diff --git a/research/paid-history-quotient/README.md b/research/paid-history-quotient/README.md
new file mode 100644
index 0000000000..f1c72c7c8e
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+++ b/research/paid-history-quotient/README.md
@@ -0,0 +1,100 @@
+# Additional paid returns with retained observations
+
+This package gives an exact finite-dimensional audit of additional paid returns.
+It includes every supplied initial observation before it applies a lower bound.
+A correctly versioned observation can remove a charge. A copy of an old version
+need not remove it. The complete kappa calculation does not change.
+
+## Statement and proof
+
+Let R be a finite-dimensional unital matrix algebra over Q, with dimension r.
+A field row is a d by dN rational matrix on N named independent input fields.
+Let H be the left R-module generated by the supplied initial rows. Let T be
+the left R-module generated by all required final rows. Set
+
+```
+Delta = dim_Q(H + T) - dim_Q(H).
+m >= ceil(Delta / r).
+```
+
+Here m counts additional paid calls. Each call returns one actual new field
+row at the same capacity. The call can act on a row that uses earlier paid
+returns. Ordinary operations are left actions by R and scalar row combinations.
+They preserve the current history module. Each actual paid return t adds Rt,
+whose dimension is at most r. Keep a conceptual copy of every paid return,
+even after physical overwrite or clear. After m returns the history dimension
+is at most dim(H)+mr. It contains H+T at completion. Subtraction proves the bound.
+No commutation or R-linearity of a paid operator is needed. All rows must denote
+linear maps on the same independent input space. Target rows include required
+dirty returns; zero scratch has no independent input column.
+
+The bound is necessary. Delta=0 means that the target rows lie in the supplied
+ordinary closure. It does not prove that the required storage, scalar operations,
+ownership, access, or final erasure are free. The dimension bound need not be a
+sharp return count for a restricted paid alphabet.
+
+The code forms R from the identity and the supplied ordinary generators. It
+repeatedly left-multiplies new basis elements by each generator. Stabilization
+occurs after at most d squared independent elements. The resulting span contains
+every generator word, so products omitted from an initial generator list cannot
+create a false positive charge. Exact rational elimination computes H and H+T.
+There is no floating-point rank decision. The direct tool accepts JSON with
+`dimension`, distinct `inputs`, `ordinary_generators`, `initial`, and `targets`.
+Each initial or target row has d rows and d times N columns. Use integer values
+or rational strings. Example: d=2, inputs=["x"], no ordinary generators,
+initial=[I_2], targets=[diag(1,-1)] gives Delta=1 and m>=1. Supplying the target
+as an initial observation gives Delta=0.
+
+## Evidence and reproduction
+
+```sh
+python3 -B scripts/research_package.py --package research/paid-history-quotient
+python3 -B research/paid-history-quotient/audit.py /path/to/interface.json
+```
+
+Python 3.12 and its standard library suffice. The registered normal and optimized
+checks cover all cached subsets for one through five distinct inputs, 96 literal
+programs with feedback and overwrite, and all 1152 prefixes. Controls distinguish
+old and current versions, require full ordinary closure, and reject ambiguous
+input names, malformed matrices, floats and zero denominators. Every check uses
+an explicit exception. The written proof gives the general finite-matrix result;
+the finite tests do not prove an all-width native statement.
+
+## Scope and overlap
+
+This answers an algebraic part of the retained-observation question in
+[PR #5446](https://github.com/CrocSwap/integer-mult-bounds/pull/5446), at
+`0d0bcf3fa261c9591e7e60ed5926c6e04d360f9e`. Its
+[pointwise proof](https://github.com/Clear1nsight/integer-mult-bounds/blob/0d0bcf3fa261c9591e7e60ed5926c6e04d360f9e/docs/research/standalone-history-scalar-ceiling/proofs/pointwise-lemma-v1.md)
+supplies the actual-return history principle and the distinction from a parent
+cut with observations. This package supplies a finite rational computation for
+an explicit initial module. It neither repeats its tensor-width theorem nor
+claims to have extracted the current physical caller.
+
+[PR #5427](https://github.com/CrocSwap/integer-mult-bounds/pull/5427) tests specific
+frozen formal-word observations. [PR #5441](https://github.com/CrocSwap/integer-mult-bounds/pull/5441)
+checks finite-memory endpoint tables and routing ranks. Those are different
+contracts. The overlap search covered 319 open PRs, 4715 Discussion records,
+and all 7402 tracked in-scope notes on canonical main
+`32890bc5de034ab8fc93ac88ead7279a6a416d04`, priority revision 7. Arbitrary diffs,
+unpublished work, and exhaustive historical PR bodies remain outside coverage.
+The selected baseline helper stopped on omitted sparse source inputs; no
+baseline or selected cold-replay pass is claimed.
+
+## Open next questions
+
+Extract one actual boundary's complete current rows and ordinary generators.
+Verify source versions, phases, ownership, access and final returns, then apply
+the audit. This could prevent a duplicate charge or certify a necessary added
+cost. The missing physical row and availability ledger is the blocker. Do not
+replace it with a list of desired observations. Semisimple component bounds
+may strengthen the dimension ratio; that is a separate algebraic task.
+
+## Credit
+
+Chris R / Clear1nsight and the retained PR5425/PR5446 authors supply the paid-history
+argument and application question. Classical module dimension and rational
+elimination supply the mathematical method. This package does not claim those
+principles as new. The exact source is linked above; no predecessor code is
+imported or changed. Ali Romman with substantial OpenAI Codex assistance.
+Apache-2.0. No independent human review or proof-kernel acceptance is claimed.
diff --git a/research/paid-history-quotient/SOURCES.json b/research/paid-history-quotient/SOURCES.json
new file mode 100644
index 0000000000..1a711914dd
--- /dev/null
+++ b/research/paid-history-quotient/SOURCES.json
@@ -0,0 +1,27 @@
+{
+  "schema_version": 1,
+  "base_commit": "32890bc5de034ab8fc93ac88ead7279a6a416d04",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "fc3d4ba105e1f097d0d7d051a308281e86ab1446",
+      "sha256": "66974efdced4083c00025f97d95d83909a226cdf801edb9301c195c6641f6903"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    }
+  ],
+  "external_provenance": [
+    {
+      "repository": "Clear1nsight/integer-mult-bounds",
+      "commit": "0d0bcf3fa261c9591e7e60ed5926c6e04d360f9e",
+      "path": "docs/research/standalone-history-scalar-ceiling/proofs/pointwise-lemma-v1.md",
+      "git_blob": "5aeadc6617e81e66b3918eab4e248ba263cdd2a9",
+      "usage": "Read-only proof precedent; no runtime import."
+    }
+  ]
+}
diff --git a/research/paid-history-quotient/TASK.json b/research/paid-history-quotient/TASK.json
new file mode 100644
index 0000000000..5c74a2f267
--- /dev/null
+++ b/research/paid-history-quotient/TASK.json
@@ -0,0 +1,95 @@
+{
+  "schema_version": 1,
+  "task_id": "formal-interface-lemma",
+  "priority_revision": 7,
+  "base_commit": "32890bc5de034ab8fc93ac88ead7279a6a416d04",
+  "original_card": {
+    "id": "formal-interface-lemma",
+    "question": "Can one uncovered interface implication be formalized with exact caller and certificate bindings?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/gate-operand-parity-lean/README.md",
+      "research/staged544-certified-endpoint/README.md",
+      "research/unconditional-batched-crt/README.md"
+    ],
+    "methods": [
+      "Formalize a missing boundary lemma identified by the selected review.",
+      "Audit a formal theorem-to-certificate binding independently."
+    ],
+    "acceptance_gate": "State the exact formalized proposition and remaining assumptions; avoid duplicate arithmetic formalizations already present.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Inventory the gate-operand parity lemma (#3807) and certified endpoint arithmetic (#3857). Neither is a full multiplication proof. For #4100, audit the compact-switch statement and its hypotheses before formalizing it.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Write one missing statement with quantified hypotheses, actual caller and existing coverage. A compact-switch lemma is a candidate only after its mathematical contract is reviewed. Establish a bounded proof plan before a separately budgeted proof-assistant build; do not rerun large Lean jobs as orientation.",
+    "bottleneck_link": "A missing interface proof can reduce the conditional gap even if it does not change kappa.",
+    "allowed_variations": [
+      "Narrow to one formal lemma or one theorem-to-certificate binding."
+    ],
+    "scope_change_triggers": [
+      "Formalizing already-covered arithmetic and claiming the full multiplication theorem."
+    ],
+    "overlap_terms": [
+      "hyperplane end",
+      "formal interface",
+      "Lean binding",
+      "3807",
+      "3857",
+      "4100",
+      "gate-operand-parity-lean",
+      "staged544-certified-endpoint",
+      "unconditional-batched-crt"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #3807: scoped gate-operand parity formalization.",
+        "evidence": "research/gate-operand-parity-lean/README.md",
+        "remaining": "Exact theorem-to-caller binding and genuinely uncovered interface lemmas.",
+        "status": "author-checked",
+        "source_commit": "4c3530f42a6abcc48e7a6bcffd6c901911dc8269",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #3857: certified real-power moment/endpoint arithmetic.",
+        "evidence": "research/staged544-certified-endpoint/README.md",
+        "remaining": "Uniform algorithmic implications, not another endpoint digit.",
+        "status": "author-checked",
+        "source_commit": "ef4315b8caa605e47a2515004845eb2d2ed71a86",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4100: compact-switch formalization proposed as future validation.",
+        "evidence": "research/unconditional-batched-crt/README.md",
+        "remaining": "Audit the statement and assumptions before a formal proof campaign.",
+        "status": "author-checked",
+        "source_commit": "337d04939571a637aeab54a0f58f54fde811663a",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Exact finite rational lower bound on additional paid returns with supplied initial observations.",
+  "scope_change_reason": "Take the PR5446 handoff as a written interface lemma and exact audit instead of a Lean formalization.",
+  "bottleneck_link": "Avoid a duplicate standalone charge at a caller boundary that already has useful paid observations.",
+  "budget_minutes": 60,
+  "stop_rule": "Prove the dimension bound and test finite chronological programs; do not construct a caller.",
+  "outcome": "completed",
+  "evidence": "README.md proof; verify.py exact cache and chronological checks.",
+  "author": "Ali Romman",
+  "ai_assistance": "Substantial OpenAI Codex assistance.",
+  "open_questions": [
+    {
+      "question": "What is the exact initial module at one actual caller boundary?",
+      "why_it_matters": "It determines which additional paid charges are necessary.",
+      "first_step": "Extract complete versioned physical rows and ordinary generators.",
+      "blocked_by": "No physical availability, ownership or source-version ledger is supplied here."
+    }
+  ]
+}
diff --git a/research/paid-history-quotient/VERIFY.json b/research/paid-history-quotient/VERIFY.json
new file mode 100644
index 0000000000..b90c62b899
--- /dev/null
+++ b/research/paid-history-quotient/VERIFY.json
@@ -0,0 +1,30 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact cache cases, complete chronological prefixes, version and closure controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "-O",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact cache cases, complete chronological prefixes, version and closure controls."
+    }
+  ]
+}
diff --git a/research/paid-history-quotient/audit.py b/research/paid-history-quotient/audit.py
new file mode 100644
index 0000000000..365d6a4ca2
--- /dev/null
+++ b/research/paid-history-quotient/audit.py
@@ -0,0 +1,112 @@
+#!/usr/bin/env python3
+"""Compute an exact lower bound for a finite paid-history interface.
+
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+import argparse
+from fractions import Fraction as Q
+import json
+from pathlib import Path
+
+
+def scalar(value):
+    if type(value) not in (int, str):
+        raise ValueError("use integers or rational strings")
+    return Q(value)
+
+
+def matrix(value, rows, columns):
+    if not isinstance(value, list) or len(value) != rows:
+        raise ValueError("invalid row count")
+    if any(not isinstance(row, list) or len(row) != columns for row in value):
+        raise ValueError("invalid column count")
+    return tuple(tuple(scalar(x) for x in row) for row in value)
+
+
+def identity(d):
+    return tuple(tuple(Q(i == j) for j in range(d)) for i in range(d))
+
+
+def multiply(a, b):
+    return tuple(tuple(sum(x * y for x, y in zip(row, col))
+                       for col in zip(*b)) for row in a)
+
+
+def flat(a):
+    return tuple(x for row in a for x in row)
+
+
+class Span:
+    def __init__(self):
+        self.rows = {}
+
+    def add(self, vector):
+        row = list(vector)
+        for pivot, basis in sorted(self.rows.items()):
+            c = row[pivot]
+            row = [x - c * y for x, y in zip(row, basis)]
+        pivot = next((i for i, x in enumerate(row) if x), None)
+        if pivot is None:
+            return False
+        c = row[pivot]
+        self.rows[pivot] = tuple(x / c for x in row)
+        return True
+
+    def __len__(self):
+        return len(self.rows)
+
+
+def algebra(d, generators):
+    basis, span = [identity(d)], Span()
+    span.add(flat(basis[0]))
+    index = 0
+    while index < len(basis):
+        for generator in generators:
+            product = multiply(generator, basis[index])
+            if span.add(flat(product)):
+                basis.append(product)
+        index += 1
+    return basis
+
+
+def audit(data):
+    d = data['dimension']
+    if type(d) is not int or d < 1:
+        raise ValueError("dimension must be a positive integer")
+    names = data['inputs']
+    if (not isinstance(names, list) or not names
+            or any(not isinstance(x, str) or not x for x in names)
+            or len(set(names)) != len(names)):
+        raise ValueError("input versions must have distinct nonempty names")
+    generators = [matrix(x, d, d) for x in data['ordinary_generators']]
+    initial = [matrix(x, d, d * len(names)) for x in data['initial']]
+    targets = [matrix(x, d, d * len(names)) for x in data['targets']]
+    ordinary = algebra(d, generators)
+    span = Span()
+    for row in initial:
+        for a in ordinary:
+            span.add(flat(multiply(a, row)))
+    initial_rank = len(span)
+    for row in targets:
+        for a in ordinary:
+            span.add(flat(multiply(a, row)))
+    delta = len(span) - initial_rank
+    return {'ordinary_dimension': len(ordinary), 'initial_dimension': initial_rank,
+            'joint_dimension': len(span), 'target_quotient_dimension': delta,
+            'additional_return_lower_bound': (delta + len(ordinary) - 1) // len(ordinary),
+            'targets_in_initial_ordinary_closure': delta == 0}
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('input', type=Path)
+    args = parser.parse_args()
+    try:
+        result = audit(json.loads(args.input.read_text(encoding='utf-8')))
+    except (ValueError, KeyError, TypeError, ZeroDivisionError) as error:
+        parser.error(str(error))
+    print(json.dumps(result, sort_keys=True))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/paid-history-quotient/verify.py b/research/paid-history-quotient/verify.py
new file mode 100644
index 0000000000..a1db712ba1
--- /dev/null
+++ b/research/paid-history-quotient/verify.py
@@ -0,0 +1,92 @@
+#!/usr/bin/env python3
+"""Check exact history bounds and full chronological examples.
+
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+from copy import deepcopy
+from itertools import combinations
+import json
+import random
+import sys
+from pathlib import Path
+sys.path.insert(0, str(Path(__file__).resolve().parent))
+from audit import audit, algebra, multiply
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def block(a, position, n):
+    return [[a[i][j % 2] if j // 2 == position else 0 for j in range(2*n)]
+            for i in range(2)]
+
+
+def encode(a):
+    return [[str(x) for x in row] for row in a]
+
+
+def main():
+    I, X, Z = [[1,0],[0,1]], [[0,1],[1,0]], [[1,0],[0,-1]]
+    cases = 0
+    for n in range(1, 6):
+        for k in range(n+1):
+            for cached in combinations(range(n), k):
+                data = {'dimension':2,'inputs':[f'v{i}@now' for i in range(n)],
+                        'ordinary_generators':[],
+                        'initial':[block(I,i,n) for i in range(n)] + [block(Z,i,n) for i in cached],
+                        'targets':[block(Z,i,n) for i in range(n)]}
+                result = audit(data)
+                require(result['target_quotient_dimension'] == n-k, 'cached quotient')
+                require(result['additional_return_lower_bound'] == n-k, 'cached lower bound')
+                cases += 1
+    E12, E21 = [[0,1],[0,0]], [[0,0],[1,0]]
+    require(len(algebra(2,[tuple(map(tuple,E12)),tuple(map(tuple,E21))])) == 4,
+            'products must be included')
+    base = {'dimension':2,'inputs':['old','new'],'ordinary_generators':[],
+            'initial':[block(I,i,2) for i in range(2)] + [block(Z,0,2)],
+            'targets':[block(Z,1,2)]}
+    require(audit(base)['additional_return_lower_bound'] == 1, 'stale version')
+    good = deepcopy(base); good['initial'].append(block(Z,1,2))
+    require(audit(good)['additional_return_lower_bound'] == 0, 'current observation')
+    closure = deepcopy(base); closure['ordinary_generators'] = [E12,E21]
+    require(audit(closure)['additional_return_lower_bound'] == 0, 'ordinary closure')
+    controls = 3
+    for key, value in [('dimension',True), ('inputs',['same','same']),
+                       ('ordinary_generators',[[[1.0,0],[0,1]]]),
+                       ('targets',[[[1,0],[0,1]]]),
+                       ('ordinary_generators',[[['1/0',0],[0,1]]])]:
+        bad=deepcopy(base); bad[key]=value
+        try: audit(bad)
+        except (ValueError, ZeroDivisionError): controls += 1
+        else: raise ValueError('accepted invalid input: '+key)
+    rng=random.Random(7301)
+    programs=prefixes=0
+    for generators in ([],[Z],[E12],[E12,E21]):
+        for _ in range(24):
+            rows=[block(I,0,2),block(I,1,2),[[0]*4 for _ in range(2)]]
+            initial=deepcopy(rows); paid=0
+            for step in range(12):
+                dst,src=rng.sample(range(3),2)
+                if step % 3 == 0:
+                    operator=[[rng.randrange(-2,3) for _ in range(2)] for _ in range(2)]
+                    rows[dst]=[list(r) for r in multiply(operator,rows[dst])]; paid+=1
+                elif step % 3 == 1:
+                    rows[dst]=[[x+y for x,y in zip(a,b)] for a,b in zip(rows[dst],rows[src])]
+                elif generators:
+                    rows[dst]=[list(r) for r in multiply(rng.choice(generators),rows[dst])]
+                else:
+                    rows[dst]=deepcopy(rows[src])
+                data={'dimension':2,'inputs':['x','y'],'ordinary_generators':generators,
+                      'initial':initial,'targets':[encode(a) for a in rows]}
+                require(audit(data)['additional_return_lower_bound'] <= paid,
+                        'history bound exceeds literal paid returns')
+                prefixes+=1
+            programs+=1
+    print('PASS_PAID_HISTORY_QUOTIENT '+json.dumps(
+        {'cache_cases':cases,'literal_programs':programs,'prefixes':prefixes,'controls':controls},sort_keys=True))
+
+
+if __name__ == '__main__':
+    main()