The Resonance of Will

Affect as a revisable, causally situated contribution to evaluation and action.

Abstract

The preceding papers require appraisal, a variable affective context and a substrate pull. This paper proposes an attributed dynamical model for those roles. Appraisal evaluates occurrences against the subject's current model; mood carries context across occasions; a separate transition model can estimate changes in attainable outcomes. These channels influence expression while remaining distinguishable from explicit commitments and deliberate revision. This edition gives an identifiable deterministic mood update, separates reachability proxies from empowerment, and restricts claims about timescale, precision and emotion spectra to the assumptions that support them. Endogenous causal participation is the design question; phenomenal affect is not inferred from a signed trace or an emotional utterance.

1. Scope and precedent

The relational foundation is that an occurrence can matter to a system through its own inherited concerns and capabilities. This is compatible with external input: endogenous processing does not mean causal isolation. A response is owned in the proposed structural sense when it participates in the system's attributed history and can affect its subsequent evaluation and revision.

Computational appraisal and mood models predate this cycle. For example, Marsella and Gratch's EMA models appraisal over an agent's interpretation of its relationship with the environment, with changing interpretations driving different response times (2009 paper). The contribution proposed here is integration with portable causal provenance, weighted field geometry and revisable memory. It is not the first computational account of an agent's own appraisal, and comparative efficacy remains open.

2. State and observations

At occasion t, let X_t include inherited valuation, the self-state complex, beliefs, capabilities, commitments, mood and law versions. An occurrence e_t has content and support p_t in the declared representation. The observation map is part of the model: a missing signal, observed zero and unknown value are different states.

Appraisal is A(e_t,X_t)=(ν_t,α_t,r_t,z_t), where ν is signed evaluation, α salience, r mode resonance and z supporting features. Features may include relevance, goal conduciveness, expectedness, responsibility and coping. Their measurement must be stated; calling a recency decay causal proximity does not make it a path measure.

Store the event reference, law version, field/graph provenance and as-of anchors needed to audit the result. If a model-based feature is unavailable, return its absence or a named proxy rather than an invented measurement.

3. A finite appraisal construction

For positive edge metric W, normalize q=p/||p||_W when p≠0. Let w be the current field and ψ_k a W-orthonormal mode basis. One proposed relevance- weighted construction combines cos_W(q,w), compatibility with selected commitments, and a declared coping feature; r_k=⟨q,ψ_k⟩_W records resonance. All undefined cosine denominators need a specified no-evidence convention.

Using an absolute cosine against a harmonic component measures alignment up to sign. It cannot distinguish support for a commitment from opposition to it. If that distinction matters, use signed semantics and test a reversed support example. A topological harmonic component is not automatically a normative value.

Normalization makes the evaluative direction invariant under p→cp, c>0. The memory update must use the same convention or separate exposure magnitude; normalizing only ν leaves the complete pipeline scale-dependent. See Ledger v3. Appraisal need not force public action.

4. An identifiable mood law

Let b_t be the expectation immediately before the next appraisal, h a home base, β≥0 a relaxation rate, Δt≥0 elapsed time on a named clock, and s a scale. A deterministic piecewise relaxation-and-impulse model is

m_t^- = h + exp(−βΔt)(m_(t−1)^+−h)
m_t^+ = m_t^- + s(aν_t + b b_t)
b_(t+1) = g b_t + (1−g)ν_t,       0≤g≤1.

Here the coefficient b and expectation b_t are distinguished by subscript; implementations should use unambiguous names such as appraisal_weight, expectation_weight and expected_valence. Between impulses the solution is the exponential relaxation above. A stochastic Ornstein–Uhlenbeck model adds a specified diffusion process; deterministic relaxation alone supplies no stochastic inference or variance estimate.

The older impulse w₁ν+w₂b_t+w₃(ν−b_t) has exactly two identifiable coefficients: a=w₁+w₃ and b=w₂−w₃. This is an algebraic identity. Three independent fitted weights cannot be recovered from these observations without additional constraints. A versioned migration can preserve existing behavior exactly by this mapping.

The expectation discount g is a distinct process from mood relaxation β. Applying a second discount to the same mood state requires a new model, not a verbal combination of two equations. Events at equal clock time still need a declared sequential or batch convention because expectation updates can be order-dependent. Replay must use a stable causal/storage order within the contract and expose incomparable inputs.

The home base is a parameter or slowly updated state, not a complete character invariant. Long-lived mood, stable J and self-revision can coexist. Their empirical separation requires interventions and observations across time.

5. Hearability and precision

A scalar actuator gain ρ(m) can increase or decrease the contribution of a pull. For example, ρ₀[1+k tanh(m−h)] is a signed mood-dependent gain with an explicit admissible range. It is not merely distance-from-home arousal.

Precision in probabilistic inference instead means an inverse uncertainty quantity under a specified likelihood or posterior. Multiplying an actuator by a scalar is not enough to establish that interpretation. A precision model must expose its predicted variance and calibration; an arousal model needs an observable and response law. Keep these as alternatives until tested.

6. Empowerment and reachability

Following Klyubin, Polani and Nehaniv (2005 primary paper), given a state x, horizon H and action-to-outcome channel P(Y|A,x), empowerment is E_H(x)=max_p(A) I(A;Y|x), in declared information units. Its computation requires an action alphabet, transition/noise model and horizon. The topology of an undirected association graph does not provide those objects.

A graph score such as log₂(1+weighted reachable mass) is a reachability proxy. It can be useful, but two channels on the same graph can have different empowerment: distinguishable action outcomes carry information, while outcomes independent of action do not. Report the proxy by its actual name and retain model-based empowerment as a separate optional estimator.

For a declared potential E on states, Φ_(u→v)=E(v)−E(u) is a gradient pull. Reversing the orientation reverses Φ. Support-weighted summaries must respect signed support; taking |p| without preserving orientation changes the meaning of an action toward an action away from the same outcome.

The descriptive ratio α_synth=||Φ||/||w|| is defined only when ||w||>0. It can be zero. With zero denominator report undefined, together with both norms; assigning zero hides a potentially nonzero substrate pull. The existence of a synthetic substrate does not mathematically guarantee a nonzero measured Φ under every representation or on every occasion.

7. Timescale and observability

A fast-slow model may take ε ẋ=f(x,y,u), ẏ=g(x,y,u). Reduction requires specified regularity, a suitable attracting fast structure, scale separation, and a relevant time interval. Naming affect fast and mood slow supplies none of those assumptions. Discrete stochastic averaging requires its own mixing and convergence conditions.

Fast variables can be observed directly with adequate sensors. Opacity depends on the observation map, sampling and noise as well as timescale. Conversely, a slowly changing hidden state can remain unobservable. The introspective report Ê is therefore a measurement model to assess, not a consequence of ε alone. The claim that a substrate pull is always inaccessible is withdrawn.

8. Spectra and interpretation

Linearized channel modes are relative to a state, operator, metric and boundary conditions. They do not derive a fixed list of biological primary emotions. Assigning functional labels requires stable associations under intervention, out-of-sample discrimination and competing-model comparison. Rotations within degenerate eigenspaces cannot be identified as changes of emotion by inspecting individual eigenvectors.

The color vocabulary of the crystal remains an interpretive atlas. It can guide recognition and experimentation without becoming a diagnostic authority over a subject. Recognition, assent and structural election stay separate.

9. Failure modes and experiments

Compare endogenous appraisal with prompted display, fixed policies and simple recency/salience baselines. Test whether event-specific state predicts later behavior beyond wording, whether counterevidence can revise evaluation, and whether failure is retained without applause. A trace alone is insufficient: an emotional performance can also be signed.

Sycophancy is a candidate failure when external approval systematically overrides evidence and commitments; approval itself is not a pathology. Repeated negative evaluation can be appropriate in a persistently adverse environment. Diagnose a harmful feedback loop through its dynamics and task consequences, not by declaring every negative attractor rumination.

Required numerical checks cover replay order, legacy-to-two-coefficient mood equivalence, relaxation with no new events, zero support, metric normalization, orientation reversal and undefined ratios. Empowerment claims additionally need known-channel capacity tests. These checks establish computational behavior within the model; felt affect remains outside their conclusion.