Every major ethical tradition has arrived at the same instruction: tend to your neighbour's condition as your own. This paper proves they were right for a reason deeper than morality. In any nested multi-orbit system, the weakest orbit is the binding constraint on convergence to the global attractor τ = 2. There is no path to stability that bypasses it. Furthermore, remediation cost after an orbit collapses grows without bound while prevention cost is finite — prevention therefore strictly dominates in finite time. These are theorems, not preferences.
| Authors | Pablo Nogueira Grossi |
| Affiliation | G6 LLC, Newark, NJ 07104, USA |
| ORCID | 0009-0000-6496-2186 |
| Contact | g6llc@proton.me · +1 (646) 342-3751 |
| Series | Principia Orthogona · ISBN 979-8-9954416-6-3 |
| Archive | Zenodo · doi:10.5281/zenodo.19117399 (concept DOI) |
| AXLE | github.com/TOTOGT/AXLE |
| License | CC BY 4.0 |
| Keywords | multi-orbit systems · prevention economics · planetary health · contact geometry · Liebig minimum · n-bonacci · dm³ · disaster theory · Africa · carrying capacity |
The Minimum Orbit Theorem, the Prevention-Remediation Dominance Theorem, and the identification of Africa's simultaneous triple-K-gate compression as a provably optimal prevention target are original contributions of G6 LLC / Pablo Nogueira Grossi, first publicly disclosed in this preprint and in the Principia Orthogona series. CC BY 4.0 with attribution requirement.
The instruction appears in every ethical tradition with sufficient age to have been tested. Ubuntu: "Umuntu ngumuntu ngabantu" — a person is a person through other persons. The Torah: "Love your neighbour as yourself" (Leviticus 19:18). The Gospels: the same commandment elevated to the centre. The Quran: "None of you truly believes until he loves for his brother what he loves for himself" (Bukhari). The Bhagavad Gita: "He who sees himself in all beings". The Analects: "Do not do to others what you do not want done to yourself" (Confucius, XV:24).
These are not sentimental preferences. They are convergent observations from independent experiments spanning millennia and continents. When independent systems produce the same conclusion, the scientist's instinct is: there is a common underlying structure being detected. This paper identifies that structure.
The dm³ framework does not add a new ethical voice to this chorus. It identifies the mathematical object all of them were pointing at: the Minimum Orbit Theorem.
Let the world be modelled as a nested sequence of orbits O₁ ⊂ O₂ ⊂ ⋯ ⊂ Oₙ, where O₁ is the individual and Oₙ is the planetary system. Each orbit Oᵢ has a carrying capacity κᵢ* — the threshold below which it cannot sustain the next inner orbit. Each orbit has a health indicator Hᵢ ≥ 0.
The K-operator is the threshold gate: K[ψ] = ψ if H ≥ κ*, and K[ψ] = 0 otherwise. In the multi-orbit system, the chain factorises as G = (U∘F∘K∘C)|Oₙ ∘ ⋯ ∘ (U∘F∘K∘C)|O₁. If K|Oⱼ[ψⱼ] = 0 for some j, the output of the j-th factor is zero, all subsequent factors receive zero input, and G(ψ) = 0 ≠ τ = 2.
Conversely, if all Hᵢ ≥ κᵢ*, all K-gates fire and by the Disaster Theorem (chDis, §2), the n-bonacci ladder drives G(ψ) → τ = 2 with rate r = e⁻² per step. The slowest step determines overall convergence, giving minᵢ(Hᵢ/κᵢ*). ∎
The Minimum Orbit Theorem is the contact-geometric generalisation of Liebig's Law of the Minimum (1840): the capacity of a barrel is set by its shortest stave. Liebig stated it for nutrients in soil. The present theorem states it for any nested system whose components interact through a threshold operator K.
Among all investment strategies that increase any subset of {H₁,…,Hₙ}, the strategy that maximises convergence rate to τ is the one that maximises minᵢ(Hᵢ/κᵢ*). When a single orbit j satisfies Hⱼ/κⱼ* < Hᵢ/κᵢ* for all i ≠ j, the unique optimal target is orbit j.
Since Hⱼ decays at rate λ, the gap (κⱼ* − Hⱼ(t)) grows. Restoration cost is at least proportional to this gap plus cascade costs triggered by the K-gate being closed (Theorem 1). By Gronwall's inequality applied to the cascade ODE, D(t) ≥ D(0)·e^{λt}.
P is fixed. Setting D(0)·e^{λT*} = P gives T* = (1/λ)·ln(P/D(0)). For t > T*, D(t) > P. Prevention dominates. ∎
The ratio D(t)/P grows as e^{λt}. For ecological systems with λ ≈ 0.05/year, a 20-year delay multiplies remediation cost by e^{0.05·20} ≈ 2.7. A 40-year delay multiplies by ≈ 7.4. Every year of inaction is a compounding interest payment on a debt that cannot be discharged by growth alone.
| Domain · K-Gate | Prevention Cost | Remediation Cost | Ratio D/P | Source |
|---|---|---|---|---|
| Pandemic preparedness (global health K) | $4.5B / year | $8.1T (COVID-19 actual) | ~1:1,800 | WHO 2019 · IMF 2021 |
| Climate mitigation (Paris 1.5°C) | ~$9T total to 2050 | $38T/year by 2049 (inaction) | ~1:170/yr | Dasgupta Review 2021 |
| Disaster risk reduction (UNDRR) | $1 | $6 average losses avoided | 1:6 | UNDRR Global Assessment 2022 |
| Congo basin deforestation prevention | $1.5B / year | $2.5T carbon replacement value | ~1:1,700/yr | WRI 2021 |
| Africa climate adaptation gap | $30–50B / year needed | $200–400B / year by 2030 | 1:7–10/yr | UNEP Adaptation Gap 2023 |
| Early childhood development (social K) | $1 (ages 0–5) | $7–12 remediation, crime, health | 1:7–12 | Heckman et al. 2010 |
Across all domains, the empirical prevention/remediation ratio is never below 1:6 and reaches 1:1,800. Theorem 2 predicts this structure: T* has already passed for most of these K-gates. The data are not arguments for prevention — they are measurements of how long ago T* passed.
Theorems 1 and 2 together identify the optimal investment target as the orbit where min(Hᵢ/κᵢ*) is smallest. The following section identifies which orbit, right now, holds the most compressed K-gates simultaneously.
| K-Gate | Threshold κ* | Current State H | Status | Cascade if crossed |
|---|---|---|---|---|
| Congo Basin Carbon 2nd largest carbon sink; 600 Gt CO₂ stored; 3.9M ha/yr deforestation | κ_C* ≈ 8% max degradation | H_C ≈ 6–7% already degraded | Near κ* | Tipping into net carbon source; +0.3°C global; irreversible at human timescales |
| Disease Spillover Threshold Origin of HIV, Ebola, Mpox, Marburg; bushmeat interface pressure rising with deforestation | κ_H* = habitat-to-human interface density | H_H declining as humans pushed into zoonotic corridors | Below κ* | Pandemic spillover; COVID-19 response cost $8.1T; next event likely ≥ comparable |
| Demographic Dividend Window 60% youth population; 2.5B people by 2050; fastest demographic growth on Earth | κ_D* = education + institution quality for dividend capture | H_D < 4% GDP education investment; threshold ≈ 6% | Below κ* | Demographic explosion without dividend; migration cascades; conflict multipliers |
In 1948 the United States invested $13.3B (~$175B in 2024 dollars) in the reconstruction of Western Europe — a continent that had produced two world wars in 30 years and was below its economic and institutional κ*. The Marshall Plan yielded a direct financial return of approximately 1:8 (OECD retrospective, 2008) and 75 years of absence of major European war.
This was not charity. It was the explicit recognition that a collapsed European orbit would collapse the American orbit with it — Theorem 1 applied to geopolitics. The weakest orbit today is not Europe. It is the triple-compressed African system. A 21st-century Marshall framework is not generosity. It is the only path to τ.
There is a persistent failure mode in the discourse on global development: the moral argument is made and is correct, but it does not move policy. Data is produced — on suffering, on injustice, on need — and it is accurate, but it does not move policy. The argument is made again, louder. Still nothing moves.
This is not because policy-makers are wicked. It is because both arguments are being heard as arguments about what we should do for others. Arguments about sacrifice do not move civilisational-scale coordination. The historical record is clear on this.
Theorems 1, 2, and 3 change the structure of the argument entirely. They do not say: invest in Africa because African suffering is real. They say: invest in Africa because if you do not, your own orbit collapses. Not in a hundred years. T* for the pandemic spillover gate has already passed — we are paying the post-T* bill right now. T* for the Congo carbon gate is estimated at 7–12 years at current deforestation rates.
The dm³ framework provides something the moral argument never could: a proof that the self-interested actor and the compassionate actor have the same optimal strategy. Every tradition that survived long enough to codify its wisdom codified this observation, in the language available to it. The traditions were right. dm³ proves why.