From 751e7b1f34a761062291bb88824d57e9af05a008 Mon Sep 17 00:00:00 2001
From: zxv <zxv@zxv>
Date: Tue, 27 Oct 2026 23:19:14 -0400
Subject: [PATCH 1/3] Align current CI navigation with selected PR6030

---
 docs/ci-verification.md         | 19 ++++++++++++++++++-
 docs/research/current-status.md |  2 +-
 2 files changed, 19 insertions(+), 2 deletions(-)

diff --git a/docs/ci-verification.md b/docs/ci-verification.md
index 006bf5a3d5..63ba3e9aa8 100644
--- a/docs/ci-verification.md
+++ b/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)
diff --git a/docs/research/current-status.md b/docs/research/current-status.md
index a84ac12213..c49a245adc 100644
--- a/docs/research/current-status.md
+++ b/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.

From 2c218eda719770e92c3256091d07c18520f0867e Mon Sep 17 00:00:00 2001
From: zxv <zxv@zxv>
Date: Wed, 28 Oct 2026 00:20:39 -0400
Subject: [PATCH 2/3] Construct source-bound stochastic compute and uncompute
 with parity restoration

---
 .../reversible-stochastic-shears/README.md    | 197 ++++++++++++++++++
 .../reversible-stochastic-shears/SOURCES.json |  36 ++++
 .../reversible-stochastic-shears/TASK.json    |  95 +++++++++
 .../reversible-stochastic-shears/VERIFY.json  |  34 +++
 .../reversible-stochastic-shears/verify.py    | 160 ++++++++++++++
 5 files changed, 522 insertions(+)
 create mode 100644 research/reversible-stochastic-shears/README.md
 create mode 100644 research/reversible-stochastic-shears/SOURCES.json
 create mode 100644 research/reversible-stochastic-shears/TASK.json
 create mode 100644 research/reversible-stochastic-shears/VERIFY.json
 create mode 100644 research/reversible-stochastic-shears/verify.py

diff --git a/research/reversible-stochastic-shears/README.md b/research/reversible-stochastic-shears/README.md
new file mode 100644
index 0000000000..cd34b124e1
--- /dev/null
+++ b/research/reversible-stochastic-shears/README.md
@@ -0,0 +1,197 @@
+# Stochastic scalar prefixes can restore dirty inputs with a parity log
+
+A grid-valued stochastic ADD is an exactly invertible shear when its inverse
+uses the complemented coin. A rounded signed halving instead needs one parity
+bit; keeping that bit suffices, and recomputing it during reversal clears it.
+Together these give a compute–accumulate–uncompute interface for the exact
+9,589-event scalar prefix studied by
+[Nick Petalas](https://github.com/petalas)'s
+[PR #6026](https://github.com/CrocSwap/integer-mult-bounds/pull/6026).
+
+The source has 84 signed halvings. The protocol uses **84 clean residue bits
+and 11,713 read-only random bits per real stream sample**, restores all 748
+arbitrary dirty working values, returns the residue log to zero, and leaves
+the random tape unchanged. Its forward arithmetic law is exactly the source's
+fresh stochastic rounding law. Thus its source-relative forward error bound
+is retained under the source's assumptions; that concentration result is cited,
+not independently replayed or strengthened here.
+
+This is a logical scalar-grid ABI, not a native rounded supplier, physically
+reversible bit-level implementation, complete caller invoice or new κ. Physical
+random-bit preparation/access, guard widths, integer arithmetic, temporary
+cleanup and the later native MOVE remain payable and unimplemented.
+
+## Exact stochastic rounding and an inverse shear
+
+Normalize the grid spacing to one. For an integer numerator $n$, dyadic
+denominator $d=2^q$, and a fixed coin integer $0\leq U<d$, divide
+$n=dk+r$ with $0\leq r<d$, using Euclidean division even for negative $n$.
+Define $S(n,d,U)$ to be $k$ plus one when $U<r$ and $k$ otherwise. Uniform
+$U$ gives exactly unbiased stochastic rounding of $n/d$.
+
+Two identities hold for every integer input and every coin:
+
+$$
+S(dy+n,d,U)=y+S(n,d,U),
+$$
+
+$$
+\boxed{S(-n,d,d-1-U)=-S(n,d,U).}
+$$
+
+For the second identity, if $r=0$ both sides are exact integers. Otherwise
+$-n=d(-k-1)+(d-r)$, and the condition $d-1-U<d-r$ is equivalent to $U\geq r$.
+This is precisely the complement of $U<r$. The proof includes all signs and
+ties. Complementing a $q$-bit coin gives $d-1-U$ without another random draw.
+
+For distinct source and destination coordinates, implement a dyadic ADD by
+
+$$
+y\leftarrow y+S(sx,d,U).
+$$
+
+It is a translation in $y$ for fixed $x,U$. The inverse is the same stochastic
+ADD with coefficient $-s/d$ and complemented coin. The first identity shows
+that this is exactly the source's rule of rounding the whole updated grid value.
+No lost fractional residue needs to be saved for a shear, because its source
+coordinate remains available.
+
+Reversing a complete word recovers that source at each corresponding inverse
+step by induction. Coins may be fresh across **forward** events and reused only
+at their matched inverse. The independent-forward concentration hypothesis is
+therefore preserved. Reusing one coin across different forward events is not
+justified by this construction.
+
+## Signed halving and its necessary local bit
+
+For a SCALE coefficient $\sigma/2$, with $\sigma$ equal to plus or minus one,
+save $r=(\sigma x)\bmod2$ in a clean bit and set $y=S(\sigma x,2,U)$. Recover
+
+$$
+x=\sigma\bigl(2y-2[U<r]+r\bigr),
+$$
+
+where the bracket is one when its condition holds and zero otherwise. This
+follows immediately from $\sigma x=2k+r$ and $y=k+[U<r]$. Recompute the recovered
+parity and XOR it into the saved bit, clearing the log without discarding an
+unknown value. The random coin is read, not overwritten.
+
+One bit is necessary for this **isolated** rounded halving when the coin is
+returned unchanged and no other value retains the input distinction: for fixed
+$U=1$, inputs zero and one both round to zero. A side state with only one value
+cannot distinguish them. The parity bit supplies the required two states.
+This is not a proof that the complete source word needs all 84 bits; later
+structure might permit reuse or infer some residues.
+
+SCALE by minus one is exact and self-inverse. SCALE by two is exact on the
+forward grid, and its inverse divides the reached even value by two. These
+are logical operations on reached grid states, not an all-state physical gate
+synthesis. A finite implementation must provide its magnitude/guard format.
+
+## Complete source-bound compute–accumulate–uncompute protocol
+
+The pinned historical #5599 prefix has 9,358 ADDs and 231 SCALEs on 748 public
+roles, all at its empty common frame before the first MOVE. Of the SCALEs,
+115 are minus one, 32 are two, 18 are one half and 66 are minus one half.
+There are 7,421 potentially rounded events, with total denominator logarithm
+11,713. These source counts are independently reconstructed by replay.
+
+Supply arbitrary grid-valued working inputs $x$, disjoint arbitrary dirty
+output accumulators $y$, an independent uniform random tape, and 84 clean
+residue bits. Give each potentially rounded forward event its own fixed coin
+block, even when its particular value happens to be on the grid.
+
+1. Execute the source prefix with its exact fresh stochastic rounding law,
+   saving each signed-half parity in its assigned clean bit.
+2. Add the resulting 748 grid components to the output accumulators.
+3. Reverse every event. Use complemented matched coins for ADDs; use saved
+   parities for signed-half recovery and clear each bit by recomputation.
+
+Let $\widetilde F_U$ be the forward rounded map at the fixed tape $U$. The
+complete endpoint is
+
+$$
+(x,y,U,0)\longmapsto(x,y+\widetilde F_U(x),U,0).
+$$
+
+The output accumulators are never read by the reverse pass, so they can begin
+arbitrarily dirty. Every working input and the entire tape are restored for
+every coin string, not merely in expectation or with high probability. The
+forward output's probability law is unchanged under uniform independent coins;
+its source-relative error guarantee applies at the copied output. There is no
+claim that the rounded map equals the exact Gaussian or native operator.
+
+The logical ledger is 9,589 forward macro-updates, 748 integer output ADDs and
+9,589 reverse macro-updates, totaling 19,926 macros. The tape contains 11,713
+bits and is read twice; the 84 owned parity bits are returned clean. A reverse
+halving macro includes parity-dependent integer correction and log clearing.
+**These are macro/resource counts, not a bit-operation or tape-movement tariff.**
+The physical sampler, temporary arithmetic storage and complete native caller
+bill are unknown, not zero.
+
+## Evidence and reproduction
+
+```
+python3.12 -B scripts/research_package.py --package research/reversible-stochastic-shears
+python3.12 -B scripts/check_research_commit.py --package research/reversible-stochastic-shears
+```
+
+The verifier authenticates the unchanged prefix source and proof, reconstructs
+all source opcodes/counts, checks exact unbiased marginals, grid translation and
+the antithetic identity for signed numerators and denominators through 16,
+enumerates signed-half inverses and a two-role protocol with changing sources,
+and executes the entire source-bound protocol on four dirty-input families and
+four deterministic coin tapes. Those tapes check pathwise restoration, not a
+Monte Carlo tail estimate. Missing residue, uncomplemented inverse coins and
+invalid coefficient/tape domains have rejecting controls. Normal and optimized
+modes write no package files; arbitrary-size and all-coin uniformity follows
+from the written inverse identities and chronological induction.
+
+The replay uses exact Python integers. Its finite fixtures cap values at 256
+bits; this is a verifier bound, not a proved physical format for arbitrary input.
+No proof assistant, selected replay, native MOVE, overflow analysis or independent
+human review is claimed. The current selected #6030 word is a different source;
+this result deliberately retains the exact #6026/#5599 historical prefix.
+
+Main was refreshed on 2026-10-28 UTC at
+721c03822c421b7f7fcbb2598993b947c4d88419, priority 8. The navigation repair from
+[PR #6050](https://github.com/CrocSwap/integer-mult-bounds/pull/6050) is included.
+The optional full-history preflight is unavailable in this shallow checkout;
+all declared pins use the reachable base. Frozen sources and selection stay unchanged.
+
+## Significance for researchers
+
+Unbiased forward rounding and exact dirty restoration are compatible at this
+scalar boundary when inverse coins are coupled correctly and irreversible
+halvings retain their residue information. The result supplies an explicit
+logical lifecycle for the random tape and residue log, rather than relying on
+fresh rounding in the inverse. It closes this scalar interface step while
+leaving the physical reversible sampler, native execution and full cost open.
+
+## Open next questions
+
+Can the logical macros be lowered to a source-compatible reversible integer
+arithmetic and tape program with bounded scratch and a competitive full bill?
+Start with one signed-half macro and an explicit input/guard format. This
+package supplies no such bit-level lowering or random-tape preparation.
+
+Can some of the 84 parity bits be reused or inferred at the actual word's
+chronological cuts? Start with one half/double segment and prove exact recovery
+for all dirty grid inputs and coins. The one-bit local necessity result does
+not establish global log optimality.
+
+Can the next actual native MOVE preserve a useful complete output error bound?
+First supply its all-column numerical contract and the output consumer. The
+common-frame scalar protocol alone does not answer that question.
+
+## Credit
+
+Prepared by [zxv](https://github.com/zxv) with substantial OpenAI Codex assistance.
+[Nick Petalas](https://github.com/petalas)'s #6026 supplies the stochastic law,
+exact historical-prefix influence proof and round/restore question.
+[Ali Romman](https://github.com/aliromman)'s #5599 supplies the unchanged word,
+with the retained helper, controller, geometry, BIT and precision lineage in
+its sources. Antithetic sampling, residue retention and compute/uncompute are
+classical techniques. The scoped contribution is the exact coupled inverse ABI,
+source-bound 84-bit parity ledger and dirty-output accumulation endpoint here.
+Apache-2.0; predecessor files and attribution remain intact.
diff --git a/research/reversible-stochastic-shears/SOURCES.json b/research/reversible-stochastic-shears/SOURCES.json
new file mode 100644
index 0000000000..f3eedb2da3
--- /dev/null
+++ b/research/reversible-stochastic-shears/SOURCES.json
@@ -0,0 +1,36 @@
+{
+  "schema_version": 1,
+  "base_commit": "751e7b1f34a761062291bb88824d57e9af05a008",
+  "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-helper-subspace-splits/WORD.json.gz",
+      "usage": "input",
+      "git_blob": "e325091e3051afe37a2d9c82f85afe9028510edb",
+      "sha256": "bbb072bced5824a8d9094c49be2fc51fbfdd05fb2b7dab8106e1275053075099"
+    },
+    {
+      "path": "research/p104-stochastic-prefix/PROOF.md",
+      "usage": "input",
+      "git_blob": "0db45191676cb037fca76e7a781e02670fcc4078",
+      "sha256": "db1640be6eade5955320c106ef4d0666742c195560402d0067fa0ee3648bbeca"
+    },
+    {
+      "path": "research/p104-stochastic-prefix/README.md",
+      "usage": "input",
+      "git_blob": "9c98f2bcdb13c28fd4c106dff8a9c61eabc41716",
+      "sha256": "ccf869519cf234eb492557d1ea559e89773fc7c093457d8e555e6471abca3fac"
+    }
+  ]
+}
diff --git a/research/reversible-stochastic-shears/TASK.json b/research/reversible-stochastic-shears/TASK.json
new file mode 100644
index 0000000000..3e834725f6
--- /dev/null
+++ b/research/reversible-stochastic-shears/TASK.json
@@ -0,0 +1,95 @@
+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 8,
+  "base_commit": "751e7b1f34a761062291bb88824d57e9af05a008",
+  "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": "Coupled-coin stochastic scalar compute/accumulate/uncompute ABI on the exact PR6026 historical prefix, with 84 clean parity bits and preserved forward law.",
+  "scope_change_reason": "Broader lateral search targets the precision_tape arithmetic interface rather than another helper invoice. This is a logical scalar protocol, not a full native supplier or all-size physical proof.",
+  "bottleneck_link": "Resolves the scalar dirty-restoration/random-tape lifecycle step while retaining the forward stochastic process; native bit costs and later MOVE remain open.",
+  "budget_minutes": 20,
+  "stop_rule": "Stop after exact inverse identities, source-bound prefix replay and a complete logical resource ledger; no physical sampler campaign.",
+  "outcome": "completed",
+  "evidence": "README.md universal inverse proof and endpoint; verify.py exact local probabilities and full prefix protocols.",
+  "author": "zxv (https://github.com/zxv), with substantial OpenAI Codex assistance",
+  "ai_assistance": "Substantial OpenAI Codex assistance.",
+  "open_questions": [
+    {
+      "question": "Can these macros be physically lowered with bounded arithmetic scratch and full tape cost?",
+      "why_it_matters": "Logical restoration alone does not admit a supplier.",
+      "first_step": "Lower one signed-half macro with an exact guard/precision format.",
+      "blocked_by": "No bit-level reversible arithmetic or random-tape preparation."
+    },
+    {
+      "question": "Can chronological reuse reduce the 84-bit parity log?",
+      "why_it_matters": "May reduce owned resource cost.",
+      "first_step": "Prove recovery across one actual half/double segment.",
+      "blocked_by": "Local one-bit necessity does not establish global optimality."
+    },
+    {
+      "question": "Can the next native MOVE and consumer preserve a useful error budget?",
+      "why_it_matters": "Needed for a full caller.",
+      "first_step": "Bind its numerical all-column contract and propagate copied-output errors.",
+      "blocked_by": "Only the common-frame scalar prefix is covered."
+    }
+  ]
+}
diff --git a/research/reversible-stochastic-shears/VERIFY.json b/research/reversible-stochastic-shears/VERIFY.json
new file mode 100644
index 0000000000..67c8606b93
--- /dev/null
+++ b/research/reversible-stochastic-shears/VERIFY.json
@@ -0,0 +1,34 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/p104-helper-subspace-splits/WORD.json.gz",
+    "research/p104-stochastic-prefix/PROOF.md",
+    "research/p104-stochastic-prefix/README.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact stochastic marginals and inverse identities, signed-half parity recovery, changing-source tiny protocols, full 9589-event source-bound compute/copy/uncompute and rejection controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Same exact checks with assertions disabled."
+    }
+  ]
+}
diff --git a/research/reversible-stochastic-shears/verify.py b/research/reversible-stochastic-shears/verify.py
new file mode 100644
index 0000000000..1c02889cbc
--- /dev/null
+++ b/research/reversible-stochastic-shears/verify.py
@@ -0,0 +1,160 @@
+"""Source-bound stochastic compute/accumulate/uncompute on an integer grid."""
+from fractions import Fraction as Q
+from itertools import product
+from pathlib import Path
+from random import Random
+import gzip
+import hashlib
+import json
+
+HERE=Path(__file__).resolve().parent
+ROOT=HERE.parents[1]
+WORD='research/p104-helper-subspace-splits/WORD.json.gz'
+
+
+def need(ok,msg):
+    if not ok:raise ValueError(msg)
+
+
+def sr(n,d,u):
+    need(d>0 and d&(d-1)==0 and 0<=u<d,'dyadic coin domain')
+    k,r=divmod(n,d)
+    return k+int(u<r)
+
+
+def source():
+    pins=json.loads((HERE/'SOURCES.json').read_text())
+    for row in pins['files']:
+        if row['usage']=='input':
+            raw=(ROOT/row['path']).read_bytes()
+            need(hashlib.sha256(raw).hexdigest()==row['sha256'],'source bytes')
+    raw=(ROOT/WORD).read_bytes();need(len(raw)<=1<<20,'compressed bound')
+    with gzip.open(ROOT/WORD,'rb') as f:raw=f.read((1<<22)+1)
+    need(len(raw)<=1<<22,'unpacked bound')
+    w=json.loads(raw);need(w['public']==748 and w['frames'][3]==[],'source boundary')
+    cut=next(i for i,e in enumerate(w['program']) if e[0]=='MOVE')
+    need(cut==9589,'source scalar prefix')
+    es=[]
+    for e in w['program'][:cut]:
+        op=e[0];need(op in ('ADD','SCALE') and e[-1]==3,'scalar opcode/chart')
+        a=Q(e[-2]);n,d=a.numerator,a.denominator
+        dst=e[1];src=e[2] if op=='ADD' else None
+        need(0<=dst<748 and d in (1,2,4) and abs(a)<=16,'coefficient/role')
+        if op=='ADD':need(0<=src<748 and src!=dst,'shear source')
+        else:need(a in (Q(-1),Q(2),Q(1,2),Q(-1,2)),'supported SCALE')
+        es.append((op,dst,src,n,d))
+    return es
+
+
+def forward(initial,events,tape):
+    need(len(events)==len(tape),'tape length')
+    x=list(initial);logs=[]
+    for (op,dst,src,n,d),u in zip(events,tape):
+        if op=='ADD':x[dst]+=sr(n*x[src],d,u)
+        else:
+            if d==2:logs.append((n*x[dst])%2)
+            x[dst]=sr(n*x[dst],d,u)
+        need(abs(x[dst]).bit_length()<=256,'finite fixture magnitude')
+    return x,logs
+
+
+def backward(current,events,tape,logs):
+    x=list(current);logs=list(logs);j=len(logs)
+    for (op,dst,src,n,d),u in reversed(list(zip(events,tape))):
+        if op=='ADD':x[dst]+=sr(-n*x[src],d,d-1-u)
+        elif d==2:
+            need(n in (-1,1),'half sign');j-=1;r=logs[j]
+            x[dst]=n*(2*x[dst]-2*int(u<r)+r)
+            logs[j]^=(n*x[dst])%2
+            need(logs[j]==0,'recomputed residue clears log')
+        elif n==-1:x[dst]=-x[dst]
+        elif n==2:
+            need(x[dst]%2==0,'exact inverse on reached doubled image')
+            x[dst]//=2
+        else:raise ValueError('unsupported inverse SCALE')
+    need(j==0 and not any(logs),'all owned residue bits restored')
+    return x,logs
+
+
+def protocol(initial,out,events,tape):
+    need(isinstance(tape,tuple),'read-only coin tape')
+    saved_tape=tuple(tape)
+    result,logs=forward(initial,events,tape)
+    target=[y+z for y,z in zip(out,result)]
+    back,cleared=backward(result,events,tape,logs)
+    need(back==list(initial),'arbitrary dirty data restored')
+    need(tuple(tape)==saved_tape and not any(cleared),'tape/log boundary')
+    return target,max((abs(v).bit_length() for v in result),default=0)
+
+
+def rejected(fn):
+    try:fn()
+    except ValueError:return
+    raise ValueError('negative control accepted')
+
+
+def main():
+    identities=0
+    for d in (1,2,4,8,16):
+        for n in range(-16,17):
+            need(sum(sr(n,d,u) for u in range(d))==n,'exact unbiased marginal')
+            for u in range(d):
+                need(sr(-n,d,d-1-u)==-sr(n,d,u),'antithetic identity')
+                for y in range(-4,5):
+                    need(sr(d*y+n,d,u)==y+sr(n,d,u),'grid translation')
+                identities+=1
+    halves=0
+    for x,sign,u in product(range(-64,65),(-1,1),(0,1)):
+        es=[('SCALE',0,None,sign,2)]
+        y,logs=forward([x],es,(u,));back,log=backward(y,es,(u,),logs)
+        need(back==[x] and log==[0],'signed half recovery')
+        halves+=1
+    tiny=[('ADD',0,1,1,4),('SCALE',1,None,-1,2),('ADD',1,0,-1,2),
+          ('SCALE',0,None,2,1),('SCALE',1,None,-1,1)]
+    tiny_cases=0
+    for a,b in product(range(-3,4),repeat=2):
+        total=[0,0];exact=[Q(a),Q(b)]
+        for op,dst,src,n,d in tiny:
+            exact[dst]=exact[dst]+Q(n,d)*exact[src] if op=='ADD' else Q(n,d)*exact[dst]
+        for u,v,w in product(range(4),range(2),range(2)):
+            protocol([a,b],[5,-7],tiny,(u,v,w,0,0));tiny_cases+=1
+            values,_=forward([a,b],tiny,(u,v,w,0,0))
+            total=[x+y for x,y in zip(total,values)]
+        need([Q(x,16) for x in total]==exact,'full tiny-word unbiased forward law')
+    events=source()
+    ops={op:sum(e[0]==op for e in events) for op in ('ADD','SCALE')}
+    bits=sum(d.bit_length()-1 for _,_,_,_,d in events)
+    h=sum(op=='SCALE' and d==2 for op,_,_,_,d in events)
+    rounded=sum(d>1 for _,_,_,_,d in events)
+    need(ops=={'ADD':9358,'SCALE':231} and bits==11713 and h==84 and rounded==7421,'source ledger')
+    rng=Random(6026)
+    fixtures=[[0]*748,[((i*73)%251)-125 for i in range(748)],
+              [(-1 if i%2 else 1)*((1<<30)+104729*i) for i in range(748)],
+              [rng.randrange(-10**6,10**6) for _ in range(748)]]
+    full=peak=0
+    for initial in fixtures:
+        target=[i*17-12345 for i in range(748)]
+        for mode in range(4):
+            tape=tuple(0 if mode==0 else d-1 if mode==1 else i%d if mode==2
+                       else hashlib.sha256(str(i).encode()).digest()[0]%d
+                       for i,(_,_,_,_,d) in enumerate(events))
+            _,size=protocol(initial,target,events,tape);peak=max(peak,size);full+=1
+    # Same uncomplemented coin is an actual failing inverse, not a guarantee.
+    y=sr(1,2,0);bad=y+sr(-1,2,0)
+    need(bad!=0,'uncomplemented-coin counterexample')
+    need(sr(0,2,1)==sr(1,2,1),'one side bit necessary for local halving')
+    rejected(lambda:need(2*sr(1,2,0)==1,'discarded half residue'))
+    es=[('SCALE',0,None,1,2)];y,log=forward([1],es,(0,));log[0]=0
+    rejected(lambda:need(backward(y,es,(0,),log)[0]==[1],'corrupt residue'))
+    rejected(lambda:sr(1,3,0))
+    rejected(lambda:sr(1,2,2))
+    rejected(lambda:forward([1],[],(0,)))
+    print(json.dumps(dict(status='PASS_REVERSIBLE_STOCHASTIC_PREFIX',local_coin_identities=identities,
+                         signed_halves=halves,tiny_protocols=tiny_cases,full_prefix_protocols=full,
+                         events=9589,random_bits=bits,clean_residue_bits=h,
+                         forward_reverse_and_copy_macros=2*len(events)+748,
+                         peak_fixture_value_bits=peak,
+                         scope='logical scalar-grid ABI; physical sampler/native costs open; no new kappa'),sort_keys=True))
+
+
+if __name__=='__main__':main()

From 2711ca28367c5bbbb54ce9601928e885d87ca165 Mon Sep 17 00:00:00 2001
From: zxv <zxv@zxv>
Date: Wed, 28 Oct 2026 00:30:47 -0400
Subject: [PATCH 3/3] Distinguish a restored random tape from fresh independent
 randomness

---
 research/reversible-stochastic-shears/README.md | 6 ++++++
 research/reversible-stochastic-shears/verify.py | 4 ++++
 2 files changed, 10 insertions(+)

diff --git a/research/reversible-stochastic-shears/README.md b/research/reversible-stochastic-shears/README.md
index cd34b124e1..829b3ebf4b 100644
--- a/research/reversible-stochastic-shears/README.md
+++ b/research/reversible-stochastic-shears/README.md
@@ -121,6 +121,12 @@ forward output's probability law is unchanged under uniform independent coins;
 its source-relative error guarantee applies at the copied output. There is no
 claim that the rounded map equals the exact Gaussian or native operator.
 
+The returned tape is unchanged bit for bit, but the retained rounded output can
+be correlated with it. It is not automatically a fresh independent random
+resource for a later adaptive call. Such a call needs new independent coin
+blocks or a new conditional-error proof; restoration alone does not replenish
+randomness. The verifier includes a two-shear reused-tape bias counterexample.
+
 The logical ledger is 9,589 forward macro-updates, 748 integer output ADDs and
 9,589 reverse macro-updates, totaling 19,926 macros. The tape contains 11,713
 bits and is read twice; the 84 owned parity bits are returned clean. A reverse
diff --git a/research/reversible-stochastic-shears/verify.py b/research/reversible-stochastic-shears/verify.py
index 1c02889cbc..94106a084e 100644
--- a/research/reversible-stochastic-shears/verify.py
+++ b/research/reversible-stochastic-shears/verify.py
@@ -142,6 +142,10 @@ def main():
     # Same uncomplemented coin is an actual failing inverse, not a guarantee.
     y=sr(1,2,0);bad=y+sr(-1,2,0)
     need(bad!=0,'uncomplemented-coin counterexample')
+    reused=sum(sr(sr(1,2,u),2,u) for u in (0,1))
+    fresh=sum(sr(sr(1,2,u),2,v) for u,v in product((0,1),repeat=2))
+    need(Q(reused,2)==Q(1,2) and Q(fresh,4)==Q(1,4),
+         'restored tape is not fresh for an adaptive second shear')
     need(sr(0,2,1)==sr(1,2,1),'one side bit necessary for local halving')
     rejected(lambda:need(2*sr(1,2,0)==1,'discarded half residue'))
     es=[('SCALE',0,None,1,2)];y,log=forward([1],es,(0,));log[0]=0