hypotheses and is consistent with an explicit feasible lift.
+
+## Scope and remaining obligations
+
+This is a new local precision contract for the cited code. It is not a
+claim about every reachable state of a native caller. A caller could avoid
+the displayed input, allow mode defects, keep extra residue information,
+or round to a finer grid. Those are changed contracts.
+
+The theorem gives neither the global optimal mode-preserving radius nor
+an efficient rounder. The A^-1 factorization motivates preserving M^T x;
+it does not imply that a complete inverse implementation must preserve
+those modes exactly. No physical correlated-rounding algorithm, native
+restoration, tape/storage tariff, complete error propagation, supplier
+admission, new kappa or selected-frontier update follows. All five framework
+assumptions remain. Independent human and formal proof-kernel review are
+unclaimed.
+
+## Credit
+
+Dillon Amburgey supplies the actual code and mode representation. JoyboyBrian's
+[PR5069](https://github.com/CrocSwap/integer-mult-bounds/pull/5069) supplies
+the source-bound fixed-grid inverse context. Dugongue's coded producer and
+the retained native, controller and compiler authors retain their credit.
+Ali Romman's [PR5838](https://github.com/CrocSwap/integer-mult-bounds/pull/5838)
+separates independently adverse completed defects from possible physical
+rounding correlations. Its word and error calculation are not inputs here.
+The lattice and finite-box arguments are elementary classical methods.
+The source-bound witness, exact fiber radius and new contract are proved here.

research/mode-preserving-rounding/README.md

@@ -0,0 +1,89 @@
+# Exact mode preservation can double the coordinate rounding radius
+
+This scoped mathematical proof with an exact finite certificate was produced by Nick Petalas.
+
+For the actual eight-mode matrix M of Dillon Amburgey's
+[PR4933](https://github.com/CrocSwap/integer-mult-bounds/pull/4933), put
+
+    x = (e_0 + e_7 + e_20 + e_111)/2.
+
+Among integer outputs z preserving all eight mode sums M^T z=M^T x,
+the minimum infinity-norm rounding error is exactly **one**. The output
+e_48 attains it. Every integer output at distance less than one is excluded
+by z_0-z_7=0 and z_0+z_7=1. Thus arbitrary joint rounding and correlated
+randomness cannot retain the usual half-unit error on all feasible mode fibers.
+Mode equality is required for every returned output, rather than only in expectation.
+Translation and scaling give the same factor-two obstruction on every grid.
+
+[PROOF.md](PROOF.md) states the boundary, literal five-row identity and
+universal argument. This is a new rounding contract, not a replay of a
+predecessor theorem. It does not assert that every mode fiber has radius one
+or that an actual caller must preserve these modes. Complete kappa is unchanged.
+
+## Reproduce
+
+From the repository root, CPython 3.12 and the standard library suffice:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/mode-preserving-rounding/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/mode-preserving-rounding/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/mode-preserving-rounding
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/mode-preserving-rounding --check-source-history
+```
+
+Other systems may substitute their CPython 3.12 executable. Direct checks
+print `PASS_MODE_PRESERVING_ROUNDING`, including sixteen excluded corners,
+91 translated/scaled attaining fixtures and nine rejection controls.
+The preceding JSON is a fresh receipt binding the source and package files
+by SHA256. No receipt is handwritten or stored. Verification writes nothing.
+Normal and optimized modes run the same explicit checks. The complete export
+with its declared source input also supports Git-free direct replay.
+
+## Scope and credit
+
+The matrix is the coded inverse's source, not a claim that this proposed
+mode-preserving rounder is installed in the selected construction. The
+uniform-transfer-audit card is changed to this precision side result.
+The selected binding control passes at conditional kappa .003125804;
+it is not an operator replay. [TASK.json](TASK.json) records the original
+card, changed scope and next questions. [SOURCES.json](SOURCES.json) pins
+the exact live mode table and historical provenance.
+
+Credit Dillon Amburgey / PR4933 for M and A^-1=I-MM^T/52,
+JoyboyBrian / PR5069 for the fixed-grid inverse context, Dugongue for the
+coded producer, and their retained native/controller/compiler lineage.
+Ali Romman / PR5838 distinguishes possible rounding correlations from its
+adverse-defect model; that calculation is not rerun. Classical lattice and
+finite-box arguments retain their credit. New work is Apache-2.0.
+Independent human and formal proof-kernel review are unclaimed.
+
+The result supplies no global rounding-radius optimum, physical correlated
+rounder, complete error ledger, restoration, precision/storage/tape bill,
+supplier admission or complete multiplication theorem. The five framework
+assumptions and selected frontier retain their status.
+
+## Open next questions
+
+1. What is the least uniform radius on all nonempty integer fibers of this
+   exact mode map? Start with a fundamental domain of its kernel lattice;
+   the lower bound here gives no global upper construction.
+2. Can a concrete physical caller exploit approximate mode preservation
+   rather than exact preservation? Start with one reached state and its
+   permitted mode-defect budget; reachability, full propagation and tape
+   costs are still missing.
+
+## Verifier bounds
+
+The verifier reads at most one MiB per file and at most 32 package files.
+There are eight modes, 112 owners, sixteen corners and 91 scaled fixtures.
+No recursion, dynamic code execution, subprocess or output-file write is
+used. Functions stay below 60 lines, and errors remain enabled under -O.
+Python allocates managed integers, Fractions and containers. This is a
+Power of Ten rule 3 deviation bounded by those fixed domains and a registered
+30-second timeout. A future arbitrary-dimension implementation must reassess
+storage before changing those limits. Warning-strict normal/optimized replay
+provides the available diagnostics; no new static-analyzer configuration or
+physical-memory bound is claimed.
+Rule 9 has a bounded test-control deviation: `reject` calls only the four
+locally named functions `parse_modes`, `validate_modes`, `vector` and
+`same_modes`, with nine fixed fixtures. Unexpected exceptions propagate.

research/mode-preserving-rounding/SOURCES.json

@@ -0,0 +1,24 @@
+{
+  "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/coded-native-caller/NORMALIZERS.json",
+      "usage": "input",
+      "git_blob": "bca92c222c395d3d75c047fbb678f3cd0947c60f",
+      "sha256": "ae8aad06aa5d1811d725895b3ab5a4f4daa9dcdd88294aa0d5f723e78bb1e0da"
+    }
+  ]
+}

research/mode-preserving-rounding/TASK.json

@@ -0,0 +1,110 @@
+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 8,
+  "base_commit": "2485805a6d47f36ea83ddde6327d453ac82d44dc",
+  "original_card": {
+    "id": "uniform-transfer-audit",
+    "question": "Can one of the selected record's five inherited assumptions be proved at its actual caller?",
+    "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"
+    ],
+    "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": "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_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": "Choose one assumption ID (native_all_columns, owned_erasure, compiler, precision_tape, analytic_recovery), one quantified implication and exact dependency closure. Check current handoffs, including #5625 row reentry and #5707 root-prefix lowering, without treating their partial scope as the whole theorem. Prove or falsify that implication with a caller-bound argument; stop at the chosen obligation. Finite tests or root digits do not discharge an all-size statement.",
+    "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": [
+      "native_all_columns",
+      "owned_erasure",
+      "compiler",
+      "precision_tape",
+      "analytic_recovery"
+    ],
+    "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": "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 a feasible eight-mode fiber of the actual PR4933 code has exact integer rounding radius one, excluding every smaller joint-rounding radius.",
+  "scope_change_reason": "Original precision side question: correlated mode-preserving rounding replaces an audit of a selected all-size implication. No predecessor result is reverified as the deliverable.",
+  "bottleneck_link": "Possible correlated rounding requires its own local error set. Exact preservation of the coded inverse modes cannot be assumed compatible with a half-grid coordinate error; a physical caller and full error invoice remain separate.",
+  "budget_minutes": 60,
+  "stop_rule": "Ship the first verified feasible-fiber witness and exact radius proof; leave the global optimum and physical caller to Open next questions.",
+  "outcome": "completed",
+  "evidence": "Written two-equation integrality proof; exact live-source five-row relation; all sixteen integer corners; 91 translated/scaled attaining fixtures and nine rejection controls.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "What is the least uniform radius on all nonempty integer fibers of this exact mode map?",
+      "why_it_matters": "The witness gives a necessary radius but no global attaining rounder.",
+      "first_step": "Describe a fundamental domain of the integer kernel lattice and bound its infinity-norm covering radius.",
+      "blocked_by": "A global upper construction is absent."
+    },
+    {
+      "question": "Can a concrete physical caller exploit approximate mode preservation?",
+      "why_it_matters": "Exact mode preservation has a larger coordinate error on this fiber; a different permitted error set may avoid it.",
+      "first_step": "Specify one reached state and its permitted mode-defect budget, then propagate it through the actual inverse.",
+      "blocked_by": "Caller reachability, full error propagation and tape costs are unspecified."
+    }
+  ],
+  "claim_type": "Scoped mathematical proof with an exact finite certificate",
+  "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19299863",
+  "question_source": "original",
+  "predecessors": [
+    {
+      "pr": 4933,
+      "author": "Dillon Amburgey / dillona",
+      "head": "d969c458c24278e0fd37e5b4c0ff8a53a496b430",
+      "use": "Actual eight-mode matrix and cited inverse factorization."
+    },
+    {
+      "pr": 5069,
+      "author": "JoyboyBrian",
+      "use": "Fixed-grid arithmetic context; not replayed."
+    },
+    {
+      "pr": 5838,
+      "author": "Ali Romman / aliromman",
+      "use": "Distinguishes a correlated physical error set from independently adverse defects; no source computation reused."
+    }
+  ]
+}

research/mode-preserving-rounding/VERIFY.json

@@ -0,0 +1,36 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/coded-native-caller/NORMALIZERS.json"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 30,
+      "checks": "Source-bound feasible fiber, all sixteen integer corners, exact radius attainment, 91 scaled/translated fixtures and nine rejection controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 30,
+      "checks": "The same complete explicit checks under optimized Python, with generated source/package hash receipt."
+    }
+  ]
+}