Lag: the separation of consequential encounters

Lag separates an initiating change from a response or return. Through that separation, manifestation has a before and an after.

Specify the two events before naming their lag: contact and usable response, departure and return, or one layer receiving and another acting. Event ordering is available before a chosen clock supplies measured duration.

A delayed response can encounter a changed situation. Retention can carry an earlier difference across the interval, and comparison can use it. These require connected operations together; delay alone is not memory or an observer.

Mark an event, the state it affects and the later point at which its consequence becomes available. A time series or phase diagram can show the separation when its event definitions and sampling are declared. A phase offset and a time delay require a stated relation before converting between them.

The printed subjective-duration formula can be made explicit as a weighted path through a declared state representation. Let r(t) contain specified measured or modeled variables, with a fixed positive-definite metric M and a stated clock interval. Define v_M(t)=sqrt(r_dot(t)^T M r_dot(t)). The original v(t)=|r_dot(t)| is the Euclidean case. Coordinate units, normalization, sampling and smoothing matter: a raw sum of unlike biological variables is not an established distance.

Let C_g(t)>=0 denote an assigned sustained-instability load, kappa>=0, and require kappa*C_g to be dimensionless. The quantity Q=integral (1+kappa*C_g(t))*v_M(t) dt is weighted path length. It measures how much of the selected path was traversed under that weight. For the same path, a pointwise increase in load cannot decrease Q; it does not follow that every fearful state produces that change. C_g here is not the library's phase-coherence measure. In normalized coordinates Q is a score; a calibrated factor with units seconds per score could turn it into a candidate duration estimate. The original T_hat notation is preserved in provenance, with its missing observational mapping now made explicit.

Use the assigned route (0,0) to (3,0) to (3,4), with segment weights 1 and 2. The unweighted route length is 7 and Q=3+8=11. Taking 3 and 4 seconds on its segments gives elapsed time 7 seconds. Taking 6 and 8 seconds on the same weighted route gives elapsed time 14 seconds and still Q=11. Waiting at the intermediate point adds elapsed time but no score while the selected state is stationary. An assigned calibration of 0.5 seconds per score yields 5.5 seconds in both moving cases; these are arithmetic examples, not measured psychological results.

The same preservation can be proved for a smooth increasing change of traversal parameter when both the path and its load are carried along that change. The chain-rule speed factor cancels the inverse interval factor. If load instead evolves differently with physical time, it is no longer the same weighted path and the conclusion need not hold. The score is also coordinate-consistent only when the metric is carried with the representation. Rescaling the first coordinate by 2 while keeping Euclidean distance changes the example to 14. Using the corresponding metric diag(1/4,1) restores 11. Projection onto only the first coordinate yields 3 because the second segment becomes invisible.

Geometry can show the traveled route, its ordering and changing weight. Matrix notation specifies which coordinate differences contribute to its length. A closed return can accumulate traveled length even when endpoint displacement is zero. Neither a stable visible state nor a shorter projected path proves that the body has stopped changing or experiencing time. The integral is a defined candidate measurement, not a physical time law or a demonstrated account of all perceived duration. Its empirical use needs actual recorded states, justified metric and load, a calibration task and comparison with alternative explanations.

The older chapter links lag, mismatch, correction and oscillation. Those relations need a specified feedback law and conditions. The scalar delayed-return reading now solves the constant-coefficient law with instantaneous damping and one retarded contribution, distinguishing positive support from signed correction. The unrestricted earlier delay theorem still needs its own state, meaning of lag and continuation criterion: neither zero delay nor positive delay by itself decides oscillatory stability, chaos or experience.

A delayed return is a relation between marked events and a state that can still use what returns. The current source distinguishes the interval from the retained condition that carries an earlier difference across it. One process may have reception, integration and return delays with different endpoints. An interval can permit a retained difference to shape a later response, or can outlast the support required to use it. The source's return-before-loss comparison concerns those declared completion events; the delay parameter in a differential equation must be connected to them before the two become one measured quantity.

For the chapter's selected constant-coefficient law, write y'=-lambda*y(t)+a*y(t-tau), with lambda>0, real a, finite tau>=0 and a continuous initial history on [-tau,0]. Here y is first a signed state or error. It may be nonnegative support when a>=0 and the history is nonnegative, because that model preserves nonnegativity. Calling it a norm additionally requires the vector-to-scalar closure: in a Euclidean state rho'= -lambda*rho+(x/rho) dot R at rho>0. Only the radial component of the incoming vector changes the norm; a general return is not automatically a*rho(t-tau).

Substitution of exp(s*t) gives the characteristic equation s+lambda-a*exp(-s*tau)=0. If |a|<lambda, all finite delays give exponential decay. An explicit bound is |y(t)|<=H*exp(-eta*t), where H bounds the initial history and eta>0 satisfies eta+|a|*exp(eta*tau)<lambda. For positive support this shows loss of support toward zero; for an error it shows recovery toward the selected reference. At a=lambda, constant histories give constant solutions and the zero state is not asymptotically stable. At a>lambda, a positive real characteristic root exists for every delay: positive return already exceeds the linear loss. Delay has not created a first oscillatory instability from a previously stable zero state. At a=-lambda, every finite delay remains exponentially stable, with no uniform claim as delay grows without bound.

Oscillatory loss of a stable reference is explicit for sufficiently strong negative feedback. Put a=-b, b>lambda. The first threshold is tau_c=acos(-lambda/b)/sqrt(b^2-lambda^2), with angular frequency omega=sqrt(b^2-lambda^2). The zero state decays exponentially below this delay. At the threshold a simple imaginary pair supports nondecaying sine/cosine modes; above it unstable modes grow. Each conjugate pair crosses from the left half-plane to the right. With lambda=1 and b=2, tau_c=2*pi/(3*sqrt(3)), approximately 1.2092 in the selected time unit. A negative coefficient can describe correction to a signed deviation; it cannot silently become a generally valid law for nonnegative magnitude. Bounded attracting oscillation beyond the crossing requires additional nonlinear dynamics, while the linear law itself supplies the threshold and growth modes.

These results preserve the useful source route from displacement through delayed restoring return to repeated redirection, and specify when one chosen equation actually realizes it. They distinguish decay, neutral persistence, growing support and corrective oscillation. The existing delayed phase-lock reading has another exact feedback placement and its own local coupling window. Neither equation is automatically the current phase engine: its stored residue, phase law, delay mechanism and identity criterion must be matched separately. The characteristic derivation for this scalar family is complete; the broader claim that all delay creates oscillation, identity or experience has not been substituted for it.

World-to-body and body-to-observer processing are two possible boundaries for examining a nested response. Their actual routes need to be identified. The old expression that adds reality, memory, emotion and attention is an explanatory list, not an arithmetic equation with compatible units.

Correction has to be strong enough to meet a mismatch and timely enough to act on a useful difference. One specified phase model earns both limits: δ̇(t) = Δω − K sin δ(t−τ), where K > 0, τ > 0 and the uncoupled mismatch Δω is constant. Phase is in radians; K and Δω have inverse-time units. The model needs an initial phase history on [−τ,0], not only its value at one instant.

A locked phase obeys sinδ* = Δω/K. On the principal branch K > |Δω| gives cosδ* > 0. Linearizing a small phase error gives ė(t) = −a e(t−τ), with a = K cosδ* = √(K²−Δω²). Its characteristic equation is λ + a exp(−λτ) = 0. At the first imaginary crossing aτ = π/2, and the crossing is toward instability. The strict local exponential-stability window is therefore |Δω| < K < √[Δω² + (π/(2τ))²]. Too little correction cannot balance the mismatch; excessively strong correction to the delayed difference destabilizes this locked branch. The endpoints are nonhyperbolic and are excluded from this test. With τ = 0 this model has no finite upper K bound of this kind.

For a fixed connected symmetric network, the declared delayed law ẋ = −kLx(t−τ) gives another exact reading, where L is its nonnegative-weight graph Laplacian and k > 0. Each disagreement mode with positive eigenvalue λⱼ obeys the scalar equation with a = kλⱼ. All disagreement modes decay exponentially when kλmax(L)τ < π/2. The zero mode records a common state and is not an error to remove. This shows how the pattern of coupling can restrict the delay margin; it does not transfer the result unchanged to directed, changing or adaptive networks.

For Δω = 1 and τ = 0.5, the window is 1 < K < 3.2969083095. In the saved finite-horizon experiments, K = 0.5 does not lock, K = 2 returns after a small disturbance toward δ* = π/6, and K = 4 loses that local lock while its late phase remains bounded and oscillatory. The last result distinguishes the failed locking relation from destruction of every possible continuation. It is a measured result of that experiment, not a proof of global boundedness for all histories.

This gives an explicit coupling-strength window for one delay equation. It differs from an amplitude band, an observer's accessible range and the source's return-before-loss interval margin. Different delay placement, instantaneous damping, adaptive weights or nested feedback can change the result. The existing engine has its own particular mechanisms to analyze on their own terms; this benchmark does not resolve its uncomputed continuity margin or the older unrestricted delay theorem.

Distinguish an external event, the signal available to the observer and its retained interpretation. A reader may compare the same node through different histories or attention. The interface can expose that distinction without claiming that a delayed canvas sample constitutes felt experience.