See the Equations — Mêtior (MEΩ)

Define MEΩ from first principles, show how we compute price, weights, returns, and risk in MEΩ units, with exact checks so anyone can audit the math.

Offline demo context

This page is frozen to the bundled snapshot dated 2025-10-08. No live calls run in demo mode; all caps and weights below come from the checked-in JSON/CSV (M_world = 108.4T USD, kappa = 1e-6, P_USD^MEIc = 108.4M USD).

  • CNY 42.4928T (39.2%)
  • XAU 22.1136T (20.4%)
  • USD 21.4632T (19.8%)
  • EUR 16.8020T (15.5%)
  • BTC 2.2764T (2.1%)
  • JPY 1.7344T (1.6%)
  • XAG 1.0840T (1.0%)
  • ETH 0.4336T (0.4%)

1) Notation

SymbolMeaning
j ∈ C(t)One monetary species (USD/EUR/JPY/CHF M2, XAU, XAG, BTC, ETH, …)
PjUSD(t)USD price of species j (FX, spot, or coin price)
Qj(t)Quantity/stock: M2 for fiat, above-ground stock (oz/tonnes) for metals, free-float for crypto
MCjUSD(t)= PjUSD(t) · Qj(t), USD "market cap" of species j
Mworld(t)= Σj MCjUSD(t), Aggregated "world money"
κScale constant (default 10-6)
PUSDMEΩ(t)USD price of 1 MEΩ
wj(t)Weight of species j in MEΩ (Σj wj = 1)

2) Construction of MEΩ (numéraire)

Market caps

MCjUSD(t) = PjUSD(t) · Qj(t)

World pool

Mworld(t) = Σj∈C(t) MCjUSD(t)

Scale & price

PUSDMEΩ(t) = κ · Mworld(t) with κ = 10-6

Example: if Mworld = 174 trn USD, then PUSDMEΩ ≈ 174 million USD.

Weights

wj(t) = MCjUSD(t) / Mworld(t)

(always sums to 1)

Invariants to test (unit checks)

  • • Σj wj(t) ≈ 1
  • • PUSDMEΩ(t) = κ Σj MCjUSD(t)

3) Returns in MEΩ (currency-neutral)

MEΩ-denominated price of any asset i

i(t) = PiUSD(t) / PUSDMEΩ(t)

Log-return in MEΩ

ri,MEΩ(t) = Δ ln P̃i(t) = Δ ln(PiUSD(t) / PUSDMEΩ(t))

This removes global money drift; you are measuring relative to the whole money universe.

Discrete computation (daily)

r_i_MEΩ[t] = log(P_i_USD[t]/P_MEΩ_USD[t]) 
            - log(P_i_USD[t-1]/P_MEΩ_USD[t-1])

4) Adaptive weights (replicator dynamic)

Continuous-time form (for intuition)

dwj = wj[rj(λ) - Σk wk rk(λ)] dt

rj(λ) = clip(rj, -5σj, +5σj)

Why clip? To keep the update Lipschitz under jumps; avoids blow-ups.

Discrete update (what we actually run)

r_clip[j] = clamp(r[j], -5*σ[j], +5*σ[j])
w[j] ← w[j] + w[j]*(r_clip[j] - Σ_k w[k]*r_clip[k])
normalize(w)   # divide by Σ_j w[j]

5) Risk in MEΩ: σ, VaR/CVaR

Realised volatility (rolling window W)

σi(t) = stdev(ri,MEΩ[t-W+1 : t])

GARCH(1,1) on ri,MEΩ (optional)

σt2 = ω + α rt-12 + β σt-12

Tail risk (EVT, POT-GPD) & CVaR

Estimate tail with GPD on excesses above threshold; compute VaRα and CVaRα in MEΩ units.

6) GINα — Global Interest-Rate Neutral Alpha (in MEΩ)

Definition

GINαt = [(1 + RtMEΩ) / (1 + rf,tMEΩ + πtMEΩ)] - 1 - carryt

Where:

  • RtMEΩ: portfolio return in MEΩ
  • rf,tMEΩ: global risk-free in MEΩ (MEΩ-weighted yields + storage/staking)
  • πtMEΩ: global inflation proxy in MEΩ
  • carryt: storage (metals) / staking (crypto) − fees

Purpose: isolate skill, not currency or rate cycles.

7) Pricing under the MEΩ numéraire (optional)

Change of numéraire: With MEΩ as numéraire, discounted asset S̃t = St/PMEΩ is a martingale under QMEΩ.

BS PDE (sketch) in MEΩ units

∂V/∂t + ½σ²S̃² ∂²V/∂S̃² = rfMEΩ V

Implementation usually uses Monte Carlo or local-vol; this is included for completeness.

8) Token mechanics (if/when tokenized)

Supply

QMEΩ = 1/κ (e.g., 106)

Oracle

Posts (Mworld, PUSDMEΩ, {wj}) from open feeds.

Integrity

Merkle/zk proofs over inputs; MEΩ is a measurement unit, not a claim on reserves.

9) Governance math (minimally sufficient, deterministic)

Inclusion rule

  • • Fiat: public M2
  • • Metals: auditable stock × spot
  • • Crypto: free-float mcap
  • Threshold: MCjUSD / Mworld ≥ 1% (policy band OK)

Cadence

Daily recompute; no human overrides.

Delisting

Stale data > 60 days or liquidity failure ⇒ set wj → 0 and renormalize.

Disclosure

Persist (date, symbol, weight, meo_usd, m_world_usd) daily.

10) Implementation recipe (discrete-time, vectorised)

  1. Pull data (FRED M2, LBMA/FRED spot, CoinGecko caps, FX)
  2. Compute caps: MC = P · Q; sum to Mworld
  3. Price: PUSDMEΩ = κ Mworld
  4. Weights: w = MC / Σ MC
  5. MEΩ returns: ri,MEΩ = Δ ln(PiUSD / PMEΩ)
  6. Risk: realised σ / GARCH; CVaR via EVT
  7. Publish: CSV/JSON & audit rows

11) Verification & audit

Unit checks (must pass)

abs(sum(w)-1) < 1e-9
P_meo_usd == kappa * sum(MC_usd) within 1 bp
len(r_MEΩ) == len(P) and finite

Audit SQL (DuckDB/SQLite)

-- Latest disclosed basket
SELECT date, symbol, ROUND(weight*100,2) AS pct, 
       meo_usd, m_world_usd
FROM benchmarks
WHERE date = (SELECT MAX(date) FROM benchmarks)
ORDER BY weight DESC
LIMIT 10;

Example API payload

{
  "date": "2025-10-08",
  "meo_usd": 108400000,
  "m_world_usd": 108400000000000,
  "weights": [
    {"symbol": "CNY", "w": 0.392, "mc_usd": 42492800000000},
    {"symbol": "XAU", "w": 0.204, "mc_usd": 22113600000000},
    {"symbol": "USD", "w": 0.198, "mc_usd": 21463200000000},
    {"symbol": "EUR", "w": 0.155, "mc_usd": 16802000000000},
    {"symbol": "BTC", "w": 0.021, "mc_usd": 2276400000000},
    {"symbol": "JPY", "w": 0.016, "mc_usd": 1734400000000},
    {"symbol": "XAG", "w": 0.01, "mc_usd": 1084000000000},
    {"symbol": "ETH", "w": 0.004, "mc_usd": 433600000000}
  ]
}

12) Edge cases & numerics

  • Data lag: forward-fill ≤ 60 days; beyond ⇒ drop & renorm
  • Missing FX: if PjUSD is NaN, exclude j for the day
  • Clipping: r(λ) = clip(r, -5σ, +5σ) ensures stable updates
  • Rounding: keep floats for math; display with fixed decimals; ledger can use Decimal for sizing

13) Stability lemmas (short sketches)

Clip is 1-Lipschitz

|clip(x) - clip(y)| ≤ |x - y|

⇒ preserves Lipschitzness when composed with returns

Simplex invariance

Replicator update keeps wj ≥ 0 and Σj wj = 1 after renorm.

Drift control

V(w) = Σj log(1/wj)

Foster–Lyapunov shows negative drift away from boundaries under clipped returns.

(Full proofs in the whitepaper; we keep the page practical.)

14) Worked numeric example (sanity)

Given (mock):

  • • Mworld = 108.4 trn USD, κ = 10-6
  • • Then PUSDMEΩ = 108.4 million USD
  • • BTC cap = 2.26 trn ⇒ wBTC = 2.26/108.4 ≈ 2.1%
  • • Asset i price rises 3%, but PMEΩ rises 1%
  • • ⇒ ri,MEΩ ≈ ln(1.03/1.01) ≈ 1.98%

TL;DR (for this page)

  • Price: PUSDMEΩ = κ Σj MCjUSD
  • Weights: wj = MCj / Σ MC
  • Returns: ri,MEΩ = Δ ln(Pi / PMEΩ)
  • Risk: σ/CVaR on MEΩ returns
  • GINα: strips global rates & inflation (in MEΩ)
  • Governance: deterministic rules; daily disclosure; no discretion