Discharge

Formal definition

A Discharge is a triple $\mathcal{D} = (d, a, t)$:

$$\mathcal{D} = (d : \mathrm{Duty},\; a : \mathrm{Act},\; t \in T)$$

satisfying the discharge condition:

$$\mathrm{discharged}(d, a, t) \iff \sigma_t(a) = a \;\wedge\; \Delta_t(a) = a \iff a \in H^*_t$$

where $H^*_t = \mathrm{Fix}(\sigma_t) \cap \mathrm{Fix}(\Delta_t)$ is the doubly-quiescent stable sublattice at history $t$.

Components:

• $d$ is the duty being discharged — a Duty $(a_{\mathrm{bearer}}, p, N, \varphi, r)$ specifying what must be done
• $a$ is the performance act — the act performed by $d$’s bearer in satisfaction of $\varphi$
• $t \in T$ is the settling history — the point in the history at which discharge is evaluated; the performance must settle at $t$

Three invariants. $\mathcal{D}$ is a discharge iff it satisfies:

1. Execution settlement: $\Delta_t(a) = a$. The act has been performed according to fixed procedure — the execution path is complete, the required steps have been taken, the act is procedurally finished. Without this, the duty is not yet performed.

2. Meaning settlement: $\sigma_t(a) = a$. The performance is recognized by the governing normative system $N$ and by the correlative party $r$ as constituting satisfaction of $\varphi$. Without this, the act may have been performed but is not yet acknowledged as discharge. A payment that the creditor does not recognize as payment is not discharge; a filing the court does not record is not discharge.

3. Joint settlement: both 1 and 2 must hold simultaneously. $a \in H^*_t$. This is why discharge requires the fixed point of both nuclei — a duty can be:

   • Executed but unrecognized ($\Delta_t(a) = a$, $\sigma_t(a) \neq a$): the act was done but not acknowledged — procedurally complete, semantically open
   • Acknowledged but unexecuted ($\sigma_t(a) \neq a$, $\Delta_t(a) = a$ — this case doesn’t arise cleanly; acknowledgement of an unperformed act is not discharge)
   • Fully discharged ($a \in H^*_t$): both settled simultaneously

The execution/recognition gap

The gap between execution settlement and meaning settlement is the most practically significant feature of discharge. Consider:

Scenario	$\Delta_t(a) = a$?	$\sigma_t(a) = a$?	Discharged?
Act performed and recognized	Yes	Yes	Yes
Act performed, recognition pending	Yes	No	No
Act performed, recognition refused	Yes	No	No — dispute arises
Act not performed	No	—	No

The execution/recognition gap creates the space for duty-discharge disputes: the bearer claims $a \in H^*_t$; the correlative party claims $\sigma_t(a) \neq a$ (the performance does not satisfy $\varphi$). The resolution requires the governing normative system $N$ to adjudicate — applying $\sigma_t$ to determine whether the act counts.

Discharge vs. satisfaction vs. performance

These three terms are commonly conflated but are formally distinct:

• Performance: the bearer does act $a$ — the execution, regardless of recognition
• Satisfaction: the correlative party $r$ accepts $a$ as meeting their claim — the recognition, regardless of whether $a$ was formally performed
• Discharge: $a \in H^*_t$ — both execution-settled and meaning-settled; the duty is legally/normatively extinguished

Discharge is the strongest condition: it requires both performance and satisfaction to be simultaneously settled. A payment satisfies the creditor (recognition) and discharges the debt (joint settlement) simultaneously. A substitute performance satisfies only if the normative system $N$ accepts substitution ($\sigma_t$ maps the substitute act to the same fixed point as the specified act).

Discharge by operation of law

In some cases, discharge occurs without performance: statute of limitations, frustration of contract, impossibility, merger. These are meaning-closures ($\sigma_t$) applied to the duty itself rather than to a performance act — the duty is moved to its settled state by the system’s recognition that it no longer applies, not by the bearer’s act.

Formally: $\sigma_t(d) = d^*$ where $d^* \in H^*_t$ is the discharged state of the duty, achieved without a corresponding performance act $a$. This is the exceptional case: discharge is normally $(d, a, t)$ with a non-trivial performance; discharge by operation of law is the degenerate case where $a = \varnothing$ and $\sigma_t$ acts directly on $d$.

Relation to the nuclear quartet

The discharge condition $a \in H^*_t$ places the performance in the Process position of the nuclear quartet — the doubly-closed position where both meaning and execution are settled. An unperformed duty corresponds to the Inquiry position (neither closed); a procedurally specified but unexecuted duty corresponds to Procedure (meaning-closed only); a performed but unrecognized act corresponds to Derivation (execution-closed only).

Open questions

• Whether partial discharge is coherent — whether a divisible duty can be partially discharged by partial performance, and what the formal condition for partial discharge is.
• Whether the meaning-settlement condition $\sigma_t(a) = a$ is sufficient when $N$ is contested — whether discharge requires recognition by a specific party (the correlative $r$) or by the normative system $N$ more generally.
• The formal relationship between discharge and the Clerk’s authentication function $A$ — whether the clerk’s entry of a judgment is the $\sigma_t$ operation that converts an executed judicial decision into a discharged legal obligation.