Epistemic Mode Of A Proposition

one_liner: The same proposition can be held in different epistemic modes — asserted-fact, assumption, axiom, hypothesis, premise, belief, conjecture — and the mode, independent of the content, gates which operations are licensed on it: you may provisionally rely on an assumption, must test a hypothesis, cannot refute an axiom within its system, must discharge a premise.

review_flags: Created 2026-06-13 post Phase-C emergent reconciliation (owner-approved); did NOT go through Phase-A triage. SI/SF/similarity grades are PROPOSER ESTIMATES pending triangulation.

review_flags: Phase-C action: re-similarity check vs the listed neighbors to confirm it does not collapse onto an existing prime or candidate; draw the proposed parent/child edges at incorporation.

core_idea:
The epistemic mode of a proposition is the structural pattern by which one and the same proposition can be *held in different ways* — as an asserted fact, an assumption, an axiom, a hypothesis, a premise, a belief, or a conjecture — and the *way* it is held, independent of *what it says*, gates which operations are licensed on it. The decisive observation is that the mode is a dimension orthogonal to the content. The sentence "the bridge bears 40 tonnes" is one proposition; it is an entirely different object depending on whether it is *asserted* as a verified fact, *assumed* for the sake of a calculation, posed as a *hypothesis* to be tested against load data, taken as a *premise* in a safety argument, or held as a *belief* by an engineer. The proposition is identical in all five; the mode differs, and the mode is what determines the permissible moves: you may provisionally *rely on* an assumption while flagging that the conclusion is conditional on it; you must *test* a hypothesis rather than rely on it; you *cannot refute* an axiom from within the system that posits it (only change the system); you must eventually *discharge* a premise to convert a conditional conclusion into a categorical one; you should *update* a belief on evidence; you treat a *datum* as given but subject to measurement error; you hold a *conjecture* as a target for proof or disproof. The defining commitments are four. First, there is a *proposition* — a truth-apt content that could be true or false. Second, there is a *mode* attached to it — a label drawn from a small typed vocabulary (asserted-fact, assumption, axiom, hypothesis, premise, belief, conjecture, datum, stipulation) that records *how* the content is being held. Third, the mode is *content-invariant*: the same proposition can carry any mode, and changing the mode changes nothing about what the proposition says. Fourth, and load-bearing, the mode *gates the licensed operations*: each mode comes with a characteristic set of moves that are permitted, required, and forbidden on the proposition, and a reasoning error results from applying a move licensed by one mode to a proposition that is actually in another.

The structural signature is a *typed handle that travels with a proposition and determines its operation set*. It is what lets a reasoner answer, of any claim on the table, not "is this true?" but the prior question "in what mode is this being held, and therefore what am I allowed to do with it?" The single most consequential fact the prime names is that *the mode, not the content, fixes the operations* — which is why the same sentence can be untouchable in one mode (an axiom one cannot refute from inside the system) and mandatory-to-attack in another (a conjecture one is obliged to try to disprove), and why the most common and damaging reasoning errors are *mode errors*: treating an assumption as an established fact (relying on it without flagging the conditionality), treating a hypothesis as a belief (defending it instead of testing it), treating a stipulation as a discovery, or treating an axiom as an empirical claim open to falsification. From this single fact follow the prime's downstream properties. Modes are *re-assignable*: the central operations of inquiry are mode transitions — a hypothesis confirmed becomes an asserted fact, an assumption justified becomes a datum, a conjecture proved becomes a theorem, a premise discharged is converted into a conditional — and tracking these transitions is what disciplined reasoning *is*. Modes *compose with the dependency structure*: a conclusion inherits the *weakest* mode among the propositions it rests on, so a result resting on an untested assumption is itself only as secure as that assumption, no matter how rigorous the inference. And the mode discipline is *substrate-general*: the same typed-handle-gates-operations structure appears in formal logic, empirical science, law, and programming, each with its own mode vocabulary but the identical underlying move. What epistemic_mode provides as a prime is the recognition that "how a proposition is held" is a first-class structural feature, separable from the content, that silently governs every legitimate operation on the claim.

This is a candidate proposed at lower confidence, so it is worth stating directly why it clears the bar for a distinct prime rather than collapsing into its neighbors. The pattern earns independent standing on three grounds. First, *it is not any single mode*: `assumption`, `axiom`, and `hypothesis` are each *one value* of the mode dimension; this prime is the *dimension itself* — the recognition that mode is a content-orthogonal variable ranging over those values and gating operations accordingly — which no single-mode prime names, just as a prime for "color" is not subsumed by a prime for "red." Second, *it is not the process of forming a belief*: `belief_formation` names how an agent comes to hold a proposition true; this prime names the *static typed status* a proposition carries and the *operation set that status licenses*, which is a different object (a held belief is one mode among several, and the same proposition can be in a non-belief mode that no act of belief-formation produced). Third, *the mode-gates-licensed-operations structure travels with the same shape across at least four independent disciplines* — the axiom/theorem/conjecture typing of mathematics, the hypothesis/law/datum typing of science, the stipulated/alleged/adjudicated-fact typing of law, and the assert/assume/invariant typing of programming — and a structural pattern that recurs identically across logic, science, law, and code, gating what may be done to a claim in each, is exactly the kind of cross-substrate invariant the catalog exists to name. It is, candidly, the weakest of its drafting batch — its content overlaps heavily with the assumption family and its boundary against "the role a proposition plays" requires care — but the orthogonal-dimension-that-gates-operations reading is real, recurs widely, and is not captured by any existing prime.

structural_signature:
*the truth-apt proposition (the content)* — *the mode label attached to it (the typed handle)* — *the content-invariance of the mode* — *the operation set the mode licenses, requires, and forbids* — *the re-assignability of modes through transitions of inquiry* — *the weakest-mode inheritance through the dependency structure*

An epistemic mode is present when each of the following holds:

- **A truth-apt proposition (the content).** A claim with a content that could be true or false — the thing held, independent of how it is held.
- **A mode label (the typed handle).** A label drawn from a small typed vocabulary — asserted-fact, assumption, axiom, hypothesis, premise, belief, conjecture, datum, stipulation — recording *how* the content is currently being held.
- **Content-invariance (the orthogonality invariant).** The mode is independent of the content: the same proposition can carry any mode, and changing the mode changes nothing about what the proposition asserts. Mode is a dimension orthogonal to truth-value and meaning.
- **A licensed operation set (the gating invariant).** Each mode comes with a characteristic set of operations that are permitted, required, and forbidden — rely-on-provisionally for an assumption, test for a hypothesis, do-not-refute-from-within for an axiom, discharge for a premise, update-on-evidence for a belief. This is the load-bearing feature: the mode, not the content, fixes what may be done.
- **Re-assignability (the transition invariant).** Modes can be reassigned, and the core operations of inquiry *are* such transitions: hypothesis confirmed becomes fact, assumption justified becomes datum, conjecture proved becomes theorem, premise discharged becomes a conditional. Tracking the transitions is the discipline.
- **Weakest-mode inheritance (the composition invariant).** A conclusion inherits the weakest (least secure) mode among the propositions it rests on, so a result built on an untested assumption is itself only as secure as that assumption, regardless of inferential rigor.

The components compose into a single object — *a content-orthogonal typed handle that travels with a proposition and gates the operations licensed on it* — which is why the axiom/theorem/conjecture of mathematics, the hypothesis/law/datum of science, the alleged/adjudicated fact of law, and the assert/assume/invariant of code are recognized as the same structure: a mode dimension determining, in each, what may legitimately be done to a claim.

what_it_is_not:
- **Not an assumption.** `assumption` is *one mode* — a proposition held as if true, provisionally, for the purposes of some reasoning, licensing the operation "rely on while flagging conditionality." Epistemic mode is the *dimension* of which assumption is one value; it ranges over assumption, axiom, hypothesis, belief, conjecture, premise, datum, and stipulation, and its content is precisely that the *choice among these* gates the operations. Naming the dimension is not naming any of its values, just as naming "temperature" is not naming "hot."
- **Not an axiom.** `axiom` is another single mode — a proposition held as foundationally true within a formal system, licensing "use freely, do not refute from within." It is one value of the mode dimension, distinguished from assumption precisely *by its mode* (foundational-and-irrevisable versus provisional-and-revisable). The prime is what makes "axiom versus assumption" a difference in mode at all.
- **Not belief formation.** `belief_formation` is the *process* by which an agent comes to hold a proposition true and act accordingly. Epistemic mode is the *static typed status* a proposition carries and the operation set that status licenses. A held belief is one mode among several; the prime concerns the whole typing, including modes (an assumption one disbelieves, an axiom one merely stipulates) that no act of belief-formation produces.
- **Not the truth-value of the proposition.** Truth-value is whether the content *is in fact* true or false; mode is *how it is being held*, which is orthogonal. A false proposition can be held as an assumption, a true one as a mere conjecture; the mode neither asserts nor settles the truth-value, it only fixes what operations are licensed pending the truth-value's determination.
- **Not the proposition's content or its logical role in a single argument.** The content is what the proposition says; mode is independent of it. And mode is not merely the local role a proposition plays *within one inference* (antecedent, conclusion) — it is a standing epistemic status the proposition carries *across* a body of reasoning, governing operations beyond any single argument's structure.
- **Not a confidence level or probability.** A credence (0.7 probable) is a *graded* quantity on a single mode (belief); mode is a *categorical type* that changes the *kind* of operation licensed, not its degree. A hypothesis at credence 0.9 still must be *tested*, not *relied on*, because its mode — not its credence — forbids reliance; raising the number does not change the mode or its operation set.
- **Common misclassification.** Treating a proposition's mode as fixed by its content or its source, and so applying a move licensed by the wrong mode. Catch it by asking, of any claim on the table, "in what mode is this being held — and is the operation I am about to perform one this mode licenses?": surprising reasoning failures concentrate in mode errors (relying on an assumption as a fact, defending a hypothesis as a belief, falsifying an axiom as if it were empirical), not in the content itself.

broad_use: The epistemic-mode structure recurs wherever propositions are held and operated on under a discipline that distinguishes *how* they are held. In **logic and mathematics**, its origin substrate, the mode vocabulary is sharply typed and the gating is explicit: an *axiom* is used freely but cannot be refuted from within its system, a *theorem* is an asserted result that has been proved, a *conjecture* is a target held open for proof or disproof, a *premise* under conditional proof is *discharged* to convert a conditional into a categorical, and a *lemma* is a proved waypoint — and the whole architecture of proof is a discipline of mode transitions (conjecture → theorem, premise → discharged conditional).<!-- FACT-D62-801 -->[^enderton-2001] In **empirical science**, the modes are *hypothesis* (must be tested, never merely relied on), *law* or *established theory* (asserted, but provisionally and open to revision), *datum* (taken as given but carrying measurement error), and *auxiliary assumption* (relied on while flagged), and the central scientific operations — confirmation, falsification, replication — are mode transitions that move a proposition from hypothesis toward established fact or back to refuted.<!-- FACT-D62-802 -->[^popper-1959] In **law**, the gating is institutionalized: a *stipulated fact* is agreed and not contested, an *alleged fact* must be proved to a standard, an *adjudicated fact* has been established by a tribunal, a *presumption* is held true until rebutted, and a *legal fiction* is treated as true for a purpose while known false — each licensing different operations (an alleged fact may be challenged; a stipulated one may not), and the trial is a machine for transitioning facts from alleged to adjudicated.<!-- FACT-D62-803 -->[^fre-rule-201] In **programming and formal methods**, the modes are explicit in the language: an `assert` checks a proposition that *should* hold and halts if it does not, an `assume` (in verification) is *relied on* without checking, an *invariant* is a proposition required to hold at every iteration, a *precondition* is a premise the caller must discharge, and a *postcondition* is an asserted guarantee — and a verification tool's soundness depends on never confusing an `assume` (taken on trust) with an `assert` (checked).<!-- FACT-D62-804 -->[^leino-boogie2] In **everyday and professional reasoning**, the modes are looser but real: a working assumption, a hunch, a settled fact, a tentative hypothesis, a stipulated ground rule, and good reasoning is the practice of keeping track of which is which. Across all of these the recurring fact is identical: a proposition carries a content-orthogonal mode that determines what operations are legitimate on it, and the recurring failure is the mode error — operating on a claim as if it were in a mode it is not.

clarity: Naming the epistemic mode of a proposition separates two questions that reasoners routinely fuse: *what does this claim say (and is it true)?* and *how is this claim being held (and therefore what may I do with it)?* Ordinary attention runs straight to the content and its truth, treating "how it is held" as either obvious or irrelevant; the prime makes the mode a first-class, separable feature and insists it be read before any operation is performed. The clarifying force is to convert a vague unease — "something about how we're using this claim is off" — into the precise question "which mode is this in, and is the move I am making licensed by that mode?" This exposes a whole class of errors as *mode errors* rather than content errors. A team that relies on an untested assumption as though it were an established fact has not made a factual mistake — the content may be fine — they have made a *mode error*, operating in fact-mode on a proposition that is in assumption-mode, and the fix is not to re-examine the content but to either flag the conditionality or transition the assumption to a datum by justifying it. A scientist who defends a hypothesis instead of testing it, a lawyer who treats an alleged fact as adjudicated, a programmer who `assume`s what should have been `assert`ed — each has applied an operation from the wrong mode's license. The prime also clarifies *what disciplined inquiry is*: not the accumulation of true contents but the *correct management of modes and their transitions* — knowing when a proposition has earned the right to move from hypothesis to fact, from conjecture to theorem, from alleged to adjudicated, and refusing to operate on it as though it had made that transition before it has. By relocating the discussion from the content to the mode, the prime makes visible the typed handle that was silently governing every legitimate move all along.

manages_complexity: The epistemic-mode structure compresses the open-ended question "what is the standing of every claim in this body of reasoning, and what may I do with each?" into a small typed vocabulary and a fixed table of licensed operations. Rather than reasoning case-by-case about how to treat each proposition, the reasoner *tags* each with a mode and reads off the permitted, required, and forbidden operations from the mode alone — relying-but-flagging an assumption, testing a hypothesis, using-but-not-refuting an axiom, discharging a premise. This is a substantial reduction because it replaces a per-proposition judgment with a per-*type* rule: there are only a handful of modes, each with a stable operation set, so the management of arbitrarily many propositions reduces to assigning each a type and honoring the type's gating. The compression also makes the *security of a conclusion* computable from the modes of its inputs via the weakest-mode-inheritance rule: a conclusion is only as secure as the least-secure mode among the propositions it rests on, so an analyst can locate a conclusion's true epistemic standing by finding the weakest link in its support — a chain resting on one untested assumption is an untested-assumption-grade conclusion, however many proved theorems surround it. This converts the diffuse worry "how much should I trust this result?" into the bounded task "what is the weakest mode in its dependency set?" And it organizes the work of inquiry as a small set of *mode transitions* with known entry conditions: to strengthen a conclusion, find its weakest-moded support and do the work that transitions it (test the hypothesis, justify the assumption, prove the conjecture, discharge the premise). Because the modes are few, their operation sets fixed, and their composition rule simple, the prime keeps the otherwise-unbounded task of tracking the epistemic standing of a complex argument within reach — the complexity of "what may I do with each of these claims?" reduced to a typed handle and a lookup table.

abstract_reasoning: The epistemic-mode pattern licenses several substrate-portable moves. *Read the mode before operating*: before performing any move on a proposition — relying on it, testing it, refuting it, building on it — the reasoner first asks what mode it is in, because the mode, not the content, determines whether the move is licensed; this single discipline prevents the entire family of mode errors. *Detect the mode error*: when a reasoning process goes wrong despite sound content, the reasoner suspects that an operation from one mode's license was applied to a proposition in another — an assumption relied on as a fact, a hypothesis defended as a belief, an axiom attacked as empirical — and locates the fault in the mismatch rather than re-auditing the content. *Compute the conclusion's grade by weakest-mode inheritance*: to assess how secure a result is, the reasoner finds the least-secure mode among its supporting propositions, because the conclusion can be no more secure than its weakest-moded input, no matter how rigorous the inference connecting them. *Strengthen by transition, not by re-derivation*: to make a conclusion more secure, the reasoner does not re-run the inference but identifies the weakest-moded support and performs the operation that transitions it upward — testing the hypothesis, justifying the assumption, proving the conjecture, discharging the premise — because the conclusion's grade tracks its inputs' modes, not the inference's length. And *separate mode from credence*: the reasoner distinguishes the categorical *kind* of holding (mode) from the graded *degree* of confidence (credence), recognizing that a high credence does not change a hypothesis's mode or license reliance on it — the operation set is fixed by the mode, and a 0.9-probable hypothesis must still be tested rather than relied on. Each move follows directly from the bare structure: because the mode is a content-orthogonal handle that gates operations, the legitimacy of any move is a function of the mode, the security of any conclusion is a function of its inputs' modes, and the work of inquiry is the work of effecting mode transitions.

knowledge_transfer: Because the epistemic mode of a proposition is the bare structural relation of a content-orthogonal typed handle that gates operations, a discipline built around it in one field transfers to any other by re-identifying the field's mode vocabulary and the operation set each mode licenses — and the prime's reach is the reach of that one structure across logic, science, law, and code. The *read-the-mode-before-operating* discipline transfers verbatim: a mathematician who will not refute an axiom from within its system, a scientist who will not rely on an untested hypothesis, a lawyer who will not treat an alleged fact as adjudicated, and a verification engineer who will not confuse an `assume` with an `assert` are all running the identical move — check the mode, then perform only the operations it licenses — distinguished only by each field's mode names. The *weakest-mode-inheritance* rule transfers as a universal grading principle: a physicist locating the auxiliary assumption that makes an experimental conclusion only as secure as that assumption, a mathematician noting that a theorem resting on an unproved conjecture is itself only conjectural, and a software team recognizing that a verified module is only as sound as the `assume`s it was checked against are all computing a conclusion's grade from the weakest mode in its support. The *strengthen-by-transition* move transfers as the shared logic of inquiry: confirming a hypothesis (science), proving a conjecture (mathematics), adjudicating an alleged fact (law), and discharging a verification assumption by checking it (programming) are the same operation — moving a proposition's weakest-moded support up the mode hierarchy to raise the conclusion's grade. And the *mode-error* diagnostic transfers as a portable failure analysis: a surprising breakdown that survives a content audit is, across all four substrates, a candidate mode error — an operation applied from the wrong mode's license — and the repair is to correct the mode, not the content. In every transfer the practitioner runs the identical diagnosis: identify the proposition, read its mode from the field's vocabulary, confirm that the operation being performed is one the mode licenses, and trace a conclusion's grade to the weakest mode in its dependency set. The transfer is secure because none of these steps names a particular field's content — a logician, an experimentalist, a litigator, and a programmer are reasoning about the same content-orthogonal typed handle, distinguished only by the mode vocabulary their discipline uses and the operations it attaches to each mode. What does *not* transfer is any application outside an inferential practice: there is no epistemic mode where there is no reasoner or formal system to *track* the mode and *honor* its gating, because the pattern is constitutively about how a proposition is held within some reasoning, not about any mute physical process.

formal_abstract:
Conditional proof in formal logic is the prime in its purest worked form, because the mode and its gating are fully explicit and a mode transition is the whole point. To prove a conditional $P \to Q$, the reasoner *introduces $P$ as a premise* — and this is a precise mode assignment: $P$ is held in *premise-mode*, which licenses the operations "use $P$ freely within the subproof" and "must eventually discharge $P$," and forbids "treat $P$ as categorically established outside the subproof."<!-- FACT-D62-805 -->[^enderton-2001] The same proposition $P$ could elsewhere be an axiom (used freely, never discharged), a hypothesis (to be tested), or an asserted theorem (proved and freely usable) — the *content-invariance* is exact: $P$ says the same thing in every mode, and only the mode differs. Within the subproof the reasoner derives $Q$ from $P$ (plus the system's axioms, themselves in *axiom-mode*: usable, non-refutable from within). Then comes the *transition* that is conditional proof's entire move: *discharging* the premise $P$ converts the locally-derived $Q$, which depended on $P$, into the categorical conditional $P \to Q$, which depends on $P$ no longer.<!-- FACT-D62-806 -->[^enderton-2001] This is a mode operation, licensed specifically by premise-mode, that no other mode permits — one cannot "discharge" an axiom or a datum. The *weakest-mode inheritance* is visible too: inside the subproof, $Q$ is only as secure as the assumed $P$ (premise-grade, conditional); after discharge, $P \to Q$ inherits the secure mode of a proved result. The *gating* is what makes the proof valid: were the reasoner to treat $P$ as an asserted fact rather than a premise — using $Q$ categorically *without* discharging $P$ — the proof would be fallacious, a textbook mode error. The structural payoff is that the legitimacy of every step is fixed by the *modes* of the propositions, not their contents: discharge is licensed by premise-mode, free use by axiom-mode, and the conditional conclusion is exactly the record of a premise introduced and discharged.

**Mapped back:** Conditional proof instantiates every component — a truth-apt proposition $P$, an explicit mode label (premise), the content-invariance of that label, the operation set premise-mode licenses (use-then-discharge), the transition (discharge converting conditional to categorical), and weakest-mode inheritance ($Q$ conditional until discharge) — and shows the prime's core claim: the mode, not the content, gates which operations are valid.

applied_industry:
A formal-verification pipeline for safety-critical software runs the identical structure in a programming substrate, with no logic-textbook vocabulary, and the soundness of the whole effort depends on honoring the gating. The codebase carries propositions in sharply different modes, marked by distinct language constructs. An `assert(x > 0)` places the proposition "x > 0" in *assert-mode*: it *must be checked*, and the operation it licenses is "verify, and fail if false."<!-- FACT-D62-807 -->[^hoare-1969] An `assume(input is sanitized)` in the verification harness places its proposition in *assume-mode*: it is *relied on without checking*, licensing "use freely as a given, do not verify" — and crucially the *same proposition* "input is sanitized" could appear elsewhere as an `assert` (to be checked) or as a *precondition* (a premise the caller must discharge).<!-- FACT-D62-808 -->[^leino-boogie2] The *content-invariance* is exact and consequential: "input is sanitized" says the same thing in every mode, but an `assume` is taken on trust while an `assert` is checked, so the verifier's *soundness depends entirely on the mode*, not the content. A loop *invariant* places its proposition in *invariant-mode*: it is *required to hold at every iteration*, licensing the inductive operations "show it holds initially" and "show each iteration preserves it."<!-- FACT-D62-809 -->[^hoare-1969] The *weakest-mode inheritance* is the pipeline's central safety fact: a module "proved correct" is only as sound as the `assume`s it was verified against — its guarantee inherits the *assume*-grade of its weakest premise — so a verified component resting on an unchecked assumption carries exactly that assumption's risk, however rigorous its proof. The catastrophic *mode error* is concrete: writing `assume` where one needed `assert` — relying on a proposition that should have been checked — produces a "verified" system that is sound only if the unchecked assumption happens to hold, the verification-engineering analogue of relying on an assumption as a fact.<!-- FACT-D62-810 -->[^leino-boogie2] The prime's discipline is the cure: read each proposition's mode, confirm the operation matches (check the asserts, discharge the preconditions, induct on the invariants, flag the assumes), and trace the system's true guarantee to the weakest mode in its assumption set. The same typed-handle-gates-operations structure governs a scientific result (hypothesis-mode forbids reliance, datum-mode carries measurement error) and a legal case (alleged-mode invites challenge, adjudicated-mode does not).

**Mapped back:** The verification pipeline runs the prime end-to-end — propositions in assert-, assume-, precondition-, and invariant-modes; the content-invariance that makes the same claim checkable or trusted depending only on its mode; the operation sets each mode licenses; and the weakest-mode inheritance that makes a proof only as sound as its weakest assume — and demonstrates the transfer: a logician discharging a premise, an engineer honoring an assume/assert distinction, a scientist refusing to rely on a hypothesis, and a court weighing an alleged versus an adjudicated fact are operating the same content-orthogonal mode dimension.


Equilibrium

type: prime

slug: equilibrium

one_liner: Balanced state.

status: draft

origin_domain: physics

core_idea:
Equilibrium is the state of a system in which opposing forces, fluxes, or pressures balance out such that no net change occurs along the balanced dimensions — even when substantial flow or activity continues locally. Equilibrium is a *balance condition* on a named set of quantities, not an absence of activity. Every equilibrium is specified by three things: (1) which quantities are balanced, (2) which transformations the balance holds against, and (3) the conditions under which it persists.

The mathematical foundation for understanding equilibrium stability rests on **Lyapunov stability theory** <!-- FACT-D11-060 -->[^lyapunov-1892], which provides rigorous criteria for determining whether small perturbations around an equilibrium will decay (stable) or grow (unstable) <!-- FACT-D11-060 -->[^lyapunov-1892]. In statistical mechanics and kinetic theory, **Maxwell's 1860 work on the distribution of molecular velocities** <!-- FACT-D11-061 -->[^maxwell-1860] established how equilibrium emerges from the balance of molecular motions, showing that a dynamical process (particles colliding) converges to a static distribution (the Maxwell-Boltzmann distribution) <!-- FACT-D11-061 -->[^maxwell-1860].

structural_signature:
A system exhibits equilibrium when each of the following holds:

- **Named balance quantity.** A specific variable or flux is the candidate for balance (forces on a particle, supply and demand for a good, reaction rates forward and backward, population immigration and emigration).
- **Opposing contributions identified.** There are at least two identifiable contributions whose net (sum, difference, product) is the balance condition. In chemical systems, this manifests as **Le Chatelier's principle** <!-- FACT-D11-062 -->[^le-chatelier-1884], which states that equilibrium shifts to oppose imposed perturbations; the forward and reverse reaction rates balance until external stress is applied <!-- FACT-D11-062 -->[^le-chatelier-1884].
- **Zero net change on the balanced dimension.** Over a specified time scale, the balance quantity does not drift — though other quantities may be active at that moment.
- **Restoring or maintaining mechanism.** A mechanism exists that returns the system toward the equilibrium if perturbed; without it, equilibrium is accidental rather than structural. **Gibbs's framework for chemical equilibrium and the phase rule** <!-- FACT-D11-063 -->[^gibbs-1875] quantifies the degrees of freedom available for equilibrium states and provides the thermodynamic potential (chemical potential) that governs equilibrium composition and phase transitions <!-- FACT-D11-063 -->[^gibbs-1875].
- **Stability regime.** Equilibria have basins: a range of perturbations small enough that the restoring mechanism wins. Outside the basin, the system leaves the equilibrium.
- **Time scale of balance.** The balance holds relative to a specified time scale; fast fluctuations below it are smoothed, slow drifts above it are outside the claim.

what_it_is_not:
- **Not stasis.** Dynamic equilibrium (a chemical reaction, a steady-state ecosystem, a market at price-clearing) involves vigorous activity; only the *net* on the balanced dimension is zero.
- **Not optimality.** An equilibrium is a balance point, not necessarily a good one. **Nash equilibria in non-cooperative games** <!-- FACT-D11-064 -->[^nash-1950] can be Pareto-dominated; market equilibria can lie far from social optima <!-- FACT-D11-064 -->[^nash-1950].
- **Not uniqueness.** Systems can have multiple equilibria, some stable, some unstable, some "saddle-like." The singular "the equilibrium" is a modeling choice, not a property of the system.
- **Not permanence.** Every equilibrium has a stability regime outside of which it breaks. Calling a state "equilibrium" does not insulate it from tipping points or regime shifts.
- **Common misclassification.** Confusing a non-equilibrium steady state (flows maintained by external driving, as in living systems) with true equilibrium. The former looks balanced locally but depends on external energy/mass input; removing the driver collapses it. **Walrasian general economic equilibrium** <!-- FACT-D11-065 -->[^walras-1874] provides the formal structure for market clearing across all goods and factors simultaneously, though real markets rarely achieve this state due to frictions, information asymmetries, and continuous shocks <!-- FACT-D11-065 -->[^walras-1874].

broad_use:
- **Physics and chemistry**
    - Mechanical equilibrium (net force = 0), thermal equilibrium (no net heat flow), chemical equilibrium (forward and reverse reaction rates equal).
    - Thermodynamic equilibrium as a maximum-entropy state given constraints. **Boltzmann's H-theorem** <!-- FACT-D11-066 -->[^boltzmann-1872] demonstrates that molecular systems evolve monotonically toward equilibrium through collisional redistribution of velocities, connecting microscopic mechanics to macroscopic equilibrium <!-- FACT-D11-066 -->[^boltzmann-1872].
    - **Brownian motion and equilibrium fluctuations** <!-- FACT-D11-067 -->[^smoluchowski-1906] show that equilibrium is not truly static; systems fluctuate around equilibrium points, and these fluctuations follow predictable statistical laws that encode the bath temperature <!-- FACT-D11-067 -->[^smoluchowski-1906].
- **Biology and ecology**
    - Homeostasis as a controlled equilibrium of physiological variables (body temperature, blood pH, blood glucose).
    - Predator-prey equilibria in population dynamics.
    - **Multiple stable equilibria and ecological resilience** <!-- FACT-D11-068 -->[^holling-1973] reveal that ecosystems can flip between alternative stable states (e.g., clear-water vs. turbid-water lakes); resilience measures the basin of attraction around the current state and the ease of shifting between basins <!-- FACT-D11-068 -->[^holling-1973].
- **Economics and finance**
    - Market equilibrium where supply equals demand and price clears.
    - **Nash equilibrium in games** <!-- FACT-D11-064 -->[^nash-1950] where no player has a profitable unilateral deviation.
    - **General equilibrium theory and existence theorems** <!-- FACT-D11-069 -->[^arrow-debreu-1954] (Arrow-Debreu) establish conditions under which an equilibrium allocation of goods and resources exists across all markets simultaneously, a foundational result for mathematical economics <!-- FACT-D11-069 -->[^arrow-debreu-1954].
- **Engineering and control**
    - Set-point regulation in control systems; PID controllers driving outputs toward a desired equilibrium.
    - **Near-equilibrium reciprocal relations** <!-- FACT-D11-070 -->[^onsager-1931] (Onsager relations) show that dissipative systems near equilibrium exhibit symmetries in their response matrices that enable efficient design of control and regulation systems <!-- FACT-D11-070 -->[^onsager-1931].
- **Psychology and social science**
    - Cognitive equilibrium in Piaget's developmental theory.
    - Social-norm equilibria: behavior patterns stable because deviation is penalized by others.

clarity: Equilibrium names the single organizing question "what is this system pulling toward?" and insists on a precise answer: what quantity, under what opposition, with what restoring force. That decomposition turns an intuitively "balanced" system into an operational object — one that can be perturbed, analyzed, and compared to other equilibria on the same terms.

manages_complexity:
- Collapses long-run behavior to a small set of equilibrium states, summarizing trajectory details into attractor identity.
- Provides a reference frame: deviations from equilibrium become the signal of interest, not the state itself.
- Separates fast transients from slow structure; transients relax, the equilibrium structure persists.
- Enables comparative statics — changing a parameter moves the equilibrium, and the new equilibrium can be described without re-simulating the whole trajectory.
- **Bifurcation theory and dynamical systems** <!-- FACT-D11-071 -->[^strogatz-1994] extend equilibrium analysis to account for how equilibria are born, collide, and disappear as parameters change, revealing the organizing structure of system behavior across regimes <!-- FACT-D11-071 -->[^strogatz-1994].
- **Non-equilibrium structures and dissipative patterns** <!-- FACT-D11-072 -->[^glansdorff-prigogine-1971] show that systems far from equilibrium can spontaneously organize into ordered spatial and temporal patterns, challenging the intuition that equilibrium is the universal attractor <!-- FACT-D11-072 -->[^glansdorff-prigogine-1971].

abstract_reasoning:
Equilibrium trains a reasoner to ask:

- What quantity is balanced here, and what are the opposing contributions?
- What mechanism restores balance after a perturbation?
- How large is the basin of attraction before the equilibrium breaks?
- Are we at equilibrium or at a non-equilibrium steady state (maintained by external driving)?
- Are there multiple equilibria? If so, which is the system currently in, and what switches between them?
- Over what time scale is this a valid description?

knowledge_transfer:
Role mappings across domains:

- **Balance quantity** ↔ net force / net flux / net utility / net population change / residual
- **Opposing contributions** ↔ action-reaction pairs / supply-demand / forward-backward rates / birth-death / inflow-outflow
- **Restoring mechanism** ↔ Hooke's law / price adjustment / homeostatic feedback / institutional sanction / control law
- **Basin of attraction** ↔ region of stability / policy envelope / linearization neighborhood / recovery regime
- **Perturbation** ↔ shock / input / mutation / deviation / policy change
- **Regime shift** ↔ tipping point / phase transition / catastrophe / bifurcation
- **Equilibrium selection** ↔ refinement / focal point / starting condition / historical accident

prose:
_(Pass B — pending; see `tracker/v2_upgrade.md`.)_

### Catalog cross-references (v1, batches 02–29)

Solution archetypes that reference this prime (per `solution_archetypes/_catalog/catalog_v1_batches_02_29.yaml`):

**As source prime (1):** `equilibrium_restoration`

**As related prime (9):** `circulation_loop_design`, `homeostatic_regulation`, `hysteresis_management`, `incentive_compatible_rule_design`, `instability_dampening`, `network_effect_bootstrapping`, `oscillation_damping`, `price_signal_design`, `resonance_detuning`

notes: _(placeholder for later refinement)_


Equity

slug: equity

one_liner: Context-sensitive fairness.

core_idea: Equity in the legal and governance sense is a body of law and reasoning that complements rigid rule-based adjudication with discretion-based remedies aimed at fairness in the particular case, an idea Aristotle (c. 350 BCE) introduced as *epieikeia* in Book V of the *Nicomachean Ethics*. <!-- FACT-D49-091 -->[^aristotle-ne-v] Rather than applying a rule universally regardless of circumstance, equity introduces flexibility: the decision-maker (judge, arbiter, committee, algorithm) examines the specific facts, circumstances, parties, and harms involved, and tailors the remedy to achieve justice when a strictly rule-bound approach would produce an unjust or inadequate result. The concept descends from Aristotle's epieikeia (επιείκεια — "equity" or "reasonable exception"), codified as aequitas in Roman law, and systematized in English Chancery courts (12th–18th centuries), where the Court of Chancery explicitly operated as an "equity court" offering remedies unavailable at common law. Modern legal systems recognize equitable doctrines — specific performance, injunction, estoppel, constructive trust — and more broadly, equitable principles in discretionary decision-making by courts, agencies, and other bodies. The abstraction generalizes beyond formal law to any institutional context where case-by-case judgment and remedial discretion must operate alongside rule systems: algorithmic exception-handling, software engineering's case-specific patches, organizational review of policy-rule enforcement, ethical committees adjudicating borderline cases, and dispute-resolution processes where rigid rules would fail the underlying purpose of fairness, a substrate-extension that Barocas and Selbst (2016) develop in their analysis of disparate impact in algorithmic systems. <!-- FACT-D49-092 -->[^barocas-selbst-2016]

structural_signature:
Equity exhibits the following structural properties:

- **A prior rule or principle** that ordinarily applies and governs outcomes in standard cases. The rule is presumed just and efficient in the common case, much as Rawls (1971) characterizes the role of public principles of justice as default constraints on the basic structure of society. <!-- FACT-D49-093 -->[^rawls-1971]

- **Particularity of fact and context** — the specific case differs materially from the typical case in ways that matter morally or functionally. The facts are not merely different; they are relevant to whether the rule's application would be just or would defeat the rule's own purpose.

- **A discretionary judgment** about whether the rule should be applied as written, modified, or set aside in favor of an alternative remedy tailored to the particular case. This judgment is not arbitrary; it is constrained by principle, precedent, and transparency about the reasoning, an internal structure Newman (2014) maps in his lectures on equitable doctrine. <!-- FACT-D49-094 -->[^smith-2021]

- **A remedial alternative** — a different outcome, relief, or procedure designed to address the particular injustice or inadequacy that applying the standard rule would produce. The remedy fits the case rather than fitting a pre-existing template.

- **Transparency and accountability** in the exercise of discretion. Equitable judgment is ideally articulated, reasoned, and reviewable. The decision-maker explains why the standard rule is being modified and how the particular remedy serves justice, a procedural-justice requirement Thibaut and Walker (1975) document as essential to perceived legitimacy of discretionary forums. <!-- FACT-D49-095 -->[^thibaut-walker-1975]

- **Recursive application** — equitable principles can themselves be challenged on grounds of fairness, and the system accommodates meta-level appeals: "even the equitable remedy is unjust in this case, because..." This recursion is bounded but real.

what_it_is_not:
- **Not pure discretion.** Equity is discretion constrained by principle, precedent, and articulated reasoning. Arbitrary decision-making ("I decide whatever seems fair to me") is not equity; it is caprice. Equitable discretion is guided discretion.

- **Not the same as mercy or forgiveness.** Mercy is a compassionate waiving of consequence; equity is a reasoned adjustment of remedy to particular fact. A judge showing mercy suspends the rule out of compassion; a judge applying equity modifies the remedy because the rule's application would be unjust in this case.

- **Not a replacement for rule-based law.** Equity operates in complement to rules, not as a wholesale substitute. A system of pure equity with no rules would be unpredictable and prone to partiality. The rule-equity pair is a structural necessity.

- **Not identical to fairness.** Fairness is a broader value; equity is a mechanism for achieving fairness through case-by-case discretion when rules alone are insufficient. Some equitable decisions may not feel fair (if the reasoning is opaque); some apparently fair outcomes may not be equitable (if they derive from pure luck rather than case-by-case deliberation).

- **Not arbitrary exception-making.** If equitable remedies are granted inconsistently or without articulated reasoning, the system has degraded into arbitrary favoritism. Equity requires that the exceptions be principled and explainable.

- **Not soft law or non-binding.** In legal contexts, equitable remedies (injunctions, specific performance, constructive trusts) are binding and enforceable. The "soft" aspect is that the decision-making process is more flexible and discretionary than rule-application; the consequence is still hard.

broad_use:
- **English common law and its descendants.**
    - The Court of Chancery (12th century onward) developed equity as an explicit corrective to common law's rigidity. A landowner dispossessed by a technical quirk of property law could petition the Chancellor (the Crown's highest judicial officer) for an equitable remedy: specific performance (forcing the rightful owner to acknowledge the true ownership) rather than mere damages. Over centuries, Chancery developed a systematic body of equitable doctrine — trusts, constructive trusts, estoppel, injunctions, specific performance — that became the foundation of modern common-law equity. Many of these doctrines remain central to modern law.

- **Modern civil and common-law litigation.**
    - Courts in the U.S., UK, Canada, Australia, and other common-law jurisdictions routinely grant equitable remedies — preliminary and permanent injunctions, specific performance of contracts, reformation of wills or trusts to reflect intent despite technical error, equitable estoppel barring a party from enforcing a strict legal right. Family courts apply equitable principles in property division on divorce. Bankruptcy courts apply equitable doctrines (equitable subordination, equitable lien). Agency discretion in administrative law is often constrained and guided by equitable principles.

- **Arbitration and alternative dispute resolution.**
    - Arbitrators and mediators frequently operate under authority to decide "ex aequo et bono" (on grounds of justice and fairness), giving them explicit discretion to depart from strict legal rules. Labor arbitration, commercial arbitration, and institutional mediation routinely invoke equitable principles.

- **Algorithmic and computational systems.**
    - Software exception-handling (catch, fallthrough, special cases) is a form of equity: the general algorithm doesn't apply; a case-specific handler produces a different outcome. Machine-learning systems with human-in-the-loop review (model output checked by a human who can overrule on case-specific grounds) instantiate equity in automated systems. Content-moderation appeals processes that allow human reviewers to override algorithmic decisions on grounds of context and particular circumstance are equitable mechanisms.

- **Organizational review and discretion.**
    - Policy-enforcement decisions (discipline, hiring, promotion, granting of exceptions to policy) often operate through formal rules plus equitable review: the policy applies in the standard case, but a review committee can examine particular circumstances (hardship, extenuating factors, unintended consequences of rule application) and modify the remedy or make an exception. Academic grade appeals, employee discipline reviews, and benefit-eligibility disputes frequently invoke this structure.

- **Ethical committees and professional discipline.**
    - Medical ethics committees, institutional review boards, and professional disciplinary boards operate as equitable forums: they evaluate conduct against general standards but recognize that case-particular facts matter. A physician's violation of standard protocol might warrant different discipline if done in an emergency under constraints unknown at the time the protocol was written.

clarity: Equity clarifies by naming the structural need for discretion when rules are insufficient. It distinguishes between (1) the rules that apply in standard cases, (2) the recognition that particular cases can differ in morally relevant ways, and (3) the mechanisms for adjusting remedy to particular fact without wholesale abandonment of rules, a conceptual disambiguation Espinoza (2007) develops in distinguishing equity from formal equality. <!-- FACT-D49-096 -->[^espinoza-2007] The framework also clarifies the tension inherent in any governed system: rules provide consistency, predictability, and protection against partiality, but they also produce unjust results in edge cases. Equity acknowledges this tension directly rather than denying it or pretending that rules can be written tightly enough to avoid all hard cases. The clarifying force is to separate two questions: (1) What does the rule say? and (2) Would applying the rule here be just? Both questions can be answered rigorously; the second is not less rigorous than the first, merely less mechanical.

manages_complexity: Equity manages complexity by introducing flexibility into otherwise rigid systems without abandoning rules entirely. A legal system that applied rules mechanically to every case would produce impossible results (unjust outcomes, absurdities); a system with no rules would be unpredictable and favor the powerful. Equity finds the middle ground: rules apply by default, but case-particular discretion is available when the rule would fail its own purpose. This reduces system complexity by avoiding the need to anticipate and write rules for every possible circumstance; instead, the system trusts decision-makers to exercise principled discretion when needed, a stance Sen (2009) defends in his comparative (rather than transcendental) approach to justice. <!-- FACT-D49-097 -->[^sen-2009] The management is imperfect: equitable discretion introduces unpredictability and the possibility of partiality, and the art is to structure equitable processes (reasoned articulation, precedent, review) to minimize these harms while preserving flexibility.

abstract_reasoning:
Equity trains a reasoner to ask:

- Does the standard rule apply here, or are the particular facts materially different in a morally or functionally relevant way?
- If the rule applies mechanically, would the outcome be just? Would it serve the rule's own underlying purpose, or would it defeat it?
- What remedy would be just in this particular case? What form should relief take?
- Can the discretionary choice be articulated and reasoned? What principle governs the exception?
- How is the equitable decision reviewable? What check prevents abuse of discretion?
- Could the rule itself be reformed to avoid the need for equitable exception in future cases, or is the exception structural and recurring?

These questions apply across contexts: formal legal proceedings, organizational policy reviews, algorithmic exception-handling, and ethical committee deliberations all benefit from asking them.

knowledge_transfer:
Role mappings across domains:

- **Rule / standard principle** ↔ algorithm / policy / protocol / standard procedure / standing order, a domain-mapping pattern Walzer (1983) develops in his account of justice as differing across distinct social spheres. <!-- FACT-D49-098 -->[^walzer-1983]
- **Particular case** ↔ exception / edge case / hardship / emergency / unusual circumstance
- **Discretionary judgment** ↔ human review / exception-handler / escalation / appeal
- **Remedy tailored to the case** ↔ custom solution / workaround / override / case-specific relief
- **Transparency of equitable reasoning** ↔ articulation of exception / documentation of override / reasoned explanation of departure from standard
- **Precedent and constraint on discretion** ↔ case law / prior exceptions / principle-based guidance / reversibility standards

A judge granting an injunction to prevent irreparable harm, a software engineer writing a special-case handler for an unusual input, an organizational review committee recommitting a disciplinary decision on grounds of hardship, and a medical ethics committee approving a protocol deviation under emergency circumstances are all exercising equity in different substrates. The structures are isomorphic: rule meets case that doesn't fit; discretion is exercised to tailor remedy; the choice is articulated and reviewable; future cases are guided by precedent.

prose:
_(Pass B — pending; see `tracker/v2_upgrade.md`.)_

### Catalog cross-references (v1, batches 02–29)

Solution archetypes that reference this prime (per `solution_archetypes/_catalog/catalog_v1_batches_02_29.yaml`):

**As source prime (2):** `satiation_aware_allocation`, `stratified_treatment`

**As related prime (14):** `adaptive_threshold_recalibration`, `adverse_selection_filtering`, `commons_governance`, `controlled_randomization`, `credible_signaling`, `equivalence_class_consolidation`, `gradient_guided_intervention`, `hidden_type_screening`, `overoptimization_guardrail`, `pareto_focus`, `price_signal_design`, `public_goods_provision`, `skin_in_the_game_alignment`, `variance_reduction`


Equivalence-Preserving Rewriting

one_liner: Transforming an object into a behaviourally equivalent but operationally different form under an explicit equivalence relation, then selecting among rewrites by an orthogonal cost criterion.

core_idea:
Equivalence-preserving rewriting is the structural move of transforming an expression, specification, derivation, or document into a *behaviourally equivalent but operationally different* form under an explicit equivalence relation, then selecting among the candidate rewrites by a cost criterion that the equivalence relation itself does not adjudicate. The pattern has three load-bearing parts: an *object* with a specified meaning (a query, a program fragment, a proof step, an algebraic term, a sentence, a clause); an *equivalence relation* that names which transformations are allowed because they preserve that meaning (relational-algebra equivalence, operational equivalence, logical equivalence, meaning-preserving paraphrase, unchanged-legal-effect rewording); and a *cost criterion* orthogonal to the equivalence (estimated execution time, instruction count, proof length, expression depth, readability, ambiguity reduction) under which one of the many admissible rewrites is chosen.

What makes the move structurally distinct from optimisation, simplification, or rewording-in-general is that the equivalence and the cost are *factored*: the equivalence relation defines a *space of safe moves*, and the cost criterion picks one point in that space. The discipline lives in keeping the two separate. Slip in a transformation outside the equivalence, and the rewrite no longer means what the original meant; slip in cost-driven shortcuts that the equivalence happens to permit but the user did not intend to permit, and the system silently changes behaviour the user expected to hold. The factoring also exposes a recurring failure mode keyed to the equivalence's grain: an equivalence relation that is *too coarse* admits rewrites that change behaviour users care about, and one that is *too fine* excludes rewrites that would be safe. The canonical instance is the compiler optimisation that is "correct" under the language-standard equivalence but "wrong" under the programmer's intuitive model — the equivalence relation is doing work the user did not notice.<!-- FACT-D62-826 -->[^wang-etal-2013] The disciplined remedy is to *write the equivalence down*, because an implicit equivalence drifts under maintenance and accumulates "obvious" rewrites it never actually permitted.

structural_signature:
*the object with a specified meaning* — *the equivalence relation defining the space of safe moves* — *the cost criterion orthogonal to the equivalence* — *the admissible-rewrite space the equivalence carves out* — *the within-space selection the cost performs* — *the grain of the equivalence (too coarse admits surprises, too fine forbids safe moves)*

A configuration exhibits equivalence-preserving rewriting when each of the following holds:

- **An object with specified meaning.** Something carries a meaning to be held invariant: a query, a program fragment, a proof step, an algebraic term, a sentence, a clause, a circuit.
- **An equivalence relation.** An explicit relation names which transformations are admissible because they preserve that meaning — relational-algebra equivalence, operational equivalence, logical equivalence, meaning-preserving paraphrase, unchanged-legal-effect rewording.
- **A cost criterion.** A separate measure, orthogonal to the equivalence and not adjudicated by it, ranks admissible forms: estimated execution time, instruction count, proof length, readability, gate count.
- **An admissible-rewrite space.** The equivalence relation carves out a space of safe moves — the set of forms behaviourally equivalent to the original — within which any choice preserves meaning.
- **A within-space selection.** The cost criterion picks one point in that space, factoring the move into safe-space construction (the equivalence) and within-space selection (the cost), which must be kept separate.
- **A grain.** The equivalence has a grain that is the locus of failure: too coarse, it admits rewrites that change behaviour users care about ("correct by the spec, wrong by the user"); too fine, it excludes rewrites that would be safe. An implicit equivalence drifts under maintenance.

The components compose so that the equivalence relation is the primary object of scrutiny — the cost is where visible work happens, but correctness lives in the equivalence — and the disciplined architecture is generate-verify-select with a swappable cost model over a declared, written-down equivalence.

what_it_is_not:
- **Not `transformation` in general.** A transformation may
  change meaning freely; this prime is the *meaning-preserving*
  subclass, constrained by an explicit equivalence relation
  and a separate cost criterion. The constraint is the whole
  point.
- **Not `relation` as such.** `relation` is the bare
  abstraction of a relationship between elements; this prime
  uses one specific relation (an equivalence) as a tool to
  carve a safe-move space, plus an orthogonal cost. It is an
  application, not the abstraction.
- **Not `refinement`.** Refinement moves toward a *more
  specified or improved* object, changing what it commits to;
  equivalence-preserving rewriting holds meaning *fixed* and
  changes only operational form. One narrows; the other
  relabels within an equivalence class.
- **Not `optimization` generically.** Optimization is search
  for a best point under a cost; this prime *factors* that
  search into a safe-space construction (the equivalence) and
  a within-space selection (the cost), which generic
  optimization fuses.
- **Not `canonical_form`.** A canonical form is a single
  distinguished representative of an equivalence class; this
  prime is the *cost-driven navigation* among many admissible
  forms, which need not converge on any canonical one.
- **Common misclassification.** Calling a "cleanup,"
  "simplification," or "rewording" equivalence-preserving
  without stating the relation. If the preservation invariant
  is unwritten, the cost work runs over an undefined safe
  space; catch it by demanding "preserved under what
  relation?" before any rewrite is trusted.

broad_use:
- **Database query optimisation**: relational-algebra rewrites — selection push-down, join reordering, predicate rewriting — transform a declarative query into operationally different but result-equivalent plans, and the optimiser picks the lowest estimated cost.<!-- FACT-D62-827 -->[^selinger-etal-1979]
- **Compiler optimisation**: loop unrolling, dead-code elimination, common-subexpression elimination, instruction selection, peephole rewrites — each a program-equivalence-preserving transformation scored by an execution-cost model.<!-- FACT-D62-828 -->[^aho-etal-1986]
- **Theorem proving**: normal-form rewriting in term-rewriting systems, rewriting a proof for shorter or more standard form, tactic-based proof restructuring under preserved logical content.<!-- FACT-D62-829 -->[^baader-nipkow-1998]
- **Algebra and symbolic computation**: factoring, expanding, simplifying, rewriting integrals into standard form — symbolic-equivalence rewrites scored by length, depth, or standard-form-ness.
- **Natural-language paraphrase**: meaning-preserving rewordings scored by clarity, register, or length; controlled-language editing for technical writing.
- **Legal drafting**: restating a clause to clarify, consolidate cross-references, or remove ambiguity without changing its legal effect; consolidation acts and uniform codes are large-scale instances.<!-- FACT-D62-830 -->[^xanthaki-2013]
- **Hardware synthesis**: equivalent-circuit rewrites under logical equivalence, scored by gate count, depth, or power.<!-- FACT-D62-831 -->[^hachtel-somenzi-1996]

clarity: Naming the move forces two questions that ordinary language conflates: *under what equivalence relation are these rewrites admissible?* and *what cost are we minimising over the admissible set?* A "simplification" that does not state its equivalence relation is undisciplined; a "rewording" that does not state what is preserved is a covert content change. The clarifying separation is between the safe-space construction (the equivalence) and the within-space selection (the cost), which ordinary usage of words like "optimise," "clean up," or "restate" fuses and thereby hides. The factoring also surfaces the grain problem explicitly: an equivalence that is too coarse admits behaviour-changing rewrites, one that is too fine forbids safe ones, and the dangerous case is the equivalence quietly doing work the user did not authorise — "correct by the spec, wrong by the user." Once the move is named, every claimed equivalence-preserving transformation invites the audit "what exactly is preserved, and by what relation?", which converts a vague appeal to having kept the meaning into a checkable commitment.

manages_complexity: The pattern manages complexity by *separating the safe-space construction from the cost minimisation*. Generating one good rewrite is hard; checking that a rewrite preserves meaning is often much easier and is mechanisable; choosing among many checked-safe rewrites by cost is a separate, equally mechanisable problem. This factoring is what makes industrial query optimisers, production compilers, and proof assistants tractable: the equivalence checker is a verifier, the cost-driven search is a planner, and the two halves can be improved independently. A second compression follows: the equivalence relation, once made explicit, is *reusable across many cost criteria* — the same relational-algebra equivalences serve cost models for spinning disk, SSD, distributed shuffle, and in-memory engines, and the same logical-equivalence laws serve proof shortening, proof clarification, and prover-friendly rewriting.<!-- FACT-D62-832 -->[^garcia-molina-etal-2009] The compression is that a database engineer, a compiler writer, a proof engineer, and a legal drafter all maintain a *rewrite-rule database* under a *declared equivalence* and a *swappable cost model*, so the architecture learned in one substrate transfers as a template in the next, and re-tuning a system to a new objective becomes a matter of swapping the cost model rather than rebuilding the safe space.

abstract_reasoning: Recognising the pattern enables several reasoning moves. *Equivalence as design choice*: which equivalence relation one commits to determines what rewrites are admissible and where surprise can leak in, so choosing the wrong equivalence is the silent root cause of "correct by the spec, wrong by the user" defects. *Verification asymmetry*: it is often easier to check that a rewrite preserves meaning than to generate it, which is why search-then-verify architectures are common — generate candidates freely, filter by the equivalence checker, then score by cost. *Cost-equivalence orthogonality*: changing the cost criterion changes which rewrite wins without changing the equivalence, so a system architected around the factoring can be re-tuned by swapping cost models. *Composable rewrites*: chains of admissible rewrites compose under the same equivalence, which underwrites the rewrite-rule databases that optimisers, term-rewriting systems, and drafting style guides all maintain. *Equivalence drift*: when the equivalence is implicit it drifts under maintenance, accepting "obvious" rewrites it did not actually permit, so the discipline is to write it down and check rewrites against it. The non-obvious move the prime licenses is to treat the equivalence relation as the primary object of scrutiny — the cost criterion is where the visible work happens, but the equivalence is where correctness lives, and most failures trace to an equivalence that was too coarse, too fine, or never written down.

knowledge_transfer: The factoring transfers across substrates that share no surface vocabulary. From *compiler to legal drafting*: the discipline of declaring the preservation invariant explicitly before allowing a rewrite ports directly — a consolidation project that articulates the "preserved legal effects" relation up front and screens proposed rewrites against it behaves more like a compiler optimiser than like ad-hoc redrafting. From *database optimiser to natural-language paraphrase*: the search-then-verify architecture — generate many candidate paraphrases, filter by a meaning-equivalence check, pick by stylistic cost — transfers to controlled-language tooling and assisted-editing pipelines. From *term rewriting to product design*: the idea that a normal form exists, a canonical rewrite to which all admissible rewrites should converge, underwrites style guides, code formatters, and design-system standardisation. The role mappings transfer directly — *object* ↔ query / program / proof / term / sentence / clause / circuit; *equivalence relation* ↔ relational-algebra equivalence / operational equivalence / logical equivalence / paraphrase / preserved legal effect / logic equivalence; *cost criterion* ↔ estimated I/O / instruction count / proof length / readability / clarity / gate count; *candidate generator and checker* ↔ rewrite-rule database plus verifier. The transferred and non-obvious lesson is that "preservation" and "preference" are two separable disciplines, and conflating them is the recurring source of defects: the legal team that optimises a statute for clarity without first writing down what legal effects must be preserved is making the same error as the compiler that applies a standard-legal but intuition-violating optimisation. A practitioner who has internalised the factoring in one substrate carries into every other the same two questions — what relation makes these rewrites safe, and what orthogonal cost picks among them — and the same architecture, generate-verify-select with a swappable cost model over a declared equivalence, which is exactly what makes the move transfer intact from query plans to statutory consolidation.

formal_abstract: The database query optimiser is the prime's defining formal instance, where the factoring of equivalence from cost is fully mechanised. The *object* is a declarative SQL query with a specified result-set meaning. The *equivalence relation* is relational-algebra equivalence: the set of rewrites — selection push-down, join reordering, predicate simplification, subquery flattening — that provably preserve the result set under set semantics.<!-- FACT-D62-833 -->[^selinger-etal-1979] This relation carves out the *admissible-rewrite space*, the family of physical plans all guaranteed to return the same rows. The *cost criterion* is orthogonal to that relation: estimated I/O, CPU, and memory under a cost model parameterised by table statistics, index availability, and the storage engine. The *within-space selection* is the optimiser's search choosing the lowest-estimated-cost plan. The architecture is exactly generate-verify-select: rewrite rules generate candidates, each *provably* inside the equivalence so correctness is never at issue, and the cost model ranks them. The prime's grain warning is concrete here: if the equivalence relation is stated for set semantics but the query expects ordered or duplicate-preserving (bag) semantics, a "correct" reorder can change behaviour the user relied on — the equivalence was *too coarse*.<!-- FACT-D62-834 -->[^garcia-molina-etal-2009] And the reusability point is sharp: the *same* relational-algebra equivalences serve completely different cost models — spinning disk, SSD, distributed shuffle, in-memory — so re-tuning to new hardware is swapping the cost model over a fixed, declared equivalence, not rebuilding the safe space.<!-- FACT-D62-835 -->[^garcia-molina-etal-2009] **Mapped back:** the SQL query is the object, relational-algebra equivalence is the declared relation defining safe moves, estimated cost is the orthogonal criterion, and the optimiser is the generate-verify-select architecture — with grain failures tracing to a set-versus-bag mismatch in the equivalence, never to the cost model.

applied_industry: Two applied instances share the factoring across substrates with no shared vocabulary. First, legal drafting and statutory consolidation: the *object* is a clause or statute with a specified *legal effect*. The *equivalence relation* is "unchanged legal effect" — the rewrites (renaming a defined term, promoting a recital into an operative clause, splitting a tangled section, consolidating cross-references) that a drafter asserts preserve the obligations, rights, and liabilities.<!-- FACT-D62-836 -->[^xanthaki-2013] The *cost criterion* is orthogonal: readability, brevity, ambiguity reduction. The prime's central discipline — *write the equivalence down* — is the load-bearing move: a consolidation project that first articulates the "preserved legal effects" relation and screens each proposed rewrite against it behaves like a compiler optimiser, while one that "cleans up for clarity" without stating what must be preserved is making the classic error — optimising the cost while silently slipping outside the equivalence and changing the law. Second, natural-language paraphrase tooling: the *object* is a sentence with a meaning to hold invariant; the *equivalence relation* is meaning-preserving paraphrase; the *cost criterion* is clarity, register, or length. The portable architecture is search-then-verify: generate many candidate rewordings freely, filter by a meaning-equivalence check, then score by stylistic cost — the same shape as the query optimiser, imported into controlled-language and assisted-editing pipelines. In both, conflating "preservation" with "preference" is the recurring defect: the legal team that optimises for clarity without writing down the preserved effects makes the same mistake as a compiler applying a standard-but-intuition-violating optimisation. **Mapped back:** clauses and sentences are objects; unchanged-legal-effect and meaning-preserving-paraphrase are the declared equivalences; clarity and brevity are the orthogonal costs; and the generate-verify-select architecture with a written-down equivalence transfers intact from the optimiser to statutory consolidation to paraphrase tooling.


Equivalence Principle

one_liner: Gravity indistinguishable from acceleration.

core_idea: The equivalence principle asserts that <!-- FACT-D14-001 -->[^einstein-1907] *the local indistinguishability* of gravitational and inertial acceleration is a fundamental symmetry of nature. Within a sufficiently small region of spacetime, <!-- FACT-D14-002 -->[^einstein-1907] *the free-fall reference frame* renders gravity undetectable: all physics reduces to special relativity in Riemann normal coordinates where the metric and its first derivatives vanish. This principle originates <!-- FACT-D14-003 -->[^galileo-1638] with Galileo's observation that all bodies fall with identical acceleration, was sharpened by Newton through the identity of gravitational and inertial mass, and was elevated by Einstein (1907) to a cornerstone of general relativity. Einstein's insight was that gravity is not a force in flat spacetime but a manifestation of <!-- FACT-D14-004 -->[^einstein-1916] *the spacetime curvature* itself: freely-falling bodies follow geodesics, and the geometry of spacetime determines inertial motion universally. The principle manifests in three strengths: weak equivalence principle (WEP, asserting <!-- FACT-D14-005 -->[^newton-1687] *the inertial-gravitational mass identity*), Einstein equivalence principle (EEP, adding local Lorentz invariance), and strong equivalence principle (SEP, extending to self-gravitating systems). The principle clarifies why gravity couples to all forms of energy-momentum with <!-- FACT-D14-006 -->[^einstein-1916] *the universal coupling property* and why no test particle can be shielded from gravitational acceleration.

structural_signature: The equivalence principle comprises six structural elements. First, <!-- FACT-D14-007 -->[^touboul-2017] *the inertial-gravitational mass identity* (m_i = m_g for all bodies, confirmed to ~10^−15 precision). Second, *the local indistinguishability*: in a small neighborhood of any spacetime event, coordinate transformations (Riemann normal coordinates) eliminate the gravitational field to first order. Third, *the free-fall reference frame*: any observer in free fall experiences a locally inertial environment where special-relativistic physics applies exactly. Fourth, *the universal coupling property*: gravity couples identically to all matter species and fields, making it a geometric property rather than a species-dependent interaction. Fifth, <!-- FACT-D14-008 -->[^misner-thorne-wheeler-1973] *the geodesic equation*, which describes the motion of test particles purely in terms of spacetime geometry (d²x^μ/dτ² + Γ^μ_ρσ (dx^ρ/dτ)(dx^σ/dτ) = 0). Sixth, <!-- FACT-D14-009 -->[^wald-1984] *the tidal-force separation*: over finite regions, tidal forces (second-order curvature effects) reveal the true gravitational field and distinguish gravity from uniform acceleration, establishing the principle's essential *locality condition*. These elements form the conceptual foundation for interpreting gravity as geometry: the Einstein field equations emerge as the unique consistent way to couple matter to spacetime curvature while preserving the equivalence principle's predictions.

what_it_is_not:
**Common misclassification**: Treating the
equivalence principle as a claim that
"acceleration is the same as gravity"
globally. The equivalence is *local*: a
uniformly-accelerated frame and a uniform
gravitational field are locally
indistinguishable, but real gravitational
fields are *not* uniform — tidal effects
(relative acceleration of nearby geodesics)
distinguish gravity from acceleration over
extended regions. The principle is a statement
about local frames, not a global identity.

**Not identical to Mach's principle**: see
*mach_s_principle* — Mach's principle is a
global claim about inertial structure being
determined by matter. The equivalence
principle is a local claim about
indistinguishability of gravity and
acceleration. They are logically independent,
though both motivated Einstein.

**Not trivially implied by m_i = m_g**: while
the weak equivalence principle is often
summarized by the equality of inertial and
gravitational mass, the Einstein equivalence
principle adds local Lorentz invariance and
local position invariance, going beyond mass
equality to constrain all physics in local
frames. Distinguishing WEP, EEP, and SEP is
essential for discussing tests and violations.

**Not without empirical caveats**: the
equivalence principle is tested with
extraordinary precision for standard matter
(MICROSCOPE: ∼10^(−15) level), but possible
violations for dark matter, quantum-mechanical
test bodies, or in extreme regimes are still
investigated. "The equivalence principle is
exactly true" is an empirical inference, not a
theorem.

**Not incompatible with all alternative
gravity theories**: many alternatives to GR
(Brans-Dicke, scalar-tensor theories,
chameleon models) satisfy the weak and Einstein
principles but violate the strong form.
Classifying theories by which form they
respect is a key part of modern gravity
phenomenology.

**Not a derivation of GR's field equations**:
the equivalence principle implies that gravity
is geometric, but by itself does not fix the
Einstein field equations. Additional input
(general covariance, linearization in the weak
field, correspondence with Newtonian gravity)
is needed. The principle is necessary, not
sufficient.

Cross-references: see *mach_s_principle*
(independent foundational principle); see
*inertia* (the operational partner of
gravitation in the WEP); see *gravitation*
(the phenomenon reinterpreted geometrically);
see *relativity* (the framework in which the
principle is made precise).

broad_use:
The equivalence principle appears in
gravitation (general relativity, geometric
interpretation, geodesic motion as free fall),
in experimental gravity (Eötvös experiment,
lunar laser ranging, MICROSCOPE, torsion
balances, atom interferometry), in alternative
theories of gravity (as a criterion for
classifying which theories preserve or violate
it), in cosmology (as a guide to the couplings
of scalar fields and dark sectors), in
quantum mechanics in gravitational fields
(COW experiment, quantum tests of WEP), in
general covariance discussions, and by analogy
in philosophy of science as a paradigm of a
"principle theory" (Einstein's classification).
It recurs wherever the distinction between
inertial and gravitational phenomena is at
issue.

clarity:
The equivalence principle clarifies why the
gravitational "force" differs structurally
from other forces: gravity affects all bodies
identically because it acts via the geometry of
space-time, not through a coupling that
depends on particle species. This explains
why gravity couples to energy-momentum rather
than to a specific charge, why free-falling
frames are locally inertial, and why the
trajectory of test bodies is determined by
geometry alone. It also organizes empirical
tests of gravity into a clean hierarchy
(WEP/EEP/SEP).

manages_complexity:
The construct manages the complexity of
gravitational theory by reducing the problem
of "how gravity couples to everything" to a
geometric statement: gravity is the curvature
of the space-time on which all matter and
fields propagate, with universal coupling. This
single insight replaces a potentially
catalog-like theory (separate couplings for
each particle and interaction) with a uniform
geometric framework.

abstract_reasoning:
Equivalence-principle reasoning proceeds by
asking whether a proposed interaction acts
universally (i.e., like gravity) or
species-specifically (like electromagnetism); by
designing precision null experiments (Eötvös-
type, atom-interferometer) to bound violations;
by evaluating alternative theories against the
hierarchy WEP ⊂ EEP ⊂ SEP; and by using the
principle constructively in theory-building —
deriving geodesic motion, tidal tensor
structure, and curvature-matter coupling from
local Minkowski structure in freely-falling
frames.

knowledge_transfer:
| Role | Weak form (WEP) | Einstein form (EEP) | Strong form (SEP) | Test form |
|---|---|---|---|---|
| Claim | All bodies fall with same acceleration | Local physics = SR in free-fall frame | Local gravity-physics universal for self-gravitating bodies | Composition/position/velocity independence |
| Key equality | m_i = m_g | Local Lorentz + local position invariance | Extension to gravitating bodies | Composition independence of a/g |
| Tested by | Eötvös, MICROSCOPE, lunar ranging | Clock comparisons, Hughes-Drever | Nordtvedt effect in lunar orbit | Many forms |
| Currently bounded at | ~10^(−15) fractional | ~10^(−7)–10^(−12) for various tests | ~10^(−13) for Nordtvedt | Various |
| Violated by | Hypothetical fifth-force models | Lorentz-violating theories | Brans-Dicke, scalar-tensor | Various alternatives |

Equivalence-principle reasoning transfers to
modern gravitational phenomenology (tests of
GR, searches for fifth forces and quintessence),
to effective-field-theory treatments of gravity,
to dark matter searches via precision tests,
and to the conceptual analysis of geometric vs
force-based descriptions in other contexts
(gauge-theoretic curvature, fiber-bundle
geometry). The structural core is "universal
coupling = geometric origin"; what varies is
the substrate being so-interpreted.

prose: *(Pass B — deferred.)*

notes:
Held at High confidence. Foundational
principle of general relativity, with clear
hierarchy (WEP/EEP/SEP) and sharp empirical
status. Entry emphasizes locality, the
logical independence from Mach's principle,
and the non-sufficiency for fixing a unique
theory of gravity. Pairs structurally with
mach_s_principle (#130) as complementary GR
motivators — related but not a tight pair, as
they are logically independent and tested
differently.