diff --git a/CONTRIBUTING.md b/CONTRIBUTING.md
index fe54f23f19..5185786f2e 100644
--- a/CONTRIBUTING.md
+++ b/CONTRIBUTING.md
@@ -63,6 +63,12 @@ For a parameter improvement, supply exact rational choices, the full dependency
 argument, and an updated patch against the pinned source. Explain whether the
 change stays within the parameter ceiling or changes one of its hypotheses.
 
+Before extending a recursive cost edit, compare its complete paid and target
+histograms with the [signed invoice preflight](docs/research/recursive-invoice-preflight.md).
+It can certify uniform concave-moment gains or losses without a root search.
+Its accounting contract is declared; source correctness and supplier admission
+still require their own evidence.
+
 Keep `upstream/`, frozen source archives and pinned evidence unchanged. When
 changing generated artifacts, edit their generators and include the regenerated
 certificates and patches in the same change. Run the checks affected by your
diff --git a/docs/research/recursive-invoice-preflight.md b/docs/research/recursive-invoice-preflight.md
new file mode 100644
index 0000000000..d55f6c136c
--- /dev/null
+++ b/docs/research/recursive-invoice-preflight.md
@@ -0,0 +1,108 @@
+# Compare the complete recursive invoice before extending a construction
+
+The same local edit can save scalar instructions while worsening its recursive
+bill. Conversely, owning a helper can remove paid work while also surrendering
+a productive target. Compare **both** paid jobs and target credits first:
+
+```sh
+python3 -B scripts/compare_recursive_invoices.py OLD.json NEW.json
+```
+
+This standard-library tool uses integer arithmetic, without a root search or
+contribution replay. It reports an exact certificate for every increasing
+concave rank tariff with f(0)=0, including r^(1-C), 0<C<1. The method is classical
+concave order, already used by mightyOil's
+[T16 audit](../../research/t16-moment-order-audit/README.md) and its predecessors.
+The shared interface adds signed target accounting; it is not a new theorem.
+The concurrent [source-bound pricing oracle #5391](https://github.com/CrocSwap/integer-mult-bounds/pull/5391)
+checks compatibility of reviewed physical, invoice, closure and funding receipts.
+This comparator is an earlier arithmetic screen, not a replacement for that
+evidence join or its trusted policies.
+
+## Input and meaning
+
+Both files use this schema, with identical declared accounting contracts:
+
+```json
+{
+  "schema": "integer-mult-recursive-invoice/v1",
+  "contract": {
+    "model": "mixed-target-unnormalized-power-v1",
+    "rank_semantics": "H_r costs r^(1-C); debit surrendered productive fields",
+    "ambient_rank": 145,
+    "boundary": "complete unit before the same mixed-target compiler"
+  },
+  "paid": {"1": 1, "8": 1},
+  "targets": {"9": 1}
+}
+```
+
+Ranks are canonical positive integer strings; counts are nonnegative integers.
+The contract identifies the common pricing and consumer boundary. A change in
+ownership can change targets inside that boundary, but requires its own legal
+construction and complete debit. Different rank meanings, normalization rules
+or transfer contracts require a separately justified comparison. Matching text
+does not prove applicability.
+
+The tool hashes the exact input bytes. It does **not** derive the invoice from
+a circuit, detect an omitted target debit, or authenticate an author's source
+claim. Keep the source word, extraction command and exact head with the invoice.
+The [worked package](../../research/owned-relation-exhaustion/README.md)
+independently reconstructs its complete source invoice before comparing it.
+
+| Result | What it establishes | Next action |
+|---|---|---|
+| `uniformly_worse` | Larger paid-minus-target cost for every power 0<t<1. | Stop this raw-moment improvement claim; identify a changed charge or contract before expanding it. |
+| `equal` | Identical net raw moment at every rank tariff. | Report scalar/compiler/implementation gains separately. Same invoice can still improve a funded compiler endpoint. |
+| `uniformly_better` | Smaller paid-minus-target cost for every power 0<t<1. | Price the complete candidate and prove its source, ownership, phase and caller obligations. |
+| `inconclusive` | Concave order does not decide the comparison. | Use a scoped exact power-moment calculation; mixed signs alone establish no crossing or failure. |
+
+For general concave tariffs the order is weak: a linear tariff can give equality.
+The strict power conclusion requires a nonzero signed difference. These results
+do not imply circuit equivalence, an admitted supplier or a compiler endpoint.
+
+## Recent discriminating examples
+
+* [#5350](https://github.com/CrocSwap/integer-mult-bounds/pull/5350): the filtered
+  151-ADD compression preserves the recursive invoice. Its unfiltered variant
+  splits one H4 into two H2 jobs, a strict loss for every 0<t<1. Scalar event count
+  alone chooses the wrong variant.
+* [#5377](https://github.com/CrocSwap/integer-mult-bounds/pull/5377) and
+  [#5383](https://github.com/CrocSwap/integer-mult-bounds/pull/5383): removing H1+H8
+  and debiting H9 gives a strict gain, 1+8^t-9^t>0. Leaving the H9 target credited
+  produces a much larger, unsupported budget.
+* [#5387](https://github.com/CrocSwap/integer-mult-bounds/pull/5387) and
+  [#5390](https://github.com/CrocSwap/integer-mult-bounds/pull/5390): compiler
+  refinements retain the same recursive invoice. #5390 also reports a smaller
+  sufficient compiler at a matched exponent. `equal` is therefore not a reason
+  to dismiss a compiler or implementation contribution.
+
+Run the source-bound current examples:
+
+```sh
+python3 -B scripts/compare_recursive_invoices.py \
+  research/owned-relation-exhaustion/examples/source.json \
+  research/owned-relation-exhaustion/examples/exchange-128.json
+python3 -B scripts/compare_recursive_invoices.py \
+  research/owned-relation-exhaustion/examples/source.json \
+  research/owned-relation-exhaustion/examples/h4-split.json
+```
+
+The second is a declared single-H4 edit to the complete source invoice, illustrating
+the unfiltered retiming's algebra; it is not a newly emitted physical supplier.
+
+## Certificate and bounds
+
+Set d_r=Δpaid_r-Δtargets_r. The report gives K(k)=Σ d_r min(r,k) at every support
+breakpoint. Between breakpoints K is linear. If all K are nonnegative, every
+increasing concave cost change is nonnegative; reversing the signs gives the
+reverse order. The discrete-slope proof is in the script and the worked package.
+No equal-rank-mass assumption is required.
+
+Inputs are bounded to 1 MiB, 4,096 keys per histogram, ranks at most 10^9 and counts
+at most 10^60. The implementation sweeps sparse support; it does not allocate an
+array of ambient width. Duplicate keys, floats, booleans and differing contracts
+fail with a diagnostic and exit code 2. Valid comparisons, including inconclusive
+ones, exit 0: this is an arithmetic report, not an automatic acceptance gate.
+
+Prepared for zxv with substantial OpenAI Codex assistance. Apache-2.0.
diff --git a/research/owned-relation-exhaustion/.gitattributes b/research/owned-relation-exhaustion/.gitattributes
new file mode 100644
index 0000000000..58f48eebd7
--- /dev/null
+++ b/research/owned-relation-exhaustion/.gitattributes
@@ -0,0 +1,4 @@
+# Preserve exact retained source exports, including original line endings.
+source/** -text
+WITNESS.json -text
+source/INVOICE.json whitespace=cr-at-eol
diff --git a/research/owned-relation-exhaustion/PROOF.md b/research/owned-relation-exhaustion/PROOF.md
new file mode 100644
index 0000000000..0010ef3f34
--- /dev/null
+++ b/research/owned-relation-exhaustion/PROOF.md
@@ -0,0 +1,128 @@
+# Signed invoice order and the fixed centre-exchange ceiling
+
+## 1. The reusable order certificate
+
+Let d_r=(P'_r-P_r)-(T'_r-T_r), with finite positive integer rank support and signed
+integer coefficients. Put K(k)=Σ_r d_r min(r,k). For an increasing concave tariff
+f with f(0)=0 and largest rank R, write s_j=f(j)-f(j-1). Direct telescoping gives
+
+    Σ_r d_r f(r) = s_R K(R) + Σ_{k=1}^{R-1} (s_k-s_{k+1}) K(k).
+
+All coefficients are nonnegative. Thus nonnegative K means no improvement in
+paid-minus-target cost; nonpositive K means no worsening. Conversely, each
+min(r,k) is an allowed tariff, so these signs characterize the whole tariff
+class. K is linear between support breakpoints, including the segment from zero
+to the first breakpoint. Checking just the reported knots is sufficient.
+
+For f(r)=r^t, 0<t<1, s_R>0 and every curvature coefficient is positive. A nonzero
+one-sided K therefore gives a strict sign. If every K vanishes, d vanishes.
+Unequal rank mass is allowed; K(R) explicitly carries its linear contribution.
+This is classical concave order, not a new theorem. mightyOil's retained T16
+audit supplies the closely related equal-mass predecessor implementation.
+
+Two useful controls have equal rank mass but opposite strict signs:
+
+* H4→2H2 gives d_2=2,d_4=-1, K(2)=2,K(4)=0: uniformly worse.
+* Remove paid H1+H8 and target H9: d_1=d_8=-1,d_9=1, K(1)=K(8)=-1,K(9)=0:
+  uniformly better. For powers, the margin gain is g(C)=1+8^(1-C)-9^(1-C)>0.
+
+## 2. Exact current-source inventory and exchange family
+
+The retained final #4610 word has 3,298 paid events per orientation, including
+exactly eight H8 centre events and no other H8 paid events. The retained p16/a7
+controller has sixteen backward and two forward local invocations: 144 original
+centre slots. `source_invoice()` separately rebuilds the complete paid histogram
+and targets from this word and controller, including copied queries, whole idles,
+all helper endpoints and main diagonal outputs. It agrees exactly with the
+final source invoice. Its rank surplus is 752.
+
+Define the following restricted optimistic family, for integers 0≤k≤144:
+
+    P(k) = P(0) - k H1 - k H8
+    T(k) = T(0) - k H9.
+
+There is at most one exchange per original centre slot and **all other recursive
+charges and target credits remain fixed**. Giving every exchange free preparation,
+storage, copying and completion only favors it. A scalar basis choice or schedule
+cannot create a 145th original centre slot within this definition. No assumption
+is made that all 144 exchanges have a physical realization.
+
+The margin is M_k(C)=Σ_r(T_r(k)-P_r(k))r^(1-C). By strict concavity,
+M_k(C)=M_0(C)+k g(C), with g(C)>0 for 0<C<1. Therefore M_144 dominates this entire
+family. The raw-root brackets proved by directed rational intervals are:
+
+| Exchanges | Complete raw crossing |
+|---|---|
+| 0 | (.00302177587077, .00302177587078) |
+| 16, #5377 bill | (.00302238736347, .00302238736348) |
+| 128, #5383/#5387 bill | (.00302667476807, .00302667476808) |
+| 144, optimistic whole-family maximum | (.00302728824993, .00302728824994) |
+
+Consequently the remaining **raw** gain above the 128-exchange crossing is less
+than .00000061348187. Even the all-144 margin at C=.0034 is below -91.8985594569.
+The #5387 claimed endpoint .003026674765 is within .00000000000308 of the
+128-exchange raw upper bracket. These are ceilings on this recursive invoice,
+not supplier admissions or a claim that every compiler achieves its raw root.
+
+`budget.py` proves strict decrease on [0,.01] by bounding the derivative
+Σ_r(P_r-T_r)log(r)r^(1-C) throughout that interval, not just at sample points.
+Each root has strictly positive and negative directed margins at its bracket
+endpoints. A 1,024-cell directed interval cover proves M_144<0 on [.01,1]. For
+each cell [a,b], it bounds targets at a and paid terms at b, using monotonicity
+of every power term. The largest upper bound is below -1,134. Thus there is no
+later positive window; the brackets give global raw ceilings for 0<C<1 in the
+stated family. Other k are dominated by 144.
+
+Every endpoint is rational. Logarithms use range reduction to [1,2] and the
+positive atanh series with an explicit geometric tail; exponentials use Taylor
+series, a geometric tail and repeated squaring. Integer endpoints round
+outwards at scale 10^50. Exact integral and square-root cases calibrate the
+implementation. This is an independently written arithmetic engine, not reuse
+of the predecessor's pricing code. The classical interval formulas are shared.
+
+## 3. Independent fixed-eight formal-row census
+
+The eight entry relations and their first physical realization are Ali Romman's
+#5383. Exact rational row reduction gives pivots
+225,227,229,230,241,245,257,288. Substitution leaves the 224 main inputs and all
+nonpivot helper inputs independent. The census walks every source event in
+both orientations, checks its frame progression and paid rank, and follows
+the retained scalar-row convention: paid inclusion changes the frame, while
+native warp symbols act on the row and cancel only adjacent inverse pairs.
+
+At every boundary it records a nonzero row's exact physical frame identifier
+and rational projective normal form. Every historical row is retained without
+charging slots or enforcing liveness. This favors possible snapshot reuse.
+Before each paid job it checks whether the row is zero or proportional to any
+earlier row already in that job's output frame. Since the source event list is
+finite and every boundary is visited, the census is exhaustive in this grammar.
+
+| Orientation / entry contract | Paid jobs | Zero rows | Proportional single-snapshot matches |
+|---|---:|---:|---:|
+| Forward / original inputs | 3,298 | 0 | 0 |
+| Backward / exact eight relations | 3,298 | 8 | 8 |
+
+All eight nonzero matches are precisely the previously supplied centre
+witnesses, each unique with factor +1. All zero jobs are the supplied later
+climbs. Relations of the same span give the same quotient regardless of basis.
+Wrong relation sign, a dropped actual ADD and a wrong snapshot sign fail.
+
+This is a **formal-row** absence result, not a proof of physical operator
+inequality. Concrete native operators obey further identities; this traversal
+uses only the retained free reduction. It also excludes arbitrary linear
+combinations of saved rows, changed scalar programs, extra entry relations,
+new preparations, alternate frame equivalences and different caller boundaries.
+It does not itself verify the physical construction's phase or erasure ABI.
+
+## 4. Scope and reopening conditions
+
+The budget and census have different scopes. The exchange ceiling allows any
+basis or schedule that keeps exactly the declared complete invoice family; the
+census fixes the eight relations and unchanged source event history, regardless
+of whether a proposed edit has that invoice. Neither implies that all productive
+helper ideas are exhausted. Removing different ranks, surrendering different
+targets, using scalar combinations or native relations, changing contact heights,
+or changing the controller requires a fresh complete signed bill and proof.
+The selected construction and its five framework assumptions are unchanged.
+
+Prepared for zxv with substantial OpenAI Codex assistance. Apache-2.0.
diff --git a/research/owned-relation-exhaustion/README.md b/research/owned-relation-exhaustion/README.md
new file mode 100644
index 0000000000..8ce9f222a9
--- /dev/null
+++ b/research/owned-relation-exhaustion/README.md
@@ -0,0 +1,119 @@
+# Signed recursive-invoice preflight and a current exchange-family ceiling
+
+**Reusable arithmetic tool, scoped budget proof and independent formal-row audit;
+no new supplier or selected exponent.** Canonical base:
+`1ec1119972c9aefe42525172d07b332fcff81cb4`, selected conditional κ=.0030217758.
+
+The practical result is a cheap check before extending a construction:
+compare its complete **paid and target** histograms. The new shared CLI gives an
+exact certificate of improvement, equality, worsening or inconclusive concave
+order, covering every power r^(1-C), 0<C<1. It needs no root search, native word
+replay or compiler. See the [guide](../../docs/research/recursive-invoice-preflight.md).
+The mathematics is classical and credited; the new work is the portable signed
+interface, its independent tests and its current source-bound application.
+Concurrent [#5391](https://github.com/CrocSwap/integer-mult-bounds/pull/5391) supplies
+the later receipt-compatibility/funding oracle. This preflight does not duplicate
+its evidence join or claim to fund a candidate from declared histograms.
+
+## Why this is timely
+
+Recent contributions give a sharp contrast: #5350's
+largest scalar deletion worsens its recursive bill, #5383 improves it by owning
+relations while surrendering helper output credit, and #5387/#5390 tighten the
+compiler with that same bill. A scalar count, a paid-only histogram or another decimal
+root does not distinguish these cases.
+
+The newest centre-paired relation method also has a small, provable remaining
+budget. Optimistically grant **all 144** original exchanges
+`−H1−H8 paid / −H9 targets`; its complete raw crossing is below
+**.00302728824994**. The 128-exchange #5383/#5387 bill crosses in
+**(.00302667476807,.00302667476808)**. Thus this exact lever has less than
+**.00000061348187** raw saving left, and even its optimistic maximum misses .0034.
+The latest #5390 claimed endpoint is within 5.6×10^-15 of this fixed-profile raw
+upper bracket. All basis choices or schedules with this same invoice obey that budget.
+
+This gives an immediate stopping rule for a .0034-oriented continuation of this
+exchange family: change the paid ranks, target debits or caller before building
+another endpoint. It does not retire other ownership or relation methods.
+The [proof](PROOF.md) and [directed receipt](RESULT.json) retain that distinction.
+
+## Independent audit of the fixed eight relations
+
+An independent interpreter checks every one of the source's 3,298 paid events
+in each orientation. After the exact eight backward relations, it finds exactly
+the eight known zero climbs and eight known proportional same-frame centre
+snapshots. It finds no additional such opportunity. The unchanged forward
+contract has neither. All historical snapshots are favorably available; each
+actual nonzero match is unique with factor +1.
+
+This is exhaustion **in the retained formal free-word grammar**. New native
+identities, scalar combinations, relation spans, preparation programs and event
+graphs are outside the census. It does not establish physical non-equality.
+
+[Ali Romman's live claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/5125#discussioncomment-19154639)
+covers additional reuse after the eight relations and was discovered during the
+final overlap check. This census is an **independent audit**, not a priority claim
+for that construction/search. The quantitative centre-exchange cap is separate
+and applies to the proposed
+[forward extension](https://github.com/CrocSwap/integer-mult-bounds/discussions/5125#discussioncomment-19154367)
+only if its complete bill stays in the stated family.
+
+## Reproduce
+
+Python 3.12, standard library, from the repository root:
+
+```sh
+python3 -B tests/test_compare_recursive_invoices.py
+python3 -B scripts/research_package.py --package research/owned-relation-exhaustion
+python3 -B scripts/compare_recursive_invoices.py \
+  research/owned-relation-exhaustion/examples/source.json \
+  research/owned-relation-exhaustion/examples/exchange-128.json
+```
+
+The registered replay executes normal and optimized interpreters, reconstructs
+the full source invoice, checks exact root brackets and a rational interval
+nonfunding cover, reruns the independent formal-row census, rejects five budget/
+census defects and compares the complete deterministic receipt. The CLI tests
+include 625 exhaustive signed small profiles, direct discrete-slope tariffs,
+unequal masses, changed contracts and malformed-input controls.
+
+All scientific inputs are self-contained committed exports with exact hashes in
+[SOURCES.json](SOURCES.json), including #5383's unchanged witness JSON. The replay
+does not require that pending PR's Git objects or a network fetch. The shared CLI
+is an explicit replay input. Normal/O success corroborates the written proof;
+it is not independent human or proof-kernel acceptance.
+Executed environments and the shallow-history limitation are recorded in
+[VALIDATION.md](VALIDATION.md).
+
+Full physical/native phase replay of the source word, compiler and BIT supplier,
+physical tape/precision execution, analytic decomposition and the full selected
+cold replay are not rerun. No supplier
+admission, practical speedup or selected-record update is claimed.
+
+## Open next questions
+
+* Can a changed relation or scalar combination remove **noncentre** paid work
+  after all lost targets are debited? Follow the active claim, pick one actual
+  source event, and prove its complete signed delta first. The formal census
+  does not exclude native identities or combinations, and the exchange cap
+  does not cover a changed paid profile.
+* Can a current generator emit this schema from its reviewed exact word? Begin
+  with one opt-in exporter and independent invoice reconstruction. That would
+  reduce manual transcription; the current CLI hashes declarations and cannot
+  detect an author omitting a target debit.
+
+## Credit
+
+Cao Yifan/intqwq supplied #4610's final native word and invoice. Ali Romman
+supplied #5377/#5383's relations, witnesses, physical construction and first
+complete positive bill; Dugongue supplied #5387's reserve refinement. Chris R /
+Clear1nsight supplied #5241/#5384's independent admission and pricing evidence.
+@ahrazzle's #5350 retiming supplies the scalar/recursive counterexample.
+mightyOil's T16 moment-order audit and its credited predecessors supply prior
+use of classical concave order; no novelty of that theorem is claimed.
+Cinna Davis, zxv, Dugongue, Jacob Alan Sussman, ToikY, Rohan Arun, Jeremy Kothe
+and the retained native, controller, compiler, BIT and analytic lineage retain
+their source credit. Repository LICENSE and NOTICE apply to the retained exports.
+
+Prepared for zxv with substantial OpenAI Codex GPT-6.1 Sol assistance, including
+three explicitly authorized research subagents. New material is Apache-2.0.
diff --git a/research/owned-relation-exhaustion/RESULT.json b/research/owned-relation-exhaustion/RESULT.json
new file mode 100644
index 0000000000..d32d6184f5
--- /dev/null
+++ b/research/owned-relation-exhaustion/RESULT.json
@@ -0,0 +1,716 @@
+{
+  "budget": {
+    "all_144_margin_at_0034": [
+      "-91.89855945693865431851670931173980037757561294366366",
+      "-91.89855945693865431851670931173980037757561272657922"
+    ],
+    "cases": {
+      "0": {
+        "derivative_on_0_to_001": [
+          "-271812.88041939139297914708268222547352434328795265125192",
+          "-218499.04067359336310283459883079247351214779211988165316"
+        ],
+        "margin_at_lower": [
+          "0.00000000162731982417840201140494288027966767182656",
+          "0.00000000162731982417840201140494288027966789127918"
+        ],
+        "margin_at_upper": [
+          "-0.00000000084490360521630087930642241439567383052174",
+          "-0.00000000084490360521630087930642241439567361130012"
+        ],
+        "paid_calls": 64404,
+        "rank_surplus": 752,
+        "raw_root_bracket": [
+          ".00302177587077",
+          ".00302177587078"
+        ],
+        "target_fields": 10928
+      },
+      "128": {
+        "derivative_on_0_to_001": [
+          "-271367.20449395235154122526900145436851166657438009043528",
+          "-218152.19575138707954283643172343030446950716973683776836"
+        ],
+        "margin_at_lower": [
+          "0.00000000112178593736422327404878786475774574899218",
+          "0.00000000112178593736422327404878786475774596713922"
+        ],
+        "margin_at_upper": [
+          "-0.00000000134640025311558435308367428097774660789150",
+          "-0.00000000134640025311558435308367428097774638974816"
+        ],
+        "paid_calls": 64148,
+        "rank_surplus": 752,
+        "raw_root_bracket": [
+          ".00302667476807",
+          ".00302667476808"
+        ],
+        "target_fields": 10800
+      },
+      "144": {
+        "derivative_on_0_to_001": [
+          "-271311.49500327247136148504229135798038508198518352033320",
+          "-218108.84013611129409783666083501003333917709193895728276"
+        ],
+        "margin_at_lower": [
+          "0.00000000235880280085021607912595033471364694361860",
+          "0.00000000235880280085021607912595033471364716107464"
+        ],
+        "margin_at_upper": [
+          "-0.00000000010887873052410130755761586084483062626164",
+          "-0.00000000010887873052410130755761586084483040851688"
+        ],
+        "paid_calls": 64116,
+        "rank_surplus": 752,
+        "raw_root_bracket": [
+          ".00302728824993",
+          ".00302728824994"
+        ],
+        "target_fields": 10784
+      },
+      "16": {
+        "derivative_on_0_to_001": [
+          "-271757.17092871151279940685597212908539775869875608114984",
+          "-218455.68505831757765783482794237220238181771432200116756"
+        ],
+        "margin_at_lower": [
+          "0.00000000132334469180651264131051541739362082148936",
+          "0.00000000132334469180651264131051541739362104038272"
+        ],
+        "margin_at_upper": [
+          "-0.00000000114837408601719137920928840484126109551074",
+          "-0.00000000114837408601719137920928840484126087652704"
+        ],
+        "paid_calls": 64372,
+        "rank_surplus": 752,
+        "raw_root_bracket": [
+          ".00302238736347",
+          ".00302238736348"
+        ],
+        "target_fields": 10912
+      }
+    },
+    "controls": {
+      "invented_145th_centre": true,
+      "omitted_H9_debits_change_answer": true
+    },
+    "global_nonfunding_cover": {
+      "cells": 1024,
+      "interval": [
+        ".01",
+        "1"
+      ],
+      "largest_margin_upper": "-1134.90457549950975081012951005747003205218439116026304"
+    },
+    "remaining_raw_headroom_upper": "61348187/100000000000000",
+    "scope": "At most one -H1-H8 paid/-H9 target exchange per original centre; all other inventory fixed. Raw moments only.",
+    "source_invoice_sha256": "b4f720a7ccbb5b46e472eaf85dc8739a40598986f19b37d26c4f9ecfbd31f0de",
+    "status": "PASS_FIXED_CENTRE_EXCHANGE_BUDGET"
+  },
+  "census": {
+    "negative_controls": {
+      "omitted_source_ADD_11331": "Exactly the known zero paid rows",
+      "wrong_cached_global_sign": "Unique exact chronological same-frame snapshot, including sign",
+      "wrong_relation_sign": "Exactly the known zero paid rows"
+    },
+    "orientations": {
+      "backward": {
+        "entry_relation_applied": true,
+        "event_counts": {
+          "adapter": 448,
+          "center": 8,
+          "climb": 3290,
+          "phase": 10,
+          "scalar": 17453,
+          "warp": 240
+        },
+        "historical_snapshots_favorably_retained": true,
+        "nonzero_proportional_snapshot_matches": [
+          {
+            "earlier_snapshots": [
+              {
+                "event": 11205,
+                "factor": "1",
+                "role": [
+                  "L",
+                  92
+                ]
+              }
+            ],
+            "event": 11332,
+            "row_terms": 56,
+            "source_event": [
+              "center",
+              0,
+              0,
+              254,
+              8
+            ]
+          },
+          {
+            "earlier_snapshots": [
+              {
+                "event": 11218,
+                "factor": "1",
+                "role": [
+                  "L",
+                  57
+                ]
+              }
+            ],
+            "event": 11358,
+            "row_terms": 56,
+            "source_event": [
+              "center",
+              0,
+              0,
+              274,
+              8
+            ]
+          },
+          {
+            "earlier_snapshots": [
+              {
+                "event": 11231,
+                "factor": "1",
+                "role": [
+                  "L",
+                  81
+                ]
+              }
+            ],
+            "event": 11384,
+            "row_terms": 56,
+            "source_event": [
+              "center",
+              0,
+              0,
+              619,
+              8
+            ]
+          },
+          {
+            "earlier_snapshots": [
+              {
+                "event": 11244,
+                "factor": "1",
+                "role": [
+                  "L",
+                  224
+                ]
+              }
+            ],
+            "event": 11410,
+            "row_terms": 56,
+            "source_event": [
+              "center",
+              0,
+              0,
+              279,
+              8
+            ]
+          },
+          {
+            "earlier_snapshots": [
+              {
+                "event": 11257,
+                "factor": "1",
+                "role": [
+                  "L",
+                  11
+                ]
+              }
+            ],
+            "event": 11436,
+            "row_terms": 56,
+            "source_event": [
+              "center",
+              0,
+              0,
+              621,
+              8
+            ]
+          },
+          {
+            "earlier_snapshots": [
+              {
+                "event": 11270,
+                "factor": "1",
+                "role": [
+                  "L",
+                  240
+                ]
+              }
+            ],
+            "event": 11462,
+            "row_terms": 56,
+            "source_event": [
+              "center",
+              0,
+              0,
+              280,
+              8
+            ]
+          },
+          {
+            "earlier_snapshots": [
+              {
+                "event": 11283,
+                "factor": "1",
+                "role": [
+                  "L",
+                  9
+                ]
+              }
+            ],
+            "event": 11488,
+            "row_terms": 56,
+            "source_event": [
+              "center",
+              0,
+              0,
+              624,
+              8
+            ]
+          },
+          {
+            "earlier_snapshots": [
+              {
+                "event": 11306,
+                "factor": "1",
+                "role": [
+                  "L",
+                  288
+                ]
+              }
+            ],
+            "event": 11514,
+            "row_terms": 56,
+            "source_event": [
+              "center",
+              0,
+              0,
+              281,
+              8
+            ]
+          }
+        ],
+        "paid_profile": {
+          "1": 2201,
+          "2": 412,
+          "3": 510,
+          "4": 127,
+          "5": 40,
+          "8": 8
+        },
+        "paid_total": 3298,
+        "zero_paid_rows": [
+          {
+            "event": 15584,
+            "source_event": [
+              "climb",
+              "warp_retimed",
+              92,
+              254,
+              1,
+              1
+            ]
+          },
+          {
+            "event": 15870,
+            "source_event": [
+              "climb",
+              "warp_retimed",
+              9,
+              624,
+              1,
+              1
+            ]
+          },
+          {
+            "event": 15876,
+            "source_event": [
+              "climb",
+              "warp_retimed",
+              11,
+              621,
+              1,
+              1
+            ]
+          },
+          {
+            "event": 17245,
+            "source_event": [
+              "climb",
+              "warp_retimed",
+              240,
+              280,
+              1,
+              1
+            ]
+          },
+          {
+            "event": 17256,
+            "source_event": [
+              "climb",
+              "warp_retimed",
+              81,
+              619,
+              1,
+              1
+            ]
+          },
+          {
+            "event": 17264,
+            "source_event": [
+              "climb",
+              "warp_retimed",
+              57,
+              274,
+              1,
+              1
+            ]
+          },
+          {
+            "event": 17580,
+            "source_event": [
+              "climb",
+              "warp_retimed",
+              288,
+              281,
+              1,
+              1
+            ]
+          },
+          {
+            "event": 17722,
+            "source_event": [
+              "climb",
+              "warp_retimed",
+              224,
+              279,
+              1,
+              1
+            ]
+          }
+        ]
+      },
+      "forward": {
+        "entry_relation_applied": false,
+        "event_counts": {
+          "adapter": 448,
+          "center": 8,
+          "climb": 3290,
+          "phase": 10,
+          "scalar": 17453,
+          "warp": 240
+        },
+        "historical_snapshots_favorably_retained": true,
+        "nonzero_proportional_snapshot_matches": [],
+        "paid_profile": {
+          "1": 2201,
+          "2": 412,
+          "3": 510,
+          "4": 127,
+          "5": 40,
+          "8": 8
+        },
+        "paid_total": 3298,
+        "zero_paid_rows": []
+      }
+    },
+    "pivots": [
+      225,
+      227,
+      229,
+      230,
+      241,
+      245,
+      257,
+      288
+    ],
+    "scope": "Fixed eight entry relations, unchanged source event boundaries and free-word grammar. Unlimited historical scalar snapshots in the same physical output frame are allowed. No extra paid zero row or proportional single-snapshot equality remains beyond the eight known backward centre/zero pairs. New relations, scalar linear combinations, native identities beyond free reduction, changed event graph and physical admission are not covered.",
+    "source_compressed_sha256": "7679daa4ec6a40f56fc7f0202379a6f454310e94945aba4f48beaaaafb838d05",
+    "source_payload_sha256": "cfcd36b40326d7a5815e1db5313444379b3fb756b27d5201c561574c085cc0dd",
+    "status": "PASS_FIXED_EIGHT_RELATION_FREE_WORD_ZERO_SNAPSHOT_SATURATION",
+    "witness_sha256": "24599011da06325e1012619b2b84a4dd3aceb90ab42d3caaf6414879e7d16591"
+  },
+  "scope": "Exact histogram order and this fixed exchange family; independent fixed-eight formal-row census. No new supplier or selected-frontier change.",
+  "signed_comparisons": {
+    "0": {
+      "assurance": {
+        "contract": "Exact matching user declarations; mathematical applicability is not verified.",
+        "inconclusive": "Mixed hinge signs do not establish a power-moment crossing or failure.",
+        "input_binding": "SHA-256 identifies the exact input bytes read, when loaded by this CLI.",
+        "scope": "Histogram arithmetic only. Histogram correctness, circuit legality, ownership, phases, precision, supplier admission and compiler endpoints are not validated."
+      },
+      "classification": "equal",
+      "contract": {
+        "ambient_rank": 145,
+        "boundary": "complete p16/a7 unit before the retained mixed-target compiler",
+        "model": "mixed-target-unnormalized-power-v1",
+        "rank_semantics": "H_r has raw tariff r^(1-C), with surrendered productive fields debited"
+      },
+      "delta_paid": {},
+      "delta_targets": {},
+      "hinges": [
+        {
+          "rank": 0,
+          "value": 0
+        }
+      ],
+      "method_credit": "Classical concave order/majorization; mightyOil's t16-moment-order-audit and its retained predecessors. No new theorem claim.",
+      "power_moment": {
+        "exponent": "0 < t < 1",
+        "new_minus_old_sign": "identically_zero",
+        "tariff": "r^t"
+      },
+      "quantity": "(new paid - new targets) - (old paid - old targets)",
+      "rank_mass": {
+        "delta": {
+          "paid": 0,
+          "paid_minus_targets": 0,
+          "targets": 0
+        },
+        "new": {
+          "paid": 239134,
+          "paid_minus_targets": -752,
+          "targets": 239886
+        },
+        "old": {
+          "paid": 239134,
+          "paid_minus_targets": -752,
+          "targets": 239886
+        }
+      },
+      "schema": "integer-mult-recursive-invoice-comparison/v1",
+      "signed_net_delta": {},
+      "tariff_class": "increasing concave f on nonnegative ranks, with f(0)=0",
+      "uniform_order": "weak; some concave tariffs can give equality"
+    },
+    "128": {
+      "assurance": {
+        "contract": "Exact matching user declarations; mathematical applicability is not verified.",
+        "inconclusive": "Mixed hinge signs do not establish a power-moment crossing or failure.",
+        "input_binding": "SHA-256 identifies the exact input bytes read, when loaded by this CLI.",
+        "scope": "Histogram arithmetic only. Histogram correctness, circuit legality, ownership, phases, precision, supplier admission and compiler endpoints are not validated."
+      },
+      "classification": "uniformly_better",
+      "contract": {
+        "ambient_rank": 145,
+        "boundary": "complete p16/a7 unit before the retained mixed-target compiler",
+        "model": "mixed-target-unnormalized-power-v1",
+        "rank_semantics": "H_r has raw tariff r^(1-C), with surrendered productive fields debited"
+      },
+      "delta_paid": {
+        "1": -128,
+        "8": -128
+      },
+      "delta_targets": {
+        "9": -128
+      },
+      "hinges": [
+        {
+          "rank": 0,
+          "value": 0
+        },
+        {
+          "rank": 1,
+          "value": -128
+        },
+        {
+          "rank": 8,
+          "value": -128
+        },
+        {
+          "rank": 9,
+          "value": 0
+        }
+      ],
+      "method_credit": "Classical concave order/majorization; mightyOil's t16-moment-order-audit and its retained predecessors. No new theorem claim.",
+      "power_moment": {
+        "exponent": "0 < t < 1",
+        "new_minus_old_sign": "strictly_negative",
+        "tariff": "r^t"
+      },
+      "quantity": "(new paid - new targets) - (old paid - old targets)",
+      "rank_mass": {
+        "delta": {
+          "paid": -1152,
+          "paid_minus_targets": 0,
+          "targets": -1152
+        },
+        "new": {
+          "paid": 237982,
+          "paid_minus_targets": -752,
+          "targets": 238734
+        },
+        "old": {
+          "paid": 239134,
+          "paid_minus_targets": -752,
+          "targets": 239886
+        }
+      },
+      "schema": "integer-mult-recursive-invoice-comparison/v1",
+      "signed_net_delta": {
+        "1": -128,
+        "8": -128,
+        "9": 128
+      },
+      "tariff_class": "increasing concave f on nonnegative ranks, with f(0)=0",
+      "uniform_order": "weak; some concave tariffs can give equality"
+    },
+    "144": {
+      "assurance": {
+        "contract": "Exact matching user declarations; mathematical applicability is not verified.",
+        "inconclusive": "Mixed hinge signs do not establish a power-moment crossing or failure.",
+        "input_binding": "SHA-256 identifies the exact input bytes read, when loaded by this CLI.",
+        "scope": "Histogram arithmetic only. Histogram correctness, circuit legality, ownership, phases, precision, supplier admission and compiler endpoints are not validated."
+      },
+      "classification": "uniformly_better",
+      "contract": {
+        "ambient_rank": 145,
+        "boundary": "complete p16/a7 unit before the retained mixed-target compiler",
+        "model": "mixed-target-unnormalized-power-v1",
+        "rank_semantics": "H_r has raw tariff r^(1-C), with surrendered productive fields debited"
+      },
+      "delta_paid": {
+        "1": -144,
+        "8": -144
+      },
+      "delta_targets": {
+        "9": -144
+      },
+      "hinges": [
+        {
+          "rank": 0,
+          "value": 0
+        },
+        {
+          "rank": 1,
+          "value": -144
+        },
+        {
+          "rank": 8,
+          "value": -144
+        },
+        {
+          "rank": 9,
+          "value": 0
+        }
+      ],
+      "method_credit": "Classical concave order/majorization; mightyOil's t16-moment-order-audit and its retained predecessors. No new theorem claim.",
+      "power_moment": {
+        "exponent": "0 < t < 1",
+        "new_minus_old_sign": "strictly_negative",
+        "tariff": "r^t"
+      },
+      "quantity": "(new paid - new targets) - (old paid - old targets)",
+      "rank_mass": {
+        "delta": {
+          "paid": -1296,
+          "paid_minus_targets": 0,
+          "targets": -1296
+        },
+        "new": {
+          "paid": 237838,
+          "paid_minus_targets": -752,
+          "targets": 238590
+        },
+        "old": {
+          "paid": 239134,
+          "paid_minus_targets": -752,
+          "targets": 239886
+        }
+      },
+      "schema": "integer-mult-recursive-invoice-comparison/v1",
+      "signed_net_delta": {
+        "1": -144,
+        "8": -144,
+        "9": 144
+      },
+      "tariff_class": "increasing concave f on nonnegative ranks, with f(0)=0",
+      "uniform_order": "weak; some concave tariffs can give equality"
+    },
+    "16": {
+      "assurance": {
+        "contract": "Exact matching user declarations; mathematical applicability is not verified.",
+        "inconclusive": "Mixed hinge signs do not establish a power-moment crossing or failure.",
+        "input_binding": "SHA-256 identifies the exact input bytes read, when loaded by this CLI.",
+        "scope": "Histogram arithmetic only. Histogram correctness, circuit legality, ownership, phases, precision, supplier admission and compiler endpoints are not validated."
+      },
+      "classification": "uniformly_better",
+      "contract": {
+        "ambient_rank": 145,
+        "boundary": "complete p16/a7 unit before the retained mixed-target compiler",
+        "model": "mixed-target-unnormalized-power-v1",
+        "rank_semantics": "H_r has raw tariff r^(1-C), with surrendered productive fields debited"
+      },
+      "delta_paid": {
+        "1": -16,
+        "8": -16
+      },
+      "delta_targets": {
+        "9": -16
+      },
+      "hinges": [
+        {
+          "rank": 0,
+          "value": 0
+        },
+        {
+          "rank": 1,
+          "value": -16
+        },
+        {
+          "rank": 8,
+          "value": -16
+        },
+        {
+          "rank": 9,
+          "value": 0
+        }
+      ],
+      "method_credit": "Classical concave order/majorization; mightyOil's t16-moment-order-audit and its retained predecessors. No new theorem claim.",
+      "power_moment": {
+        "exponent": "0 < t < 1",
+        "new_minus_old_sign": "strictly_negative",
+        "tariff": "r^t"
+      },
+      "quantity": "(new paid - new targets) - (old paid - old targets)",
+      "rank_mass": {
+        "delta": {
+          "paid": -144,
+          "paid_minus_targets": 0,
+          "targets": -144
+        },
+        "new": {
+          "paid": 238990,
+          "paid_minus_targets": -752,
+          "targets": 239742
+        },
+        "old": {
+          "paid": 239134,
+          "paid_minus_targets": -752,
+          "targets": 239886
+        }
+      },
+      "schema": "integer-mult-recursive-invoice-comparison/v1",
+      "signed_net_delta": {
+        "1": -16,
+        "8": -16,
+        "9": 16
+      },
+      "tariff_class": "increasing concave f on nonnegative ranks, with f(0)=0",
+      "uniform_order": "weak; some concave tariffs can give equality"
+    }
+  },
+  "source_reconstruction": {
+    "backward_invocations": 16,
+    "forward_invocations": 2,
+    "local_jobs_each_orientation": 3298,
+    "original_centre_slots": 144,
+    "rank_surplus": 752
+  },
+  "status": "PASS_SIGNED_INVOICE_PREFLIGHT_AND_SCOPED_CENTRE_EXHAUSTION"
+}
diff --git a/research/owned-relation-exhaustion/SOURCES.json b/research/owned-relation-exhaustion/SOURCES.json
new file mode 100644
index 0000000000..410bd32ccc
--- /dev/null
+++ b/research/owned-relation-exhaustion/SOURCES.json
@@ -0,0 +1,71 @@
+{
+  "schema_version": 1,
+  "base_commit": "1ec1119972c9aefe42525172d07b332fcff81cb4",
+  "files": [
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b288ae44032fd84dd5564243422836cbcd29835e",
+      "sha256": "f9ed52bb642cd1e1486cf2743557ac5fa23cf1514a3277d786a644e39a565038"
+    },
+    {
+      "path": "research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz",
+      "usage": "provenance",
+      "git_blob": "46d929df83817b85a5387bcdb9745e0627d6366e",
+      "sha256": "7679daa4ec6a40f56fc7f0202379a6f454310e94945aba4f48beaaaafb838d05"
+    },
+    {
+      "path": "research/kappa-round6-compiler-tightening-20261022/ARITHMETIC.json",
+      "usage": "provenance",
+      "git_blob": "0fdb65f946d557c13d8e2781b9151e2a8ddee5c3",
+      "sha256": "b4f720a7ccbb5b46e472eaf85dc8739a40598986f19b37d26c4f9ecfbd31f0de"
+    },
+    {
+      "path": "research/joint-helper-lattice/CONTROLLER.json",
+      "usage": "provenance",
+      "git_blob": "49a2f0675819f9deeea27ed77be2c2b6ff2e87e5",
+      "sha256": "1f5d6aa267e4e66048818714b5d3b5eae17a58abe91d15deeaa34574f7391a5f"
+    }
+  ],
+  "retained_exports": [
+    {
+      "local_path": "source/ACTUAL-WORD.json.gz",
+      "source_pr": 4610,
+      "originating_commit": "20bb93cd990d2ff683e4d0fc56b87a42cd5d3296",
+      "original_path": "research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz",
+      "sha256": "7679daa4ec6a40f56fc7f0202379a6f454310e94945aba4f48beaaaafb838d05",
+      "source_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/4610"
+    },
+    {
+      "local_path": "source/INVOICE.json",
+      "source_pr": 4610,
+      "originating_commit": "20bb93cd990d2ff683e4d0fc56b87a42cd5d3296",
+      "original_path": "research/kappa-round6-compiler-tightening-20261022/ARITHMETIC.json",
+      "sha256": "b4f720a7ccbb5b46e472eaf85dc8739a40598986f19b37d26c4f9ecfbd31f0de",
+      "source_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/4610"
+    },
+    {
+      "local_path": "source/CONTROLLER.json",
+      "source_pr": 4470,
+      "originating_commit": "1ec1119972c9aefe42525172d07b332fcff81cb4",
+      "original_path": "research/joint-helper-lattice/CONTROLLER.json",
+      "sha256": "1f5d6aa267e4e66048818714b5d3b5eae17a58abe91d15deeaa34574f7391a5f",
+      "source_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/4470"
+    },
+    {
+      "local_path": "WITNESS.json",
+      "source_pr": 5383,
+      "originating_commit": "4bd6efa51a495a7c7b7754915cf0244e7e60969e",
+      "original_path": "research/kappa-joint-relations-20261023/WITNESS.json",
+      "sha256": "24599011da06325e1012619b2b84a4dd3aceb90ab42d3caaf6414879e7d16591",
+      "source_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/5383"
+    }
+  ],
+  "export_note": "All scientific exports are committed locally. No pending foreign Git object or network is needed for replay. The PR5383 export is pinned by its local SHA256 and by census.py; its original source attribution is retained."
+}
diff --git a/research/owned-relation-exhaustion/TASK.json b/research/owned-relation-exhaustion/TASK.json
new file mode 100644
index 0000000000..c3d6c40d63
--- /dev/null
+++ b/research/owned-relation-exhaustion/TASK.json
@@ -0,0 +1,106 @@
+{
+  "schema_version": 1,
+  "task_id": "independent-moment",
+  "priority_revision": 7,
+  "base_commit": "1ec1119972c9aefe42525172d07b332fcff81cb4",
+  "original_card": {
+    "id": "independent-moment",
+    "question": "Does a new paid-profile or mixture-ceiling claim have independent source-bound coverage of its actual changed ledger?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/staged544-mixed-fourth-entry/README.md",
+      "research/staged-fourth-mixture-ceiling/README.md",
+      "research/r437-two-window-carrier/README.md"
+    ],
+    "methods": [
+      "Independently reconstruct one changed paid ledger or pooled dual with exact source and model binding.",
+      "Check a genuinely uncovered moment implication with a different enclosure method and semantic controls."
+    ],
+    "acceptance_gate": "Demonstrate independence, source binding, positive cases and semantic failure controls. Name the existing checks and the missing coverage. More digits or another equivalent evaluation of an unchanged root is not by itself new verification coverage.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Choose one uncovered implication in #4088 literal mixed fill, #4091 pooled dual/model binding or #4067 changed frame/host bill. #4091 pins an earlier #4088 head: validate word, capacity and stock equivalence before applying that ceiling to the later fill.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Independently reconstruct one changed ledger or pooled certificate with a positive case and a semantic failure control. Compare exact sources and name what existing checks miss. Keep this separate from supplier-boundary-audit (ordinary/outer dispatch) and uniform-transfer-audit (all-size proof). Stop after the chosen implication; more digits of an unchanged root are not additional mechanism coverage.",
+    "bottleneck_link": "Independent pricing prevents unsupported frontier claims and can reveal genuine accounting slack.",
+    "allowed_variations": [
+      "Reconstruct either a source ledger or a numeric enclosure with a different method."
+    ],
+    "scope_change_triggers": [
+      "Importing the primary function and calling its output independent verification.",
+      "Calling an aggregate histogram or source archive an independently accepted physical supplier."
+    ],
+    "overlap_terms": [
+      "independent moment",
+      "rank histogram",
+      "exact enclosure",
+      "2979",
+      "rank7 joint endframes",
+      "paired extraction",
+      "joint lowering",
+      "4088",
+      "4091",
+      "4067",
+      "staged544-mixed-fourth-entry",
+      "staged-fourth-mixture-ceiling",
+      "r437-two-window-carrier"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4088: mixed local/owner/phase ledger and moment reported.",
+        "evidence": "research/staged544-mixed-fourth-entry/README.md",
+        "remaining": "Independent binding of changed construction to paid histogram, distinct from full admission.",
+        "status": "author-checked",
+        "source_commit": "72d7efc181edb21806a878f174847c96dd410ab0",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4091: sharp pooled ceiling proposed for fixed THREE/FOUR words and inventory.",
+        "evidence": "research/staged-fourth-mixture-ceiling/README.md",
+        "remaining": "Validate dual, exhaustive pattern scope and exact source equivalence before applying to newer fills.",
+        "status": "author-checked",
+        "source_commit": "a29bdd8f56cd18b805917775cdec09ff935eddc4",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4067: paid (2,2) to (1,3) exchange and fixed OQ4 host reported.",
+        "evidence": "research/r437-two-window-carrier/README.md",
+        "remaining": "Independent full cost delta; remaining caller/precision/stock obligations are separate.",
+        "status": "author-checked",
+        "source_commit": "a95f11c52e3ca1e4c5171b3c20d8f4b0815aee2e",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "scope_change_reason": "Explicit donor assignment to identify a significant reusable finding and prepare a focused contribution; replace historical scalar-root orientation with a shared signed-invoice screen and current centre-exchange budget audit.",
+  "actual_scope": "Exact signed paid-and-target concave-order CLI; source-derived all144 centre-exchange ceiling; independent formal-row census for the fixed eight PR5383 relations.",
+  "bottleneck_link": "Reject uniformly losing recursive edits before construction and compiler work; quantify the remaining headroom of the newest active helper-exchange method.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after exact histogram/cap/census checks and a focused draft; do not build a new supplier or change research allocations/frontier.",
+  "outcome": "completed",
+  "evidence": "RESULT.json and PROOF.md; shared CLI tests; exact-source independent census.",
+  "author": "Prepared for zxv with OpenAI Codex assistance",
+  "ai_assistance": "Substantial OpenAI Codex GPT-6.1 Sol assistance on main and three explicitly authorized research subagents.",
+  "open_questions": [
+    {
+      "question": "Can a changed ownership relation or scalar combination remove noncentre paid work after charging every lost target?",
+      "why_it_matters": "The exact H1+H8-H9 family cannot reach .0034; changed paid ranks escape this ceiling.",
+      "first_step": "Follow Ali Romman's active Discussion5125 claim, name one source paid event and prove its complete signed invoice delta.",
+      "blocked_by": "Existing formal zero/single-snapshot matches are exhausted; new combinations or native identities require a separate source-bound proof."
+    },
+    {
+      "question": "Can the signed-invoice schema be emitted directly by a current construction generator?",
+      "why_it_matters": "The CLI authenticates input bytes but cannot detect an author omitting a surrendered target.",
+      "first_step": "Add an opt-in exporter to one changed supplier after its exact-head review, preserving target and paid provenance.",
+      "blocked_by": "Requires author cooperation or a separately reviewed source adapter; no automatic supplier admission is claimed."
+    }
+  ]
+}
diff --git a/research/owned-relation-exhaustion/VALIDATION.md b/research/owned-relation-exhaustion/VALIDATION.md
new file mode 100644
index 0000000000..1a0d2323c9
--- /dev/null
+++ b/research/owned-relation-exhaustion/VALIDATION.md
@@ -0,0 +1,31 @@
+# Executed checks and limits
+
+Executed on macOS arm64 with standard-library CPython 3.12.8
+(`/opt/local/bin/python3.12`) and CPython 3.14.5 (`python3`). No third-party
+dependencies, Lean, Docker or hosted workflows were launched.
+
+* Ten shared-CLI tests pass in normal and optimized modes on both versions.
+  The tests include 625 exact signed histogram comparisons against a separate
+  discrete-slope tariff enumeration and the changed-contract/input controls.
+* Direct source-bound package replay passes normal and optimized modes on both
+  versions, with identical committed RESULT.json. The registered Python 3.12
+  runner passes both commands and all metadata/source pins.
+* Two additional Codex agents critically reviewed the signed-order proof,
+  interval formulas/global cover, source reconstruction and scope. They found
+  no blocking defect. This is AI review, not independent human/kernel acceptance.
+* Committed-byte checks pass, including the unchanged CRLF invoice export.
+  The official 21-file export at `ff9137e2a103654d9fa8f3f950c7a0f8afa4a79b`
+  passes transport integrity and optimized scientific replay with an empty PATH
+  in a read-only extracted directory. Later edits only add/clarify documentation;
+  the final publication export is checked separately.
+* Direct ancestry from canonical base
+  `1ec1119972c9aefe42525172d07b332fcff81cb4` to the contribution passes.
+  The optional `--check-source-history` preflight was attempted but cannot run
+  because this checkout is shallow. Full-history verification is not claimed.
+  All consumed scientific exports, including the pending witness, are committed
+  package-local; no foreign Git object is needed for scientific replay.
+
+The exact reusable commands are in VERIFY.json and README.md. The full selected
+cold replay, physical/native phase lowering, complete compiler execution, tape/
+precision implementation, BIT supplier and inherited analytic foundations are
+outside this contribution. No source profile or selected record is changed.
diff --git a/research/owned-relation-exhaustion/VERIFY.json b/research/owned-relation-exhaustion/VERIFY.json
new file mode 100644
index 0000000000..d97ba2ec7c
--- /dev/null
+++ b/research/owned-relation-exhaustion/VERIFY.json
@@ -0,0 +1,32 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "scripts/compare_recursive_invoices.py"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Independent complete source invoice reconstruction, directed rational global exchange cap, signed fixtures, formal quotient census and five failure controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "The same checks with assertions disabled; exact deterministic receipt comparison."
+    }
+  ]
+}
diff --git a/research/owned-relation-exhaustion/WITNESS.json b/research/owned-relation-exhaustion/WITNESS.json
new file mode 100644
index 0000000000..19665f9eac
--- /dev/null
+++ b/research/owned-relation-exhaustion/WITNESS.json
@@ -0,0 +1,434 @@
+[
+  {
+    "relation": [
+      [
+        230,
+        "1"
+      ],
+      [
+        324,
+        "-1"
+      ],
+      [
+        326,
+        "1"
+      ],
+      [
+        328,
+        "1"
+      ],
+      [
+        333,
+        "-1"
+      ]
+    ],
+    "event": 11332,
+    "frame": 254,
+    "cached_event": 11205,
+    "cached_role": 92,
+    "zero_event": 15584,
+    "zero_role": 92,
+    "copy_factor": "1",
+    "difference_factor": "1/4",
+    "zero_factor": "-1/4"
+  },
+  {
+    "relation": [
+      [
+        225,
+        "1"
+      ],
+      [
+        384,
+        "-1"
+      ],
+      [
+        388,
+        "-1"
+      ],
+      [
+        390,
+        "1"
+      ],
+      [
+        391,
+        "1"
+      ],
+      [
+        392,
+        "1"
+      ],
+      [
+        397,
+        "-1"
+      ]
+    ],
+    "event": 11358,
+    "frame": 274,
+    "cached_event": 11218,
+    "cached_role": 57,
+    "zero_event": 17264,
+    "zero_role": 57,
+    "copy_factor": "1",
+    "difference_factor": "1/4",
+    "zero_factor": "-1/4"
+  },
+  {
+    "relation": [
+      [
+        225,
+        "1"
+      ],
+      [
+        227,
+        "-1"
+      ],
+      [
+        229,
+        "1"
+      ],
+      [
+        231,
+        "-2"
+      ],
+      [
+        232,
+        "1"
+      ],
+      [
+        234,
+        "-1"
+      ],
+      [
+        235,
+        "-1"
+      ],
+      [
+        236,
+        "-1"
+      ],
+      [
+        237,
+        "-1"
+      ],
+      [
+        238,
+        "2"
+      ],
+      [
+        239,
+        "1"
+      ],
+      [
+        384,
+        "-1"
+      ],
+      [
+        388,
+        "-1"
+      ],
+      [
+        390,
+        "1"
+      ],
+      [
+        391,
+        "1"
+      ],
+      [
+        392,
+        "1"
+      ],
+      [
+        397,
+        "-1"
+      ]
+    ],
+    "event": 11384,
+    "frame": 619,
+    "cached_event": 11231,
+    "cached_role": 81,
+    "zero_event": 17256,
+    "zero_role": 81,
+    "copy_factor": "1",
+    "difference_factor": "1/4",
+    "zero_factor": "-1/4"
+  },
+  {
+    "relation": [
+      [
+        227,
+        "1"
+      ],
+      [
+        229,
+        "1"
+      ],
+      [
+        232,
+        "-1"
+      ],
+      [
+        234,
+        "1"
+      ],
+      [
+        235,
+        "1"
+      ],
+      [
+        236,
+        "-1"
+      ],
+      [
+        237,
+        "1"
+      ],
+      [
+        239,
+        "-1"
+      ],
+      [
+        324,
+        "-1"
+      ],
+      [
+        326,
+        "1"
+      ],
+      [
+        328,
+        "1"
+      ],
+      [
+        333,
+        "-1"
+      ]
+    ],
+    "event": 11410,
+    "frame": 279,
+    "cached_event": 11244,
+    "cached_role": 224,
+    "zero_event": 17722,
+    "zero_role": 224,
+    "copy_factor": "1",
+    "difference_factor": "-1/4",
+    "zero_factor": "1/4"
+  },
+  {
+    "relation": [
+      [
+        241,
+        "1"
+      ],
+      [
+        243,
+        "-1"
+      ],
+      [
+        245,
+        "1"
+      ],
+      [
+        247,
+        "-1"
+      ],
+      [
+        252,
+        "-1"
+      ],
+      [
+        253,
+        "-1"
+      ],
+      [
+        254,
+        "1"
+      ],
+      [
+        255,
+        "1"
+      ]
+    ],
+    "event": 11436,
+    "frame": 621,
+    "cached_event": 11257,
+    "cached_role": 11,
+    "zero_event": 15876,
+    "zero_role": 11,
+    "copy_factor": "1",
+    "difference_factor": "1/4",
+    "zero_factor": "-1/4"
+  },
+  {
+    "relation": [
+      [
+        225,
+        "1"
+      ],
+      [
+        241,
+        "-1"
+      ],
+      [
+        243,
+        "1"
+      ],
+      [
+        245,
+        "1"
+      ],
+      [
+        247,
+        "-1"
+      ],
+      [
+        252,
+        "-1"
+      ],
+      [
+        253,
+        "1"
+      ],
+      [
+        254,
+        "1"
+      ],
+      [
+        255,
+        "-1"
+      ],
+      [
+        324,
+        "-1"
+      ],
+      [
+        326,
+        "1"
+      ],
+      [
+        328,
+        "1"
+      ],
+      [
+        333,
+        "-1"
+      ],
+      [
+        384,
+        "-1"
+      ],
+      [
+        388,
+        "-1"
+      ],
+      [
+        390,
+        "1"
+      ],
+      [
+        391,
+        "1"
+      ],
+      [
+        392,
+        "1"
+      ],
+      [
+        397,
+        "-1"
+      ]
+    ],
+    "event": 11462,
+    "frame": 280,
+    "cached_event": 11270,
+    "cached_role": 240,
+    "zero_event": 17245,
+    "zero_role": 240,
+    "copy_factor": "1",
+    "difference_factor": "-1/4",
+    "zero_factor": "1/4"
+  },
+  {
+    "relation": [
+      [
+        257,
+        "1"
+      ],
+      [
+        259,
+        "-1"
+      ],
+      [
+        261,
+        "1"
+      ],
+      [
+        263,
+        "-1"
+      ],
+      [
+        268,
+        "-1"
+      ],
+      [
+        269,
+        "-1"
+      ],
+      [
+        270,
+        "1"
+      ],
+      [
+        271,
+        "1"
+      ]
+    ],
+    "event": 11488,
+    "frame": 624,
+    "cached_event": 11283,
+    "cached_role": 9,
+    "zero_event": 15870,
+    "zero_role": 9,
+    "copy_factor": "1",
+    "difference_factor": "1/4",
+    "zero_factor": "-1/4"
+  },
+  {
+    "relation": [
+      [
+        288,
+        "1"
+      ],
+      [
+        294,
+        "-1"
+      ],
+      [
+        296,
+        "-1"
+      ],
+      [
+        297,
+        "-1"
+      ],
+      [
+        300,
+        "1"
+      ],
+      [
+        303,
+        "1"
+      ]
+    ],
+    "event": 11514,
+    "frame": 281,
+    "cached_event": 11306,
+    "cached_role": 288,
+    "zero_event": 17580,
+    "zero_role": 288,
+    "copy_factor": "1",
+    "difference_factor": "1/4",
+    "zero_factor": "-1/4"
+  }
+]
diff --git a/research/owned-relation-exhaustion/budget.py b/research/owned-relation-exhaustion/budget.py
new file mode 100644
index 0000000000..e5aaf0dddd
--- /dev/null
+++ b/research/owned-relation-exhaustion/budget.py
@@ -0,0 +1,205 @@
+#!/usr/bin/env python3
+"""Directed rational arithmetic for the fixed centre-exchange invoice family.
+
+Independent implementation; classical atanh/Taylor interval formulas. Construction
+credit: Cao Yifan #4610, Ali Romman #5377/#5383, and retained native/compiler lineage.
+Prepared with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+from fractions import Fraction as F
+from functools import lru_cache
+from hashlib import sha256
+from pathlib import Path
+import json
+
+HERE = Path(__file__).resolve().parent
+S = 10**50
+SOURCE_SHA = 'b4f720a7ccbb5b46e472eaf85dc8739a40598986f19b37d26c4f9ecfbd31f0de'
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def up(a, b):
+    return -((-a) // b)
+
+
+def point(x):
+    x = F(x)
+    return x.numerator*S//x.denominator, up(x.numerator*S, x.denominator)
+
+
+def add(a, b):
+    return a[0]+b[0], a[1]+b[1]
+
+
+def scale(a, x):
+    x = F(x)
+    values = [F(a[0])*x, F(a[1])*x]
+    return min(values).__floor__(), max(values).__ceil__()
+
+
+def mul(a, b):
+    values = [x*y for x in a for y in b]
+    return min(values)//S, up(max(values), S)
+
+
+def reciprocal(a):
+    need(a[0] > 0, 'Positive reciprocal denominator')
+    return S*S//a[1], up(S*S, a[0])
+
+
+@lru_cache(None)
+def log_reduced(q):
+    q = F(q)
+    need(1 <= q <= 2, 'Log reduction range')
+    z = (q-1)/(q+1)
+    term, total = z, F(0)
+    for j in range(64):
+        total += 2*term/(2*j+1)
+        term *= z*z
+    # All omitted terms are nonnegative, denominators at least 129.
+    tail = 2*term/(129*(1-z*z))
+    return point(total)[0], point(total+tail)[1]
+
+
+@lru_cache(None)
+def log_rank(r):
+    need(type(r) is int and r > 0, 'Positive integer rank')
+    e = r.bit_length()-1
+    return add(scale(log_reduced(F(2)), e), log_reduced(F(r, 2**e)))
+
+
+def exp_positive(a):
+    need(0 <= a[0] <= a[1], 'Nonnegative exponential argument')
+    if a[1] >= S:
+        half = scale(a, F(1, 2))
+        value = exp_positive(half)
+        return mul(value, value)
+    result, term = point(1), point(1)
+    for j in range(1, 49):
+        term = scale(mul(term, a), F(1, j))
+        result = add(result, term)
+    first = scale(mul(term, a), F(1, 49))
+    ratio = scale(a, F(1, 50))
+    tail = mul(first, reciprocal((S-ratio[1], S-ratio[0])))
+    return result[0], result[1]+tail[1]
+
+
+@lru_cache(None)
+def power(r, c):
+    c = F(c)
+    need(0 <= c <= 1, 'Saving in [0,1]')
+    return scale(reciprocal(exp_positive(scale(log_rank(r), c))), r)
+
+
+def margin(targets, paid, c):
+    result = point(0)
+    for histogram, sign in [(targets, 1), (paid, -1)]:
+        for r, n in histogram.items():
+            result = add(result, scale(power(r, F(c)), sign*n))
+    return result
+
+
+def derivative(targets, paid, lo, hi):
+    # d/dc (T-P) = sum(P-T) log(r) r**(1-c).
+    result = point(0)
+    for histogram, sign in [(paid, 1), (targets, -1)]:
+        for r, n in histogram.items():
+            value = (power(r, F(hi))[0], power(r, F(lo))[1])
+            result = add(result, scale(mul(log_rank(r), value), sign*n))
+    return result
+
+
+def decimal(a):
+    def convert(x):
+        prefix = '-' if x < 0 else ''
+        x = abs(x)
+        return prefix+str(x//S)+'.'+str(x%S).zfill(50)
+    return [convert(x) for x in a]
+
+
+def exchange(targets, paid, count):
+    need(type(count) is int and 0 <= count <= 144, 'At most 144 original centre slots')
+    t, p = dict(targets), dict(paid)
+    p[1] -= count
+    p[8] -= count
+    t[9] -= count
+    need(min([*t.values(), *p.values()]) >= 0, 'No overspent paid or target inventory')
+    return t, p
+
+
+def run():
+    raw = (HERE/'source/INVOICE.json').read_bytes()
+    need(sha256(raw).hexdigest() == SOURCE_SHA, 'Exact original final-4610 invoice bytes')
+    source = json.loads(raw)
+    targets = {int(r): n for r, n in source['targets'].items()}
+    paid = {int(r): n for r, n in source['paid'].items()}
+    need(paid[8] == 144, 'All original centre jobs counted')
+    brackets = {0: ('.00302177587077', '.00302177587078'),
+                16: ('.00302238736347', '.00302238736348'),
+                128: ('.00302667476807', '.00302667476808'),
+                144: ('.00302728824993', '.00302728824994')}
+    cases = {}
+    for count, (lo, hi) in brackets.items():
+        t, p = exchange(targets, paid, count)
+        lower, upper = margin(t, p, F(lo)), margin(t, p, F(hi))
+        need(lower[0] > 0 > upper[1], 'Strict root bracket '+str(count))
+        # Monotonicity near the crossing; global nonfunding below handles the
+        # later range without incorrectly requiring a globally negative derivative.
+        slope = derivative(t, p, F(0), F(1, 100))
+        need(slope[1] < 0, 'Strictly decreasing margin on [0,.01] '+str(count))
+        cases[str(count)] = {
+            'raw_root_bracket': [lo, hi],
+            'margin_at_lower': decimal(lower), 'margin_at_upper': decimal(upper),
+            'derivative_on_0_to_001': decimal(slope),
+            'paid_calls': sum(p.values()), 'target_fields': sum(t.values()),
+            'rank_surplus': sum(r*n for r,n in t.items())-sum(r*n for r,n in p.items())}
+    t, p = exchange(targets, paid, 144)
+    bit_boundary = margin(t, p, F('.0034'))
+    need(bit_boundary[1] < 0, 'Even all centre exchanges miss .0034')
+    # A directed interval cover proves that no second positive window occurs.
+    # For c in [lo,hi], every r**(1-c) lies between its endpoint values.
+    # All smaller exchange counts have <= this best-case margin by concavity.
+    cells = []
+    for j in range(1024):
+        lo = F(1,100)+F(99*j,102400)
+        hi = F(1,100)+F(99*(j+1),102400)
+        bound = point(0)
+        for histogram, sign, endpoint in [(t, 1, lo), (p, -1, hi)]:
+            for r,n in histogram.items():
+                bound = add(bound, scale(power(r, endpoint), sign*n))
+        need(bound[1] < 0, 'Global nonfunding interval '+str(j))
+        cells.append(bound[1])
+    # Independent exact calibrations of logarithm/exponential arithmetic.
+    for r in (1, 2, 4, 8, 9, 16, 64, 145):
+        for c, value in [(F(0), r), (F(1), 1)]:
+            a = power(r, c)
+            need(a[0] <= value*S <= a[1], 'Exact integral-power calibration')
+    for r, root in [(4,2), (9,3), (16,4), (64,8)]:
+        a = power(r, F(1,2))
+        need(a[0] <= root*S <= a[1], 'Exact square-root calibration')
+    rejected = {}
+    try:
+        exchange(targets, paid, 145)
+    except ValueError:
+        rejected['invented_145th_centre'] = True
+    bad_t, bad_p = exchange(targets, paid, 128)
+    bad_t[9] += 128
+    need(margin(bad_t, bad_p, F('.0034'))[0] > 0,
+         'Omitted target debits falsely fund the bit boundary')
+    rejected['omitted_H9_debits_change_answer'] = True
+    need(len(rejected) == 2, 'Both budget controls executed')
+    return {'status': 'PASS_FIXED_CENTRE_EXCHANGE_BUDGET',
+            'source_invoice_sha256': SOURCE_SHA, 'cases': cases,
+            'all_144_margin_at_0034': decimal(bit_boundary),
+            'global_nonfunding_cover': {'interval': ['.01','1'], 'cells': 1024,
+                'largest_margin_upper': decimal((0,max(cells)))[1]},
+            'remaining_raw_headroom_upper': str(F(brackets[144][1])-F(brackets[128][0])),
+            'controls': rejected,
+            'scope': 'At most one -H1-H8 paid/-H9 target exchange per original centre; all other inventory fixed. Raw moments only.'}
+
+
+if __name__ == '__main__':
+    print(json.dumps(run(), indent=2, sort_keys=True))
diff --git a/research/owned-relation-exhaustion/census.py b/research/owned-relation-exhaustion/census.py
new file mode 100644
index 0000000000..861536a2c0
--- /dev/null
+++ b/research/owned-relation-exhaustion/census.py
@@ -0,0 +1,217 @@
+#!/usr/bin/env python3
+"""Fixed-relation zero and literal/proportional snapshot census.
+
+New independent source traversal, with the free-word convention credited to
+Cinna Davis/zxv and Cao Yifan's retained PR4610 word. Relations and their first
+physical realization are Ali Romman's PR5383. Prepared with OpenAI Codex
+assistance, Apache-2.0. This census does not import predecessor evaluators.
+"""
+from collections import Counter, defaultdict
+from copy import deepcopy
+from fractions import Fraction as Q
+from hashlib import sha256
+from pathlib import Path
+import argparse
+import gzip
+import json
+
+HERE = Path(__file__).resolve().parent
+SOURCE = HERE / 'source/ACTUAL-WORD.json.gz'
+WITNESS = HERE / 'WITNESS.json'
+PAYLOAD = 'cfcd36b40326d7a5815e1db5313444379b3fb756b27d5201c561574c085cc0dd'
+WITNESS_SHA = '24599011da06325e1012619b2b84a4dd3aceb90ab42d3caaf6414879e7d16591'
+EXPECTED_PROFILE = {1: 2201, 2: 412, 3: 510, 4: 127, 5: 40, 8: 8}
+
+
+def need(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def add(destination, source, coefficient):
+    for key, value in source.items():
+        value = destination.get(key, Q(0)) + coefficient * value
+        if value:
+            destination[key] = value
+        else:
+            destination.pop(key, None)
+
+
+def relation_basis(witnesses):
+    basis = {}
+    for witness in witnesses:
+        pairs = witness['relation']
+        row = {h: Q(value) for h, value in pairs}
+        need(len(row) == len(pairs) and all(224 <= h < 632 for h in row), 'Helper-only relation')
+        for pivot, previous in sorted(basis.items()):
+            add(row, previous, -row.get(pivot, Q(0)))
+        need(row, 'Independent relation')
+        pivot = min(row)
+        coefficient = row[pivot]
+        row = {h: value / coefficient for h, value in row.items()}
+        for previous in basis.values():
+            add(previous, row, -previous.get(pivot, Q(0)))
+        basis[pivot] = row
+    need(len(basis) == 8, 'Exactly eight fixed independent relations')
+    return dict(sorted(basis.items()))
+
+
+def normal(row):
+    """Exact projective key, retaining signs separately as the leading value."""
+    need(row, 'Nonzero projective row')
+    first = row[min(row)]
+    return tuple((key, value / first) for key, value in sorted(row.items())), first
+
+
+def traverse(source, witnesses, direction, omit_scalar=None):
+    basis = relation_basis(witnesses)
+    rows = {('L', i): {(('L', i), ()): Q(1)} for i in range(632)}
+    rows['C', 0] = {(('C', 0), ()): Q(1)}
+    if direction == 'backward':
+        for pivot, row in basis.items():
+            rows['L', pivot] = {(('L', h), ()): -value for h, value in row.items() if h != pivot}
+    frames = {('L', i): f for i, f in enumerate(source['canonical']['start'])}
+    frames['C', 0] = None
+    # Keeping every historical boundary value favors the putative replacement:
+    # it ignores snapshot slots, liveness, COPY, scalar and restoration costs.
+    seen = defaultdict(list)
+    def remember(event, role, frame):
+        if rows[role]:
+            key, coefficient = normal(rows[role])
+            seen[frame, key].append((event, role, coefficient))
+    for i, frame in enumerate(source['canonical']['start']):
+        remember(-1, ('L', i), frame)
+    paid, events = Counter(), Counter()
+    zero_rows, matches = [], []
+    for position, event in enumerate(source['programs'][direction]):
+        kind = event[0]
+        events[kind] += 1
+        role, frame = None, None
+        if kind == 'scalar':
+            operation, frame = event[1:]
+            role = tuple(operation[1])
+            if position == omit_scalar:
+                need(operation[0] == 'add', 'Omission control names an ADD')
+                continue
+            if operation[0] == 'clear':
+                rows[role] = {}
+                frames[role] = frame
+            else:
+                need(frames[role] == frame, 'Scalar destination frame')
+                if operation[0] in ('copy', 'add'):
+                    other = tuple(operation[2])
+                    need(frames[other] == frame, 'Scalar source frame')
+                    if operation[0] == 'copy':
+                        rows[role] = dict(rows[other])
+                    else:
+                        add(rows[role], rows[other], Q(operation[3], operation[4]))
+                else:
+                    need(operation[0] == 'scale', 'Known scalar opcode')
+                    rows[role] = {key: Q(operation[2], operation[3]) * value for key, value in rows[role].items()}
+        elif kind == 'warp':
+            _, target, operator, old, frame = event
+            role = tuple(target)
+            need(frames[role] == old, 'Native input frame')
+            symbol = (1 if operator > 0 else -1) * (abs(operator) + (10**6 if direction == 'backward' else 0))
+            rows[role] = {(variable, word[1:] if word and word[0] == -symbol else (symbol,) + word): value
+                          for (variable, word), value in rows[role].items()}
+            frames[role] = frame
+        elif kind in ('climb', 'center'):
+            if kind == 'climb':
+                _, _, index, old, frame, rank = event
+                role = 'L', index
+            else:
+                _, index, old, frame, rank = event
+                role = 'C', index
+            need(frames[role] == old, 'Paid input frame')
+            need(rank == abs(len(source['canonical']['frames'][old]) - len(source['canonical']['frames'][frame])), 'Paid rank')
+            paid[rank] += 1
+            if not rows[role]:
+                zero_rows.append({'event': position, 'source_event': event})
+            else:
+                key, coefficient = normal(rows[role])
+                prior = seen.get((frame, key), [])
+                if prior:
+                    matches.append({'event': position, 'source_event': event, 'row_terms': len(rows[role]),
+                        'earlier_snapshots': [{'event': i, 'role': list(r), 'factor': str(coefficient / c)} for i, r, c in prior]})
+            frames[role] = frame
+        else:
+            need(kind in ('phase', 'adapter'), 'Known source event')
+        if role is not None and frame is not None:
+            remember(position, role, frame)
+    need([frames['L', i] for i in range(632)] == source['canonical']['final'], 'All final frames')
+    need(dict(paid) == EXPECTED_PROFILE, 'Complete source paid census')
+    return {'paid_profile': dict(sorted(paid.items())), 'paid_total': sum(paid.values()),
+            'event_counts': dict(events), 'zero_paid_rows': zero_rows,
+            'nonzero_proportional_snapshot_matches': matches,
+            'historical_snapshots_favorably_retained': True,
+            'entry_relation_applied': direction == 'backward'}
+
+
+def verify_census(result, witnesses, direction):
+    need(result['paid_total'] == 3298, 'All paid events visited')
+    zeros = [row['event'] for row in result['zero_paid_rows']]
+    expected_zeros = sorted(w['zero_event'] for w in witnesses) if direction == 'backward' else []
+    need(zeros == expected_zeros, 'Exactly the known zero paid rows')
+    matches = result['nonzero_proportional_snapshot_matches']
+    need([m['event'] for m in matches] == ([w['event'] for w in witnesses] if direction == 'backward' else []), 'Exactly the known matched paid sites')
+    for match, witness in zip(matches, witnesses):
+        need(match['earlier_snapshots'] == [{'event': witness['cached_event'], 'role': ['L', witness['cached_role']],
+                                         'factor': str(Q(witness['copy_factor']))}], 'Unique exact chronological same-frame snapshot, including sign')
+
+
+def run(controls=True):
+    raw = SOURCE.read_bytes()
+    need(sha256(gzip.decompress(raw)).hexdigest() == PAYLOAD, 'Pinned complete source payload')
+    source = json.loads(gzip.decompress(raw))
+    witness_raw = WITNESS.read_bytes()
+    need(sha256(witness_raw).hexdigest() == WITNESS_SHA, 'Exact PR5383 witness bytes')
+    witnesses = json.loads(witness_raw)
+    directions = {}
+    for direction in ('forward', 'backward'):
+        directions[direction] = traverse(source, witnesses, direction)
+        verify_census(directions[direction], witnesses, direction)
+    rejected = {}
+    if controls:
+        mutated = deepcopy(witnesses)
+        mutated[0]['relation'][1][1] = str(-Q(mutated[0]['relation'][1][1]))
+        altered = traverse(source, mutated, 'backward')
+        try:
+            verify_census(altered, witnesses, 'backward')
+        except ValueError as error:
+            rejected['wrong_relation_sign'] = str(error)
+        else:
+            raise ValueError('Wrong relation accepted')
+        altered = traverse(source, witnesses, 'backward', omit_scalar=11331)
+        try:
+            verify_census(altered, witnesses, 'backward')
+        except ValueError as error:
+            rejected['omitted_source_ADD_11331'] = str(error)
+        else:
+            raise ValueError('Omitted source ADD accepted')
+        mutated = deepcopy(witnesses)
+        mutated[0]['copy_factor'] = '-1'
+        try:
+            verify_census(directions['backward'], mutated, 'backward')
+        except ValueError as error:
+            rejected['wrong_cached_global_sign'] = str(error)
+        else:
+            raise ValueError('Wrong cached sign accepted')
+    return {'status': 'PASS_FIXED_EIGHT_RELATION_FREE_WORD_ZERO_SNAPSHOT_SATURATION',
+            'source_compressed_sha256': sha256(raw).hexdigest(), 'source_payload_sha256': PAYLOAD,
+            'witness_sha256': sha256(WITNESS.read_bytes()).hexdigest(), 'pivots': list(relation_basis(witnesses)),
+            'orientations': directions, 'negative_controls': rejected,
+            'scope': 'Fixed eight entry relations, unchanged source event boundaries and free-word grammar. Unlimited historical scalar snapshots in the same physical output frame are allowed. No extra paid zero row or proportional single-snapshot equality remains beyond the eight known backward centre/zero pairs. New relations, scalar linear combinations, native identities beyond free reduction, changed event graph and physical admission are not covered.'}
+
+
+if __name__ == '__main__':
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--no-controls', action='store_true')
+    parser.add_argument('--out', type=Path)
+    arguments = parser.parse_args()
+    result = run(not arguments.no_controls)
+    output = json.dumps(result, indent=2, sort_keys=True) + '\n'
+    if arguments.out:
+        arguments.out.write_text(output)
+    else:
+        print(output, end='')
diff --git a/research/owned-relation-exhaustion/examples/exchange-128.json b/research/owned-relation-exhaustion/examples/exchange-128.json
new file mode 100644
index 0000000000..f93e0fbf87
--- /dev/null
+++ b/research/owned-relation-exhaustion/examples/exchange-128.json
@@ -0,0 +1,40 @@
+{
+  "schema": "integer-mult-recursive-invoice/v1",
+  "contract": {
+    "model": "mixed-target-unnormalized-power-v1",
+    "rank_semantics": "H_r has raw tariff r^(1-C), with surrendered productive fields debited",
+    "ambient_rank": 145,
+    "boundary": "complete p16/a7 unit before the retained mixed-target compiler"
+  },
+  "paid": {
+    "1": 41170,
+    "2": 7416,
+    "3": 9180,
+    "4": 2286,
+    "5": 720,
+    "8": 16,
+    "16": 448,
+    "24": 448,
+    "32": 448,
+    "40": 448,
+    "48": 448,
+    "56": 448,
+    "64": 448,
+    "72": 224
+  },
+  "targets": {
+    "4": 504,
+    "7": 126,
+    "8": 846,
+    "9": 5740,
+    "17": 448,
+    "25": 448,
+    "33": 448,
+    "41": 448,
+    "49": 448,
+    "57": 448,
+    "65": 448,
+    "73": 224,
+    "145": 224
+  }
+}
diff --git a/research/owned-relation-exhaustion/examples/exchange-144.json b/research/owned-relation-exhaustion/examples/exchange-144.json
new file mode 100644
index 0000000000..fef046daab
--- /dev/null
+++ b/research/owned-relation-exhaustion/examples/exchange-144.json
@@ -0,0 +1,39 @@
+{
+  "schema": "integer-mult-recursive-invoice/v1",
+  "contract": {
+    "model": "mixed-target-unnormalized-power-v1",
+    "rank_semantics": "H_r has raw tariff r^(1-C), with surrendered productive fields debited",
+    "ambient_rank": 145,
+    "boundary": "complete p16/a7 unit before the retained mixed-target compiler"
+  },
+  "paid": {
+    "1": 41154,
+    "2": 7416,
+    "3": 9180,
+    "4": 2286,
+    "5": 720,
+    "16": 448,
+    "24": 448,
+    "32": 448,
+    "40": 448,
+    "48": 448,
+    "56": 448,
+    "64": 448,
+    "72": 224
+  },
+  "targets": {
+    "4": 504,
+    "7": 126,
+    "8": 846,
+    "9": 5724,
+    "17": 448,
+    "25": 448,
+    "33": 448,
+    "41": 448,
+    "49": 448,
+    "57": 448,
+    "65": 448,
+    "73": 224,
+    "145": 224
+  }
+}
diff --git a/research/owned-relation-exhaustion/examples/exchange-16.json b/research/owned-relation-exhaustion/examples/exchange-16.json
new file mode 100644
index 0000000000..ec9c01129d
--- /dev/null
+++ b/research/owned-relation-exhaustion/examples/exchange-16.json
@@ -0,0 +1,40 @@
+{
+  "schema": "integer-mult-recursive-invoice/v1",
+  "contract": {
+    "model": "mixed-target-unnormalized-power-v1",
+    "rank_semantics": "H_r has raw tariff r^(1-C), with surrendered productive fields debited",
+    "ambient_rank": 145,
+    "boundary": "complete p16/a7 unit before the retained mixed-target compiler"
+  },
+  "paid": {
+    "1": 41282,
+    "2": 7416,
+    "3": 9180,
+    "4": 2286,
+    "5": 720,
+    "8": 128,
+    "16": 448,
+    "24": 448,
+    "32": 448,
+    "40": 448,
+    "48": 448,
+    "56": 448,
+    "64": 448,
+    "72": 224
+  },
+  "targets": {
+    "4": 504,
+    "7": 126,
+    "8": 846,
+    "9": 5852,
+    "17": 448,
+    "25": 448,
+    "33": 448,
+    "41": 448,
+    "49": 448,
+    "57": 448,
+    "65": 448,
+    "73": 224,
+    "145": 224
+  }
+}
diff --git a/research/owned-relation-exhaustion/examples/h4-split.json b/research/owned-relation-exhaustion/examples/h4-split.json
new file mode 100644
index 0000000000..4a7a9c6697
--- /dev/null
+++ b/research/owned-relation-exhaustion/examples/h4-split.json
@@ -0,0 +1,40 @@
+{
+  "schema": "integer-mult-recursive-invoice/v1",
+  "contract": {
+    "model": "mixed-target-unnormalized-power-v1",
+    "rank_semantics": "H_r has raw tariff r^(1-C), with surrendered productive fields debited",
+    "ambient_rank": 145,
+    "boundary": "complete p16/a7 unit before the retained mixed-target compiler"
+  },
+  "paid": {
+    "1": 41298,
+    "2": 7418,
+    "3": 9180,
+    "4": 2285,
+    "5": 720,
+    "8": 144,
+    "16": 448,
+    "24": 448,
+    "32": 448,
+    "40": 448,
+    "48": 448,
+    "56": 448,
+    "64": 448,
+    "72": 224
+  },
+  "targets": {
+    "4": 504,
+    "7": 126,
+    "8": 846,
+    "9": 5868,
+    "17": 448,
+    "25": 448,
+    "33": 448,
+    "41": 448,
+    "49": 448,
+    "57": 448,
+    "65": 448,
+    "73": 224,
+    "145": 224
+  }
+}
diff --git a/research/owned-relation-exhaustion/examples/source.json b/research/owned-relation-exhaustion/examples/source.json
new file mode 100644
index 0000000000..b27318644f
--- /dev/null
+++ b/research/owned-relation-exhaustion/examples/source.json
@@ -0,0 +1,40 @@
+{
+  "schema": "integer-mult-recursive-invoice/v1",
+  "contract": {
+    "model": "mixed-target-unnormalized-power-v1",
+    "rank_semantics": "H_r has raw tariff r^(1-C), with surrendered productive fields debited",
+    "ambient_rank": 145,
+    "boundary": "complete p16/a7 unit before the retained mixed-target compiler"
+  },
+  "paid": {
+    "1": 41298,
+    "2": 7416,
+    "3": 9180,
+    "4": 2286,
+    "5": 720,
+    "8": 144,
+    "16": 448,
+    "24": 448,
+    "32": 448,
+    "40": 448,
+    "48": 448,
+    "56": 448,
+    "64": 448,
+    "72": 224
+  },
+  "targets": {
+    "4": 504,
+    "7": 126,
+    "8": 846,
+    "9": 5868,
+    "17": 448,
+    "25": 448,
+    "33": 448,
+    "41": 448,
+    "49": 448,
+    "57": 448,
+    "65": 448,
+    "73": 224,
+    "145": 224
+  }
+}
diff --git a/research/owned-relation-exhaustion/source/ACTUAL-WORD.json.gz b/research/owned-relation-exhaustion/source/ACTUAL-WORD.json.gz
new file mode 100644
index 0000000000..46d929df83
Binary files /dev/null and b/research/owned-relation-exhaustion/source/ACTUAL-WORD.json.gz differ
diff --git a/research/owned-relation-exhaustion/source/CONTROLLER.json b/research/owned-relation-exhaustion/source/CONTROLLER.json
new file mode 100644
index 0000000000..49a2f06758
--- /dev/null
+++ b/research/owned-relation-exhaustion/source/CONTROLLER.json
@@ -0,0 +1,5224 @@
+{
+  "ambient_rank": 145,
+  "boundaries": {
+    "left": [
+      [
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "1",
+        "0"
+      ],
+      [
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "1",
+        "0",
+        "0",
+        "0"
+      ],
+      [
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "1",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0"
+      ],
+      [
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "1",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0"
+      ],
+      [
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "1",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0",
+        "0"
+      ],
+      [
+        "0",
+        "0",
+        "0",
+        "0",
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diff --git a/research/owned-relation-exhaustion/source/INVOICE.json b/research/owned-relation-exhaustion/source/INVOICE.json
new file mode 100644
index 0000000000..0fdb65f946
--- /dev/null
+++ b/research/owned-relation-exhaustion/source/INVOICE.json
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+      "global_gamma_power_slack",
+      "actual_logarithmic_gamma_guard",
+      "source_axis_superpolynomial",
+      "gaussian_above_main",
+      "new_semantic_guard_with_C1_one"
+    ],
+    "closed_uniform": [
+      "new_complex_uniform_saving_above_b"
+    ],
+    "old_BIT": [
+      "actual_bit_above_effective"
+    ],
+    "unfunded_endpoint": [
+      "main_transform_above_kappa",
+      "global_coordinate_router_above_kappa",
+      "fused_gaussian_above_kappa",
+      "group_bit_router_above_kappa",
+      "boundary_algebraic_matrix_above_kappa"
+    ]
+  },
+  "nonrecursive": {
+    "local_routes": 4802000,
+    "local_scalar": 158168,
+    "raw_scalar_upper": 181098496
+  },
+  "ordinary_BIT": {
+    "admitted": "17/5000",
+    "after_cover": "69/20000",
+    "atom": "1/250",
+    "before_cover": "3459/1000000",
+    "ordinary_saving": "8590884599/2500000000000",
+    "prime_rule": "odd q > 2^26*N^5 avoiding all fixed denominators and rank witnesses"
+  },
+  "outer": {
+    "functional_checks": {
+      "actual_new_BIT_stock_sublinear": "174299999999253/175000000000000",
+      "fixed_halo_explicit_metadata_branch": "13475000000001/175000000000",
+      "new_BIT_row_stock_sublinear_in_active_group": "697899999997009/700000000000000",
+      "new_semantic_group_guard": "3/700000000000"
+    },
+    "margins": {
+      "boundary_algebraic_matrix": "1480670144797884756937/490000000000000000000000",
+      "fused_gaussian": "1480670144793654270811/490000000000000000000000",
+      "global_coordinate_router": "264405383/87500000000",
+      "group_bit_router": "185083768186706783851/61250000000000000000000",
+      "group_scans": "349999999999/350000000000",
+      "main_transform": "1480670144097884756937/490000000000000000000000",
+      "microbox_face_corrections": "13475000000001/175000000000",
+      "polynomial_products": "699999999999/700000000000",
+      "sparse_union_movement": "200543170144797884756937/490000000000000000000000"
+    },
+    "parameters": {
+      "a": "264405383/87500000000",
+      "b": "423048613/140000000000",
+      "beta": "699999999997/700000000000",
+      "bit_saving": "17/5000",
+      "cap_xi": "445543170144097884756937/490000000000000000000000",
+      "complex_uniform_saving": "1057621533/350000000000",
+      "d": "1/700000000000",
+      "epsilon": "699999999999/700000000000",
+      "kappa": "15108879/5000000000",
+      "lambda_": "199395644839/200000000000",
+      "lambda_prime": "697884756937/700000000000",
+      "layout": "1/700000000000",
+      "q": "2115243063/700000000000",
+      "r0": "1/7000000000000",
+      "stop": "1/700000000000",
+      "tested_saving": "302177581/100000000000"
+    },
+    "strict_checks": {
+      "actual_bit_above_effective": "33094617/87500000000",
+      "actual_logarithmic_gamma_guard": "1/700000000000",
+      "actual_logarithmic_group_guard": "44456829856952115243063/245000000000000000000000",
+      "alpha_logarithm_absorption": "1/7000000000000",
+      "boundary_algebraic_matrix_above_kappa": "2797884756937/490000000000000000000000",
+      "cap_above_band": "200543170144097884756937/490000000000000000000000",
+      "cap_below_cell": "44456829855202115243063/490000000000000000000000",
+      "cap_below_phase_cell_power": "534456829855902115243063/490000000000000000000000",
+      "complex_above_effective": "1/700000000000",
+      "complex_below_one_over_32": "3951951387/140000000000",
+      "complex_endpoint": "1/350000000000",
+      "epsilon_below_one": "1/700000000000",
+      "epsilon_positive": "699999999999/700000000000",
+      "fine_prime_halo_umbrella": "7/2",
+      "fused_gaussian_above_kappa": "2793654270811/490000000000000000000000",
+      "gaussian_above_main": "347884756937/245000000000000000000000",
+      "global_coordinate_router_above_kappa": "1/175000000000",
+      "global_gamma_power_slack": "9/7000000000000",
+      "group_below_dimension": "1/350000000000",
+      "group_bit_router_above_kappa": "436706783851/61250000000000000000000",
+      "group_gauge_power_slack": "44456829856917115243063/245000000000000000000000",
+      "group_key_fits_record": "1/350000000000",
+      "group_positive": "699999999997/700000000000",
+      "group_scans_above_kappa": "348942378469/350000000000",
+      "halo_covers_generic_inverse_radius": "5",
+      "improvement_over_user_baseline": "645402690400418148631/500000000000000000000000",
+      "lambda_above_internal": "1/1400000000000",
+      "lambda_above_sigma": "3/1400000000000",
+      "lambda_above_tau": "1/1400000000000",
+      "lambda_prime_above_lambda": "1/1400000000000",
+      "lambda_prime_above_leaf": "279576951387/98000000000000000000000",
+      "leaf_saving_above_effective": "139576951387/98000000000000000000000",
+      "main_transform_above_kappa": "2097884756937/490000000000000000000000",
+      "membership_arithmetic_fits_record": "1/700000000000",
+      "microbox_face_corrections_above_kappa": "3368617797309/43750000000",
+      "microbox_halo_volume": "55300000000003/700000000000",
+      "new_complex_uniform_saving_above_b": "1/700000000000",
+      "new_semantic_guard_with_C1_one": "1/700000000000",
+      "polynomial_products_above_kappa": "697884756939/700000000000",
+      "positive_layout_slack": "1/700000000000",
+      "q_below_effective": "1/700000000000",
+      "q_below_internal": "1/700000000000",
+      "q_below_leaf": "279576951387/98000000000000000000000",
+      "q_positive": "2115243063/700000000000",
+      "short_record_fallback": "139576951387/140000000000",
+      "source_axis_superpolynomial": "1/700000000000",
+      "sparse_union_movement_above_kappa": "199062500002797884756937/490000000000000000000000",
+      "stop_below_one": "699999999999/700000000000",
+      "stop_positive": "1/700000000000",
+      "strassen_integer_power_gap": "1951441519231",
+      "support_router_span": "3/700000000000",
+      "theta_positive_polynomial_power": "2",
+      "union_header_fits_record": "3/700000000000",
+      "union_tag_work_per_visit_fits_record": "3/700000000000"
+    }
+  },
+  "paid": {
+    "1": 41298,
+    "16": 448,
+    "2": 7416,
+    "24": 448,
+    "3": 9180,
+    "32": 448,
+    "4": 2286,
+    "40": 448,
+    "48": 448,
+    "5": 720,
+    "56": 448,
+    "64": 448,
+    "72": 224,
+    "8": 144
+  },
+  "paid_calls": 64404,
+  "productive_fields": 10928,
+  "rank_mass_surplus": 752,
+  "raw_root_interval": [
+    "302177587077/100000000000000",
+    "151088793539/50000000000000"
+  ],
+  "scope": "Exact changed-profile arithmetic only. Requires separately admitted changed native programs, paid transports and absolute phases. Retained compiler/native/precision/tape/analytic/recovery assumptions.",
+  "source_sha256": {
+    "research/fresh-helper-boundary/native.py": "9b55a2d3c3972148fae9d112e745d9244968166bd89bd900c938f429fc76ab29",
+    "research/higher-moment-strictification/verify.py": "0ce7e7137422d5ffeda22085c81fa88471bbe963100fdad62e659f1e3bf89326",
+    "research/joint-helper-lattice/CONTROLLER.json": "1f5d6aa267e4e66048818714b5d3b5eae17a58abe91d15deeaa34574f7391a5f",
+    "research/joint-helper-lattice/native_joint.py": "50869b1bcad7493e6b4cb2a4cd5d2fd9a193a5921100483fa2c3f2c999ddfd46",
+    "research/joint-helper-lattice/verify.py": "8eb13f77a94315f811612cb557a17edeecb6b9d9a75dfa238b112dd6520c336f",
+    "research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/bit-compression/carrier-reuse/reference/joint_pair_dag.py": "8a0de761642d21919e969475c3f13e5d01a4bae7343c1e1f37c586d94b676a85",
+    "research/kappa-rebuild/architecture/mixed_target_probe.py": "011c3c9d0cf81103dc84d709a0e36793eb7c5ae3f23c85eac27650ef13afa04a",
+    "research/kappa-rebuild/bit/complex_controller_factor.py": "1de9aaf918e6a4f2f5448754c1f3c3da3b85e6257849ebd097af09859c4e5497",
+    "research/kappa-rebuild/bit/mixed_productive_bit.py": "b32291b0db7fc0d332ec444c70746ab0fb03351d2e50546f8bcb64060e3ff95e",
+    "research/kappa-round6-compiler-tightening-20261022/arithmetic.py": "960864e5427a1ca2c319a7b2dee292b6fe50bd5964830031e611591af10dac38",
+    "research/pr4359-p7-padding/dependencies/compiler-refill/endpoint.py": "d097e55f06c3f244f2074ffd7c1a0a5f3c4b4b6a6accf9619d1dde5c0ae85e15",
+    "research/query-thinned-controllers/RESULT.json": "6ebfc8c0f09166f4834e9d7427410cb31aefff2cd4d4ce588c8e52c9c70e9a0f",
+    "research/query-thinned-controllers/controller.py": "8b462d660487b1be4940357c4c7522967b1bfaede3c02b1c66109b85acae0fd2"
+  },
+  "status": "PASS_CHANGED_PROFILE_ARITHMETIC_NATIVE_ADMISSION_SEPARATE",
+  "strictification": {
+    "alpha": {
+      "lower": "0.999999999936687383304469699479",
+      "upper": "0.999999999936687383304469699480"
+    },
+    "cutoff": 20,
+    "delta": {
+      "lower": "0.958285308639504859158849914083",
+      "upper": "0.958285308639504859158849914084"
+    },
+    "floor": 21,
+    "gamma": {
+      "lower": "0.956217509753795293454516855483",
+      "upper": "0.956217509753795293454516855484"
+    },
+    "normalized_paid_upper": "104857600097656251/104857601073741824",
+    "power": 1024,
+    "tail_offset": "1/5",
+    "tested": "302177581/100000000000"
+  },
+  "targets": {
+    "145": 224,
+    "17": 448,
+    "25": 448,
+    "33": 448,
+    "4": 504,
+    "41": 448,
+    "49": 448,
+    "57": 448,
+    "65": 448,
+    "7": 126,
+    "73": 224,
+    "8": 846,
+    "9": 5868
+  }
+}
diff --git a/research/owned-relation-exhaustion/verify.py b/research/owned-relation-exhaustion/verify.py
new file mode 100644
index 0000000000..d901f2c1dd
--- /dev/null
+++ b/research/owned-relation-exhaustion/verify.py
@@ -0,0 +1,117 @@
+#!/usr/bin/env python3
+"""Source-bound signed-invoice examples, exchange cap and independent reuse audit."""
+from collections import Counter
+from hashlib import sha256
+from pathlib import Path
+import gzip
+import importlib.util
+import json
+import sys
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+sys.path.insert(0, str(HERE))
+import budget
+import census
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def load_comparator():
+    path = ROOT/'scripts/compare_recursive_invoices.py'
+    spec = importlib.util.spec_from_file_location('invoice_order', path)
+    module = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(module)
+    return module
+
+
+def source_invoice(c):
+    word = json.loads(gzip.decompress((HERE/'source/ACTUAL-WORD.json.gz').read_bytes()))
+    ctrl = json.loads((HERE/'source/CONTROLLER.json').read_bytes())
+    original = json.loads((HERE/'source/INVOICE.json').read_bytes())
+    can = word['canonical']
+    frames = can['frames']
+    need((can['v'], can['h'], ctrl['pairs'], ctrl['prefix_pairs'], ctrl['ambient_rank'])
+         == (112,9,16,7,145), 'Pinned rank conventions and caller')
+    helpers = Counter(len(frames[can['final'][i]])-len(frames[can['start'][i]]) for i in range(224,632))
+    need(min(helpers) > 0, 'Productive helper target ranks')
+    local = Counter({int(r): n for r,n in c['orientations']['forward']['paid_profile'].items()})
+    need(local == Counter({int(r):n for r,n in c['orientations']['backward']['paid_profile'].items()}), 'Both local paid profiles')
+    for direction in word['programs'].values():
+        centres = [e for e in direction if e[0]=='center']
+        need(len(centres)==8 and all(e[-1]==8 for e in centres), 'Eight original H8 centre slots')
+        need(all(e[-1]!=8 for e in direction if e[0]=='climb'), 'All original H8 jobs are centres')
+    paid, targets, events, directions = Counter(), Counter(), Counter(), Counter()
+    for event in ctrl['physical_events']:
+        kind = event[0]
+        events[kind] += 1
+        if kind == 'LOCAL':
+            directions[bool(event[3])] += 1
+            paid.update(local)
+            targets.update(helpers)
+        elif kind == 'QUERY_SHEAR':
+            paid[1] += 112
+        elif kind == 'IDLE':
+            need(event[2] and len(set(event[2]))==len(event[2]), 'Distinct whole-idle stages')
+            paid[8*len(event[2])] += 224
+        else:
+            need(kind in ('ADD','SIGNED_SWAP'), 'Known whole-caller event')
+    for diagonal in ctrl['diagonal']:
+        need(len(diagonal)==34 and diagonal[0] in (-1,1) and all(t in (0,1) for t in diagonal[1:]), 'Main output rank convention')
+        targets[1+8*sum(diagonal[1:])] += 112
+    need(events['LOCAL']==18 and directions==Counter({True:16,False:2}), '16 backward and two forward invocations')
+    need(paid == Counter({int(r):n for r,n in original['paid'].items()}), 'Independently rebuilt complete paid invoice')
+    need(targets == Counter({int(r):n for r,n in original['targets'].items()}), 'Independently rebuilt complete target invoice')
+    return targets, paid
+
+
+def invoice(targets, paid, module):
+    return {'schema': module.SCHEMA,
+        'contract': {'model':'mixed-target-unnormalized-power-v1',
+            'rank_semantics':'H_r has raw tariff r^(1-C), with surrendered productive fields debited',
+            'ambient_rank':145,
+            'boundary':'complete p16/a7 unit before the retained mixed-target compiler'},
+        'paid':{str(r):n for r,n in sorted(paid.items()) if n},
+        'targets':{str(r):n for r,n in sorted(targets.items()) if n}}
+
+
+def run():
+    manifest = json.loads((HERE/'SOURCES.json').read_bytes())
+    for row in manifest['retained_exports']:
+        need(sha256((HERE/row['local_path']).read_bytes()).hexdigest()==row['sha256'], 'Retained source: '+row['local_path'])
+    c = census.run()
+    targets, paid = source_invoice(c)
+    module = load_comparator()
+    original = invoice(targets, paid, module)
+    examples = {}
+    for count in (0,16,128,144):
+        t,p = budget.exchange(targets, paid, count)
+        candidate = invoice(t,p,module)
+        report = module.compare_invoices(original,candidate)
+        need(report['classification']==('equal' if count==0 else 'uniformly_better'), 'Signed target-debit comparison')
+        path = HERE/'examples'/('source.json' if count==0 else f'exchange-{count}.json')
+        need(json.loads(path.read_bytes())==candidate, 'Exact committed example invoice')
+        examples[str(count)] = report
+    split = invoice(targets,paid,module)
+    split['paid']['4'] -= 1
+    split['paid']['2'] += 2
+    need(module.compare_invoices(original,split)['classification']=='uniformly_worse', 'Scalar-saving H4 split loses')
+    need(json.loads((HERE/'examples/h4-split.json').read_bytes())==split, 'Committed split example')
+    b = budget.run()
+    result = {'status':'PASS_SIGNED_INVOICE_PREFLIGHT_AND_SCOPED_CENTRE_EXHAUSTION',
+        'source_reconstruction':{'local_jobs_each_orientation':3298,'backward_invocations':16,'forward_invocations':2,
+            'original_centre_slots':144,'rank_surplus':752},
+        'signed_comparisons':examples,'budget':b,'census':c,
+        'scope':'Exact histogram order and this fixed exchange family; independent fixed-eight formal-row census. No new supplier or selected-frontier change.'}
+    # JSON object keys are strings, including the census's integer rank keys.
+    return json.loads(json.dumps(result))
+
+
+if __name__=='__main__':
+    result = run()
+    expected = json.loads((HERE/'RESULT.json').read_bytes())
+    need(result==expected, 'Deterministic committed receipt')
+    print('PASS_SIGNED_INVOICE_PREFLIGHT_AND_SCOPED_CENTRE_EXHAUSTION')
diff --git a/scripts/compare_recursive_invoices.py b/scripts/compare_recursive_invoices.py
new file mode 100644
index 0000000000..fda2a852e9
--- /dev/null
+++ b/scripts/compare_recursive_invoices.py
@@ -0,0 +1,218 @@
+#!/usr/bin/env python3
+# SPDX-License-Identifier: Apache-2.0
+"""Compare declared recursive invoices by exact signed concave order.
+
+Usage: python3 -B scripts/compare_recursive_invoices.py OLD.json NEW.json
+
+Each input has schema ``integer-mult-recursive-invoice/v1``, an identical
+contract object with model, rank_semantics, ambient_rank and boundary, and
+paid/targets objects mapping canonical positive decimal ranks to nonnegative
+integer counts. Contracts are declarations, not verified mathematical premises.
+
+For d = (new paid - old paid) - (new targets - old targets), set
+K(k) = sum_r d[r] min(r,k). The sign of K at every support knot determines
+the order for every increasing concave tariff f with f(0)=0. Indeed, on the
+integer ranks 0..R, write s_j=f(j)-f(j-1). Then
+
+  sum_r d[r] f(r) = s_R K(R) + sum_(k=1..R-1) (s_k-s_(k+1)) K(k).
+
+All coefficients on the right are nonnegative. Conversely, the increasing
+concave tariffs min(r,k) witness every required hinge. Between support knots
+K is linear. For f(r)=r**t, 0<t<1, all the displayed slope coefficients are
+positive, so a nonzero one-sided hinge certificate gives strict power order.
+Other concave tariffs can give equality; the uniform classification is weak.
+
+This is classical concave-order/majorization arithmetic, already used by
+mightyOil's research/t16-moment-order-audit and its credited predecessors.
+The new interface retains both paid jobs and surrendered target credits.
+Prepared with substantial OpenAI Codex assistance. No new theorem is claimed.
+"""
+
+import argparse
+from hashlib import sha256
+import json
+from pathlib import Path
+import re
+import sys
+
+
+SCHEMA = "integer-mult-recursive-invoice/v1"
+REPORT_SCHEMA = "integer-mult-recursive-invoice-comparison/v1"
+MAX_BYTES = 1024 * 1024
+MAX_SUPPORT = 4096
+MAX_RANK = 10**9
+MAX_COUNT = 10**60
+MAX_TEXT = 4096
+RANK_KEY = re.compile(r"[1-9][0-9]{0,9}\Z", re.ASCII)
+CONTRACT_KEYS = {"model", "rank_semantics", "ambient_rank", "boundary"}
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def _integer(text):
+    # Check before int(): do not rely on a particular Python digit limit.
+    digits = text.removeprefix("-")
+    require(len(digits) <= 61, "integer token exceeds 61 digits")
+    value = int(text)
+    require(abs(value) <= MAX_COUNT, "integer token exceeds magnitude 10^60")
+    return value
+
+
+def _pairs(pairs):
+    result = {}
+    for key, value in pairs:
+        require(key not in result, "duplicate JSON object key")
+        result[key] = value
+    return result
+
+
+def _not_integer(_text):
+    raise ValueError("floating-point and nonfinite JSON numbers are unsupported")
+
+
+def validate_invoice(value, label="invoice"):
+    """Validate and copy an invoice; remove zero counts, preserving string keys."""
+    require(type(value) is dict, label + ": expected an object")
+    require(value.keys() == {"schema", "contract", "paid", "targets"},
+            label + ": expected exactly schema, contract, paid and targets")
+    require(value["schema"] == SCHEMA, label + ": unsupported schema")
+    contract = value["contract"]
+    require(type(contract) is dict and contract.keys() == CONTRACT_KEYS,
+            label + ": contract needs model, rank_semantics, ambient_rank and boundary")
+    for key in ("model", "rank_semantics", "boundary"):
+        item = contract[key]
+        require(type(item) is str and 0 < len(item) <= MAX_TEXT and item.strip(),
+                label + ": contract " + key + " must be a nonempty bounded string")
+    ambient = contract["ambient_rank"]
+    require(type(ambient) is int and 1 <= ambient <= MAX_RANK,
+            label + ": ambient_rank must be an integer in 1..10^9")
+    result = {"schema": SCHEMA, "contract": dict(contract)}
+    for side in ("paid", "targets"):
+        histogram = value[side]
+        require(type(histogram) is dict and len(histogram) <= MAX_SUPPORT,
+                label + ": " + side + " must have at most 4096 rank keys")
+        clean = {}
+        for key, count in histogram.items():
+            require(type(key) is str and RANK_KEY.fullmatch(key) is not None,
+                    label + ": " + side + " rank keys must be canonical positive decimals")
+            rank = int(key)
+            require(rank <= ambient, label + ": rank exceeds declared ambient_rank")
+            require(type(count) is int and 0 <= count <= MAX_COUNT,
+                    label + ": counts must be integers in 0..10^60")
+            if count:
+                clean[key] = count
+        result[side] = dict(sorted(clean.items(), key=lambda item: int(item[0])))
+    return result
+
+
+def parse_invoice(raw, label="invoice"):
+    require(type(raw) is bytes and len(raw) <= MAX_BYTES,
+            label + ": input exceeds 1 MiB")
+    try:
+        value = json.loads(raw.decode("utf-8"), object_pairs_hook=_pairs,
+                           parse_int=_integer, parse_float=_not_integer,
+                           parse_constant=_not_integer)
+    except (UnicodeError, RecursionError) as exc:
+        raise ValueError(label + ": invalid UTF-8 or excessive JSON nesting") from exc
+    return validate_invoice(value, label)
+
+
+def load_invoice(path):
+    path = Path(path)
+    with path.open("rb") as source:
+        raw = source.read(MAX_BYTES + 1)
+    invoice = parse_invoice(raw, str(path))
+    return invoice, {"path": str(path), "sha256": sha256(raw).hexdigest(),
+                     "bytes": len(raw)}
+
+
+def _difference(new, old):
+    ranks = sorted({int(r) for r in new} | {int(r) for r in old})
+    return {str(r): new.get(str(r), 0) - old.get(str(r), 0) for r in ranks
+            if new.get(str(r), 0) != old.get(str(r), 0)}
+
+
+def _mass(histogram):
+    return sum(int(rank) * count for rank, count in histogram.items())
+
+
+def _masses(invoice):
+    paid, targets = _mass(invoice["paid"]), _mass(invoice["targets"])
+    return {"paid": paid, "targets": targets, "paid_minus_targets": paid - targets}
+
+
+def compare_invoices(old, new):
+    """Return exact arithmetic evidence; input correctness is a declared premise."""
+    old, new = validate_invoice(old, "old"), validate_invoice(new, "new")
+    require(old["contract"] == new["contract"], "declared invoice contracts differ")
+    paid = _difference(new["paid"], old["paid"])
+    targets = _difference(new["targets"], old["targets"])
+    net = _difference(paid, targets)
+    # A sparse sweep avoids both an ambient-rank array and a quadratic scan.
+    hinges = [{"rank": 0, "value": 0}]
+    prefix_mass, remaining_count = 0, sum(net.values())
+    for rank, count in net.items():
+        k = int(rank)
+        prefix_mass += k * count
+        remaining_count -= count
+        hinges.append({"rank": k, "value": prefix_mass + k * remaining_count})
+    values = [entry["value"] for entry in hinges]
+    if not net:
+        classification, power_sign = "equal", "identically_zero"
+    elif all(value <= 0 for value in values):
+        classification, power_sign = "uniformly_better", "strictly_negative"
+    elif all(value >= 0 for value in values):
+        classification, power_sign = "uniformly_worse", "strictly_positive"
+    else:
+        classification, power_sign = "inconclusive", "not_determined"
+    old_mass, new_mass = _masses(old), _masses(new)
+    return {
+        "schema": REPORT_SCHEMA,
+        "classification": classification,
+        "contract": old["contract"],
+        "delta_paid": paid,
+        "delta_targets": targets,
+        "signed_net_delta": net,
+        "hinges": hinges,
+        "rank_mass": {"old": old_mass, "new": new_mass,
+                      "delta": {key: new_mass[key] - old_mass[key] for key in old_mass}},
+        "quantity": "(new paid - new targets) - (old paid - old targets)",
+        "tariff_class": "increasing concave f on nonnegative ranks, with f(0)=0",
+        "uniform_order": "weak; some concave tariffs can give equality",
+        "power_moment": {"tariff": "r^t", "exponent": "0 < t < 1",
+                         "new_minus_old_sign": power_sign},
+        "assurance": {
+            "contract": "Exact matching user declarations; mathematical applicability is not verified.",
+            "input_binding": "SHA-256 identifies the exact input bytes read, when loaded by this CLI.",
+            "scope": "Histogram arithmetic only. Histogram correctness, circuit legality, ownership, "
+                     "phases, precision, supplier admission and compiler endpoints are not validated.",
+            "inconclusive": "Mixed hinge signs do not establish a power-moment crossing or failure.",
+        },
+        "method_credit": "Classical concave order/majorization; mightyOil's t16-moment-order-audit "
+                         "and its retained predecessors. No new theorem claim.",
+    }
+
+
+def main(argv=None):
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("old", help="old invoice JSON")
+    parser.add_argument("new", help="new invoice JSON")
+    args = parser.parse_args(argv)
+    try:
+        old, old_evidence = load_invoice(args.old)
+        new, new_evidence = load_invoice(args.new)
+        report = compare_invoices(old, new)
+        report["inputs"] = {"old": old_evidence, "new": new_evidence}
+    except (OSError, ValueError, TypeError, RecursionError) as exc:
+        print(json.dumps({"status": "INVALID_RECURSIVE_INVOICE", "error": str(exc)},
+                         ensure_ascii=True), file=sys.stderr)
+        return 2
+    print(json.dumps(report, indent=2, sort_keys=True, ensure_ascii=True))
+    return 0
+
+
+if __name__ == "__main__":
+    raise SystemExit(main())
diff --git a/tests/test_compare_recursive_invoices.py b/tests/test_compare_recursive_invoices.py
new file mode 100644
index 0000000000..35d797b4c5
--- /dev/null
+++ b/tests/test_compare_recursive_invoices.py
@@ -0,0 +1,155 @@
+# SPDX-License-Identifier: Apache-2.0
+"""Independent finite tariff checks and real algebra controls for signed invoices."""
+
+import copy
+from hashlib import sha256
+import importlib.util
+import itertools
+import json
+from pathlib import Path
+import subprocess
+import sys
+import tempfile
+import unittest
+
+
+SCRIPT = Path(__file__).resolve().parents[1] / "scripts/compare_recursive_invoices.py"
+SPEC = importlib.util.spec_from_file_location("compare_recursive_invoices", SCRIPT)
+M = importlib.util.module_from_spec(SPEC)
+SPEC.loader.exec_module(M)
+
+
+def invoice(paid=None, targets=None):
+    return {"schema": M.SCHEMA,
+            "contract": {"model": "declared-complete-recursive-invoice",
+                         "rank_semantics": "one Gaussian child of integer rank r",
+                         "ambient_rank": 145,
+                         "boundary": "same arbitrary independent inputs and charged outputs"},
+            "paid": {str(r): n for r, n in (paid or {}).items()},
+            "targets": {str(r): n for r, n in (targets or {}).items()}}
+
+
+class InvoiceComparisonTests(unittest.TestCase):
+    def test_real_h4_split_is_uniformly_worse(self):
+        result = M.compare_invoices(invoice({4: 1}), invoice({2: 2}))
+        self.assertEqual(result["classification"], "uniformly_worse")
+        self.assertEqual(result["signed_net_delta"], {"2": 2, "4": -1})
+        self.assertEqual(result["hinges"], [{"rank": 0, "value": 0},
+                                             {"rank": 2, "value": 2},
+                                             {"rank": 4, "value": 0}])
+        self.assertEqual(result["rank_mass"]["delta"]["paid_minus_targets"], 0)
+        self.assertEqual(result["power_moment"]["new_minus_old_sign"], "strictly_positive")
+
+    def test_owned_relation_debits_surrendered_h9_target(self):
+        result = M.compare_invoices(invoice({1: 1, 8: 1}, {9: 1}), invoice())
+        self.assertEqual(result["classification"], "uniformly_better")
+        self.assertEqual(result["delta_targets"], {"9": -1})
+        self.assertEqual(result["signed_net_delta"], {"1": -1, "8": -1, "9": 1})
+        self.assertEqual([row["value"] for row in result["hinges"]], [0, -1, -1, 0])
+        self.assertEqual(result["power_moment"]["new_minus_old_sign"], "strictly_negative")
+
+    def test_scalar_only_and_equal_net_moments(self):
+        self.assertEqual(M.compare_invoices(invoice({4: 1}), invoice({4: 1}))["classification"], "equal")
+        result = M.compare_invoices(invoice(), invoice({7: 2}, {7: 2}))
+        self.assertEqual(result["classification"], "equal")
+        self.assertEqual(result["delta_paid"], {"7": 2})
+        self.assertEqual(result["delta_targets"], {"7": 2})
+
+    def test_mixed_hinges_are_inconclusive(self):
+        result = M.compare_invoices(invoice({2: 3}), invoice({1: 4}))
+        self.assertEqual(result["classification"], "inconclusive")
+        self.assertEqual(result["power_moment"]["new_minus_old_sign"], "not_determined")
+
+    def test_unequal_mass_and_large_gaps_need_no_expanded_rank_array(self):
+        result = M.compare_invoices(invoice(), invoice({145: 1}))
+        self.assertEqual(result["classification"], "uniformly_worse")
+        self.assertEqual(result["rank_mass"]["delta"]["paid"], 145)
+        value = invoice({M.MAX_RANK: 1})
+        value["contract"]["ambient_rank"] = M.MAX_RANK
+        zero = copy.deepcopy(value)
+        zero["paid"] = {}
+        self.assertEqual(len(M.compare_invoices(zero, value)["hinges"]), 2)
+
+    def test_exhaustive_independent_discrete_slope_tariffs(self):
+        # Every extreme tariff min(r,k), including linear r, occurs among these
+        # slopes. Direct cumulative evaluation is independent of hinge code.
+        slopes = [s for s in itertools.product(range(3), repeat=4)
+                  if all(s[i] >= s[i + 1] for i in range(3))]
+        tariffs = [list(itertools.accumulate(s)) for s in slopes]
+        for delta in itertools.product(range(-2, 3), repeat=4):
+            old = invoice({r: -d for r, d in enumerate(delta, 1) if d < 0})
+            new = invoice({r: d for r, d in enumerate(delta, 1) if d > 0})
+            # Add matching but nonconstant paid/target entries to exercise the
+            # signed presentation without changing its actual tariff value.
+            new["targets"] = {str(r): r for r in range(1, 5)}
+            old["targets"] = dict(new["targets"])
+            values = [sum(d * f for d, f in zip(delta, tariff)) for tariff in tariffs]
+            expected = ("equal" if not any(delta) else
+                        "uniformly_better" if max(values) <= 0 else
+                        "uniformly_worse" if min(values) >= 0 else "inconclusive")
+            self.assertEqual(M.compare_invoices(old, new)["classification"], expected, delta)
+
+    def test_changed_contract_rejected(self):
+        for key, value in (("model", "other"), ("rank_semantics", "different rank"),
+                           ("ambient_rank", 146), ("boundary", "different ownership")):
+            old, new = invoice(), invoice()
+            new["contract"][key] = value
+            with self.assertRaisesRegex(ValueError, "contracts differ"):
+                M.compare_invoices(old, new)
+
+    def test_histogram_and_contract_domain_rejections(self):
+        for histogram in ({"01": 1}, {"0": 1}, {"+1": 1}, {"1.0": 1},
+                          {" 1": 1}, {"١": 1}, {"146": 1}, {"1": True},
+                          {"1": -1}, {"1": 1.0}, {"1": 10**61},
+                          {"1" * 10000: 1}, {True: 1}):
+            value = invoice()
+            value["paid"] = histogram
+            with self.assertRaises(ValueError):
+                M.validate_invoice(value)
+        for contract in ({}, {**invoice()["contract"], "ambient_rank": True},
+                         {**invoice()["contract"], "boundary": " "},
+                         {**invoice()["contract"], "model": "x" * (M.MAX_TEXT + 1)}):
+            value = invoice()
+            value["contract"] = contract
+            with self.assertRaises(ValueError):
+                M.validate_invoice(value)
+
+    def test_json_duplicate_numbers_and_resource_controls(self):
+        good = json.dumps(invoice()).encode()
+        cases = [good.replace(b'"paid": {}', b'"paid": {"1":1,"1":2}'),
+                 good.replace(b'"paid": {}', b'"paid": {"1":' + b'9' * 5000 + b'}'),
+                 good.replace(b'"paid": {}', b'"paid": {"1":NaN}'),
+                 good.replace(b'"paid": {}', b'"paid": {"1":1e5}'),
+                 b'x' * (M.MAX_BYTES + 1), b'\xff', b'[' * 2000]
+        for raw in cases:
+            with self.assertRaises(ValueError):
+                M.parse_invoice(raw)
+        too_many = invoice()
+        too_many["contract"]["ambient_rank"] = M.MAX_RANK
+        too_many["paid"] = {str(r): 0 for r in range(1, M.MAX_SUPPORT + 2)}
+        with self.assertRaises(ValueError):
+            M.validate_invoice(too_many)
+
+    def test_cli_binds_exact_bytes_and_rejects_cleanly(self):
+        with tempfile.TemporaryDirectory() as folder:
+            old_path, new_path = Path(folder) / "old.json", Path(folder) / "new.json"
+            old_raw = (json.dumps(invoice({1: 1, 8: 1}, {9: 1})) + "\n").encode()
+            old_path.write_bytes(old_raw)
+            new_path.write_text(json.dumps(invoice()))
+            completed = subprocess.run([sys.executable, "-I", "-B", str(SCRIPT),
+                                        str(old_path), str(new_path)], capture_output=True, text=True)
+            self.assertEqual(completed.returncode, 0, completed.stderr)
+            report = json.loads(completed.stdout)
+            self.assertEqual(report["classification"], "uniformly_better")
+            self.assertEqual(report["inputs"]["old"]["sha256"], sha256(old_raw).hexdigest())
+            new_path.write_text('{"paid":{"1":' + '9' * 5000 + '}}')
+            failed = subprocess.run([sys.executable, "-I", "-O", "-B", str(SCRIPT),
+                                     str(old_path), str(new_path)], capture_output=True, text=True)
+            self.assertEqual(failed.returncode, 2)
+            self.assertEqual(failed.stdout, "")
+            self.assertEqual(json.loads(failed.stderr)["status"], "INVALID_RECURSIVE_INVOICE")
+            self.assertNotIn("Traceback", failed.stderr)
+
+
+if __name__ == "__main__":
+    unittest.main()