κ — the rate
that replaces interest.

The benchmark the Islamic world already tried —
and still needs.

LIBOR ended in 2023–24. Every Islamic instrument that quietly benchmarked to it — murabaha mark-ups, ijara rentals, sukuk coupons — now needs a replacement that is both Shariah-valid and market-determined, not submission-based. It does not yet exist at term. In 2011 the IDB, AAOIFI and Thomson Reuters launched exactly this — the Islamic Interbank Benchmark Rate (IIBR). It was discontinued for three reasons that read like a spec sheet for what κ fixes by construction.

Malaysia's Myor-i proved the measured-not-quoted approach works — but it is overnight-only, ringgit-only, with no term structure. κ's differentiator is exactly that: a decoupled, measured curve, across assets and currencies, with a published index running since 2015. Its natural institutional form is a benchmark administrator — an auditable methodology over federated data, the way SOFR and CF Benchmarks are run — not a bank, and not an inflationary token. Everything below is the evidence and the theory under that one claim. The sovereign chain, if it ever comes, is the maximalist someday-version downstream of the benchmark — not the product.

Not the scaffolding — the engine.

For fifty years, Islamic finance was an exercise in form-matching: it accepted the conventional financial engine as an optimisation of nature and built legalistic, contract-level scaffolding around it to satisfy the prohibitions. This work doesn't modify the scaffolding. It swaps out the engine.

Conventional finance prices time with r — a reward for the sheer, passive passage of time, time as an empty vessel that grows simply by existing. That is the root of riba al-nasiah. Ground the whole stack in κ (convergence intensity) instead and time is priced by the velocity of real economic convergence and real-world loss — by what happens, not by waiting. The pricing of a derivative, the yield curve of a sovereign bond, the premium of an insurance pool, the value of money, the stability of a macroeconomy, even a validator's paycheck end up humming to the same measurable frequency: a risk-bearing event intensity. Interest, it turns out, was an artifact of historical design — not an unalterable rule of mathematics.

The fifty-year gap

Since the first modern Islamic bank opened in 1975, the industry faced one question it could never answer — not “what contract form is permissible” (scholars settled that centuries ago), but “what is the fair price of this contract, computed without r?” Every tool of modern finance begins with a discount rate: present value is e^(−rT), the bond curve bootstraps off r, the option prices under the r-drift measure. Remove r and the whole quantitative toolbox returns undefined. So the industry kept its conviction in the contract and surrendered it in the price — a sukuk coupon benchmarked to SOFR, a markup read off LIBOR. Usmani said it aloud in 2008: most sukuk are compliant in form, conventional in substance. Not a scandal — a symptom. The mathematics to fix the number did not exist. Here are the five empty slots, and what now fills each.

Everyone held a piece. No one held the bridge.

The gap survived fifty years not because anyone was foolish, but because the pieces sat in literatures that never read each other.

Interest is a modelling choice — not a mathematical necessity.

Ackerer, Hugonnier & Jermann (Mathematical Finance, 2025) prove that a perpetual futures contract has two parameters — a convergence intensity κconvergence intensity — the rate at which a real event resolves (a default, a drought, a price converging) and an interest parameter ιthe interest parameter — the term that rewards the mere passage of time; set it to zero and riba al-nasiah is gone — and that at ι = 0 the contract still has a unique, arbitrage-free price: the expected spot value at a random stopping time τ ∼ Exp(κ). The interest term is removable; the price is not. This program asks how far that single fact carries — from one contract to an entire monetary system — and answers with real data at every step.

The Separation Theorem

The perpetual's stopping time is, exactly, the Duffie–Singleton default timeDuffie–Singleton (1999): the standard reduced-form way to price defaultable debt — default arrives at a random time governed by a hazard rate λ. That identifies κ (convergence intensity) with the credit hazard λλ — the instantaneous probability of default per unit time in a reduced-form credit model. The theorem shows κ and λ are the same object — so the machinery of credit pricing transfers wholesale, with the interest input set to zero. Credit risk and time-preference, long entangled in the single number r, come apart: you can keep the first and zero the second without anything breaking.

The same type of object in all three — a stopping-time intensity identified with zero interest input — but spanning four orders of magnitude in size, from year-scale credit to hour-scale funding.

What it shows: the three channels never share inputs, yet each returns the same kind of object — a stopping-time intensity with no interest term. Their sizes differ by ~20,000× because the events resolve on wildly different clocks: a default takes years (κ ≈ 0.08/yr), a crop failure years (≈ 0.20/yr), a perp basis converges in hours (≈ 1,400/yr). One primitive, measured wherever events resolve.

The κ-index, 2015 – 2026

A monthly riba-free benchmark, computed from the live spreads of every quoted USD sukuk — 273 bonds, 95 issuers, 17 countries. The closest thing to a measured interest-free rate: credit-ordered, mean-reverting (half-life 1.4–4.3 months), non-negative by construction, and it caught both the 2015–16 oil shock and the March-2020 blowout in real time.

Full-universe Cbonds pull (2026-06): 273 bonds with usable daily G-spread histories, 178,089 bond-days, δ=0.4. The panel grows from 5 bonds in 2015 to 239 by 2026 — early months are thinner.

Investment grade vs high yield

Split the universe by rating and κ does what a credit benchmark should: high yield rides 2–3× above investment grade throughout, blows out to 11.4% in the COVID crash, and both compress after 2023.

κ̄ by sovereign — and it mean-reverts

Long-run κ̄ from roll-down-free constant-maturity series (7y, 2018–2026): a clean sovereign credit spectrum, Malaysia to Turkey. Each label shows the bias-corrected mean-reversion half-life.

What it shows: κ behaves like a real rate, not a fitted curiosity. It is credit-ordered (0.89% Malaysia → 7.44% Turkey), it mean-reverts with a half-life of months (the value after each bar), and it stays non-negative — the Feller condition holds in all 12/12 series. A measured, stationary, riba-free rate.

The receipts

It doesn't refute the interest-rate world.
It contains it.

The sharpest objection a referee can raise is that κ is parasitic on the interest rate — that we are secretly reading r back out of bond prices. Answering it produced the strongest result in the corpus: κ (convergence intensity) inverts from prices alone, and the wedge between the price-only estimate and the Treasury curve turns out to be the Treasury's own κ — the reserve sovereign's event intensity — recovered to within about fourteen basis points.

What it shows: compute κ from each bond's own price with no Treasury curve at all, and it still recovers the spread-based credit ordering — rank correlation rising to 0.84 (sovereign) / 0.94 (corporate) as the recovery δ drops to ~0.10, the level actually observed on defaulted sukuk. The referee's deepest objection — “κ is parasitic on r” — fails on its own terms: r was a convenience, never a dependency.

One argument in seven movements

Companions ground it in the classical canon (al-Ghazālī, Ibn Taymiyyah, Ibn Khaldūn) and state the product-layer boundary; a frontier trilogy answers liveness, measurement integrity, and the safe asset.

A door with a lock, not a wall with a sign. The chain is Phase 3 — a 3–5 year horizon. What runs today is the Arbitrum testnet protocol; the consensus layer is the destination, stated as a destination.

A clean baseline, not a finished verdict.

The boundary is the point. A clean mathematical baseline invites the scholarly community to engage — not a compromised approximation dressed as compliance.

The wager

Start with the thing both systems share and neither can repeal: the pain is conserved. Drought, default, pandemic, recession destroy real value under any monetary constitution. Once you accept that both the interest system and the κ-system are internally consistent — the whole point of this work — the comparison was never “pain vs no pain.” It collapses to a narrower axis: where the pain lands, who carries it, and whether they are allowed to see it.

On that axis the two systems are anesthesia versus daylight. The interest system's deepest machinery makes the bill invisible at the moment of impact. A crisis hits and the central bank prints — no visible loss, but every cash-holder pays a sliver forever through debasement, a tax with no vote and no name. A bank fails and the bailout spreads the loss across taxpayers who never see the line item. A government overborrows and financial repression melts the debt out of pensions over twenty quiet years instead of one honest default. The pain is fully paid every time — just diffused, delayed, and renamed, with costs spread thin so no one organises against them and benefits concentrated so someone always organises for them.

The κ-system strips the anesthesia by construction. Money visibly reprices the week its backing is destroyed, in everyone's own unit of account. A crisis draws down a granary everyone watched being filled — and if the society saved, it survives the winter; if it didn't, the damage is ~3.6× worse and everyone knows whose discipline failed. Sovereign recklessness prints in the sovereign's own κ, in the open. Nothing is forgiven, nothing is hidden, every bill is itemised.

And here the proofs stop. Everything else in the corpus was provable because it was a question of consistency — does pricing survive at ι=0, is the price level determinate, is honesty incentive-compatible for a validator. But “will a people choose the honest system” is a question about preference under temptation, a different category. History runs against it: every hard-money constitution humanity ever adopted — the gold standard most famously — was abandoned the first time a war or depression made the honest bill unbearable and the printing press was sitting right there.

And it always was. The riba prohibition itself — fourteen centuries before any of these equations — was this exact wager: that visible, shared, honest risk-bearing serves a community better than invisible, asymmetric, compounding extraction, even though the extraction feels painless and the honesty feels hard. It was never an optimisation theorem. It was a command — because it was always a question of what a people would choose against their own short-term comfort. The corpus didn't change that wager. It did something narrower, and for the first time complete: it removed the last excuse. Before, you could say “honest finance is impossible — the math needs r.” Now the math is done, the data sits under it, the machine runs.

No equation answers that. The work's only job was to make sure that when the answer comes, it is the real question being answered — not “is it possible,” which is now settled, but “do you choose it.” That is why it is a wager and not a proof: the one line in all of this that points outward, at people, rather than inward, at the math.