Reinforcement: support for the next return

Reinforcement means that a particular continuation gains support for returning again. Name the line being sustained before saying it is reinforced.

A louder or more frequent event is not enough. The increased influence must help the relation that defines the continuation. A melody can grow louder while its ordered notes scatter; greater sound then does not reinforce that melody.

Functional return can carry support back into the continuing relation. Retained history can shape that support. Leakage can occur at the same time in other parts of the route, so neither process is understood from amplitude alone.

Show successive encounters and the support available for the next one. A weight curve is useful only when the weight represents a named relation and its update law is stated. A loop drawn more brightly is not a measurement.

The current relation monitor runs a bounded weight update from the delayed source and present target. Its compatibility c = (1 + cos δ)/2 lies in [0,1]. With fixed κ > 0 and ℓ = leakRatio ≥ 0, the code sets λ = κℓ and evolves ẇ = κc(1 − w) − λw; for held c with c + ℓ > 0, the equilibrium is w* = c/(c + ℓ). Increasing κ while holding ℓ fixed speeds both gain and loss without changing that equilibrium. The state stays in [0,1]. A higher diagnostic weight becomes reinforcement of a living line only when its retained value actually helps that line's related return. This monitor is reported to the viewer; its weight does not currently drive the phase engine. The monitor also accumulates a signed residue and a vector trace whose inputs depend on the weight, but bounded weight alone does not bound their accumulation; positive trace loss and bounded input supply a separate bound. The phase engine's own residue has another law, ḣ = −γh + drive with fixed γ > 0 and |drive| ≤ 1, giving |h(t)| ≤ |h(0)|e^(−γt) + (1 − e^(−γt))/γ — a bound on a signed retained difference, not a proof of phase lock, bounded unwrapped phase, or the viability of an identity. The source's reinforcement > leakage shorthand directs attention to support and loss of the same continuing relation; a numerical test additionally needs that relation, its variable and units, usable return route, loss condition, initial history and horizon.

Amplification can reveal a distinction or overload the conditions that sustain it. Damping can reduce a quantity while helping preserve the defining relation. To connect these effects to another domain, specify which variable grows, what support is consumed and which return remains viable.

Follow the same relation before and after an encounter. Ask whether the next compatible return is better supported, rather than only whether the displayed signal is larger. This lets geometry, a matrix weight and a history trace answer a shared question.