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The Law of Legitimacy — An Algebra of Real Cost
The Law of Legitimacy — An Algebra of Real Cost
Abstract
This article proposes a simple but rarely explicit law:
the legitimacy of an act, a system, or an institution depends directly on its ability to assume the real cost of the power it exercises.
When this cost is displaced, deferred, or transferred, debt accumulates, and reality inevitably demands payment.
This article formalizes the law, proposes a minimal algebraic framework, and shows how it helps explain contemporary phenomena ranging from technical systems to public institutions.
1. Why Speak of Legitimacy Today?
The term legitimacy is often confused with:
- legality,
- consent,
- popularity,
- efficiency,
- or economic performance.
Yet these criteria repeatedly fail to explain why systems that are legal, efficient, and sometimes popular eventually come to be rejected, bypassed, or abandoned.
The thesis defended here is the following:
Legitimacy is a structural property, not an opinion.
It depends on an objective relationship between power and cost.
2. Power and Cost: Basic Definitions
2.1 Power (P)
By power, we mean any capacity to act upon reality:
- producing a technical system,
- organizing an institution,
- imposing a rule,
- transmitting a norm,
- durably influencing behavior.
Power is neither moral nor immoral.
It is a fact.
2.2 Assumed Cost (C)
Cost is what the agent exercising power agrees to pay personally and concretely, including:
- irreversible real time,
- sustained effort,
- discipline,
- renunciation,
- direct responsibility,
- exposure to real consequences.
2.3 Transferred Cost (T)
Transferred cost is the portion of the real price that is not assumed by the agent, but displaced:
- onto other individuals,
- onto subordinates or users,
- into the future,
- onto anonymous or abstract actors.
3. The Fundamental Law
Any real power can be decomposed into assumed cost and transferred cost:
$$ P = C + T $$
This relation is not normative.
It describes an observable invariant.
When transferred cost increases, debt accumulates, even if the system appears to function.
4. Legitimacy and Cost Transfer
Legitimacy (L) is not a symbolic reward, but:
the capacity of a system to endure without permanent corrective constraint
(control, coercion, justification, or violence).
The central relation can be expressed as:
$$ L = f(C, T) $$
with:
$$ \frac{\partial L}{\partial C} > 0 \qquad \frac{\partial L}{\partial T} < 0 $$
In other words:
- the more a system assumes its costs, the more legitimate it is;
- the more it transfers its costs, the more its legitimacy erodes.
A system may remain legal, efficient, or profitable while steadily losing legitimacy.
5. Debt and the Return of Reality
5.1 Accumulation of Debt
A transferred cost never disappears. It accumulates as debt:
$$ D(t) = \int T(t),dt $$
This debt may remain invisible for a long time.
5.2 The Return of Reality
When debt exceeds a critical threshold, reality enforces payment:
- crisis,
- rupture,
- collapse of trust,
- mass rejection,
- symbolic or material violence.
This return is generally non-linear:
$$ R = g(D) \qquad g \text{ is convex} $$
This is why systems often appear stable…
until they suddenly are not.
6. Non-Transferable Costs (C*)
Some costs cannot be displaced without irreversible loss. We denote them C*.
Typical examples include:
- trust,
- system auditability,
- educational transmission at the right moment,
- foundational coherence,
- health and sleep.
A non-transferable cost that is not paid at the right time
cannot be correctly paid later.
Violating a C* produces a brutal collapse of legitimacy that no later compensation can repair.
7. How to Identify Costs
Costs are not measured only in monetary units. They are detected through traces.
7.1 Traces of Assumed Cost
- irreversible time invested,
- real fatigue,
- explicit renunciations,
- personal exposure to consequences.
7.2 Traces of Transferred Cost
- imposed delays and frictions,
- anonymization of the cost bearer,
- multiplication of intermediaries,
- permanent need for justification.
7.3 Simple Test for C*
Can this cost be paid later without qualitative loss?
If the answer is no, it is a non-transferable cost.
8. A Practical Tool: The Legitimacy Balance Sheet
At the end of a cycle (project, reform, decision), ask four questions:
- What cost did I assume directly? (C)
- What cost did I transfer? (T)
- What cannot be recovered? (C*)
- How will reality make me pay? (R)
If any answer is vague, legitimacy has already begun to erode.
9. Scope and Limits
This law:
- does not guarantee success,
- does not remove tragedy,
- does not replace decision-making.
But it does provide one essential capability:
preventing the confusion of temporary efficiency with durable legitimacy.
10. Conclusion
The law of legitimacy recalls a fundamental constraint that is often forgotten:
No one can sustainably exercise a power whose real cost they refuse to pay.
This law does not tell us what to desire.
It tells us what must be assumed.
In a world increasingly built on transfer, insurance, and compensation, it offers a simple criterion to distinguish what still holds from what is already hollow.
11. A Concrete Numerical Case — Power, Debt, and Legitimacy
To make the law of legitimacy fully operational, we now consider a concrete, quantified case drawn from a technical domain where cost transfer and the return of reality are especially visible:
a deterministic, auditable kernel / SoC project.
The goal is to deliver a credible v1 in 12 weeks.
Here, power $P$ represents the real value delivered: functional capability and structural robustness.
Decomposing Power
We decompose total power into two components:
$$ P = P_f + P_q $$
Where:
- $P_f$ = functional power (features: shell, networking, drivers, etc.)
- $P_q$ = qualitative power (determinism, invariants, auditability, tests, documentation)
Assume that a credible v1 requires:
- $P_f = 60$
- $P_q = 40$
Thus:
$$ P = 100 $$
Cost Allocation
As stated by the fundamental law:
$$ P = C + T $$
Where:
- $C$ = cost paid now (discipline, structure, tests, documentation)
- $T$ = cost transferred (technical debt, latent bugs, ambiguity, “we’ll fix it later”)
Scenario A — Speed First, Structure Later
You focus on visible features and postpone structural rigor.
Costs actually paid:
- $C_f = 55$
- $C_q = 10$
Total assumed cost:
$$ C = 65 $$
The system therefore actually delivers:
$$ P’ = C = 65 $$
Yet the project is announced as $P = 100$.
The transferred cost (debt) is:
$$ T = P - C = 35 $$
Debt and Return of Reality
Assume the return of reality (bugs, regressions, instability, loss of trust) grows faster than linearly:
$$ R = k \cdot T^2 $$
With an illustrative calibration $k = 0.02$:
$$ R = 0.02 \cdot 35^2 = 24.5 $$
The effective power after reality catches up is:
$$ P_{\text{eff}} = P - R = 75.5 $$
You delivered something — but not the v1 that was promised.
Scenario B — Paying the Non-Transferable Cost
You deliver fewer visible features, but secure the structural invariants.
Costs actually paid:
- $C_f = 45$
- $C_q = 30$
Total assumed cost:
$$ C = 75 $$
Transferred cost:
$$ T = 100 - 75 = 25 $$
Return of reality:
$$ R = 0.02 \cdot 25^2 = 12.5 $$
Effective power:
$$ P_{\text{eff}} = 87.5 $$
You ship less surface-level functionality, but a tenable and explainable system.
Introducing Legitimacy Explicitly
We now introduce legitimacy as a structural function of cost allocation:
$$ L = f(C, T) \qquad \text{with} \qquad \frac{\partial L}{\partial C} > 0, \quad \frac{\partial L}{\partial T} < 0 $$
In this domain, a key non-transferable cost $C^*$ is auditability and determinism.
We impose a structural constraint:
- If $C_q < 25$, the system is not auditability-grade.
- Below this threshold, legitimacy collapses.
We model this as:
$$ L = \begin{cases} 1 & \text{if } C_q \ge 25 \ 0.3 & \text{if } C_q < 25 \end{cases} $$
Legitimacy Outcomes
- Scenario A: $C_q = 10 < 25 \Rightarrow L = 0.3$
- Scenario B: $C_q = 30 \ge 25 \Rightarrow L = 1$
We define useful power as:
$$ U = L \cdot P_{\text{eff}} $$
Results:
-
Scenario A: $$ U = 0.3 \cdot 75.5 = 22.65 $$
-
Scenario B: $$ U = 1 \cdot 87.5 = 87.5 $$
Interpretation
Scenario A appears fast but produces a non-transmissible artifact.
When legitimacy collapses, most of the apparent power evaporates.
Scenario B progresses more slowly but produces a system that:
- can be audited,
- can be transmitted,
- can be trusted,
- can endure.
Failing to pay a non-transferable cost does not merely delay delivery — it destroys value.
Generalization
The same algebra applies beyond technical systems:
- $C$ = real presence, effort, responsibility
- $T$ = deferred load, compensation, substitution
- $R$ = conflicts, breakdowns, exhaustion
- $C^*$ = moments or properties that cannot be replaced later
Some costs do not come back as debt.
They come back as loss of legitimacy.