diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index d806ee45b..cfd6258e7 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -167,7 +167,7 @@ jobs:
       # fresh against the restored libs, which is cheap.
       - name: Define the hex-dev cache + build target sets
         run: |
-          echo "HEX_LIB_TARGETS=HexBasic HexTruncatedSeries HexTruncatedSeriesMathlib HexArith HexPoly HexPolyFast HexMvPoly HexModArith HexGF2 HexPolyZ HexRoots HexResultant HexInterval HexIntervalExperiment HexIntervalMathlib HexIntervalMathlibExperiment HexIntervalReplayProbe HexIntervalMathlibReplayProbe HexRealRootsMathlibReplayProbe HexRCFProofProbe HexPolyFp HexGFqRing HexGFqField HexBerlekamp HexHensel HexConway HexGFq HexPrimality HexIntFactor HexPrimalityKernelProbe HexIntFactorKernelProbe HexMvGcdKernelProbe HexBerlekampZassenhaus HexRealRoots HexMatrix HexRowReduce HexDeterminant HexBareiss HexHermite HexSmith HexCharPoly HexMinPoly HexPolySmith HexGramSchmidt HexLLL HexMatrixMathlib HexHermiteMathlib HexSmithMathlib HexSmithTests HexCharPolyMathlib HexMinPolyMathlib HexPolySmithMathlib HexGramSchmidtMathlib HexLLLMathlib HexBerlekampZassenhausMathlib HexBerlekampZassenhausMathlibProofProbe HexBerlekampMathlibProofProbe HexPrimalityMathlib HexIntFactorMathlib HexPolyFpMathlib HexFactorizationModules HexRealRootsMathlib HexGF2Mathlib HexGFqMathlib HexMvPolyMathlib HexMvPolyMathlibProofProbe HexRCF HexRCFTests HexRootsMathlib HexResultantMathlib HexNumberField HexNumberFieldMathlib HexNumberFieldTower HexNumberFieldTowerMathlib HexReleaseTests HexReleaseExamples HexAggregateCheck" >> "$GITHUB_ENV"
+          echo "HEX_LIB_TARGETS=HexBasic HexTruncatedSeries HexTruncatedSeriesMathlib HexArith HexPoly HexPolyFast HexMvPoly HexModArith HexGF2 HexPolyZ HexRoots HexResultant HexInterval HexIntervalExperiment HexIntervalMathlib HexIntervalMathlibExperiment HexIntervalReplayProbe HexIntervalMathlibReplayProbe HexRealRootsMathlibReplayProbe HexRCFProofProbe HexPolyFp HexGFqRing HexGFqField HexBerlekamp HexHensel HexConway HexGFq HexPrimality HexIntFactor HexPrimalityKernelProbe HexPrimalityElabProbe HexIntFactorKernelProbe HexMvGcdKernelProbe HexBerlekampZassenhaus HexRealRoots HexMatrix HexRowReduce HexDeterminant HexBareiss HexHermite HexSmith HexCharPoly HexMinPoly HexPolySmith HexGramSchmidt HexLLL HexMatrixMathlib HexHermiteMathlib HexSmithMathlib HexSmithTests HexCharPolyMathlib HexMinPolyMathlib HexPolySmithMathlib HexGramSchmidtMathlib HexLLLMathlib HexBerlekampZassenhausMathlib HexBerlekampZassenhausMathlibProofProbe HexBerlekampMathlibProofProbe HexPrimalityMathlib HexIntFactorMathlib HexPolyFpMathlib HexFactorizationModules HexRealRootsMathlib HexGF2Mathlib HexGFqMathlib HexMvPolyMathlib HexMvPolyMathlibProofProbe HexRCF HexRCFTests HexRootsMathlib HexResultantMathlib HexNumberField HexNumberFieldMathlib HexNumberFieldTower HexNumberFieldTowerMathlib HexReleaseTests HexReleaseExamples HexAggregateCheck" >> "$GITHUB_ENV"
           echo "HEX_EXE_TARGETS=hextruncatedseries_bench hexarith_bench hexpoly_bench hexpolyfast_bench hexpolysmith_bench hexmvpoly_bench hexmvgcd_bench hexsparsepoly_bench hexpolyz_bench hexpolyfp_bench hexmodarith_bench hexmodular_bench hexgf2_bench hexgfqring_bench hexgfqfield_bench hexgfq_bench hexhensel_bench hexprimality_bench hexprimality_policy_probe hexintfactor_bench hexberlekamp_bench hexbz_bench hexconway_bench hexmatrix_bench hexstrassen_compare hexdeterminant_bench hexbareiss_bench hexcharpoly_bench hexminpoly_bench hexgramschmidt_bench hexhermite_bench hexsmith_bench hexrealroots_bench hexrcf_bench hexroots_bench hexresultant_bench hexnumberfield_bench hexnumberfieldtower_bench hexinterval_decision_bench hexroots_demo hex_interval_representation_spike hex_interval_center_spike hex_interval_scale_spike hex_interval_scheduler_spike hex_interval_policy_frontier_spike hex_arith_floor hexlll_bench hexlll_gram_bench hexlll_external_reduction hexbz_factor_service" >> "$GITHUB_ENV"
       # Shared build. The libraries, bench exes, conformance #guard drivers, and
       # emit-fixture exes are all elaborated here so the two verification tails
diff --git a/HexPrimality/Elab.lean b/HexPrimality/Elab.lean
index 57e705836..3f11a5184 100644
--- a/HexPrimality/Elab.lean
+++ b/HexPrimality/Elab.lean
@@ -28,9 +28,10 @@ numeral `e`; `primality n` adds `this : Hex.Nat.Prime n`;
 `primality h : n` names it `h`.
 
 For reproducible syntax with no seed argument, the elaborator uses
-`Rand.ofSeed n` and `defaultPrimeFuel n`; the lower `primeCert?` API remains
-explicitly seeded, and diagnostics report the seed and fuel if certificate
-search exhausts its budget. The companion library may register an
+`Rand.ofSeed n` and `primalityFuel n`, the measured policy cap over
+`defaultPrimeFuel n`; the lower `primeCert?` API remains explicitly seeded,
+and diagnostics report the seed and fuel if certificate search exhausts its
+budget. The companion library may register an
 additional `@[tactic primalityTac]` handler for `Nat.Prime` goal shapes;
 handlers registered later run first and defer here by throwing
 `unsupportedSyntax`.
@@ -89,11 +90,54 @@ meta def checkClosed (tactic : String) (e : Expr) : MetaM Unit := do
     throwError "{tactic}: the argument{indentExpr e}\
         \nmust not contain free or meta variables"
 
-/-- Bit-length ceiling on tactic inputs: the kernel replay uses
-`O(K log n)` modular and bounded ordinary multiplications plus `O(K)`
-factor-subject comparisons, and beyond this size certificate search itself
-is the bottleneck to fix first. -/
-meta def primalityBitBudget : Nat := 8192
+/-- Supported bit-length ceiling for every elaboration-time certificate route.
+The 512-bit boundary is the largest measured fresh-module rung; changing it
+requires new end-to-end search, reification, and kernel-replay evidence. -/
+meta def primalityBitBudget : Nat := 512
+
+/-- Maximum recursive fuel passed to elaboration-time certificate search.
+At the supported bit ceiling `defaultPrimeFuel` is 1038; the round 1040 cap
+makes the resource bound explicit and prevents later changes to the default
+from silently widening the tactic policy. -/
+meta def primalityFuelBudget : Nat := 1040
+
+/-- Maximum Brent restarts at one partial-factor worklist entry on every
+elaboration-time certificate route. -/
+meta def primalityRhoRestartBudget : Nat := 2
+
+/-- Maximum Brent cycle steps per restart on every elaboration-time
+certificate route. -/
+meta def primalityRhoStepBudget : Nat := 1 <<< 15
+
+/-- The explicit rho allocation shared by every elaboration-time route. -/
+meta def primalitySearchBudget : Hex.Nat.PrimeCertBudget :=
+  ⟨primalityRhoRestartBudget, primalityRhoStepBudget⟩
+
+/-- The fuel selected by every elaboration-time certificate route. -/
+meta def primalityFuel (n : Nat) : Nat :=
+  min (Hex.Nat.defaultPrimeFuel n) primalityFuelBudget
+
+/-- Whether an input is admitted by the common elaboration policy. -/
+meta def withinPrimalityBudget (n : Nat) : Bool :=
+  decide (n.log2 + 1 ≤ primalityBitBudget)
+
+/-- Enforce the common input-size policy before any certificate search. -/
+meta def checkPrimalityPolicy (tactic : String) (n : Nat) : MetaM Unit := do
+  let bits := n.log2 + 1
+  unless withinPrimalityBudget n do
+    throwError "{tactic}: input has {bits} bits; the enforced policy supports \
+        at most {primalityBitBudget} bits; raising the ceiling requires new \
+        end-to-end benchmark evidence"
+
+/-- Report bounded-search exhaustion without inviting an unbounded fallback. -/
+meta def throwPrimalityExhausted {α : Type} (tactic : String) (n attempts fuel : Nat) :
+    MetaM α :=
+  throwError "{tactic}: certificate search for {n} exhausted after {attempts} \
+      attempts (seed {n}, recursive fuel {fuel}; policy maximum \
+      {primalityFuelBudget} fuel at {primalityBitBudget} bits, \
+      {primalityRhoRestartBudget} rho restarts with \
+      {primalityRhoStepBudget} steps each); no total primality decision was \
+      attempted"
 
 /-- Run the certificate search and emit the checked proof term with `head`
 applied to the subject, the reified certificate, and the `Eq.refl true`
@@ -107,10 +151,10 @@ meta def provePrimeWith (head : Name) (tactic : String) (n : Nat)
         \nevaluates to {n} but is not definitionally transparent to the \
         elaborator (an imported definition without `@[expose]`?); the kernel \
         could not check the emitted certificate against it"
-  if n.log2 + 1 > primalityBitBudget then
-    throwError "{tactic}: {n} has more than {primalityBitBudget} bits; \
-        raising `primalityBitBudget` is a separate, benchmarked change"
-  match Hex.Nat.primeCert? n (Hex.Rand.ofSeed n) (Hex.Nat.defaultPrimeFuel n) with
+  checkPrimalityPolicy tactic n
+  let fuel := primalityFuel n
+  match Hex.Nat.Internal.primeCertCountedWith? primalitySearchBudget n
+      (Hex.Rand.ofSeed n) fuel with
   | .error f =>
       match f.stop with
       | .composite =>
@@ -122,11 +166,9 @@ meta def provePrimeWith (head : Name) (tactic : String) (n : Nat)
           | none =>
               throwError "{tactic}: {n} is not prime"
       | .exhausted =>
-          throwError "{tactic}: certificate search for {n} exhausted its \
-              budget after {f.attempts} attempts (seed {n}, fuel \
-              {Hex.Nat.defaultPrimeFuel n}); the factorization of n - 1 may \
-              be out of reach"
-  | .ok (c, _) =>
+          throwPrimalityExhausted tactic n f.attempts fuel
+  | .ok success =>
+      let c := success.cert
       -- Untrusted-search self-check before emitting anything.
       unless c.raw.subject == n && Hex.Nat.checkPrime c.raw do
         throwError "{tactic}: internal error: the found certificate fails \
diff --git a/HexPrimality/SPEC/hex-primality.md b/HexPrimality/SPEC/hex-primality.md
index f5e8feecf..c9bca7a9b 100644
--- a/HexPrimality/SPEC/hex-primality.md
+++ b/HexPrimality/SPEC/hex-primality.md
@@ -782,6 +782,12 @@ structure NextPrimeFailure where
 
 def defaultPrimeFuel (n : Nat) : Nat
 
+structure PrimeCertBudget where
+  rhoRestarts : Nat
+  rhoSteps    : Nat
+
+def defaultPrimeCertBudget : PrimeCertBudget
+
 structure Internal.PrimeCertSuccess (n : Nat) where
   cert     : CheckedPrimeCert n
   attempts : Nat
@@ -790,6 +796,14 @@ structure Internal.PrimeCertSuccess (n : Nat) where
 def Internal.primeCertCounted? (n : Nat) (r : Rand) (fuel : Nat) :
     Except PrimeCertFailure (Internal.PrimeCertSuccess n)
 
+def Internal.primeCertCountedWith? (budget : PrimeCertBudget)
+    (n : Nat) (r : Rand) (fuel : Nat) :
+    Except PrimeCertFailure (Internal.PrimeCertSuccess n)
+
+theorem Internal.primeCertCountedWith?_composite {budget n r fuel f}
+    (hresult : Internal.primeCertCountedWith? budget n r fuel = .error f)
+    (hstop : f.stop = .composite) : ¬ Prime n
+
 theorem Internal.primeCertCounted?_composite {n r fuel f}
     (hresult : Internal.primeCertCounted? n r fuel = .error f)
     (hstop : f.stop = .composite) : ¬ Prime n
@@ -902,9 +916,62 @@ it replays `O(k log n)` modular multiplications on GMP-backed `Nat`, and never
 the search.
 
 For reproducible syntax with no seed argument, the elaborator uses
-`Rand.ofSeed n`; the lower `primeCert?` API remains explicitly seeded,
-and diagnostics report the seed and attempts if certificate search
-exhausts its fuel.
+`Rand.ofSeed n`; the lower `primeCert?` API remains explicitly seeded.
+
+**The supported positive-certificate elaboration policy is at most 512 input
+bits, at most 1040 recursive certificate-search fuel, at most 2 Brent restarts
+per partial-factor worklist entry, and at most 32768 Brent cycle steps per
+restart.** Every elaboration-time certificate route uses
+`primalityFuel n = min (defaultPrimeFuel n) 1040`; at the exact 512-bit boundary
+the default contributes 1038. The recursive fuel bounds certificate-construction
+depth. Partial factorization is bounded separately by `2 * n.log2 + 8` worklist
+steps at each certificate node and the rho restart/cycle allocation above. The
+fixed witness budget remains 32 candidates per factor entry. The exact attempt
+counter includes rho restarts and witness candidates and can therefore exceed
+the recursive fuel used at a node. Inputs above 512 bits are rejected before
+Miller--Rabin or certificate search. Search exhaustion reports the seed,
+selected recursive fuel, exact attempt count, and all enforced search maxima,
+and explicitly says that no total decision was attempted.
+
+The ceiling is the largest rung with both compiled and kernel evidence. The
+exact-boundary prime `100297^22 * 2^146 + 1`, namely
+`9521691625768090263084389838561930764813603239089634545416648725957969250257409112878363599328138633827640729385461401574761860536478435114675541614002177`, is 512 bits. Its table factors alone do not meet the
+Pocklington threshold: the accepted core and companion routes use bounded rho
+work to discover the above-table factor `100297`, finish after 31 counted
+attempts, reify its recursive certificate, and replay it in the kernel. The
+same prepared certificate is checked by `HexPrimalityKernelProbe` and the
+native `runDecision`, `runCertSearch`, and `runChecker` families. The paired
+fresh-module sweep also exercises the non-smooth 512-bit probable-prime input
+`11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123`,
+which reaches bounded rho work in a recursive child and exhausts after 11
+attempts at fuel 1038, and the 513-bit value `2^512`, which is rejected before
+search. Both core and companion routes have a 10-second absolute fresh-module
+wall-clock budget on the designated benchmark host. The harness compares the
+largest raw candidate wall time in every substantive sample set against that
+budget and makes any failure invalidate `release_quality`; it does not use the
+reference-subtracted tactic delta for this contract. This single end-to-end
+budget includes Lake startup and build-graph traversal, importing, compiled
+search, compiled self-check, reification, and kernel replay. Rejection and
+exhaustion may be indistinguishable from their import-only controls, but their
+absolute wall times remain budget-gated.
+The paired null controls remain in the record to classify those deltas; their
+spread is not a release gate for this absolute-only contract. Per-arm CPU and
+SMT-sibling interference checks remain hard gates for every accepted sample.
+The raw paired samples and host provenance are committed at
+`reports/bench-results/hex-primality-elaborator-policy-issue-9779-chungus2.json`.
+They reproduce with:
+
+```bash
+python3 scripts/bench/primality_elab_sweep.py --samples 6 \
+  --shared-host --expected-host chungus2 --cpu 22 --timeout 30 \
+  --warm-timeout 600 --max-pair-retries 32 \
+  --output reports/bench-results/hex-primality-elaborator-policy-issue-9779-chungus2.json
+```
+
+`lake build HexPrimalityElabProbe` is the untimed build-only reproduction.
+`primality` calls only `primeCertCountedWith?` with the explicit allocation
+above: it never calls the total `isPrime`, whose exact trial fallback is
+intentionally unbounded.
 
 The companion adds `Nat.Prime n` through that correspondence. Bare
 `primality` uses the certificate route directly. Pinned Mathlib's
@@ -912,8 +979,15 @@ The companion adds `Nat.Prime n` through that correspondence. Bare
 retain Mathlib's trial-division behavior; the later Hex registration cannot
 transparently pre-empt it. A module explicitly opts into the supported Hex
 policy with `use_hex_primality_norm_num`. Under that policy, numerals below
-`2^24` use a guarded alias of Mathlib's trial extension and larger numerals
-use bounded Hex certificate search. The opt-in erasure is local to the
+`2^24` use a guarded alias of Mathlib's trial extension and larger positive
+proofs use the same 512-bit/1040-fuel/2-restart/32768-step Hex certificate
+policy as `primality`. After a fixed-tier composite verdict, negative proofs
+use a separate finite factor search of 16 restarts, each bounded by
+`rhoInnerFuel n`; a found factor is dynamically revalidated before proof
+emission, and exhaustion only declines the goal. It never starts total trial
+division. Above 512 bits the opt-in extension declines both positive and
+negative goals; `norm_num` reports an unsolved goal rather than silently
+restoring Mathlib's total trial decision. The opt-in erasure is local to the
 module and does not persist when that module is imported.
 
 The `2^24` boundary comes from fresh one-goal modules on the pinned
@@ -999,7 +1073,7 @@ in the part of a certificate tree replayed before acceptance or rejection.
 | `isPrime` worst case | `O(√n)` remainder tests | exact fallback after default search exhaustion |
 | `checkPrime`, one Pocklington level | `O(k b)` modular multiplications; `O(k b)` bounded ordinary multiplications; `O(k)` subject comparisons, divisions, and gcds | canonical subject preflight is linear on accepted and rejected lists |
 | `checkPrime`, full tree | `O(Σᵥ kᵥ bᵥ)` modular and bounded ordinary multiplications; `O(K)` subject comparisons, divisions, and gcds | `kᵥ`, `bᵥ` are the entry count and subject bit bound at each visited node; arithmetic preflight bounds each replayed child's subject below its parent, so the sum is `O(K b)` for root bit length `b` |
-| `primeCert?` | dominated by `partialFactor` | unbounded; fuel-limited |
+| `primeCert?` | dominated by `partialFactor` | bounded by recursive fuel, per-node worklist fuel, and `defaultPrimeCertBudget` rho restarts/cycle steps |
 | sieve to `N` | `O(√N + π(√N) · 32)` loop/doubling rounds | each marking round is a bit operation on an `N/3`-bit `Nat` |
 
 These are operation counts, not bit complexity; subject comparisons,
@@ -1112,8 +1186,9 @@ Policy-selection evidence, retained as input to but not a claim about Phase 4:
 
 Families:
 - **Kernel replay**, `checkPrime` on certificates for primes of `31`,
-  `61`, `123`, `256`, and `511` bits (the table-smooth ladder). Decides
-  the `powModNat`-versus-Montgomery question under "Kernel exposure".
+  `61`, `123`, `256`, `511`, and `512` bits. The lower rungs are table-smooth;
+  the 512-bit rung includes the above-table factor `100297`. Decides the
+  `powModNat`-versus-Montgomery question under "Kernel exposure".
 - **Native decision**, bounded `isPrime?` across the same bit lengths.
   The total `isPrime` gets no separate row: it differs only on the
   exhausted-search path, where its exact trial-division fallback is
diff --git a/HexPrimality/Search.lean b/HexPrimality/Search.lean
index d159b3ca8..e03ba2e57 100644
--- a/HexPrimality/Search.lean
+++ b/HexPrimality/Search.lean
@@ -230,30 +230,36 @@ def rhoDrawTrace (n : Nat) (r : Rand) : Nat × Nat × Nat :=
 
 end Internal
 
-private def rhoTry (n : Nat) : Nat → Nat → Rand →
+private def rhoTry (n innerFuel : Nat) : Nat → Nat → Rand →
     Except RhoFailure Internal.RhoSuccess
   | 0, attempts, r => .error ⟨.exhausted, attempts, r⟩
   | tries + 1, attempts, r =>
       let draw := rhoDraw n r
-      match (brentGo n draw.c (rhoInnerFuel n) (brentStart draw.start)).factor with
+      match (brentGo n draw.c (min (rhoInnerFuel n) innerFuel)
+          (brentStart draw.start)).factor with
       | some d =>
           if 1 < d then
             if d < n then
               if n % d = 0 then .ok ⟨d, attempts + 1, draw.rand⟩
-              else rhoTry n tries (attempts + 1) draw.rand
-            else rhoTry n tries (attempts + 1) draw.rand
-          else rhoTry n tries (attempts + 1) draw.rand
-      | none => rhoTry n tries (attempts + 1) draw.rand
+              else rhoTry n innerFuel tries (attempts + 1) draw.rand
+            else rhoTry n innerFuel tries (attempts + 1) draw.rand
+          else rhoTry n innerFuel tries (attempts + 1) draw.rand
+      | none => rhoTry n innerFuel tries (attempts + 1) draw.rand
 
 namespace Internal
 
 /-- A dynamically validated proper-factor candidate by batched Brent rho,
 with its exact restart count. -/
-def rhoFactorCounted? (n : Nat) (r : Rand) (fuel : Nat) :
+def rhoFactorCountedWith? (n : Nat) (r : Rand) (restarts innerFuel : Nat) :
     Except RhoFailure RhoSuccess :=
   if n < 4 then .error ⟨.invalidInput, 0, r⟩
   else if n % 2 = 0 then .ok ⟨2, 0, r⟩
-  else rhoTry n fuel 0 r
+  else rhoTry n innerFuel restarts 0 r
+
+/-- A counted rho search with the production per-restart cycle budget. -/
+def rhoFactorCounted? (n : Nat) (r : Rand) (fuel : Nat) :
+    Except RhoFailure RhoSuccess :=
+  rhoFactorCountedWith? n r fuel (rhoInnerFuel n)
 
 end Internal
 
@@ -271,9 +277,9 @@ def rhoFactor? (n : Nat) (r : Rand) (fuel : Nat) :
   | .error failure => .error failure
   | .ok success => .ok (success.factor, success.rand)
 
-private theorem rhoTry_spec {n : Nat} :
+private theorem rhoTry_spec {n innerFuel : Nat} :
     ∀ (tries attempts : Nat) (r : Rand) {success : Internal.RhoSuccess},
-      rhoTry n tries attempts r = .ok success →
+      rhoTry n innerFuel tries attempts r = .ok success →
         1 < success.factor ∧ success.factor < n ∧ success.factor ∣ n := by
   intro tries
   induction tries with
@@ -297,12 +303,14 @@ private theorem rhoTry_spec {n : Nat} :
         · exact ih _ _ h
       · exact ih _ _ h
 
-/-- A counted rho success is a validated proper factor. -/
-theorem Internal.rhoFactorCounted?_spec {n : Nat} {r : Rand} {fuel : Nat}
+/-- A rho success under explicit restart and cycle budgets is a validated
+proper factor. -/
+theorem Internal.rhoFactorCountedWith?_spec {n : Nat} {r : Rand}
+    {restarts innerFuel : Nat}
     {success : Internal.RhoSuccess}
-    (h : Internal.rhoFactorCounted? n r fuel = .ok success) :
+    (h : Internal.rhoFactorCountedWith? n r restarts innerFuel = .ok success) :
     1 < success.factor ∧ success.factor < n ∧ success.factor ∣ n := by
-  unfold Internal.rhoFactorCounted? at h
+  unfold Internal.rhoFactorCountedWith? at h
   by_cases h4 : n < 4
   · rw [if_pos h4] at h
     cases h
@@ -314,7 +322,14 @@ theorem Internal.rhoFactorCounted?_spec {n : Nat} {r : Rand} {fuel : Nat}
       change 1 < 2 ∧ 2 < n ∧ 2 ∣ n
       exact ⟨by omega, by omega, Nat.dvd_of_mod_eq_zero heven⟩
     · rw [if_neg heven] at h
-      exact rhoTry_spec fuel 0 r h
+      exact rhoTry_spec restarts 0 r h
+
+/-- A counted rho success is a validated proper factor. -/
+theorem Internal.rhoFactorCounted?_spec {n : Nat} {r : Rand} {fuel : Nat}
+    {success : Internal.RhoSuccess}
+    (h : Internal.rhoFactorCounted? n r fuel = .ok success) :
+    1 < success.factor ∧ success.factor < n ∧ success.factor ∣ n := by
+  exact Internal.rhoFactorCountedWith?_spec h
 
 /-- The one theorem about the rho primitive: a success is a validated
 proper factor. -/
@@ -433,6 +448,21 @@ private theorem insertFactor_prod (p : Nat) :
 /-- Restart budget for each rho call inside the worklist. -/
 private def rhoRestartBudget : Nat := Internal.rhoRestartCap
 
+/-- Pollard-rho work admitted at each partial-factor worklist entry during
+certificate search. This controls search resources only; every reported
+factor is still validated dynamically and every emitted certificate is
+checker-replayed. -/
+structure PrimeCertBudget where
+  /-- Maximum Brent restarts at one worklist entry. -/
+  rhoRestarts : Nat
+  /-- Maximum Brent cycle steps per restart. -/
+  rhoSteps : Nat
+deriving Repr, DecidableEq
+
+/-- The production certificate-search rho allocation used by the public API. -/
+def defaultPrimeCertBudget : PrimeCertBudget :=
+  ⟨rhoRestartBudget, 1 <<< 22⟩
+
 /-- Internal partial-factor worklist result with exact randomized work. -/
 private structure RhoPhaseResult where
   factors : List (Nat × Nat)
@@ -444,30 +474,35 @@ private structure RhoPhaseResult where
 as a claimed factor if the filter calls it prime, split it if rho finds a
 factor, and multiply it into the residual otherwise. Fuel exhaustion dumps
 the remaining stack into the residual, preserving the product exactly. -/
-private def rhoPhase :
+private def rhoPhase (budget : PrimeCertBudget) :
     Nat → List Nat → List (Nat × Nat) → Nat → Rand → Nat → RhoPhaseResult
   | 0, stack, acc, residual, r, attempts =>
       ⟨acc, listProd stack * residual, r, attempts⟩
   | _ + 1, [], acc, residual, r, attempts => ⟨acc, residual, r, attempts⟩
   | fuel + 1, m :: stack, acc, residual, r, attempts =>
-      if m = 1 then rhoPhase fuel stack acc residual r attempts
+      if m = 1 then rhoPhase budget fuel stack acc residual r attempts
       else if isProbablePrime m then
-        rhoPhase fuel stack (insertFactor m acc) residual r attempts
+        rhoPhase budget fuel stack (insertFactor m acc) residual r attempts
       else
-        match Internal.rhoFactorCounted? m r rhoRestartBudget with
+        match Internal.rhoFactorCountedWith? m r budget.rhoRestarts
+            budget.rhoSteps with
         | .ok success =>
-            rhoPhase fuel (success.factor :: m / success.factor :: stack) acc
-              residual success.rand (attempts + success.attempts)
+            rhoPhase budget fuel
+              (success.factor :: m / success.factor :: stack) acc residual
+              success.rand (attempts + success.attempts)
         | .error f =>
-            rhoPhase fuel stack acc (residual * m) f.rand (attempts + f.attempts)
+            rhoPhase budget fuel stack acc (residual * m) f.rand
+              (attempts + f.attempts)
 
 private theorem rhoPhase_prod :
-    ∀ (fuel : Nat) (stack : List Nat) (acc : List (Nat × Nat))
+    ∀ (budget : PrimeCertBudget) (fuel : Nat) (stack : List Nat)
+      (acc : List (Nat × Nat))
       (residual : Nat) (r : Rand),
       ∀ attempts : Nat,
-      prodPows (rhoPhase fuel stack acc residual r attempts).factors *
-          (rhoPhase fuel stack acc residual r attempts).residual =
+      prodPows (rhoPhase budget fuel stack acc residual r attempts).factors *
+          (rhoPhase budget fuel stack acc residual r attempts).residual =
         prodPows acc * listProd stack * residual := by
+  intro budget
   intro fuel
   induction fuel with
   | zero =>
@@ -495,7 +530,7 @@ private theorem rhoPhase_prod :
               · rename_i success hok
                 rw [ih]
                 obtain ⟨hd1, hdlt, hddvd⟩ :=
-                  Internal.rhoFactorCounted?_spec hok
+                  Internal.rhoFactorCountedWith?_spec hok
                 have hdm : success.factor *
                     (m / success.factor * listProd stack) =
                     m * listProd stack := by
@@ -517,17 +552,19 @@ with `fuel` bounding the worklist steps. Everything the search cannot split
 multiplies into the residual. Internal; certificate search is the only
 consumer, and hex-int-factor builds its own assembly over the counted rho
 adapter. -/
-private def partialFactor (n : Nat) (r : Rand) (fuel : Nat) :
+private def partialFactor (budget : PrimeCertBudget) (n : Nat) (r : Rand)
+    (fuel : Nat) :
     PartialSearch :=
   let trial := trialGo primeTable.toList [] n
-  let phase := rhoPhase fuel [trial.2] trial.1 1 r 0
+  let phase := rhoPhase budget fuel [trial.2] trial.1 1 r 0
   ⟨⟨phase.factors, phase.residual⟩, phase.rand, phase.attempts⟩
 
 /-- The product invariant: the claimed powers times the residual recover the
 input exactly. This is the one fact certificate search needs. -/
-private theorem partialFactor_prod (n : Nat) (r : Rand) (fuel : Nat) :
-    prodPows (partialFactor n r fuel).raw.factors *
-      (partialFactor n r fuel).raw.residual = n := by
+private theorem partialFactor_prod (budget : PrimeCertBudget) (n : Nat)
+    (r : Rand) (fuel : Nat) :
+    prodPows (partialFactor budget n r fuel).raw.factors *
+      (partialFactor budget n r fuel).raw.residual = n := by
   unfold partialFactor
   dsimp only
   rw [rhoPhase_prod]
@@ -551,10 +588,11 @@ partial factorization's product invariant exercised at runtime. -/
         | .error f => f.stop == .exhausted
         | _ => false)
 set_option maxRecDepth 10000 in  -- table walks inside partialFactor
-#guard (let pf := (partialFactor 720 (Rand.ofSeed 1) 32).raw
+#guard (let pf := (partialFactor defaultPrimeCertBudget 720 (Rand.ofSeed 1) 32).raw
         pf.factors.foldl (fun a x => a * x.1 ^ x.2) 1 * pf.residual == 720)
 set_option maxRecDepth 10000 in
-#guard (let pf := (partialFactor (97 * 101 * 101) (Rand.ofSeed 2) 32).raw
+#guard (let pf := (partialFactor defaultPrimeCertBudget (97 * 101 * 101)
+    (Rand.ofSeed 2) 32).raw
         pf.factors.foldl (fun a x => a * x.1 ^ x.2) 1 * pf.residual ==
           97 * 101 * 101)
 
@@ -646,7 +684,7 @@ a certified child with a searched witness. The candidate entries are sorted
 into the checker's canonical subject order, then the assembled node is
 validated by the public wrapper; neither the factorization nor this
 preprocessing is trusted. -/
-private def primeCertGo (fuel n : Nat) (r : Rand) :
+private def primeCertGo (budget : PrimeCertBudget) (fuel n : Nat) (r : Rand) :
     Except PrimeCertFailure (Counted PrimeCert) :=
   if n < 2 then .error ⟨.composite, 0, r⟩
   else if n < primeTableBound then
@@ -659,9 +697,9 @@ private def primeCertGo (fuel n : Nat) (r : Rand) :
         match fuel with
         | 0 => .error ⟨.exhausted, 0, r⟩
         | fuel + 1 =>
-            let factored := partialFactor (n - 1) r (2 * n.log2 + 8)
-            match assembleGo fuel n factored.raw.factors [] factored.attempts
-                factored.rand with
+            let factored := partialFactor budget (n - 1) r (2 * n.log2 + 8)
+            match assembleGo budget fuel n factored.raw.factors []
+                factored.attempts factored.rand with
             | .error f => .error f
             | .ok assembled =>
                 let entries := assembled.value.mergeSort fun x y =>
@@ -680,21 +718,21 @@ termination_by (fuel, 0)
 witness base per entry. A child failure is reported as exhaustion: the
 child's compositeness would only mean the untrusted factorization guessed
 wrong, never that `n` is composite. -/
-private def assembleGo (fuel n : Nat) :
+private def assembleGo (budget : PrimeCertBudget) (fuel n : Nat) :
     List (Nat × Nat) → List (Nat × Nat × PrimeCert) → Nat → Rand →
       Except PrimeCertFailure (Counted (List (Nat × Nat × PrimeCert)))
   | [], acc, attempts, r => .ok ⟨acc.reverse, attempts, r⟩
   | (q, e) :: rest, acc, attempts, r =>
-      if e = 0 then assembleGo fuel n rest acc attempts r
+      if e = 0 then assembleGo budget fuel n rest acc attempts r
       else
-        match primeCertGo fuel q r with
+        match primeCertGo budget fuel q r with
         | .error f => .error ⟨.exhausted, attempts + f.attempts, f.rand⟩
         | .ok child =>
             match witnessGo n q witnessBudget 0 child.rand with
             | .error f =>
                 .error ⟨f.stop, attempts + child.attempts + f.attempts, f.rand⟩
             | .ok witness =>
-                assembleGo fuel n rest
+                assembleGo budget fuel n rest
                   ((witness.value, e - 1, child.value) :: acc)
                   (attempts + child.attempts + witness.attempts) witness.rand
 termination_by l => (fuel, l.length + 1)
@@ -713,10 +751,12 @@ structure PrimeCertSuccess (n : Nat) where
   /-- Generator state after those attempts. -/
   rand : Rand
 
-/-- Bounded certificate search retaining exact successful-attempt metering. -/
-def primeCertCounted? (n : Nat) (r : Rand) (fuel : Nat) :
+/-- Bounded certificate search with an explicit rho allocation, retaining
+exact successful-attempt metering. -/
+def primeCertCountedWith? (budget : PrimeCertBudget) (n : Nat) (r : Rand)
+    (fuel : Nat) :
     Except PrimeCertFailure (PrimeCertSuccess n) :=
-  match primeCertGo fuel n r with
+  match primeCertGo budget fuel n r with
   | .error f => .error f
   | .ok result =>
       if hs : result.value.subject = n then
@@ -725,6 +765,11 @@ def primeCertCounted? (n : Nat) (r : Rand) (fuel : Nat) :
         else .error ⟨.exhausted, result.attempts, result.rand⟩
       else .error ⟨.exhausted, result.attempts, result.rand⟩
 
+/-- Bounded certificate search retaining exact successful-attempt metering. -/
+def primeCertCounted? (n : Nat) (r : Rand) (fuel : Nat) :
+    Except PrimeCertFailure (PrimeCertSuccess n) :=
+  primeCertCountedWith? defaultPrimeCertBudget n r fuel
+
 end Internal
 
 /-- Bounded certificate search. A success is a `CheckedPrimeCert`, so a
@@ -754,10 +799,11 @@ private theorem witnessGo_error_stop {n q : Nat} :
       · cases h
       · exact ih _ _ h
 
-private theorem assembleGo_error_stop {fuel n : Nat} :
+private theorem assembleGo_error_stop {budget : PrimeCertBudget} {fuel n : Nat} :
     ∀ (l : List (Nat × Nat)) (acc : List (Nat × Nat × PrimeCert))
       (attempts : Nat) (r : Rand) {f : PrimeCertFailure},
-      assembleGo fuel n l acc attempts r = .error f → f.stop = .exhausted := by
+      assembleGo budget fuel n l acc attempts r = .error f →
+        f.stop = .exhausted := by
   intro l
   induction l with
   | nil =>
@@ -781,8 +827,9 @@ private theorem assembleGo_error_stop {fuel n : Nat} :
             exact hfstop
           next => exact ih _ _ _ h
 
-private theorem primeCertGo_composite {fuel n : Nat} {r : Rand}
-    {f : PrimeCertFailure} (h : primeCertGo fuel n r = .error f)
+private theorem primeCertGo_composite {budget : PrimeCertBudget} {fuel n : Nat}
+    {r : Rand} {f : PrimeCertFailure}
+    (h : primeCertGo budget fuel n r = .error f)
     (hstop : f.stop = .composite) : ¬ Prime n := by
   unfold primeCertGo at h
   by_cases h2 : n < 2
@@ -825,12 +872,15 @@ private theorem primeCertGo_composite {fuel n : Nat} {r : Rand}
               cases hstop
             · split at h <;> cases h
 
-/-- A counted `.composite` failure is a verdict: the input is not prime. -/
-theorem Internal.primeCertCounted?_composite {n : Nat} {r : Rand} {fuel : Nat}
+/-- A budgeted counted `.composite` failure is a verdict: the input is not
+prime. The result is independent of the resource allocation because only the
+fixed size, table, and Miller--Rabin tiers can emit that stop reason. -/
+theorem Internal.primeCertCountedWith?_composite {budget : PrimeCertBudget}
+    {n : Nat} {r : Rand} {fuel : Nat}
     {f : PrimeCertFailure}
-    (hresult : Internal.primeCertCounted? n r fuel = .error f)
+    (hresult : Internal.primeCertCountedWith? budget n r fuel = .error f)
     (hstop : f.stop = .composite) : ¬ Prime n := by
-  unfold Internal.primeCertCounted? at hresult
+  unfold Internal.primeCertCountedWith? at hresult
   split at hresult
   · rename_i f' herr
     injection hresult with h
@@ -846,6 +896,14 @@ theorem Internal.primeCertCounted?_composite {n : Nat} {r : Rand} {fuel : Nat}
       subst h
       cases hstop
 
+/-- A counted `.composite` failure under the default allocation is a verdict:
+the input is not prime. -/
+theorem Internal.primeCertCounted?_composite {n : Nat} {r : Rand} {fuel : Nat}
+    {f : PrimeCertFailure}
+    (hresult : Internal.primeCertCounted? n r fuel = .error f)
+    (hstop : f.stop = .composite) : ¬ Prime n := by
+  exact Internal.primeCertCountedWith?_composite hresult hstop
+
 /-- A `.composite` failure is a verdict: the input is not prime. Justified
 by size, table completeness, or a failed Miller-Rabin base; never by
 anything the untrusted search merely failed to do. -/
diff --git a/HexPrimalityMathlib/NormNum.lean b/HexPrimalityMathlib/NormNum.lean
index 439b5f04d..32e3971d6 100644
--- a/HexPrimalityMathlib/NormNum.lean
+++ b/HexPrimalityMathlib/NormNum.lean
@@ -34,7 +34,9 @@ certificate through `natPrime_of_checkPrimeAt`; the kernel replays only the
 checker. A negative verdict emits a dynamically validated proper factor
 through Mathlib's `deriveNotPrime`. If bounded certificate or factor search
 is exhausted, both Hex extensions decline, rather than falling through to
-unbounded trial division.
+unbounded trial division. The certificate extension uses the same 512-bit
+input ceiling and 1040 recursive-fuel cap as the core and companion
+`primality` handlers.
 
 The tactic handler registers on the same `primality` syntax kind as the
 Mathlib-free elaborator; registration order makes this handler run first,
@@ -61,9 +63,12 @@ verdicts at and above `natPrimeCertThreshold`. -/
     let ⟨nn, pn⟩ ← deriveNat n _
     let n' := nn.natLit!
     if n' < natPrimeCertThreshold then failure
-    match Hex.Nat.primeCert? n' (Hex.Rand.ofSeed n')
-        (Hex.Nat.defaultPrimeFuel n') with
-    | .ok (c, _) =>
+    unless withinPrimalityBudget n' do failure
+    let fuel := primalityFuel n'
+    match Hex.Nat.Internal.primeCertCountedWith? primalitySearchBudget n'
+        (Hex.Rand.ofSeed n') fuel with
+    | .ok success =>
+        let c := success.cert
         -- Untrusted-search self-check before emitting anything.
         unless c.raw.subject == n' && Hex.Nat.checkPrime c.raw do failure
         let prf : Q(_root_.Nat.Prime $nn) :=
diff --git a/SPEC/benchmarking.md b/SPEC/benchmarking.md
index c0102503b..7f26aac39 100644
--- a/SPEC/benchmarking.md
+++ b/SPEC/benchmarking.md
@@ -424,6 +424,18 @@ so an isolated outlier can widen but never shrink the admission envelope.
 A control whose IQR exceeds 10% of its build magnitude invalidates release
 quality.
 
+A suite may instead preregister `absolute_only` when every substantive pair
+declares an absolute fresh-module wall-clock budget and no pair declares a
+reference-subtracted tactic budget. In that mode the raw candidate maximum,
+not a paired delta, is the release contract. Null controls remain in the
+artifact to describe and classify paired noise, but their IQR and relative
+build-magnitude coverage do not gate release quality because neither quantity
+can change the absolute conclusion. Dirty-state, provenance, timeout,
+frequency, per-arm CPU/SMT interference, and absolute-budget failures retain
+their ordinary fail-closed behavior. The harness rejects `absolute_only`
+manifests that omit an absolute budget from any substantive pair or add a
+relative tactic budget.
+
 The artifact interpolates control IQRs and conservative envelopes between
 representative build magnitudes. Outside the measured range it may scale a
 cheaper control upward by the build-time magnitude ratio, but never scales an
diff --git a/bench/HexBench/PrimalityKernel.lean b/bench/HexBench/PrimalityKernel.lean
index bf6f4cb1a..67bfbca1f 100644
--- a/bench/HexBench/PrimalityKernel.lean
+++ b/bench/HexBench/PrimalityKernel.lean
@@ -21,9 +21,9 @@ which is the SPEC's "kernel replay" bench family. Sweep it with a
 fresh-module build (delete this module's outputs and rebuild) rather than
 in-process timing.
 
-The certificates are the same table-smooth family the native bench uses:
-`n - 1 = k · 2^m` with `k` factoring over the committed table, so the
-shape is fixed and the exponentiation cost dominates.
+The certificates are the same fixed family the native bench uses. The lower
+rungs are table-smooth; the 512-bit policy rung contains the recursive
+certificate for the above-table factor `100297` discovered by bounded rho.
 -/
 
 namespace HexBench.PrimalityKernel
@@ -70,6 +70,17 @@ def cert511 : PrimeCert :=
     [(13757245211066428521, 503, .small 2),
      (10451216379200822467, 0, .small 127)]
 
+/-- The exact 512-bit elaborator-policy boundary: `100297^22 · 2^146 + 1`.
+Its certificate records the above-table factor that the bounded rho search
+discovers. -/
+def cert512 : PrimeCert :=
+  .pock 9521691625768090263084389838561930764813603239089634545416648725957969250257409112878363599328138633827640729385461401574761860536478435114675541614002177
+    [(12105408859821572020, 145, .small 2),
+     (2427313743710699239, 21,
+      .pock 100297
+        [(6478, 2, .small 2), (58864, 1, .small 3),
+         (35592, 0, .small 7), (37339, 0, .small 199)])]
+
 theorem replay31 : checkPrime cert31 = true := by decide +kernel
 
 theorem replay61 : checkPrime cert61 = true := by decide +kernel
@@ -80,6 +91,8 @@ theorem replay256 : checkPrime cert256 = true := by decide +kernel
 
 theorem replay511 : checkPrime cert511 = true := by decide +kernel
 
+theorem replay512 : checkPrime cert512 = true := by decide +kernel
+
 /-- The cube-root checker arm also replays through the kernel-facing closure. -/
 theorem replayPock3 :
     checkPrime
diff --git a/bench/HexPrimality/Bench.lean b/bench/HexPrimality/Bench.lean
index 883e80464..c08b1deb2 100644
--- a/bench/HexPrimality/Bench.lean
+++ b/bench/HexPrimality/Bench.lean
@@ -11,9 +11,10 @@ import LeanBench
 Benchmark registrations for `HexPrimality`.
 
 Three certificate families run over a fixed ladder of primes of 31, 61,
-123, 256, and 511 bits whose `n - 1` is `k · 2^m` with `k` factoring over
-the committed table, so partial factorization is deterministic trial
-division and no rho variance enters the decision or checker rows:
+123, 256, 511, and 512 bits. The lower rungs have `n - 1 = k · 2^m` with `k`
+factoring over the committed table. The 512-bit policy rung instead requires
+rho to discover the deterministic above-table factor `100297`; its prepared
+certificate keeps the checker row fixed:
 
 * `runDecision` runs the bounded decision `isPrime?` end to end;
 * `runCertSearch` runs `primeCert?` alone (the SPEC keeps this family
@@ -76,16 +77,26 @@ def cert511 : PrimeCert :=
     [(13757245211066428521, 503, .small 2),
      (10451216379200822467, 0, .small 127)]
 
+/-- The exact 512-bit elaborator-policy boundary: `100297^22 · 2^146 + 1`. -/
+def cert512 : PrimeCert :=
+  .pock 9521691625768090263084389838561930764813603239089634545416648725957969250257409112878363599328138633827640729385461401574761860536478435114675541614002177
+    [(12105408859821572020, 145, .small 2),
+     (2427313743710699239, 21,
+      .pock 100297
+        [(6478, 2, .small 2), (58864, 1, .small 3),
+         (35592, 0, .small 7), (37339, 0, .small 199)])]
+
 /-- Map a bit-size rung to its prepared input. -/
 def prepInput (bits : Nat) : Input :=
   if bits ≤ 31 then { n := 2147483647, cert := cert31 }
   else if bits ≤ 61 then { n := 1945555039024054273, cert := cert61 }
   else if bits ≤ 123 then { n := cert123.subject, cert := cert123 }
   else if bits ≤ 256 then { n := cert256.subject, cert := cert256 }
-  else { n := cert511.subject, cert := cert511 }
+  else if bits ≤ 511 then { n := cert511.subject, cert := cert511 }
+  else { n := cert512.subject, cert := cert512 }
 
 private def sizeParams : Array Nat :=
-  #[31, 61, 123, 256, 511]
+  #[31, 61, 123, 256, 511, 512]
 
 -- Every rung's prepared certificate replays, and is about its own input.
 #guard sizeParams.all fun bits =>
@@ -125,7 +136,7 @@ setup_benchmark runDecision n => n * n * n
   with prep := prepInput
   where {
     paramFloor := 31
-    paramCeiling := 511
+    paramCeiling := 512
     paramSchedule := .custom sizeParams
     maxSecondsPerCall := 5.0
     targetInnerNanos := 100000000
@@ -133,15 +144,14 @@ setup_benchmark runDecision n => n * n * n
     slopeTolerance := 0.5
   }
 
-/- Certificate search adds witness search over the same exponentiation
-primitive, so the same cubic bit-cost proxy applies; the SPEC records
-that this family's numbers say the least about the library, and the
-fixed table-smooth inputs at least remove the rho variance. -/
+/- Certificate search adds witness and, at the 512-bit rung, rho search over
+the same arithmetic primitives, so the same cubic bit-cost proxy applies; the
+SPEC records that this family's numbers say the least about the library. -/
 setup_benchmark runCertSearch n => n * n * n
   with prep := prepInput
   where {
     paramFloor := 31
-    paramCeiling := 511
+    paramCeiling := 512
     paramSchedule := .custom sizeParams
     maxSecondsPerCall := 5.0
     targetInnerNanos := 100000000
@@ -156,7 +166,7 @@ setup_benchmark runChecker n => n * n * n
   with prep := prepInput
   where {
     paramFloor := 31
-    paramCeiling := 511
+    paramCeiling := 512
     paramSchedule := .custom sizeParams
     maxSecondsPerCall := 5.0
     targetInnerNanos := 100000000
diff --git a/bench/HexPrimality/ProofProbe/Core512.lean b/bench/HexPrimality/ProofProbe/Core512.lean
new file mode 100644
index 000000000..6cb94d0e1
--- /dev/null
+++ b/bench/HexPrimality/ProofProbe/Core512.lean
@@ -0,0 +1,23 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+
+module
+
+public import HexPrimality
+
+public section
+
+namespace HexPrimality.ProofProbe
+
+/-! End-to-end core elaboration at the exact 512-bit policy ceiling. The
+above-table factor `100297` forces the accepted route through bounded rho. -/
+
+theorem core512 : Hex.Nat.Prime 9521691625768090263084389838561930764813603239089634545416648725957969250257409112878363599328138633827640729385461401574761860536478435114675541614002177 :=
+  primality 9521691625768090263084389838561930764813603239089634545416648725957969250257409112878363599328138633827640729385461401574761860536478435114675541614002177
+
+#print axioms core512
+
+end HexPrimality.ProofProbe
diff --git a/bench/HexPrimality/ProofProbe/CoreBaseline.lean b/bench/HexPrimality/ProofProbe/CoreBaseline.lean
new file mode 100644
index 000000000..9fb09a679
--- /dev/null
+++ b/bench/HexPrimality/ProofProbe/CoreBaseline.lean
@@ -0,0 +1,13 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+
+module
+
+public import HexPrimality
+
+public section
+
+/-! Import-only baseline for the core `primality` fresh-module probes. -/
diff --git a/bench/HexPrimality/ProofProbe/CoreExhausted.lean b/bench/HexPrimality/ProofProbe/CoreExhausted.lean
new file mode 100644
index 000000000..2cd8ee27e
--- /dev/null
+++ b/bench/HexPrimality/ProofProbe/CoreExhausted.lean
@@ -0,0 +1,20 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+
+module
+
+public import HexPrimality
+
+public section
+
+/-! Bounded-search exhaustion at the exact 512-bit policy ceiling. -/
+
+/--
+error: primality: certificate search for 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123 exhausted after 11 attempts (seed 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123, recursive fuel 1038; policy maximum 1040 fuel at 512 bits, 2 rho restarts with 32768 steps each); no total primality decision was attempted
+-/
+#guard_msgs in
+example : Hex.Nat.Prime 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123 :=
+  primality 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123
diff --git a/bench/HexPrimality/ProofProbe/CoreOverBudget.lean b/bench/HexPrimality/ProofProbe/CoreOverBudget.lean
new file mode 100644
index 000000000..be15da483
--- /dev/null
+++ b/bench/HexPrimality/ProofProbe/CoreOverBudget.lean
@@ -0,0 +1,20 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+
+module
+
+public import HexPrimality
+
+public section
+
+/-! Rejection immediately above the core elaborator's bit ceiling. -/
+
+/--
+error: primality: input has 513 bits; the enforced policy supports at most 512 bits; raising the ceiling requires new end-to-end benchmark evidence
+-/
+#guard_msgs in
+example : Hex.Nat.Prime 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096 :=
+  primality 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096
diff --git a/bench/HexPrimalityMathlib/ProofProbe/Mathlib512.lean b/bench/HexPrimalityMathlib/ProofProbe/Mathlib512.lean
new file mode 100644
index 000000000..8c6cdf139
--- /dev/null
+++ b/bench/HexPrimalityMathlib/ProofProbe/Mathlib512.lean
@@ -0,0 +1,19 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+
+import HexPrimalityMathlib
+
+namespace HexPrimalityMathlib.ProofProbe
+
+/-! End-to-end Mathlib elaboration at the exact 512-bit policy ceiling. The
+above-table factor `100297` forces the accepted route through bounded rho. -/
+
+theorem mathlib512 : Nat.Prime 9521691625768090263084389838561930764813603239089634545416648725957969250257409112878363599328138633827640729385461401574761860536478435114675541614002177 := by
+  primality
+
+#print axioms mathlib512
+
+end HexPrimalityMathlib.ProofProbe
diff --git a/bench/HexPrimalityMathlib/ProofProbe/MathlibBaseline.lean b/bench/HexPrimalityMathlib/ProofProbe/MathlibBaseline.lean
new file mode 100644
index 000000000..340cf9f05
--- /dev/null
+++ b/bench/HexPrimalityMathlib/ProofProbe/MathlibBaseline.lean
@@ -0,0 +1,9 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+
+import HexPrimalityMathlib
+
+/-! Import-only baseline for the Mathlib `primality` fresh-module probes. -/
diff --git a/bench/HexPrimalityMathlib/ProofProbe/MathlibExhausted.lean b/bench/HexPrimalityMathlib/ProofProbe/MathlibExhausted.lean
new file mode 100644
index 000000000..1919864fa
--- /dev/null
+++ b/bench/HexPrimalityMathlib/ProofProbe/MathlibExhausted.lean
@@ -0,0 +1,16 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+
+import HexPrimalityMathlib
+
+/-! Bounded-search exhaustion at the exact 512-bit policy ceiling. -/
+
+/--
+error: primality: certificate search for 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123 exhausted after 11 attempts (seed 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123, recursive fuel 1038; policy maximum 1040 fuel at 512 bits, 2 rho restarts with 32768 steps each); no total primality decision was attempted
+-/
+#guard_msgs in
+example : Nat.Prime 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123 := by
+  primality
diff --git a/bench/HexPrimalityMathlib/ProofProbe/MathlibOverBudget.lean b/bench/HexPrimalityMathlib/ProofProbe/MathlibOverBudget.lean
new file mode 100644
index 000000000..800a40bc4
--- /dev/null
+++ b/bench/HexPrimalityMathlib/ProofProbe/MathlibOverBudget.lean
@@ -0,0 +1,16 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+
+import HexPrimalityMathlib
+
+/-! Rejection immediately above the Mathlib handler's bit ceiling. -/
+
+/--
+error: primality: input has 513 bits; the enforced policy supports at most 512 bits; raising the ceiling requires new end-to-end benchmark evidence
+-/
+#guard_msgs in
+example : Nat.Prime 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096 := by
+  primality
diff --git a/conformance/HexPrimality/Conformance.lean b/conformance/HexPrimality/Conformance.lean
index dd39fc02d..2149d987c 100644
--- a/conformance/HexPrimality/Conformance.lean
+++ b/conformance/HexPrimality/Conformance.lean
@@ -220,6 +220,24 @@ open Hex.Nat
       failure.stop == .exhausted && failure.attempts == 36
   | .ok _ => false)
 
+-- The elaborator's explicit rho allocation reaches a deterministic bounded
+-- exhaustion on a non-smooth 512-bit probable-prime input.
+#guard (match Internal.primeCertCountedWith? ⟨2, 1 <<< 15⟩
+    9521691625768090263084389838561930764813603239089634545416648725957969250257409112878363599328138633827640729385461401574761860536478435114675541614002177
+    (Hex.Rand.ofSeed 9521691625768090263084389838561930764813603239089634545416648725957969250257409112878363599328138633827640729385461401574761860536478435114675541614002177)
+    1038 with
+  | .ok success => success.attempts == 31
+  | .error _ => false)
+
+-- The same allocation fails promptly when both bounded restarts miss.
+#guard (match Internal.primeCertCountedWith? ⟨2, 1 <<< 15⟩
+    11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123
+    (Hex.Rand.ofSeed 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123)
+    1038 with
+  | .error failure =>
+      failure.stop == .exhausted && failure.attempts == 11
+  | .ok _ => false)
+
 -- Routine gcds are genuinely batched: this fixed restart performs 95
 -- polynomial steps but only seven gcds.
 private def rhoBatchTrace : Hex.Nat.Internal.RhoTrace :=
@@ -302,6 +320,8 @@ example : Hex.Nat.Prime 97 := primality 97
 example : Hex.Nat.Prime 9973 := primality 9973
 example : Hex.Nat.Prime 10007 := primality 10007
 example : Hex.Nat.Prime 2147483647 := primality 2147483647
+example : Hex.Nat.Prime 9521691625768090263084389838561930764813603239089634545416648725957969250257409112878363599328138633827640729385461401574761860536478435114675541614002177 :=
+  primality 9521691625768090263084389838561930764813603239089634545416648725957969250257409112878363599328138633827640729385461401574761860536478435114675541614002177
 example : Hex.Nat.Prime 101 := by primality
 example : Hex.Nat.Prime 2147483647 := by primality
 example : True := by
@@ -313,6 +333,20 @@ example : True := by
 #guard_msgs in
 example : Hex.Nat.Prime 561 := primality 561
 
+/--
+error: primality: certificate search for 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123 exhausted after 11 attempts (seed 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123, recursive fuel 1038; policy maximum 1040 fuel at 512 bits, 2 rho restarts with 32768 steps each); no total primality decision was attempted
+-/
+#guard_msgs in
+example : Hex.Nat.Prime 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123 :=
+  primality 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123
+
+/--
+error: primality: input has 513 bits; the enforced policy supports at most 512 bits; raising the ceiling requires new end-to-end benchmark evidence
+-/
+#guard_msgs in
+example : Hex.Nat.Prime 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096 :=
+  primality 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096
+
 /--
 error: primality: the goal
   Prime (2 + 2)
diff --git a/conformance/HexPrimalityMathlib/Conformance.lean b/conformance/HexPrimalityMathlib/Conformance.lean
index 791e2593e..4018aeecc 100644
--- a/conformance/HexPrimalityMathlib/Conformance.lean
+++ b/conformance/HexPrimalityMathlib/Conformance.lean
@@ -50,6 +50,21 @@ example : Nat.Prime 2147483647 := by norm_num
 example : Nat.Prime 2147483647 := by primality
 example : Hex.Nat.Prime 2147483647 := by primality
 example : Nat.Prime 65537 := by primality
+example : Nat.Prime 9521691625768090263084389838561930764813603239089634545416648725957969250257409112878363599328138633827640729385461401574761860536478435114675541614002177 := by
+  primality
+
+/--
+error: primality: certificate search for 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123 exhausted after 11 attempts (seed 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123, recursive fuel 1038; policy maximum 1040 fuel at 512 bits, 2 rho restarts with 32768 steps each); no total primality decision was attempted
+-/
+#guard_msgs in
+example : Nat.Prime 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123 := by primality
+
+/--
+error: primality: input has 513 bits; the enforced policy supports at most 512 bits; raising the ceiling requires new end-to-end benchmark evidence
+-/
+#guard_msgs in
+example : Nat.Prime 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096 := by
+  primality
 
 /--
 error: primality: the goal
diff --git a/conformance/HexPrimalityMathlib/OptInConformance.lean b/conformance/HexPrimalityMathlib/OptInConformance.lean
index ade4baa84..0ea42545b 100644
--- a/conformance/HexPrimalityMathlib/OptInConformance.lean
+++ b/conformance/HexPrimalityMathlib/OptInConformance.lean
@@ -21,6 +21,8 @@ example : Nat.Prime 16777121 := by norm_num       -- last tested prime below the
 example : Nat.Prime 16777259 := by norm_num       -- first prime above the threshold
 example : ¬ Nat.Prime 16777216 := by norm_num     -- the threshold itself
 example : Nat.Prime 2147483647 := by norm_num     -- certificate-backed positive proof
+example : Nat.Prime 9521691625768090263084389838561930764813603239089634545416648725957969250257409112878363599328138633827640729385461401574761860536478435114675541614002177 := by
+  norm_num                                          -- exact 512-bit ceiling
 example : ¬ Nat.Prime 2147483649 := by norm_num   -- validated factor proof
 example : ¬ Nat.Prime 3215031751 := by norm_num   -- strong pseudoprime to bases 2, 3, 5, 7
 example : ¬ Nat.Prime 0 := by norm_num
@@ -31,7 +33,29 @@ example : ¬ Nat.Prime 1 := by norm_num
 -- after bounded Hex search is exhausted.
 /--
 error: unsolved goals
-⊢ Nat.Prime 3317044064679887385961981
+⊢ Nat.Prime
+    11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123
 -/
 #guard_msgs in
-example : Nat.Prime 3317044064679887385961981 := by norm_num
+example : Nat.Prime 11069588345001798189188705872711741673446310956174776680242876230365522527670481055399138994024099817696810905038323515123654848684366962778647276800762123 := by
+  norm_num
+
+/--
+error: unsolved goals
+⊢ Nat.Prime
+    13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096
+-/
+#guard_msgs in
+example : Nat.Prime 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096 := by
+  norm_num
+
+-- The input ceiling applies to negative goals too; the opt-in extension does
+-- not silently recover Mathlib's unbounded trial decision above the ceiling.
+/--
+error: unsolved goals
+⊢ ¬Nat.Prime
+      13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096
+-/
+#guard_msgs in
+example : ¬ Nat.Prime 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096 := by
+  norm_num
diff --git a/lakefile.lean b/lakefile.lean
index 139ea2fb4..41d701d0f 100644
--- a/lakefile.lean
+++ b/lakefile.lean
@@ -301,6 +301,17 @@ lean_lib HexPrimalityKernelProbe where
   srcDir := "bench"
   globs := #[`HexBench.PrimalityKernel]
 
+lean_lib HexPrimalityElabProbe where
+  srcDir := "bench"
+  globs := #[`HexPrimality.ProofProbe.CoreBaseline,
+    `HexPrimality.ProofProbe.Core512,
+    `HexPrimality.ProofProbe.CoreExhausted,
+    `HexPrimality.ProofProbe.CoreOverBudget,
+    `HexPrimalityMathlib.ProofProbe.MathlibBaseline,
+    `HexPrimalityMathlib.ProofProbe.Mathlib512,
+    `HexPrimalityMathlib.ProofProbe.MathlibExhausted,
+    `HexPrimalityMathlib.ProofProbe.MathlibOverBudget]
+
 lean_lib HexIntFactorKernelProbe where
   srcDir := "bench"
   globs := #[`HexBench.IntFactorKernel]
diff --git a/libraries.yml b/libraries.yml
index 411da8eb3..dfe0dc858 100644
--- a/libraries.yml
+++ b/libraries.yml
@@ -824,6 +824,7 @@ libraries:
   HexPrimalityMathlib:
     deps: [HexPrimality]
     mathlib: true
+    proof_probes: [bench/HexPrimalityMathlib/ProofProbe]
     done_through: 0
     status: active
   HexIntFactorMathlib:
diff --git a/reports/bench-results/hex-primality-elaborator-policy-issue-9779-chungus2.json b/reports/bench-results/hex-primality-elaborator-policy-issue-9779-chungus2.json
new file mode 100644
index 000000000..114db3623
--- /dev/null
+++ b/reports/bench-results/hex-primality-elaborator-policy-issue-9779-chungus2.json
@@ -0,0 +1,17494 @@
+{
+  "config": {
+    "absolute_only": true,
+    "accounting_quantization_ticks": 3,
+    "allow_busy": false,
+    "allow_dirty": false,
+    "command_template": "lake build +<module>:olean",
+    "core_interference_accounting": "measurement-cpu-foreign-plus-all-SMT-sibling-busy",
+    "cpu_affinity": [
+      22
+    ],
+    "cpu_topology": {
+      "core_id": "52",
+      "logical_cpu": 22,
+      "physical_package_id": "0",
+      "scaling_cur_freq_khz": "4449997",
+      "scaling_governor": "schedutil",
+      "thread_siblings_list": "22,70"
+    },
+    "expected_host": "chungus2",
+    "frequency_measurement": "cpufreq-time-in-state-arm-mean",
+    "import_baseline_control": null,
+    "lean_num_threads": "1",
+    "max_core_interference_ratio": 0.002,
+    "max_frequency_spread_ratio": 0.15,
+    "max_load_per_cpu": 0.5,
+    "max_null_robust_spread_ratio": 0.1,
+    "max_pair_retries": 32,
+    "measurement_cpu_foreign_accounting": "busy-minus-child-minus-runner-minus-irq-softirq",
+    "minimum_control_magnitude_ratio": 2.0,
+    "null_magnitude_factor": 3.0,
+    "order": [
+      "core-baseline-null",
+      "core-512-null",
+      "mathlib-baseline-null",
+      "core-512",
+      "mathlib-512",
+      "core-exhausted",
+      "mathlib-exhausted",
+      "core-over-budget",
+      "mathlib-over-budget"
+    ],
+    "pairing": "adjacent measured reference and candidate fresh modules; contamination retries the complete oriented pair",
+    "preflight_max_busy_ticks": 2,
+    "preflight_timeout_seconds": 300.0,
+    "preflight_window_seconds": 2.0,
+    "requested_cpu": 22,
+    "rotation": "pairs left by round index; pair orientation alternates",
+    "samples": 6,
+    "shared_host": true,
+    "timeout_seconds": 30.0,
+    "warm_command_template": "lake build +<module>:deps",
+    "warm_timeout_seconds": 600.0
+  },
+  "environment": {
+    "architecture": "x86_64",
+    "cpu_model": "AMD EPYC 9455 48-Core Processor",
+    "dependency_checkouts": {
+      "Cli": {
+        "dirty": false,
+        "head": "ab3a82db9fea14cf0fd7f5a2de650f4b534640af",
+        "path": "/home/kim/worktrees/hex-dev/hex-dev-issue-9779/.lake/packages/Cli",
+        "state_sha256": "65df434312cafbe625f7c255f523575c7ac14dd658fe7d755edcf376634406e2",
+        "status": "",
+        "tree": "9f7b5faae284be737c556b3b205043b293061fd2"
+      },
+      "LeanSearchClient": {
+        "dirty": false,
+        "head": "ba67e212be1197b84c1f1f6299488a10a3002713",
+        "path": "/home/kim/worktrees/hex-dev/hex-dev-issue-9779/.lake/packages/LeanSearchClient",
+        "state_sha256": "4554659b2b9ebedb58446b1c8d0c3b4439940cd9b4d92c081036ca51b3c4d1dc",
+        "status": "",
+        "tree": "89abb77cd7d9344beed0f5e041b444c5458674f7"
+      },
+      "MD4Lean": {
+        "dirty": false,
+        "head": "31907cc18f48a95384f99cee5582c00fb39e0f67",
+        "path": "/home/kim/worktrees/hex-dev/hex-dev-issue-9779/.lake/packages/MD4Lean",
+        "state_sha256": "371e7dd0e12c88d3bf727066e87ba96e0b3630c1b381e30f80bb86d2d3d98b01",
+        "status": "",
+        "tree": "968ae7de50111d78c47ecf0682e9de2dfd257f67"
+      },
+      "Qq": {
+        "dirty": false,
+        "head": "507746ab8f4b643ccdacb2ec4cdb5853fa9f8ab3",
+        "path": "/home/kim/worktrees/hex-dev/hex-dev-issue-9779/.lake/packages/Qq",
+        "state_sha256": "a98d09e28230aad983f781592db94732afd1c3f6ee89665f5381a4a523c24cd7",
+        "status": "",
+        "tree": "22ba3c95e36bbc7d0aa9d55677ffebf0be90cccd"
+      },
+      "aesop": {
+        "dirty": false,
+        "head": "18889deb9e83ea7420ef51c160d6f88552e744e3",
+        "path": "/home/kim/worktrees/hex-dev/hex-dev-issue-9779/.lake/packages/aesop",
+        "state_sha256": "6f95bf2c6fe0e9d21f222f7894115224789824d945de6e03ebb05077c5c6035e",
+        "status": "",
+        "tree": "ef71b806106b107d2b259d9b50077d4786aefb66"
+      },
+      "batteries": {
+        "dirty": false,
+        "head": "d54dddc581e08be364c278052863524bff7a99a9",
+        "path": "/home/kim/worktrees/hex-dev/hex-dev-issue-9779/.lake/packages/batteries",
+        "state_sha256": "d147fd1a7a237e14711c7d53bbac46f83e1df7f386e6368dde5dfcd3274aa6b9",
+        "status": "",
+        "tree": "f7460b599b386df89d85bfdf1a7465cd44ff0861"
+      },
+      "illuminate": {
+        "dirty": false,
+        "head": "6e558472c981dbee8cb9fcde92fa9593daf04228",
+        "path": "/home/kim/worktrees/hex-dev/hex-dev-issue-9779/.lake/packages/illuminate",
+        "state_sha256": "a98eb6e62a445ff4d98e457669bdcf6fbd215cfb275333232cef9833329a925a",
+        "status": "",
+        "tree": "6fe64d8f2c0a64d7b7a3647f555f465fb636a55b"
+      },
+      "importGraph": {
+        "dirty": false,
+        "head": "d8823026ac7ef130c253089d95685f9877b95323",
+        "path": "/home/kim/worktrees/hex-dev/hex-dev-issue-9779/.lake/packages/importGraph",
+        "state_sha256": "24ea70ed738b9b9a3e6f6e4fb49174b541e5fe5f285b730163cd3cba093193b4",
+        "status": "",
+        "tree": "d5d88b13966b0a4d89a5d04e5cf9f8f5bf25dfeb"
+      },
+      "lean-bench": {
+        "dirty": false,
+        "head": "fa30c2763cf523f3ac8e46dc3a1dad0845a40098",
+        "path": "/home/kim/worktrees/hex-dev/hex-dev-issue-9779/.lake/packages/lean-bench",
+        "state_sha256": "689f9504c42512f18b20bb58391f1b3026e7a114591037ba1c2e4400db003ff9",
+        "status": "",
+        "tree": "5e2c6d004439e19a8fc005e136fea4beb9f28c5b"
+      },
+      "mathlib": {
+        "dirty": false,
+        "head": "85e3a25e006c35636f0e53b0e9296caca2685bc0",
+        "path": "/home/kim/worktrees/hex-dev/hex-dev-issue-9779/.lake/packages/mathlib",
+        "state_sha256": "40b120f09ff47e9ca7d54b6aca919463d61cb27f1ad58ee07362fe7b54bf3c23",
+        "status": "",
+        "tree": "955cb3928b885a10e8deb4e1f293f8f50d82664e"
+      },
+      "plausible": {
+        "dirty": false,
+        "head": "d9598f07b1bc701f1e3aae163d2681c1fd978793",
+        "path": "/home/kim/worktrees/hex-dev/hex-dev-issue-9779/.lake/packages/plausible",
+        "state_sha256": "9bd4c1e4cd877b3bdc974e50fc27789ab6a80107ae139a051d3f467037d4e16e",
+        "status": "",
+        "tree": "81f195d87b0f1efa7b9430b6d21034ed51bd2006"
+      },
+      "proofwidgets": {
+        "dirty": false,
+        "head": "a8acbfd87375ff4abe14ce09db5b7664d383bc7f",
+        "path": "/home/kim/worktrees/hex-dev/hex-dev-issue-9779/.lake/packages/proofwidgets",
+        "state_sha256": "cb5d130298c96a2f42742dc55b5ca802ee6703fb70147c39d23a10ffaecc8687",
+        "status": "",
+        "tree": "7c0d97efaaf9c6b9faeed6c863b8c4085cc989a2"
+      },
+      "subverso": {
+        "dirty": false,
+        "head": "847084e80500726e4331dded5f17007ddaf89c31",
+        "path": "/home/kim/worktrees/hex-dev/hex-dev-issue-9779/.lake/packages/subverso",
+        "state_sha256": "1ce57a39bc7eb6e755cb1a46f70ea6022bfdf80c5436ff2108efa77c6e9918e6",
+        "status": "",
+        "tree": "653a5d4a775e9b68056f0f8b9b253da8c65c286f"
+      },
+      "verso": {
+        "dirty": false,
+        "head": "90c688db9db7280364c8bd7f8264c82a01582fd4",
+        "path": "/home/kim/worktrees/hex-dev/hex-dev-issue-9779/.lake/packages/verso",
+        "state_sha256": "a5dfe8c36b946d57874bafe6ebdd4477bf0415624f31e324f4f733587b76810d",
+        "status": "",
+        "tree": "a8e1171167bee808e429a0bf3492a96bd54c5c1a"
+      }
+    },
+    "git_commit": "1bfeadb5db02ef3749bfd1825ab0ce50e24ecd2f",
+    "git_dirty": false,
+    "gnu_time": "/run/current-system/sw/bin/time",
+    "host_after": {
+      "affinity_cpu_frequency_khz": "4452670",
+      "available_logical_cpus": 1,
+      "concurrent_lake_lean": [
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+    "HexPrimality/Search.lean": "bdf0ad50951259d58262f3fbcf365bb1464ef5ca671a860cb240d389c738bd6d",
+    "HexPrimality/Sieve.lean": "e39b7ad1b9247d05252850092d7ff8cfe41bc7a107ceb848486138e9264bda43",
+    "HexPrimality/SieveElab.lean": "f1829920ea1493485bbcf998df4995a303d643c026b5c7ba0407ac6d8f7c8b39",
+    "HexPrimality/Table.lean": "b9caff6c3e1fe3987ca3500346f0236d765c6850a2202307ed89d492e1b16fb3",
+    "HexPrimalityMathlib.lean": "f686e0874a6bb2f04d5d57929db6180f11f49a6541d7e7c395cb00b0e043294e",
+    "HexPrimalityMathlib/NormNum.lean": "1d947a65d8c9c6cf133534bd3dd509029fab33197dc0d42f0b6e80fe53843f1c",
+    "HexPrimalityMathlib/Prime.lean": "e6bf9916618d68a801d159231e0e33ff7f2e08924c939bb112b6a8b1f30c8a2f",
+    "HexPrimalityMathlib/Segment.lean": "c4570ca1f8a4fbba0112ba88a375f6824f78688e0d83071c931bd5cfe5d38bf3",
+    "bench/HexBench/PrimalityKernel.lean": "cff2adf5834c98926140021bd6fe4ffcacacd1011a81d714004be7ba97d4ee13",
+    "bench/HexPrimality/Bench.lean": "778e2be54330d1eb0bb54dea08a5d90c5d58793b883e04c281ac5398b6671db8",
+    "bench/HexPrimality/ProofProbe/Core512.lean": "102daa54ce0f550ac59d128ad9a7f041ce6e1ea0d07c08b0ecd46b9aa34123ca",
+    "bench/HexPrimality/ProofProbe/CoreBaseline.lean": "e597a9a1210518aa2253a440480bf33ffc4a92e32bbbe0f4c544d0402dbdfcf0",
+    "bench/HexPrimality/ProofProbe/CoreExhausted.lean": "d362d18417eb64ef180da6249acab2f04c38c9ad162f69984862cd861003b1dd",
+    "bench/HexPrimality/ProofProbe/CoreOverBudget.lean": "feae51a6d2d4dcddfe5b3a265e344e565501af572012e3a50d36987c498b7064",
+    "bench/HexPrimalityMathlib/ProofProbe/Mathlib512.lean": "6871b90d3879c5867eca75e51cc6610e48da1e218196bdd4e64fe40fed0dd5fa",
+    "bench/HexPrimalityMathlib/ProofProbe/MathlibBaseline.lean": "e0dffdf25b1c06b242ba509fbec20da4298d5c32a72a013d2cb2889f6310173b",
+    "bench/HexPrimalityMathlib/ProofProbe/MathlibExhausted.lean": "6e8e43b59629e34740edb52b56c4ffb4da663a3e1ae5d7de2a26270043948c57",
+    "bench/HexPrimalityMathlib/ProofProbe/MathlibOverBudget.lean": "da6200ec04943bbf614f45411ff904328d9ea0606aa7a639e988d95badad46db",
+    "lake-manifest.json": "1092fda98a3a25bc60c34032a45c68fb266ee9b63637cbed3c0039c0845677c8",
+    "lakefile.lean": "620570438ae885062fcfb2359eb97504c9bd8149505b177057301c3d55df7f25",
+    "lean-toolchain": "8190e75a201741065fe508b28955dd64dd72d090babe5f70ce6848879d68ae88",
+    "scripts/bench/fresh_module_sweep.py": "2a44028bf07807a8e5f54811bf596677288b106703caeb1f5a5a94126bbce385",
+    "scripts/bench/primality_elab_sweep.py": "56eb57268d1810a6c152030474bae72ecf3b00d32da1082b06f7133fad86973e",
+    "scripts/ci/check_benches_mathlib_free.py": "ad01bd8d9edd2f111fcc30d90b3a427a7b3301fc45d4a33bf8e59a9081008cad"
+  },
+  "validity": {
+    "exceptions": [],
+    "host_protocol": "designated-shared-host-v3",
+    "observed": {
+      "accounting_quantization_ticks": 3,
+      "expected_frequency_observations": 108,
+      "frequency_spread_ratio": 0.013902932254802769,
+      "max_aggregate_core_interference_ratio": 0.024894778709065105,
+      "max_arm_mean_frequency_khz": 3150000.0,
+      "max_attempts_per_pair": 3,
+      "max_concurrent_lake_lean_count": 8,
+      "max_core_interference_ratio": 0.002,
+      "max_cpu_pressure_some_delta_us": 196594,
+      "max_effective_core_interference_ratio": 0.033363344662757824,
+      "max_frequency_spread_ratio": 0.15,
+      "max_interference_allowance_seconds": 0.03,
+      "max_load_1m_per_cpu": 0.03045654296875,
+      "max_measurement_cpu_foreign_ratio": 0.01522585976802793,
+      "max_measurement_cpu_interrupt_ratio": 0.021964931482290714,
+      "max_preflight_wait_seconds": 8.004323860164732,
+      "max_smt_sibling_busy_ratio": 0.021959450990975234,
+      "measurement_cpu": 22,
+      "min_arm_mean_frequency_khz": 3106806.282722513,
+      "observed_frequency_observations": 108,
+      "smt_sibling_cpus": [
+        70
+      ],
+      "total_exhausted_pairs": 0,
+      "total_preflight_failures": 0,
+      "total_rejected_pair_attempts": 5,
+      "total_rejected_preflight_windows": 10,
+      "violations": []
+    },
+    "release_quality": true
+  }
+}
diff --git a/scripts/bench/fresh_module_sweep.py b/scripts/bench/fresh_module_sweep.py
index cb74d590e..699281c32 100644
--- a/scripts/bench/fresh_module_sweep.py
+++ b/scripts/bench/fresh_module_sweep.py
@@ -96,6 +96,7 @@ class SweepSpec:
     required_samples: int | None = None
     max_pair_retries: int = DEFAULT_MAX_PAIR_RETRIES
     import_baseline_control: str | None = None
+    absolute_only: bool = False
 
 
 def parse_args(
@@ -767,6 +768,17 @@ def validate_spec(spec: SweepSpec) -> None:
             )
     if spec.required_samples is not None and spec.required_samples < 1:
         raise RuntimeError("fresh-module sweep required_samples must be positive")
+    if spec.absolute_only:
+        substantive = [pair for pair in spec.pairs if not pair.null_control]
+        if any("fresh_module_budget_ms" not in pair.metadata for pair in substantive):
+            raise RuntimeError(
+                "absolute-only sweep requires a fresh-module budget on every "
+                "substantive pair"
+            )
+        if any("tactic_budget_ms" in pair.metadata for pair in substantive):
+            raise RuntimeError(
+                "absolute-only sweep cannot declare a relative tactic budget"
+            )
     target = _ExeTarget(spec.probe_target, "", spec.src_dir)
     package_root = ROOT / ".lake" / "packages"
     for root_module in measured:
@@ -1874,6 +1886,20 @@ def summarize(
                 budget_status = "passed"
             result["budget_nanos"] = budget_nanos
             result["budget_status"] = budget_status
+        fresh_budget_ms = pair.metadata.get("fresh_module_budget_ms")
+        if fresh_budget_ms is not None:
+            fresh_budget_nanos = int(fresh_budget_ms) * 1_000_000
+            max_candidate_wall_nanos = max(
+                int(sample["candidate"]["wall_nanos"])
+                for sample in result["samples"]
+            )
+            result["fresh_module_budget_nanos"] = fresh_budget_nanos
+            result["max_candidate_wall_nanos"] = max_candidate_wall_nanos
+            result["fresh_module_budget_status"] = (
+                "failed"
+                if max_candidate_wall_nanos >= fresh_budget_nanos
+                else "passed"
+            )
     return summary
 
 
@@ -2162,7 +2188,8 @@ def validity_summary(
             issue for issue in observations["violations"]
             if issue not in issues
         )
-    if args.shared_host:
+    requires_null_resolution = not spec.absolute_only
+    if args.shared_host and requires_null_resolution:
         control_magnitudes = [
             int(results[pair.name]["build_magnitude_wall_nanos"])
             for pair in spec.pairs
@@ -2177,19 +2204,20 @@ def validity_summary(
             issues.append(
                 "null-control build magnitudes are not sufficiently distinct"
             )
-    for pair in spec.pairs:
-        if not pair.null_control:
-            continue
-        spread_ratio = results[pair.name].get("null_robust_spread_ratio")
-        if (
-            spread_ratio is None
-            or float(spread_ratio) > MAX_NULL_ROBUST_SPREAD_RATIO
-        ):
-            issues.append(
-                f"{pair.name}: robust null IQR/build ratio "
-                f"{spread_ratio!r} exceeds "
-                f"{MAX_NULL_ROBUST_SPREAD_RATIO:.3f}"
-            )
+    if requires_null_resolution:
+        for pair in spec.pairs:
+            if not pair.null_control:
+                continue
+            spread_ratio = results[pair.name].get("null_robust_spread_ratio")
+            if (
+                spread_ratio is None
+                or float(spread_ratio) > MAX_NULL_ROBUST_SPREAD_RATIO
+            ):
+                issues.append(
+                    f"{pair.name}: robust null IQR/build ratio "
+                    f"{spread_ratio!r} exceeds "
+                    f"{MAX_NULL_ROBUST_SPREAD_RATIO:.3f}"
+                )
     for pair in spec.pairs:
         if pair.null_control:
             continue
@@ -2237,6 +2265,14 @@ def validity_summary(
             issue = f"{pair.name}: tactic budget {status}"
             if issue not in issues:
                 issues.append(issue)
+    for pair in spec.pairs:
+        if "fresh_module_budget_ms" not in pair.metadata:
+            continue
+        status = results[pair.name].get("fresh_module_budget_status")
+        if status != "passed":
+            issue = f"{pair.name}: fresh-module budget {status}"
+            if issue not in issues:
+                issues.append(issue)
     release_quality = (
         not args.allow_dirty
         and not args.allow_busy
@@ -2546,6 +2582,7 @@ def run_cli(
             "null_magnitude_factor": NULL_MAGNITUDE_FACTOR,
             "max_null_robust_spread_ratio":
                 MAX_NULL_ROBUST_SPREAD_RATIO,
+            "absolute_only": spec.absolute_only,
             "import_baseline_control": spec.import_baseline_control,
             "frequency_measurement":
                 "cpufreq-time-in-state-arm-mean",
diff --git a/scripts/bench/primality_elab_sweep.py b/scripts/bench/primality_elab_sweep.py
new file mode 100644
index 000000000..3f03ed316
--- /dev/null
+++ b/scripts/bench/primality_elab_sweep.py
@@ -0,0 +1,134 @@
+#!/usr/bin/env python3
+"""Measure the supported core and Mathlib ``primality`` policy routes.
+
+Every substantive probe is paired with the matching import-only module. The
+accepted probes exercise certificate search, compiled self-check, certificate
+reification, and kernel replay at the exact 512-bit ceiling. A non-smooth
+512-bit safe probable prime exercises bounded rho exhaustion at that same
+ceiling. Rejection immediately above it must occur before search. No failure
+route may fall through to a total decision.
+"""
+
+from __future__ import annotations
+
+import sys
+from pathlib import Path
+
+
+ROOT = Path(__file__).resolve().parents[2]
+sys.path.insert(0, str(ROOT))
+
+from scripts.bench.fresh_module_sweep import (  # noqa: E402
+    ProbeModule,
+    ProbePair,
+    SweepSpec,
+    run_cli,
+)
+
+
+CORE_BASELINE = ProbeModule("HexPrimality.ProofProbe.CoreBaseline")
+PROOF_AXIOMS = ("propext", "Classical.choice", "Quot.sound")
+CORE_512 = ProbeModule("HexPrimality.ProofProbe.Core512", PROOF_AXIOMS)
+MATHLIB_BASELINE = ProbeModule("HexPrimalityMathlib.ProofProbe.MathlibBaseline")
+MATHLIB_512 = ProbeModule("HexPrimalityMathlib.ProofProbe.Mathlib512", PROOF_AXIOMS)
+
+
+def policy_pair(
+    name: str,
+    route: str,
+    outcome: str,
+    reference: ProbeModule,
+    candidate: ProbeModule,
+    bits: int,
+) -> ProbePair:
+    return ProbePair(
+        name,
+        reference,
+        candidate,
+        {
+            "route": route,
+            "outcome": outcome,
+            "bits": bits,
+            "fresh_module_budget_ms": 10_000,
+        },
+    )
+
+
+SPEC = SweepSpec(
+    description=__doc__ or "primality elaborator policy sweep",
+    pairs=(
+        ProbePair(
+            "core-baseline-null",
+            CORE_BASELINE,
+            CORE_BASELINE,
+            {"route": "core", "outcome": "calibration-only", "bits": 0},
+            null_control=True,
+        ),
+        ProbePair(
+            "core-512-null",
+            CORE_512,
+            CORE_512,
+            {"route": "core", "outcome": "calibration-only", "bits": 512},
+            null_control=True,
+        ),
+        ProbePair(
+            "mathlib-baseline-null",
+            MATHLIB_BASELINE,
+            MATHLIB_BASELINE,
+            {"route": "mathlib", "outcome": "calibration-only", "bits": 0},
+            null_control=True,
+        ),
+        policy_pair("core-512", "core", "accepted", CORE_BASELINE, CORE_512, 512),
+        policy_pair(
+            "mathlib-512", "mathlib", "accepted", MATHLIB_BASELINE, MATHLIB_512, 512
+        ),
+        policy_pair(
+            "core-exhausted",
+            "core",
+            "exhausted",
+            CORE_BASELINE,
+            ProbeModule("HexPrimality.ProofProbe.CoreExhausted"),
+            512,
+        ),
+        policy_pair(
+            "mathlib-exhausted",
+            "mathlib",
+            "exhausted",
+            MATHLIB_BASELINE,
+            ProbeModule("HexPrimalityMathlib.ProofProbe.MathlibExhausted"),
+            512,
+        ),
+        policy_pair(
+            "core-over-budget",
+            "core",
+            "over-budget",
+            CORE_BASELINE,
+            ProbeModule("HexPrimality.ProofProbe.CoreOverBudget"),
+            513,
+        ),
+        policy_pair(
+            "mathlib-over-budget",
+            "mathlib",
+            "over-budget",
+            MATHLIB_BASELINE,
+            ProbeModule("HexPrimalityMathlib.ProofProbe.MathlibOverBudget"),
+            513,
+        ),
+    ),
+    probe_target="HexPrimalityElabProbe",
+    schema="hex-primality-elaborator-policy-v1",
+    measurement="paired-fresh-module-olean-wall-robust-null-v2",
+    output_stem="hex-primality-elaborator-policy",
+    extra_sources=(
+        Path("HexPrimality/SPEC/hex-primality.md"),
+        Path("bench/HexBench/PrimalityKernel.lean"),
+        Path("bench/HexPrimality/Bench.lean"),
+    ),
+    required_samples=6,
+    max_pair_retries=32,
+    absolute_only=True,
+)
+
+
+if __name__ == "__main__":
+    raise SystemExit(run_cli(SPEC, Path(__file__)))
diff --git a/scripts/bench/proof_only_runtime_exemptions/lakefile-lean-e82c6493-41d701d0.json b/scripts/bench/proof_only_runtime_exemptions/lakefile-lean-e82c6493-41d701d0.json
new file mode 100644
index 000000000..c9bb2f006
--- /dev/null
+++ b/scripts/bench/proof_only_runtime_exemptions/lakefile-lean-e82c6493-41d701d0.json
@@ -0,0 +1,6 @@
+{
+  "path": "lakefile.lean",
+  "baseline_blob": "e82c649322c5dc6574071e576e92d644683eb00a",
+  "current_blob": "41d701d0f34ff9567fa3a4986b777573a4ce6cc5",
+  "reason": "Combines the inherited HexPolyFp precompileModules elaboration-time code-generation transition with the independent build-only HexPrimalityElabProbe registration. Neither changes HexPolyFp source, hexbz_factor_service, or the service's runtime call graph."
+}
diff --git a/scripts/bench/test_fresh_module_sweep.py b/scripts/bench/test_fresh_module_sweep.py
index 3ea653e66..863ee7d9a 100644
--- a/scripts/bench/test_fresh_module_sweep.py
+++ b/scripts/bench/test_fresh_module_sweep.py
@@ -716,6 +716,41 @@ def rows(wall: int, deltas: list[int]) -> list[dict[str, object]]:
         self.assertEqual(summary["tactic"]["resolution"], "resolved")
         self.assertEqual(summary["tactic"]["budget_status"], "unresolved")
 
+    def test_summary_enforces_absolute_fresh_module_budget(self) -> None:
+        baseline = sweep.ProbeModule("Probe.Baseline")
+        candidate = sweep.ProbeModule("Probe.Candidate")
+        pair = sweep.ProbePair(
+            "absolute",
+            baseline,
+            candidate,
+            {"fresh_module_budget_ms": 100},
+        )
+        spec = sweep.SweepSpec(
+            description="absolute budget",
+            pairs=(pair,),
+            probe_target="Probe",
+            schema="test",
+            measurement="test",
+            output_stem="test",
+        )
+        rows = {pair.name: [
+            {
+                "reference": {"wall_nanos": 20_000_000, "peak_rss_kb": None},
+                "candidate": {"wall_nanos": 90_000_000, "peak_rss_kb": None},
+                "signed_wall_delta_nanos": 70_000_000,
+            },
+            {
+                "reference": {"wall_nanos": 20_000_000, "peak_rss_kb": None},
+                "candidate": {"wall_nanos": 110_000_000, "peak_rss_kb": None},
+                "signed_wall_delta_nanos": 90_000_000,
+            },
+        ]}
+        with mock.patch.object(sweep, "artifact_sizes", return_value={}):
+            summary = sweep.summarize(spec, rows)[pair.name]
+        self.assertEqual(summary["fresh_module_budget_nanos"], 100_000_000)
+        self.assertEqual(summary["max_candidate_wall_nanos"], 110_000_000)
+        self.assertEqual(summary["fresh_module_budget_status"], "failed")
+
     def test_one_sided_null_uses_zero_centred_envelope(self) -> None:
         module = sweep.ProbeModule("Probe.Baseline")
         candidate = sweep.ProbeModule("Probe.Candidate")
@@ -1502,6 +1537,35 @@ def test_shared_budget_must_exceed_interference_ceiling(self) -> None:
             "not resolvable under the admitted core-interference ceiling",
         )
 
+    def test_failed_fresh_module_budget_invalidates_record(self) -> None:
+        module = sweep.ProbeModule("Probe.Baseline")
+        pair = sweep.ProbePair(
+            "absolute",
+            module,
+            sweep.ProbeModule("Probe.Candidate"),
+            {"fresh_module_budget_ms": 100},
+        )
+        spec = sweep.SweepSpec(
+            description="absolute budget",
+            pairs=(pair,),
+            probe_target="Probe",
+            schema="test",
+            measurement="test",
+            output_stem="test",
+        )
+        args = sweep.parse_args("validity", [])
+        results = {
+            pair.name: {
+                "resolution": "resolved",
+                "fresh_module_budget_status": "failed",
+            }
+        }
+        quality, issues = sweep.validity_summary(
+            spec, args, results, None, []
+        )
+        self.assertFalse(quality)
+        self.assertIn("absolute: fresh-module budget failed", issues)
+
     def test_contention_violation_makes_release_quality_false(self) -> None:
         args = sweep.parse_args("validity", [])
         observations = {"violations": ["sibling contention"]}
@@ -1545,8 +1609,84 @@ def test_excessive_robust_null_spread_invalidates_record(self) -> None:
         self.assertFalse(quality)
         self.assertRegex("; ".join(issues), "robust null IQR/build ratio")
 
+    def test_absolute_budget_does_not_require_null_resolution(self) -> None:
+        module = sweep.ProbeModule("Probe.Baseline")
+        pairs = (
+            sweep.ProbePair(
+                "null", module, module, {}, null_control=True
+            ),
+            sweep.ProbePair(
+                "absolute",
+                module,
+                sweep.ProbeModule("Probe.Candidate"),
+                {"fresh_module_budget_ms": 100},
+            ),
+        )
+        spec = sweep.SweepSpec(
+            description="absolute budget",
+            pairs=pairs,
+            probe_target="Probe",
+            schema="test",
+            measurement="test",
+            output_stem="test",
+            absolute_only=True,
+        )
+        results = {
+            "null": {
+                "build_magnitude_wall_nanos": 1_000,
+                "null_robust_spread_ratio": 0.11,
+            },
+            "absolute": {
+                "resolution": "unresolved",
+                "fresh_module_budget_status": "passed",
+            },
+        }
+        quality, issues = sweep.validity_summary(
+            spec, sweep.parse_args("validity", []), results, None, []
+        )
+        self.assertTrue(quality)
+        self.assertEqual(issues, [])
+
 
 class HarnessValidationTests(unittest.TestCase):
+    def test_absolute_only_requires_every_absolute_budget(self) -> None:
+        module = sweep.ProbeModule("Probe.Baseline")
+        spec = sweep.SweepSpec(
+            description="invalid absolute-only sweep",
+            pairs=(sweep.ProbePair(
+                "effect", module, sweep.ProbeModule("Probe.Candidate"), {}
+            ),),
+            probe_target="Probe",
+            schema="test",
+            measurement="test",
+            output_stem="test",
+            absolute_only=True,
+        )
+        with self.assertRaisesRegex(RuntimeError, "fresh-module budget"):
+            sweep.validate_spec(spec)
+
+    def test_absolute_only_rejects_relative_budget(self) -> None:
+        module = sweep.ProbeModule("Probe.Baseline")
+        spec = sweep.SweepSpec(
+            description="mixed absolute-only sweep",
+            pairs=(sweep.ProbePair(
+                "effect",
+                module,
+                sweep.ProbeModule("Probe.Candidate"),
+                {
+                    "fresh_module_budget_ms": 100,
+                    "tactic_budget_ms": 100,
+                },
+            ),),
+            probe_target="Probe",
+            schema="test",
+            measurement="test",
+            output_stem="test",
+            absolute_only=True,
+        )
+        with self.assertRaisesRegex(RuntimeError, "relative tactic budget"):
+            sweep.validate_spec(spec)
+
     def test_shared_host_arguments_are_complete_and_exclusive(self) -> None:
         with mock.patch.object(sys, "stderr", new=io.StringIO()):
             with self.assertRaises(SystemExit):
diff --git a/scripts/bench/test_primality_policy_sweeps.py b/scripts/bench/test_primality_policy_sweeps.py
index 5b0c4c00a..610be571c 100644
--- a/scripts/bench/test_primality_policy_sweeps.py
+++ b/scripts/bench/test_primality_policy_sweeps.py
@@ -5,6 +5,8 @@
 
 import unittest
 
+from scripts.bench import fresh_module_sweep
+from scripts.bench import primality_elab_sweep as elab
 from scripts.bench import primality_policy_sweep as policy
 from scripts.bench import primality_table_sweep as table
 
@@ -54,5 +56,82 @@ def test_ladder_has_both_outcomes_and_adversarial_primes(self) -> None:
         self.assertGreater(max(chains), 10_000_000)
 
 
+class ElaboratorSweepTests(unittest.TestCase):
+    def test_manifest_covers_both_routes_and_policy_outcomes(self) -> None:
+        substantive = [pair for pair in elab.SPEC.pairs if not pair.null_control]
+        self.assertEqual(
+            {(pair.metadata["route"], pair.metadata["outcome"])
+             for pair in substantive},
+            {
+                ("core", "accepted"),
+                ("mathlib", "accepted"),
+                ("core", "exhausted"),
+                ("mathlib", "exhausted"),
+                ("core", "over-budget"),
+                ("mathlib", "over-budget"),
+            },
+        )
+        self.assertEqual(
+            {pair.metadata["fresh_module_budget_ms"] for pair in substantive},
+            {10_000},
+        )
+
+    def test_boundary_is_accepted_at_512_and_rejected_at_513(self) -> None:
+        by_name = {pair.name: pair for pair in elab.SPEC.pairs}
+        self.assertEqual(by_name["core-512"].metadata["bits"], 512)
+        self.assertEqual(by_name["mathlib-512"].metadata["bits"], 512)
+        self.assertEqual(by_name["core-over-budget"].metadata["bits"], 513)
+        self.assertEqual(by_name["mathlib-over-budget"].metadata["bits"], 513)
+        self.assertEqual(by_name["core-exhausted"].metadata["bits"], 512)
+        self.assertEqual(by_name["mathlib-exhausted"].metadata["bits"], 512)
+
+    def test_each_route_has_a_null_control(self) -> None:
+        self.assertEqual(
+            {pair.metadata["route"] for pair in elab.SPEC.pairs
+             if pair.null_control},
+            {"core", "mathlib"},
+        )
+
+    def test_release_measurement_protocol_is_preregistered(self) -> None:
+        self.assertEqual(elab.SPEC.required_samples, 6)
+        self.assertEqual(elab.SPEC.max_pair_retries, 32)
+        self.assertEqual(
+            elab.SPEC.measurement,
+            "paired-fresh-module-olean-wall-robust-null-v2",
+        )
+        fresh_module_sweep.validate_spec(elab.SPEC)
+
+    def test_every_probe_is_wired_into_lake(self) -> None:
+        lakefile = (elab.ROOT / "lakefile.lean").read_text(encoding="utf-8")
+        for module in fresh_module_sweep.probe_modules(elab.SPEC):
+            self.assertIn(f"`{module}", lakefile, module)
+
+    def test_handlers_share_the_bounded_policy(self) -> None:
+        core = (elab.ROOT / "HexPrimality/Elab.lean").read_text(encoding="utf-8")
+        companion = (elab.ROOT / "HexPrimalityMathlib/NormNum.lean").read_text(
+            encoding="utf-8"
+        )
+        self.assertIn("meta def primalityBitBudget : Nat := 512", core)
+        self.assertIn("meta def primalityFuelBudget : Nat := 1040", core)
+        self.assertIn("meta def primalityRhoRestartBudget : Nat := 2", core)
+        self.assertIn("meta def primalityRhoStepBudget : Nat := 1 <<< 15", core)
+        self.assertIn("let fuel := primalityFuel n", core)
+        self.assertIn("unless withinPrimalityBudget n' do failure", companion)
+        self.assertIn("unless withinPrimalityBudget n do", core)
+        self.assertIn("let fuel := primalityFuel n'", companion)
+        total_decision = (
+            r"\b(?:Hex\.Nat\.)?(?:isPrimeTrial|isPrime\?|isPrime)(?:\s|\()"
+        )
+        cert_handler = companion.split("def evalNatPrimeCert", 1)[1].split(
+            "/-- The `Nat.Prime` goal handler", 1
+        )[0]
+        tactic_handler = companion.split("def evalPrimalityTacNat", 1)[1].split(
+            "end Hex.PrimalityTactic", 1
+        )[0]
+        self.assertNotRegex(core, total_decision)
+        self.assertNotRegex(cert_handler, total_decision)
+        self.assertNotRegex(tactic_handler, total_decision)
+
+
 if __name__ == "__main__":
     unittest.main()
diff --git a/scripts/check_dag.py b/scripts/check_dag.py
index cf03cdacd..0ac70af6f 100644
--- a/scripts/check_dag.py
+++ b/scripts/check_dag.py
@@ -48,6 +48,7 @@
     "HexGF2BenchSupport",
     "HexBerlekampKernelProbe",
     "HexPrimalityKernelProbe",
+    "HexPrimalityElabProbe",
     "HexIntFactorKernelProbe",
     "HexMvGcdKernelProbe",
     "HexMvGcdBenchSupport",
diff --git a/scripts/libgraph.py b/scripts/libgraph.py
index 8b6f33346..89bbc403a 100644
--- a/scripts/libgraph.py
+++ b/scripts/libgraph.py
@@ -20,6 +20,7 @@
     "HexGF2BenchSupport",
     "HexBerlekampKernelProbe",
     "HexPrimalityKernelProbe",
+    "HexPrimalityElabProbe",
     "HexIntFactorKernelProbe",
     "HexMvGcdKernelProbe",
     "HexMvGcdBenchSupport",