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irsa.institute › library › r-star-index
A universal measurement framework for institutional regenerative capacity. Combines structural invariants with behavioural dimensions into a single index on [−1, 1].
Where it sits. Cross-trunk index enabling empirical comparison across domains.
Institutional performance is measured almost entirely through flows — GDP, budget outlays, annual output — and a flow says nothing about whether a system can still do this in twenty years. R* combines four structural invariants with three behavioural indicators to answer that. But the paper’s central result is negative: the index cannot be a weighted sum. Build it that way and a grant-funded system — realised recycling zero by definition — scores as fully regenerative.
The structural half asks whether the architecture permits regeneration: SΔ decoupling from fragility cycles, SΛ alignment of capital with mission cycles, Ra realised recycling, and X non-extraction. The behavioural half asks whether it is actually happening: capability gradient, volatility under shocks, mission-cycle completion.
The word doing the work is necessary. These are not seven things that contribute. Each invariant is a precondition, and the theory is therefore conjunctive — which is a claim about logical form, and it is what the arithmetic then has to respect.
The natural first attempt is a weighted sum of the two composites, architecture weighted above behaviour. It is what the source specification used, and it contradicts the theory it exists to express — because a weighted sum is compensatory: every term substitutes for every other at a fixed exchange rate. The source states that behaviour cannot compensate for degenerate architecture in the same paragraph that sets a weight at which it does.
Cases one to three are severe. The fourth is disqualifying. Ra = 0 is not an extreme parameter choice — it is a grant, the instrument this entire research programme exists to offer an alternative to. An index that certifies a grant-funded system as fully regenerative is not mis-calibrated; it is measuring something other than regeneration.
And no choice of weights repairs it. Any strictly positive weight on behaviour lets behaviour offset a zero structural score; a zero weight discards the dimension the index exists to combine. The functional form has to change — as do the two obvious substitutes, since recentring the sum or replacing it with a minimum both still score the grant positive, each keeping an arithmetic composite in which a zero is averaged away.
Regeneration requires that the architecture permit it and that the system exhibit it. The first is a gate. Structural capacity G is the harmonic mean of the four invariants, so it vanishes the moment any one of them does — which is the formal content of “each is necessary” — and the index is R* = G · B where behaviour is non-negative, and R* = B where it is not. Capability decline passes through undamped; good architecture does not soften it.
The harmonic mean was chosen over a geometric one for what happens just above zero, because in practice nothing measures at exactly zero. A pool recovering a tenth of what it deploys scored 0.562 under a fourth root — more than half of maximum structural capacity — and scores 0.308 here.
Read as a decomposition it is two questions: can it? times is it? Because G is a harmonic mean the smallest invariant dominates it, so a low score resolves to the binding constraint rather than to an average — which is what a system governed by necessary conditions needs a diagnosis to do.
The gated form separates configurations a single-dimension metric conflates — and the two in the middle are the ones that matter, because both look healthy on output.
| Configuration | Reads | What it looks like from outside |
|---|---|---|
| Sound but underfunded | small positive | Struggling. It is the improvable case. |
| Doomed, currently performing | exactly zero | Healthy. G = 0 with strong behaviour — every output measure approves. |
| Sound, in decline | negative | Healthy for now. Good architecture, capability falling. |
| Both strong | large | Actually regenerating. |
Applied to four sectors, the index returns something sharper than a spread of scores. Every conventionally financed row carries Ra ≈ 0 — and that column is not an assumption. The source states it, four times, once per sector. A zero invariant sets G = 0, so none of them is regenerating at all, and no behavioural improvement can change that while it remains so.
| Sector | As conventionally financed | Backed by a recycling instrument |
|---|---|---|
| Health | G 0.00 · R* 0.00 — not regenerating | G 0.80 · R* +0.28 — regenerative |
| Climate | G 0.00 · R* −0.15 — degenerative | G 0.84 · R* +0.25 — regenerative |
| Science | G 0.00 · R* −0.05 — degenerative | G 0.80 · R* +0.32 — regenerative |
| Civic | G 0.00 · R* −0.20 — degenerative | G 0.75 · R* +0.19 — emergent |
That is the argument delivered as a consequence rather than asserted. The instrument was not built to favour recycling capital; it was built to express four necessity conditions, and the sector table is what those conditions say about how these systems are funded today.
The negative result is not new, and the paper says so. In the literature on composite indicators the property is called compensability, and the finding is textbook: under additive aggregation the weights act as substitution rates rather than importance coefficients. The precedent that matters is not a paper but an operating index — the Human Development Index made this exact repair in 2010, moving from an arithmetic to a geometric mean for precisely this reason. R* reaches the same repair sixteen years later in a different domain.
That costs the claim of a novel result. What survives is the application, and it is the more interesting claim anyway: that capital continuity belongs to the class of dimensions for which compensation must not be admitted — a claim about institutions, not about index arithmetic, which was long settled.
🔴 And it reports no measurements. No institution, sector or country has been scored against R*. The paper specifies an instrument; every figure in it is an illustrative parameter choice. That distinction is the whole credibility of an index, so it is stated here rather than left to a footnote.
The works either side of this one in the canon order — what it builds on and what builds on it.
Where to go once you have the argument — the paper it comes from, the instrument that measures it, or the next thing it depends on.