diff --git a/research/shifted-persistence-threshold/PROOF.md b/research/shifted-persistence-threshold/PROOF.md
new file mode 100644
index 0000000000..f91657a594
--- /dev/null
+++ b/research/shifted-persistence-threshold/PROOF.md
@@ -0,0 +1,166 @@
+# A shifted sufficient threshold for the one-step comparison
+
+This scoped proof was produced by Nick Petalas. Apache-2.0.
+
+## Statement
+
+Let `d0,a,b` be positive integers, and put `D=d0*b`. Define
+
+```text
+T(a,D) = 1                                    if a>D,
+         max(2, ceil((3D-a)/(2a)))             if a<=D.
+```
+
+For every integer `m>=T(a,D)`,
+
+```text
+(m+1)^D < 2^a m^D.                                      (1)
+```
+
+Moreover, `T(a,D)<=ceil(27D/(17a))` for every positive integer `a,D`.
+At `(d0,a,b)=(1,1,5)` the new sufficient threshold is the exact minimum
+`7`, whereas the cited #7450 sufficient threshold is `8`. At `(1,2,5)`
+both sufficient thresholds are the exact minimum `4`.
+
+This is a new application of a classical elementary analytic inequality.
+No novelty is claimed for that inequality or for logarithmic differentiation.
+The general sufficient threshold is not claimed to be an exact minimum.
+
+## The shifted power inequality
+
+For real `x>0`, put `F(x)=(2x+1) log(1+1/x)`, using natural logarithms.
+Differentiation gives
+
+```text
+F'(x) = 2 log(1+1/x) - (2x+1)/(x(x+1)).
+```
+
+The function `t -> 1/t` is strictly convex on the positive real axis.
+On `[x,x+1]` its graph is strictly below the secant except at the endpoints.
+Integration therefore gives
+
+```text
+log(1+1/x) = integral_x^(x+1) dt/t
+          < (1/2)(1/x + 1/(x+1)).
+```
+
+Consequently `F'(x)<0`. Since `F(1)=3 log 2`, every real `x>1` satisfies
+`F(x)<3 log 2`. Exponentiating, for every integer `m>=2`,
+
+```text
+((m+1)/m)^(2m+1) < 8.                                  (2)
+```
+
+At `m=1` there is equality, not strict inequality. This endpoint is why
+the definition of `T` separates `a>D` and requires `m>=2` otherwise.
+
+## Deriving the threshold
+
+Suppose `m>=2` and `a(2m+1)>=3D`. Raise (2) to the positive integer power
+`D` and use that exponent inequality:
+
+```text
+((m+1)/m)^(D(2m+1)) < 2^(3D) <= 2^(a(2m+1)).
+```
+
+Taking a positive `(2m+1)`-st root proves (1). Its hypothesis holds when
+`m>=ceil((3D-a)/(2a))`. Thus the stated `T` works when `a<=D`.
+
+If `a>D`, then for every integer `m>=1`, `(m+1)/m<=2`, so
+`((m+1)/m)^D<=2^D<2^a`. Therefore `T=1` works in this case.
+
+All inequalities are strict where required. In particular, equality in
+`a(2m+1)>=3D` is allowed because (2) is strict for `m>=2`.
+
+## Comparison with the cited bound
+
+The predecessor #7450 supplies `ceil(27D/(17a))` as a sufficient threshold
+through a uniform binomial bound and the pair `3^17<2^27`. That predecessor
+result is a cited premise, not a replay target. The following comparison is
+also an elementary comparison of the two displayed formulas.
+
+When `a>D`, the old ceiling is at least one, so it is at least `T=1`.
+When `a<=D`, put `R=D/a>=1`. The old ceiling is at least two, and
+
+```text
+27R/17 - (3R-1)/2 = 3R/34 + 1/2 > 0.
+```
+
+Ceiling monotonicity then gives
+
+```text
+max(2, ceil((3R-1)/2)) <= ceil(27R/17).
+```
+
+Hence the new sufficient threshold never worsens that cited threshold.
+
+## The two sharp examples
+
+For `D=5,a=1`, the new value is `ceil(14/2)=7`. At its predecessor,
+
+```text
+7^5 = 16807 > 15552 = 2*6^5.
+```
+
+Thus (1) fails at `m=6`. The theorem proves (1) for every `m>=7`;
+directly, `8^5=32768<33614=2*7^5`. This makes seven the least integer
+threshold valid for all subsequent integers at this triple.
+
+For `D=5,a=2`, the new value is `ceil(13/4)=4`. At its predecessor,
+
+```text
+4^5 = 1024 > 972 = 4*3^5.
+```
+
+The theorem proves (1) for every `m>=4`; directly,
+`5^5=3125<4096=4*4^5`. Thus four remains the exact threshold.
+
+These examples do not make the general formula an exact minimum. For
+`D=10,a=1`, it gives fifteen, but (1) already holds at fourteen because
+`15^10=576650390625<578509309952=2*14^10`.
+
+## Persistence at the retained comparison
+
+Section 6 of the #7000 application review considers a fixed constant
+`C>=1`, an integer degree `d0` and positive rational slack `delta=a/b`.
+For positive `d0`, raising its comparison to the power `b` gives
+
+```text
+P(j): C^b (j+1)^D <= 2^(aj),       D=d0*b.
+```
+
+If `P(j)` holds and `j+1>=T(a,D)`, apply (1) at `m=j+1`:
+
+```text
+C^b (j+2)^D < 2^a C^b (j+1)^D <= 2^(a(j+1)).
+```
+
+Thus `P(j+1)` holds, and induction proves `P(k)` for every integer `k>=j`.
+When `d0=0`, this persistence is immediate without a threshold lemma.
+This establishes a smaller sufficient one-step region. It does not supply
+an initial `j` satisfying `P(j)`, compute any complete selected cutoff,
+or replace the other conditions in the review.
+
+## Evidence and limits
+
+The universal claims above follow from the written argument. The verifier
+corroborates (2) for `m=2..256`, and decreasing `F` at consecutive integers
+for `m=1..255` by exact cross-multiplication. It checks both threshold
+formulas for every `a,D=1..96`, and checks (1) at each new threshold and
+the next two integers. These are bounded finite checks of this new result.
+They do not replay a predecessor verifier or establish the analytic proof
+by enumeration. All arithmetic in the verifier is exact integer arithmetic.
+
+The comparison is an auxiliary threshold lemma, not a physical supplier,
+a new kappa, or a complete multiplication theorem. All five selected
+framework assumptions remain. Independent human and formal proof-kernel
+review are unclaimed.
+
+## Open next questions
+
+Can an elementary rational correction to the shifted bound remove the
+remaining sufficient-versus-necessary gap, such as fifteen versus fourteen
+at `D=10,a=1`? A first step is an explicit upper bound for
+`log(1+1/m)` with one more rational term and a proof that the resulting
+integer test is no worse. This may reduce the auxiliary persistence region;
+it does not determine the initial selected cutoff or settle physical costs.
diff --git a/research/shifted-persistence-threshold/README.md b/research/shifted-persistence-threshold/README.md
new file mode 100644
index 0000000000..3c353a825b
--- /dev/null
+++ b/research/shifted-persistence-threshold/README.md
@@ -0,0 +1,119 @@
+# A shifted persistence threshold that includes the missing seven-step case
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+Claim type: scoped written proof with exact finite corroboration. For
+positive integers `d0,a,b`, set `D=d0*b` and
+
+```text
+T = 1                              if a>D,
+    max(2,ceil((3D-a)/(2a)))        otherwise.
+```
+
+For every integer `m>=T`, `(m+1)^D < 2^a m^D`. The new `T` is never larger
+than `ceil(27D/(17a))`, the sufficient threshold in unpaidmoon's
+[PR #7450](https://github.com/CrocSwap/integer-mult-bounds/pull/7450).
+At `(d0,a,b)=(1,1,5)` it lowers eight to the sharp seven. At `(1,2,5)`
+the sharp four is preserved. The proof uses the classical decreasing
+function `(2x+1) log(1+1/x)`. The threshold application is the new result.
+
+[PROOF.md](PROOF.md) gives the complete argument, the exact endpoint
+exception at `m=1`, and the persistence implication. The general sufficient
+threshold is not an exact minimum. For example it gives fifteen at
+`D=10,a=1`, where fourteen already works.
+
+## Source and scope
+
+This answers Open next question 1 of unpaidmoon #7450 at
+`f9b9057f9befe744df7f6b58b61b53e77001af1f` using an `m`-dependent inequality,
+instead of refining another pair `3^p<2^q` under its uniform bound. Its
+power-pair census and persistence construction are cited, not audited or
+replayed. The question is the source; the classical convexity argument is
+self-contained. No predecessor code or data is needed to run this verifier.
+
+The retained comparison is section 6 of
+[the #7000 application review](../../docs/research/pr-intake-evidence/pr7000-application/PROOF-REVIEW.md).
+[SOURCES.json](SOURCES.json) pins that historical context, the selected
+record and the originating card at canonical base
+`e54523517d0f03e58352e647c403a5af3e525a57`. These are provenance, not live
+mathematical inputs. The source PR is external context with its exact head
+and URL recorded in [TASK.json](TASK.json).
+
+Card `uniform-transfer-audit` has changed scope to this auxiliary lemma.
+It reduces a sufficient persistence region relevant to the effective
+cutoff discussion. It neither constructs the initial successful index nor
+discharges `precision_tape` or another framework assumption. Selected
+`kappa=363005817/100000000000` is unchanged.
+
+## Reproduce
+
+Use CPython 3.12 and the standard library, from the repository root:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/shifted-persistence-threshold/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/shifted-persistence-threshold/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/shifted-persistence-threshold
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/shifted-persistence-threshold --check-source-history
+```
+
+On another machine, substitute its CPython 3.12 executable. The direct
+verifier emits `PASS_SHIFTED_PERSISTENCE_THRESHOLD` in a JSON receipt
+containing all six package-file SHA256 hashes and the exact enumerated
+counts. Redirect it outside the checkout to retain a generated receipt.
+No handwritten receipt or generated replay output is committed.
+
+The bounded checks cover 255 shifted-power cases, 255 consecutive decreases,
+9,216 parameter pairs and 27,648 one-step evaluations. In that parameter
+rectangle, 4,753 threshold pairs strictly improve. Nine semantic controls
+cover both sharp examples, strict equality, the `m=1` branch, a
+sufficient-versus-necessary distinction and an overstrong slope. Eight
+invalid parameter pairs must fail with the exact declared reason. Optimized
+execution keeps every check active because checks do not rely on `assert`.
+
+No selected cold replay, predecessor scientific suite, Lean or Docker run
+is needed for this local theorem. No hosted CI success is claimed here.
+
+## Verification design
+
+`verify.py` has a fixed finite workload, no recursion, subprocesses,
+network calls, file writes or dynamic dispatch. All functions are shorter
+than sixty lines. Positive-integer arguments are validated at the threshold
+boundary; all result checks raise explicit errors even under `-O`.
+
+Power of Ten rule 3 has a managed-runtime deviation in `verify.py`: CPython
+allocates arbitrary-precision integers during execution. The finite ranges
+bound each power-product operand below `2^10240`, and the verifier retains
+only one case at a time plus six source hashes. The registered commands
+exercise that complete bounded workload. If these ranges become configurable,
+the change must introduce explicit input and memory caps. Python has no
+compiler warning mode; both registered commands use `-W error`. The package
+does not alter shared diagnostics or repository CI configuration. Ruff and
+Mypy are unavailable on the local PATH; in-memory compilation and the
+warning-strict registered replays are the available Python diagnostics.
+
+## Related open work
+
+- [#7450](https://github.com/CrocSwap/integer-mult-bounds/pull/7450), unpaidmoon,
+  supplies the `27/17` bound and explicitly asks about the missing `m=7`
+  case. This proof includes that case and retains the same sharp four.
+- [#7286](https://github.com/CrocSwap/integer-mult-bounds/pull/7286), unpaidmoon,
+  tightens a monomial record count used elsewhere in the same review.
+  Its count and elimination model are independent of this one-step inequality.
+
+## Open next questions
+
+The final section of [PROOF.md](PROOF.md) asks for a sharper elementary
+correction where this sufficient bound exceeds the true threshold. Its
+first example is `D=10,a=1`; no external blocker is required to begin.
+Physical implementation, complete cutoffs, supplier admission and all five
+framework assumptions remain separate obligations.
+
+## Credit
+
+Credit unpaidmoon #7450 for the source question, the prior `27/17` threshold
+and its retained persistence-note lineage at
+`8e0124415604b89a1cfc842cb6dc6da96ea87ac5`. Credit Douglas Colkitt's retained
+#7000 application review for the effective-cutoff comparison. The convexity,
+logarithm and induction arguments are classical. The shifted
+threshold, its application and its exact verifier were produced by Nick
+Petalas. Independent human and formal proof-kernel review are unclaimed.
diff --git a/research/shifted-persistence-threshold/SOURCES.json b/research/shifted-persistence-threshold/SOURCES.json
new file mode 100644
index 0000000000..cfcfe0ca38
--- /dev/null
+++ b/research/shifted-persistence-threshold/SOURCES.json
@@ -0,0 +1,24 @@
+{
+  "schema_version": 1,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "75d77b328ea02ded535e7f23aee5b1e55d0bb1a8",
+      "sha256": "154c3d30455d778fd3343b6b4851f0fa4665f0f7625b0adb3675e2bafbbf6bf1"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "docs/research/pr-intake-evidence/pr7000-application/PROOF-REVIEW.md",
+      "usage": "provenance",
+      "git_blob": "3dd1af2285e597676725e85ad4f65e7ad326d695",
+      "sha256": "4501502c2ceb9759b20372435fe4beb51082c5a5e2910c1fd5dbe6edca415055"
+    }
+  ]
+}
diff --git a/research/shifted-persistence-threshold/TASK.json b/research/shifted-persistence-threshold/TASK.json
new file mode 100644
index 0000000000..c5fadef43c
--- /dev/null
+++ b/research/shifted-persistence-threshold/TASK.json
@@ -0,0 +1,111 @@
+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 8,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "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": "A self-contained shifted sufficient threshold for (m+1)^(d0*b) < 2^a m^(d0*b), with exact sharp seven and four examples.",
+  "scope_change_reason": "Answer Open next question 1 of unpaidmoon PR7450 rather than audit a predecessor or discharge a complete selected assumption.",
+  "bottleneck_link": "Auxiliary effective-cutoff persistence comparison in section 6 of the selected #7000 application review. This changes a sufficient one-step region only.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after the first proved threshold improvement and its bounded exact verifier.",
+  "outcome": "completed",
+  "evidence": "PROOF.md is the scoped universal argument; verify.py corroborates shifted powers, 9216 parameter pairs, 27648 steps and boundary/failure controls. It emits generated six-file hash-bound JSON receipts.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Can a further rational correction reduce the gap between the shifted sufficient threshold and the exact minimum, first at D=10,a=1?",
+      "why_it_matters": "A smaller sufficient region can reduce an auxiliary effective-cutoff requirement without changing the supplier.",
+      "first_step": "Bound log(1+1/m) with one more rational term and derive a sufficient integer inequality that is no worse.",
+      "blocked_by": "No external blocker; initial successful cutoff and physical costs remain separate."
+    }
+  ],
+  "claim_type": "scoped proof with exact finite corroboration",
+  "source_question": {
+    "pr": 7450,
+    "author": "unpaidmoon",
+    "head": "f9b9057f9befe744df7f6b58b61b53e77001af1f",
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/7450",
+    "location": "research/threshold-ratio-sharper-pair/README.md, Open next questions 1",
+    "usage": "external context; no predecessor computation is replayed"
+  },
+  "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19572802",
+  "environment": {
+    "interpreter": "/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12",
+    "dependencies": "standard library only",
+    "python_version": "3.12.15"
+  },
+  "overlap": {
+    "checked_at": "2026-11-12T19:17:00Z",
+    "open_prs": 768,
+    "discussion_records": 7356,
+    "own_recent_prs_in_source_check": 106,
+    "own_recent_source_matches": 0,
+    "note": "Full remote pagination; sparse local scan supplemented by origin/main README Git searches. Keyword checks are leads, not a novelty certificate.",
+    "precommit_refresh": {
+      "canonical_main": "e54523517d0f03e58352e647c403a5af3e525a57",
+      "no_new_overlapping_claim_found": true,
+      "merged_search": "Exact one-step, threshold-ratio, shifted threshold and general polynomial/exponential context; #6109 is a different source-specific compiler-grid binding."
+    }
+  }
+}
diff --git a/research/shifted-persistence-threshold/VERIFY.json b/research/shifted-persistence-threshold/VERIFY.json
new file mode 100644
index 0000000000..966a1c15c2
--- /dev/null
+++ b/research/shifted-persistence-threshold/VERIFY.json
@@ -0,0 +1,32 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Exact shifted-power and monotonicity comparisons, all a,D=1..96 threshold pairs and three steps each, sharp/boundary/overstrong controls, invalid inputs; generated package hashes. Checks remain active under optimization."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-O",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Exact shifted-power and monotonicity comparisons, all a,D=1..96 threshold pairs and three steps each, sharp/boundary/overstrong controls, invalid inputs; generated package hashes. Checks remain active under optimization."
+    }
+  ]
+}
diff --git a/research/shifted-persistence-threshold/verify.py b/research/shifted-persistence-threshold/verify.py
new file mode 100644
index 0000000000..045d0cc376
--- /dev/null
+++ b/research/shifted-persistence-threshold/verify.py
@@ -0,0 +1,120 @@
+#!/usr/bin/env python3
+"""Exact finite corroboration of the shifted persistence threshold.
+
+Produced by Nick Petalas. Apache-2.0.
+The universal argument is in PROOF.md; this program uses integers only.
+"""
+
+import hashlib
+import json
+import platform
+import sys
+from pathlib import Path
+
+
+def require(condition: bool, message: str) -> None:
+    if not condition:
+        raise ValueError(message)
+
+
+def threshold(a: int, degree: int) -> int:
+    """Return the new sufficient threshold, with D named degree."""
+    if type(a) is not int or type(degree) is not int:
+        raise ValueError("a and D must be positive integers")
+    if a < 1 or degree < 1:
+        raise ValueError("a and D must be positive integers")
+    if a > degree:
+        return 1
+    return max(2, (3 * degree - a + 2 * a - 1) // (2 * a))
+
+
+def step(a: int, degree: int, m: int) -> bool:
+    return (m + 1) ** degree < (1 << a) * m ** degree
+
+
+def check_shifted_power() -> dict[str, int]:
+    powers = 0
+    decreases = 0
+    for m in range(2, 257):
+        require((m + 1) ** (2 * m + 1) < 8 * m ** (2 * m + 1),
+                f"shifted power failed at {m}")
+        powers += 1
+    for m in range(1, 256):
+        # Cross-multiply F(m)>F(m+1), without logarithms or floats.
+        left = (m + 1) ** (4 * m + 4)
+        right = m ** (2 * m + 1) * (m + 2) ** (2 * m + 3)
+        require(left > right, f"shifted decrease failed at {m}")
+        decreases += 1
+    return {"shifted_power_cases": powers, "decrease_cases": decreases}
+
+
+def check_thresholds() -> dict[str, int]:
+    parameters = 0
+    steps = 0
+    drops = 0
+    for a in range(1, 97):
+        for degree in range(1, 97):
+            new = threshold(a, degree)
+            old = (27 * degree + 17 * a - 1) // (17 * a)
+            require(new <= old, f"threshold worsened at {a},{degree}")
+            if a <= degree:
+                require(a * (2 * new + 1) >= 3 * degree,
+                        "shifted exponent hypothesis missing")
+            for m in range(new, new + 3):
+                require(step(a, degree, m), f"step failed at {a},{degree},{m}")
+                steps += 1
+            drops += int(new < old)
+            parameters += 1
+    return {"parameter_pairs": parameters, "step_cases": steps,
+            "strict_threshold_drops": drops}
+
+
+def check_controls() -> dict[str, int]:
+    controls = (
+        (threshold(1, 5) == 7, "new first working example missing"),
+        (step(1, 5, 7) and not step(1, 5, 6), "seven not sharp"),
+        (threshold(2, 5) == 4, "second example changed"),
+        (step(2, 5, 4) and not step(2, 5, 3), "four not sharp"),
+        (not step(1, 1, 1), "strict equality was accepted"),
+        (threshold(1, 1) == 2, "m=1 equality exception missing"),
+        (threshold(2, 1) == 1 and step(2, 1, 1), "m=1 success missing"),
+        (threshold(1, 10) == 15 and step(1, 10, 14), "threshold not necessary"),
+        # A smaller, unproved 7/5 slope would propose m=8 at D=6,a=1.
+        (not step(1, 6, 8), "overstrong slope control did not reject"),
+    )
+    for condition, message in controls:
+        require(condition, message)
+    invalid = ((0, 1), (1, 0), (-1, 1), (1, -1),
+               (True, 1), (1, False), (1.0, 1), (1, "1"))
+    rejected = 0
+    for a, degree in invalid:
+        try:
+            threshold(a, degree)
+        except ValueError as error:
+            require(str(error) == "a and D must be positive integers",
+                    "wrong rejection reason")
+            rejected += 1
+        else:
+            raise ValueError("invalid input accepted")
+    return {"semantic_controls": len(controls), "invalid_input_controls": rejected}
+
+
+def main() -> None:
+    require(len(sys.argv) == 1, "usage: verify.py")
+    counts = {}
+    counts.update(check_shifted_power())
+    counts.update(check_thresholds())
+    counts.update(check_controls())
+    package = Path(__file__).resolve().parent
+    names = ("README.md", "PROOF.md", "TASK.json", "SOURCES.json",
+             "VERIFY.json", "verify.py")
+    hashes = {name: hashlib.sha256((package / name).read_bytes()).hexdigest()
+              for name in names}
+    print(json.dumps({"status": "PASS_SHIFTED_PERSISTENCE_THRESHOLD",
+                      "claim_type": "scoped proof with finite corroboration",
+                      "counts": counts, "python": platform.python_version(),
+                      "package_sha256": hashes}, sort_keys=True))
+
+
+if __name__ == "__main__":
+    main()