Joint denominator-13 decisions need fewer expected fair bits than separate calls

docs/ci-verification.md

@@ -1,6 +1,23 @@
 # CI verification
 
-## Current selected-family replay: #5599
+## Current selected-family replay: #6030
+
+The [selected record](../certificates/selected-result.json) binds conditional
+κ = 0.003125804 to the [proof review](research/pr-intake-evidence/pr6030-audit/PROOF-REVIEW.md),
+[validation receipts](research/pr-intake-evidence/pr6030-audit/VALIDATION.json),
+and [explicit source closure](../certificates/frontier6030-source.json).
+`make frontier6030-verify` follows that source-bound selected replay. The review
+separates retained normal primary macOS/Linux evidence from fresh independent
+word/invoice/native, original caller, adapted caller and optimized primary checks.
+Consult its receipts for exact commands, source heads and limits; the #5599
+checkpoint below retains its own historical replay details.
+
+The five inherited framework assumptions remain. Neither the full uniform word
+nor a complete rounded multiplier is materialized. A source/metadata check is
+not an operator replay or a formal theorem proof. Routine research submissions
+do not need to rerun this selected-family replay.
+
+## Historical selected-family replay: #5599
 
 The [selected record](../certificates/selected-result.json) binds conditional
 κ = 0.00310045 to the [proof review](research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md)

docs/research/current-status.md

@@ -1,6 +1,6 @@
 # Current reviewed status
 
-Selected conditional κ: **0.003125804 = 781451/250000000**, DreamingOfClouds'
+Selected conditional κ: **0.003125804** = 781451/250000000, DreamingOfClouds'
 PR6030 on the cumulative P104 construction. The complete caller counts 13,444
 productive fields and 66,834 paid calls. The saving is about 0.81775% above the
 preceding selected PR5599 claim 0.00310045.

research/paired-residue-sampler/README.md

@@ -0,0 +1,188 @@
+# Joint denominator-13 decisions use fewer expected fair bits
+
+For every fixed pair of nonzero residue probabilities $a/13,b/13$, an explicit
+finite-state sampler produces **two independent** Bernoulli decisions with
+optimal expected fair-bit count strictly below $7/2$. Two separate exact
+standalone samplers require expected count at least four. When both probabilities
+are $1/13$, the joint expected count is below $9/4$, saving more than 43.75% of
+the standalone expected draws.
+
+Each parameter pair has exactly 312 internal control states and 624 binary
+transitions. Including its four output labels, every target fits nine bits;
+a packed transition table uses 5,616 bits, or 702 bytes, per parameter pair.
+The implementation generates one table at a time rather than storing all 144.
+
+This is a finite exact sampling construction and an expected-bit optimum,
+not a physical inverse, worst-case latency bound, reversible cleanup protocol,
+supplier or new κ. It deliberately changes the **standalone-call** hypothesis
+of [PR #6068](https://github.com/CrocSwap/integer-mult-bounds/pull/6068).
+The underlying dyadic decision-graph method is classical; the contribution is
+this small complete denominator-13 catalogue, exact means and resource ledger.
+
+## Contract and source relevance
+
+The denominator-13 inverse in
+[PR #5069](https://github.com/CrocSwap/integer-mult-bounds/pull/5069) motivates
+rounding probabilities with this denominator. Both probabilities must be known
+**before** this sampler consumes any bits. Their intended decisions are
+independent. A second probability that depends on the first rounded result
+cannot be substituted into this batch without a new contract.
+
+No actual native caller is asserted to provide the required residue pair or
+batching window. The retained deterministic inverse is not alleged defective.
+The stochastic prefix of
+[Nick Petalas](https://github.com/petalas)'s
+[PR #6026](https://github.com/CrocSwap/integer-mult-bounds/pull/6026) supplies a
+related arithmetic context, but its adaptive updates are not automatically
+batchable, and its native/caller error analysis is not extended here.
+
+## Exact joint decision graph
+
+In output order 00, 01, 10, 11, set the integer weights
+
+$$
+(w_0,w_1,w_2,w_3)
+=((13-a)(13-b),(13-a)b,a(13-b),ab).
+$$
+
+Their sum is 169. Dividing by 169 gives exactly the product law, hence the
+required independent marginals. Write the binary digits of each probability
+as $d_{i,k}$ at depth $k\geq1$. Define
+
+$$
+n_k=2^k-\sum_{i=0}^3\lfloor2^k w_i/169\rfloor.
+$$
+
+This is the number of unfinished binary prefixes after depth $k$. At the root,
+$n_0=1$. At the next depth, double every unfinished node, label the first
+$d_{i,k}$ available children with output $i$, and retain the rest as unfinished.
+Each digit is zero or one, and
+
+$$
+n_k=2n_{k-1}-\sum_i d_{i,k}.
+$$
+
+Thus this rule never overfills a level. An executable version stores the
+unfinished-node index $j$: after drawing one bit $u$, set $j\leftarrow2j+u$,
+then subtract the four probability digits in output order. The first subtraction
+making $j$ negative returns that output; otherwise the remaining $j$ is the
+next unfinished-node index. All four digits come from exact residue doubling
+modulo 169.
+
+By induction, the completed output-$i$ prefixes at depth $k$ number exactly
+$\lfloor2^k w_i/169\rfloor$. Since $1\leq n_k\leq3$, unfinished probability
+is at most $3/2^k$, tending to zero. The resulting output law is therefore
+exactly $w_i/169$, not an approximation from finite samples.
+
+## Finite periodic controller and exact expected cost
+
+The order of two modulo 169 is 156; replay checks this with exact integer
+powers. Every weight is coprime to 13, so its residue sequence is periodic
+with that period. The digits and unfinished-node counts repeat. There are
+exactly 312 internal phase/node pairs for every one of the 144 residue pairs;
+this finite count is checked exhaustively. Two transitions per state give the
+624-entry table. No growing arithmetic precision is needed to represent the
+sampler's logical control state, although its random tape and stopping length
+have no finite worst-case bound.
+
+The exact expected draw count is
+
+$$
+E[T]=\frac{\sum_{k=0}^{155} n_k2^{-k}}{1-2^{-156}}.
+$$
+
+For $(a,b)=(1,1)$ this is
+
+$$
+E[T]=\frac{9960277917013927549978}{4649714690825496825793}<\frac94.
+$$
+
+The exact finite catalogue certifies $2<E[T]<7/2$ for all 144 nonzero residue
+pairs. These numerical bounds and state counts are finite certificates; the
+construction, law and expected-cost formula have the preceding written proof.
+No floating-point logarithms or Monte Carlo estimates are used.
+
+## Optimality and the changed standalone hypothesis
+
+For any exact fair-bit decision tree with these four output probabilities,
+completed output-$i$ mass after $k$ bits is an integer multiple of $2^{-k}$
+and cannot exceed $w_i/169$. Its completed prefix count is therefore at most
+$\lfloor2^k w_i/169\rfloor$. Its unfinished probability is at least $n_k/2^k$.
+Summing these tail bounds proves that its expected draw count is at least the
+displayed formula. This sampler saturates every depth bound, so it is optimal
+for the stated fixed joint law, with no free initial random state.
+
+A standalone non-dyadic Bernoulli sampler must leave at least one fair prefix
+unfinished at every depth, giving expectation at least two. Separate calls
+without a shared entropy reservoir therefore require at least four draws in
+expectation. The joint four-outcome tree changes that boundary; it preserves
+independence of the **decisions**, without requiring separately executed
+sampling trees. Reusing a single sampled flag for both decisions would have
+the right marginals but the wrong joint law and is rejected by a control.
+
+The sampler still has a positive unfinished probability at every finite depth.
+It chooses the unbounded-stopping option in #6068's tradeoff, not an escape
+from that worst-case theorem. A finite control table is not a finite draw cap.
+
+## Costs and unproved physical obligations
+
+The logical sampler performs one table transition per consumed fair bit, then
+returns one two-bit outcome. Per-configuration packed table storage is 702
+bytes and control needs nine bits including output labels. Selecting or building
+the table from actual data residues, table access, random-bit movement, output
+storage, stopping-record retention and reversible cleanup are additional costs.
+The returned decisions are correlated with the consumed tape, so that tape is
+not automatically a fresh reusable random resource.
+
+No source-bound physical batching window, all-column primitive, deterministic
+tariff, compiler or outer error budget is supplied. The expected bit reduction
+must not be read as a runtime speedup or an admitted exponent improvement.
+
+## Evidence and reproduction
+
+```
+python3.12 -B scripts/research_package.py --package research/paired-residue-sampler
+python3.12 -B scripts/check_research_commit.py --package research/paired-residue-sampler
+```
+
+The standard-library verifier constructs every table, checks the complete
+outcome-prefix counts for two full 156-step periods, independently enumerates
+36,864 literal eight-bit tapes, certifies the exact expected counts and resource
+bounds, and rejects corrupted output labels and false independence/latency
+claims. Normal and optimized modes write no package files. Universal probability
+and optimality come from the written depth-count argument, not the literal tapes.
+
+Canonical main was refreshed on 2026-10-28 UTC at
+721c03822c421b7f7fcbb2598993b947c4d88419, priority 8. The navigation-only repair
+from [PR #6050](https://github.com/CrocSwap/integer-mult-bounds/pull/6050) is
+included. The optional full-history check is unavailable in this shallow checkout;
+source pins use its reachable base. No selected/native cold replay, independent
+human review or formal-kernel acceptance is claimed. Frozen sources stay unchanged.
+
+## Significance for researchers
+
+The standalone two-draw lower bound is not a per-scalar amortized limit. A small
+joint interface can reduce expected randomness while preserving the exact
+independent rounding law. This supplies a concrete batching target and exposes
+the next obligation: an actual nonadaptive residue pair and a full hosting and
+cleanup bill, rather than free entropy or a claimed bounded-time sampler.
+
+## Open next questions
+
+Does one actual inverse-service stage expose two residue probabilities before
+either output is consumed? First bind that data-dependence and timing boundary,
+then price table selection, access and random-tape cleanup. No such caller is
+instantiated here.
+
+Can the table be shared or parameterized across residue pairs with less total
+physical control cost? Start with two configurations and an explicit dispatcher
+that preserves the joint law. A lower expected bit count alone does not fund it.
+
+## Credit
+
+Prepared by [zxv](https://github.com/zxv) with substantial OpenAI Codex assistance.
+The dyadic distribution-generation tree and depth-count optimality argument are
+classical and not claimed as new. [JoyboyBrian](https://github.com/JoyboyBrian)'s
+#5069/#5037 lineage supplies the inverse context;
+[Nick Petalas](https://github.com/petalas)'s #6026 supplies the stochastic-prefix
+context. Their sources and attribution remain unchanged. Apache-2.0.

research/paired-residue-sampler/SOURCES.json

@@ -0,0 +1,30 @@
+{
+  "schema_version": 1,
+  "base_commit": "3a219d0ae1f2c984d4176a08132a1abbcb5a882f",
+  "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-stochastic-prefix/PROOF.md",
+      "usage": "input",
+      "git_blob": "0db45191676cb037fca76e7a781e02670fcc4078",
+      "sha256": "db1640be6eade5955320c106ef4d0666742c195560402d0067fa0ee3648bbeca"
+    },
+    {
+      "path": "research/physical-inverse-pair/README.md",
+      "usage": "input",
+      "git_blob": "a4ed4756eaa2e2e1fe3d91f2d8c7f1d682d8185b",
+      "sha256": "32505039ae7289b6358de5ec9d83a8bb63b76140209b521c7e8c82e8e0000783"
+    }
+  ]
+}

research/paired-residue-sampler/TASK.json

@@ -0,0 +1,89 @@
+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 8,
+  "base_commit": "3a219d0ae1f2c984d4176a08132a1abbcb5a882f",
+  "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": "Exact optimal joint samplers for all 144 nonzero denominator-13 residue pairs, with finite 312-state controllers and lower expected fair-bit counts.",
+  "scope_change_reason": "Broader lateral search changes the standalone randomness boundary to a nonadaptive two-output batch. It supplies no physical inverse integration.",
+  "bottleneck_link": "Constructively escapes treating the standalone two-draw lower bound as a per-scalar amortized tariff, while retaining the unbounded worst-case stopping obligation.",
+  "budget_minutes": 10,
+  "stop_rule": "Finish the finite catalogue and exact expected-bit proof; no larger batch or physical sampler campaign.",
+  "outcome": "completed",
+  "evidence": "README.md exact DDG/law/optimality proof; verify.py all tables, periodic count certificates and literal tapes.",
+  "author": "zxv (https://github.com/zxv), with substantial OpenAI Codex assistance",
+  "ai_assistance": "Substantial OpenAI Codex assistance.",
+  "open_questions": [
+    {
+      "question": "Can an actual inverse service expose a nonadaptive residue pair economically?",
+      "why_it_matters": "Needed to realize the bit saving in a caller.",
+      "first_step": "Bind the two probabilities and timing, then price table access and cleanup.",
+      "blocked_by": "No source-bound batching window or physical host."
+    },
+    {
+      "question": "Can residue tables share a costed dispatcher?",
+      "why_it_matters": "May reduce control cost without changing the law.",
+      "first_step": "Implement two configurations and check exact output-prefix counts.",
+      "blocked_by": "Only per-configuration logical tables are supplied."
+    }
+  ]
+}

research/paired-residue-sampler/VERIFY.json

@@ -0,0 +1,33 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/p104-stochastic-prefix/PROOF.md",
+    "research/physical-inverse-pair/README.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "All 144 joint laws and 312-state tables, complete prefix counts over two periods, exact expected-bit bounds, literal tapes and rejection controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Same exact checks with assertions disabled."
+    }
+  ]
+}