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The supercurrent is the spinor-valued Noether current of rigid supersymmetry. Its spatial integral is a supercharge. Localizing the supersymmetry parameter produces ${\partial }_{\mu }\overline{ϵ}{S}^{\mu }$ in the variation of the action; a gravitino coupling cancels this term and starts the Noether gauging procedure.

The Super-Poincaré group combines Lorentz transformations and supertranslations. Its even part acts by spin-group transformations and ordinary spacetime translations; odd parameters generate the supercharges. Their graded commutator closes on translations through the Super-Poincaré algebra. A superspace tensor built from invariant coframes and Lorentz-invariant contractions is invariant under the full group.

A deterministic interpretation with a definite particle configuration guided by the wavefunction. The latter obeys the Time-dependent Schrödinger equation, while the particle follows the guidance equation. A Born rule distribution of initial positions is preserved by quantum equilibrium equivariance. The guidance law restricts admissible initial velocities, even when trajectories are rewritten as a second-order force equation.

A superselection rule partitions a representation into sectors between which the specified observable algebra has no matrix elements. Relative phases between vectors in different sectors cannot be detected by those observables, so a coherent vector spanning sectors and its sector-diagonal mixture define the same observable state. The restriction is relative to the observable algebra; it is not by itself a physical wave-function collapse. Infinite-system phases can yield disjoint representations and emergent superselection sectors even though each finite system permits global connecting operators.

The EPR argument uses correlations of an entangled state and a locality assumption to challenge the completeness of the quantum description. Its Einstein–Podolsky–Rosen criterion of reality identifies quantities predictable with certainty without disturbing a distant system as elements of physical reality.

Conformance of compiled machine code to the behaviour of the code model used to justify it. Binary correctness can extend formal verification across the compilation step; the result applies to the particular compiled artifact and its assumptions.

A mathematically precise description of permitted behaviour. A formal specification for the microkernel file-read example's service could relate an authorized request to the exact returned bytes; a formal specification of its microkernel's IPC need only describe the protected transport operation. They state different obligations.

For the summation loop verification, require a nonnegative integer input $n$ and a returned value $s$ satisfying $2s=n(n+1)$. The formal specification describes what answer is acceptable without requiring this particular loop: another implementation may satisfy the same contract. State separately whether partial correctness or total correctness is required. A Hoare triple packages the precondition, the program and the postcondition; Hoare logic supplies rules for establishing such claims.

Precision alone does not ensure a useful requirement. The postcondition $s\ge 0$ permits returning zero for every nonnegative integer input; it does not specify the intended triangular number. Conversely, an impossible precondition permits no starts, making a universal correctness claim vacuous. Check both that intended inputs are admitted and that admitted outputs express the intended task.

An abstract family of languages is a family of formal languages containing a nonempty language and closed under finite union, concatenation of formal languages, positive closure of a formal language, nonerasing monoid homomorphisms between free word monoids, inverse homomorphisms, and intersection with regular languages. Here positive closure means $L^{+}={\bigcup }_{n\ge 1}{L}^n$, and nonerasing means that the homomorphism maps every alphabet letter to a nonempty word.

A full abstract family of languages allows arbitrary word homomorphisms, including ones that erase letters. This distinction matters: erasing an unbounded amount of a computation or derivation can change the resources needed to recognize the image language. The regular languages and context-free languages are full abstract families. In particular, the closure axioms say that if $L$ is context-free and $R$ is regular, then $L\cap R$ is context-free; they do not assert closure under intersection of two arbitrary context-free languages. The framework collects the common constructions without identifying the different language classes.^[ref]

In a formal grammar, terminal symbols are symbols of completed words; a nonterminal symbol is an auxiliary symbol that may be replaced by a production. The two alphabets are disjoint.

A language is computably enumerable when a machine recognizes its members, possibly running forever on nonmembers.

A finite-state machine processes inputs using a finite set of automaton states and input-dependent automaton transitions. An automaton state records everything the machine remembers; an automaton transition says how that memory changes on the next symbol. A finite-state machine may also emit outputs. Here the emphasis is on recognizing a formal language: start in a designated automaton state, read a word from an alphabet, and accept exactly when the final automaton state is designated accepting.

Finite memory does not impose a maximum input length. A finite-state machine can read arbitrarily long words, revisiting its automaton states indefinitely. The restriction is that the part already read must be compressed into one of finitely many summaries. For divisibility by three, a word of binary digits can describe an enormous integer, yet its congruence class modulo three is all that the machine needs to retain. Two already-read words giving the same automaton state behave identically under every subsequent continuation in a DFA.

This makes finite-state machines useful whenever past details can be forgotten safely: checking whether a sequence of commands follows an allowed order, or recognizing a pattern with a regular expression. The beautiful design question is which distinctions about the past matter to the future. DFA minimization makes that question precise by merging indistinguishable states of a deterministic finite automaton. Both DFAs and NFAs recognize exactly the regular languages.

A word over an alphabet $\Sigma$ is a finite sequence of symbols from $\Sigma$. The empty word has length zero, and ${\Sigma }^{\ast }$ denotes the set of all finite words.

Newton's laws of motion consist of inertia in an inertial frame, the relation between net force and change of momentum, and equal-and-opposite pair forces. The following three children state the individual laws.

The Van der Waals force is a distance-dependent nonbonded interaction between atoms or molecules, including dispersion and dipolar contributions. In the attractive nonretarded pair model, an interaction energy $U(r)=-\beta /r^6$ gives force magnitude $6\beta /r^7$. Macroscopic integration introduces a Hamaker constant; it does not replace the sign change and molecular cutoff needed at very short distances.

Tension is a pulling force transmitted along a taut string, rope, or similar body. For an ideal uniform string with linear mass density $\mu$, constant tension $T$ sets the small transverse wave speed $c=\sqrt{T/\mu }$.

A restoring force acts toward an equilibrium when a body is displaced from it. In one dimension, taking displacement $x=0$ at equilibrium, this means $xF(x)<0$ for sufficiently small nonzero $x$. An ideal linear spring has $F(x)=-kx$ with positive spring constant $k$, producing simple harmonic motion for a constant mass without damping. More general restoring forces need not be linear.

Here gravitational weight means the force on a body due to gravity. In the Newtonian model, a body of mass $m$ in a uniform gravitational acceleration $\mathbf{g}$ has gravitational weight $\mathbf{W}=m\mathbf{g}$, with magnitude $W=mg$. Its direction follows $\mathbf{g}$; a buoyant force is a separate contribution to the body's net force. ^[ref].

A food chain orders species so that each consumer feeds on the level below and may be eaten by the level above. Basal resource growth, consumer mortality, conversion efficiency and density regulation determine whether all levels can persist.

Two consumers with different break-even requirements cannot both have bounded persistent abundance on a single well-mixed limiting resource when their per-capita growth is linear in that resource. For $\dot{Y}/Y=cX-d$ and $\dot{Z}/Z=eX-f$, differentiation of $\log Y/c-\log Z/e$ gives the nonzero constant $-d/c+f/e$ when their break-even levels differ. This excludes simultaneous bounded persistence without assuming convergence to an equilibrium. Additional resources, environmental variation or different interactions can invalidate the simple one-resource argument.

For a cooriented integrable codimension-one distribution, choose a nonvanishing defining form $\zeta$ and an auxiliary form with $d\zeta =\eta \wedge \zeta$. Changing the auxiliary form to $\eta +a\zeta$ changes $\eta \wedge d\eta$ by $-d(ad\zeta )$. Replacing $\zeta$ by $h\zeta$ changes the auxiliary form by $d\log |h|$, changing the three-form by the exact differential $d(-d\log |h|\wedge \eta )$. Stokes' theorem proves invariance on a closed oriented three-manifold. For the weak stable foliation of a curvature-minus-one surface geodesic flow, take $\zeta =\theta +\omega$, $\eta =\alpha$ to obtain $\mathrm{gv}=4{\pi }^2\chi (M)$ in the positive base/fibre orientation.