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Wittgenstein, Operationalized: A Unified Mathematical Framework for Picture Theory, Language Games, and Hinge Certainty

1. Introduction

Ordinary-language philosophy has long emphasized the ways our words work in practice—how propositions depict, how expressions are used within activities, and how certainty is anchored by background “hinges.” Yet these insights are typically presented in prose that resists direct operationalization. This paper develops a strictly testable and computationally tractable restatement of three Wittgensteinian cores—picture theory, language games, and hinge certainty—so that each becomes a target for measurement, learning, and statistical evaluation.

Motivation. Philosophical accounts of meaning and certainty are most valuable when they constrain inference, prediction, and intervention. We therefore cast (i) truth-conditions as structural correspondences decidable by constraint satisfaction, (ii) meanings as equilibrium strategies in partially observed cooperative games, and (iii) hinges as hyperpriors that move only under decisive Bayes-factor evidence. Each restatement yields concrete estimators, data requirements, and falsifiable predictions.

Thesis. Three cores admit strict restatement:

Picture theory → structural correspondence. A proposition is true just in case there exists a structure-preserving map from its syntactic “picture” to a relational model of the world. This reduces truth to a homomorphism feasibility problem (previewed by (1.1)) with an empirical fit functional over observations (previewed by (1.2)).
Language games → equilibrium strategies. “Meaning is use” becomes meaning-as-the-policy component that maximizes expected social utility in a stochastic interaction game, estimable via inverse reinforcement learning and validated by out-of-context robustness.
Hinges → hyperprior stopping. Certainty is modeled as slow-moving hyperpriors that change only when cumulative log Bayes factors exceed a switching cost; the “end of doubt” is an optimal stopping rule rather than a metaphysical boundary.

Contributions. We provide a unified mathematical and empirical program:

Structural-semantic truth as CSP/SAT. Define picture adequacy via a homomorphism condition (1.1), and an empirical fit score for partial observability (1.2). This yields decision procedures and generalization bounds for truth-as-correspondence.
Meaning-as-use via equilibrium policies. Define the meaning of a term as the optimal policy component in a partially observed game (2.1). Provide stability conditions and estimators (inverse RL / policy gradient) with diagnostics for equilibrium robustness.
Family resemblance as multi-view prototyping. Model category membership by multiple embeddings and prototype radii (3.1), show when multi-view prototypes reduce risk, and specify transfer metrics (ΔF1, nDCG, ECE) to validate gains.
Rule-following as identifiability under capacity control. Formalize “same rule” as PAC-identifiable under distributional separation (4.1) with VC/MDL regularization, explaining when rule-following is determinate and when it is not.
Private language as failure of public calibratability. Prove that purely inner referents lacking any effect on public tasks cannot meet cross-observer calibration criteria (5.1), turning the private-language argument into a testable null.
Hinges as hyperpriors with Bayes-factor stopping. Specify hinge update only when cumulative evidence surpasses a switching threshold (6.1), yielding stability claims and experimental designs for belief revision with costs.
Aspect-seeing and therapeutic resolution as algorithms. Treat aspect switches as multistable inference in an energy landscape and operationalize the therapeutic method as mismatch detection with an optimization-based refactoring criterion (8.1).

Roadmap. Section 2 reviews background in Wittgenstein scholarship alongside structural semantics, game-theoretic pragmatics, learning theory, and Bayesian decision theory. Section 3 fixes notation for observers, actions, communal games, and datasets. Sections 4–6 develop the formal cores: structural correspondence (Tractatus), meaning-as-use (Investigations), and family resemblance. Sections 7–9 treat rule-following, private language, and hinge certainty, making explicit their identifiability, calibratability, and stopping-theoretic structure. Section 10 analyzes aspect-seeing as multistable inference; Section 11 formulates the therapeutic method as a mismatch-repair algorithm. Section 12 details datasets, metrics, and experimental designs supporting each claim. Sections 13–15 discuss scope, ethics, and conclusions. Appendices provide proof sketches, algorithms, and implementation details.

2. Background and Related Work

This section situates the paper at the intersection of analytic philosophy (Wittgenstein I/II/III) and modern computational methods (model theory, game-theoretic pragmatics, learning theory, Bayesian decision, and multistable inference). We emphasize operational formalisms that admit estimation and falsification.

2.1 Wittgenstein I/II/III: From Picture to Practice to Certainty

Tractatus (Wittgenstein I). The early view treats propositions as “pictures” whose internal form mirrors possible facts. Our recasting interprets picturing as structure-preserving mapping between a syntactic picture and a relational world model; truth reduces to the existence of a homomorphism (previewed later by (4.1)–(4.2) in §4).

Philosophical Investigations (Wittgenstein II). The later view replaces essence with use: words acquire meaning through rule-governed participation in “language games.” We operationalize this with stochastic cooperative games and equilibrium policies, enabling data-driven estimation (see (5.1) in §5).

On Certainty (Wittgenstein III). Wittgenstein’s “hinges” anchor doubt and inquiry. We formalize hinges as slow-moving hyperpriors with optimal stopping governed by cumulative evidence and switching costs (cf. (9.1) in §9). The transformation converts a therapeutic diagnosis into testable decision rules.

2.2 Structural Semantics, Model Theory, and CSP

Model-theoretic semantics captures truth via satisfaction in a structure. Constraint Satisfaction Problems (CSP) and SAT provide the computational substrate: deciding whether a homomorphism exists matches CSP feasibility; empirical fit under partial observability becomes a constrained maximum-likelihood (or risk-minimization) problem. This connection yields: (i) decision procedures; (ii) sample-based generalization bounds for truth-as-correspondence.

2.3 Game-Theoretic Pragmatics, IRL, and Equilibrium Learning

Pragmatics models meaning as strategic coordination under uncertainty. We leverage Markov (partially observed) games and inverse reinforcement learning (IRL) to infer latent utilities and policies from interaction traces. Equilibrium refinements (correlated/Markov perfect) provide stability diagnostics, while policy-gradient estimators turn “meaning-as-use” into an identifiable object with cross-context robustness tests.

2.4 Prototype Theory and Multi-View Learning

Prototype and exemplar theories explain categorization without necessary-and-sufficient definitions. Multi-view learning generalizes this by combining heterogeneous embeddings (textual, visual, task-behavioral). “Family resemblance” becomes prototype tuples with view-specific radii; a majority or threshold rule over views improves calibration and transfer. This directly supports our membership rule (3.1) in §6 and predicts measurable gains (ΔF1, nDCG, ECE↓) when views are conditionally complementary.

2.5 Statistical Learning Theory: VC/Rademacher and MDL

Wittgenstein’s rule-following challenge—that no finite course of training fixes a unique rule—corresponds to non-identifiability in unconstrained hypothesis spaces. VC dimension and Rademacher complexity quantify capacity; Minimum Description Length (MDL) and structural risk minimization impose priors/penalties. Under distributional separation and capacity control, we obtain PAC-identifiability of the “same rule,” clarifying when rule-talk is determinate (see §7).

2.6 Bayesian Hyperpriors and Decision-Theoretic Stopping

Bayesian hyperpriors encode slow-changing background commitments. Sequential analysis and optimal stopping supply explicit criteria for hinge revision: switch only when cumulative log Bayes factors exceed a switching cost. This reframes “the end of doubt” as rational evidence-thresholding with testable trajectories (see (9.1) in §9).

2.7 Bistability in Vision/Cognition; Cognitive Control and Reframing

Ambiguous figures, perceptual multistability, and framing effects reveal energy landscapes with multiple attractors. Small contextual cues induce bifurcations (e.g., saddle-node, Hopf), producing S-shaped psychometric transitions and reaction-time signatures. We import these tools to model aspect-seeing (duck–rabbit) and extend them to conceptual reframing and therapeutic resolution, where optimization-based mismatch repair reduces conflict between uses (§10–§11).

2.8 Gap Filled by This Paper

Across disciplines, pieces of Wittgenstein’s program have analogs—truth via satisfaction, meaning via strategic use, certainty via background constraints—but they remain fragmented and often non-operational in philosophy or atheoretical in applied ML. This paper contributes a unified, testable formalization aligned with modern ML/AI methodology:

a CSP/SAT criterion and empirical fit for picture theory;
a game-theoretic/IRL semantics for meaning-as-use with stability metrics;
a multi-view prototype account of family resemblance with transfer guarantees;
a capacity-controlled identifiability theorem addressing rule-following;
a calibratability criterion formalizing the private-language argument;
a Bayes-factor stopping account of hinges with explicit costs;
a multistable-inference account of aspect-seeing and an algorithmic therapy for conceptual disputes.

By making each claim estimable on real data with falsifiable predictions, we turn Wittgenstein’s insights into a single operational framework that is ready for empirical scrutiny and cumulative improvement.

3. Preliminaries: Observers, Actions, and Communal Games

This section fixes the mathematical objects, measurability, and standing assumptions used throughout. All symbols are summarized again in Appendix C.

3.1 Basic Sets and Processes

Let (X) be the observation space, (A) the action space, and (S) the latent state space. We assume all are standard Borel spaces unless noted.

Stochastic kernels.
State dynamics and sensing are given by kernels (T) and (O):

• State transition: (T(s' ∣ s,a)) is a Markov kernel on (S). (3.1)
• Observation: (O(x ∣ s)) is a likelihood kernel on (X). (3.2)

Utilities and costs.
A (possibly social) utility is (u : X × A × S → ℝ). For multi-agent sections we use (u_i) for agent (i).

Policies.
A (history-dependent) policy is a stochastic kernel (\pi : A ⇐ 𝓗_t), where (𝓗_t = (x_{1:t},u_{1:t-1},a_{1:t-1})) is the public history. In stationary analyses we use (\pi(a ∣ x)) or (\pi(a ∣ x,u)).

Interaction process.
Given initial (s_1 ∼ ρ): for (t = 1,2,…),

(x_t ∼ O(· ∣ s_t))
utterance (u_t) is produced (mechanism defined below)
action (a_t ∼ \pi(· ∣ 𝓗_t))
utility (r_t = u(x_t,a_t,s_t))
(s_{t+1} ∼ T(· ∣ s_t,a_t)). (3.3)

3.2 Utterances as Tests and Transformations

We model the semantic role of utterances in two operational ways.

(i) Test semantics.
Each proposition (p) induces a partial test (\pi_p : X → {1,0,⊥}) with truth (1), falsity (0), or undefined (⊥) on current evidence. (3.4)

(ii) Transform semantics.
Directives or re-framings act as state-of-information transforms (\tau_p : X → X), e.g., re-anchoring attention/features. (3.5)

Composition.
Utterances compose as programs: if (p;q) then (\pi_{p;q}(x) = \pi_q(τ_p(x))). (3.6)

From tokens to operators.
A surface string (u) is mapped to an operator via a parser/grounder (\Gamma):
(\Gamma(u) = (\pi_u,τ_u)) with optional parameters (\θ_u). (3.7)

3.3 Community Graph and Calibration

Let (𝒪 = {1,…,N}) denote observers/tools (people, instruments, models). The community graph (G = (𝒪,E)) encodes teaching, calibration, and tool-use.

Edges and roles.
• An edge (i → j) indicates that (i) can instruct/calibrate (j).
• A tool node (k) exposes an interface (\Lambda_k) (settings, prompts), with observable I/O. (3.8)

Cross-observer mappings.
A calibration map (C_{ij}) aligns outputs of (i) and (j) on shared tasks (𝒯):
(C_{ij}) minimizes a discrepancy (Δ_{ij}) over datasets (D_{𝒯}). (3.9)

Public calibratability (preview of §8).
A referent (q) is publicly calibratable iff there exists a common (C) s.t. inter-observer variance of (C)’s outputs on shared data is small. (3.10)

3.4 Data Model and Logs

We consider timestamped interaction tuples
(D = {(x_t, u_t, a_t, r_t)}_{t=1}^T). (3.11)

Latent augmentations.
When available, we append (s_t) or weak supervision (\tilde{s}_t), and tool settings (\ζ_t) (prompts, hyperparameters). (3.12)

Task partitions.
We split (D) into train/validation/test folds respecting temporal order; counterfactual probes are generated by controlled perturbations to context (C). (3.13)

3.5 Measurability and Regularity Assumptions

A1 (Borel measurability).
(X,S,A) are standard Borel; (T(· ∣ s,a)), (O(· ∣ s)), and (\pi(· ∣ 𝓗_t)) are measurable kernels. (3.14)

A2 (Integrability).
Utilities satisfy (\mathbb{E}[|u(X,A,S)|] < ∞) under all admissible policies. (3.15)

A3 (Ergodicity or mixing, when invoked).
There exist (\beta ∈ (0,1)), (M < ∞) such that β-mixing coefficients of (𝓗_t) satisfy (\sum_t β(t) < M). (3.16)

A4 (Identifiability under capacity control).
For hypothesis classes (ℛ) (rules) and (𝒫) (policies), capacity is bounded by VC/Rademacher measures specified in §7. (3.17)

A5 (Well-posed parsing/grounding).
(\Gamma) yields measurable ((\pi_u,τ_u)) with Lipschitz constants bounded on compact subsets of (X). (3.18)

3.6 Induced Quantities Used Later

Empirical fit for picture adequacy (preview).
Given (p) and observations (x_{1:T}), define
(\mathrm{Fit}(p; x_{1:T}) = \max_h \Pr\big[\text{homomorphism constraints hold on observed fragments}\big]). (3.19)

Meaning-as-use policy component (preview).
For term (w), define
(\mathrm{Meaning}(w) ≔ π_w^* = \arg\max_{π} \mathbb{E}[u ∣ π,\ \text{community policies}]). (3.20)

Exclusivity for therapeutic diagnosis (preview).
For propositions (p,q) in context (C),
(\mathrm{Excl}(p,q ∣ C) = 1 − \mathrm{Jaccard}(\mathrm{Supp}(π_p^* ∣ C),\ \mathrm{Supp}(π_q^* ∣ C))). (3.21)

3.7 Regularization, Costs, and Priors

Capacity control.
We use penalized objectives with MDL priors (ℒ(θ) + λ·\mathrm{MDL}(θ)) to ensure identifiability and prevent rule overfitting. (3.22)

Hinge hyperpriors (preview).
Background commitments (𝓗) are encoded as hyperpriors over model parameters (θ); switches occur only when cumulative evidence exceeds an explicit cost (see §9). (3.23)

3.8 Notation and Conventions

• Boldface structures (𝔚, 𝔓) denote relational models and pictures.
• Scripts (𝒢, 𝒪, 𝒯) denote games, observer sets, and task sets.
• Time indices (t) are discrete; expectations are over the generative process defined by (3.3).
• All norms are (ℓ_2) unless noted; distances on probability measures use total variation or Wasserstein-1 as specified in context.

These preliminaries make later sections self-contained: §4 uses (3.19) for truth-as-correspondence; §5–§6 build on (3.20); §11 employs (3.21) to drive mismatch detection and repair.

4. Truth as Structural Correspondence (Recasting the Tractatus)

This section operationalizes the Tractarian claim that propositions “picture” facts by casting truth as the existence of a structure-preserving map. The upshot is a decision problem reducible to CSP/SAT, with an empirical fit functional usable under partial observability and finite data.

4.1 Structures and Pictures

Let the world be a finite (or σ-finite) relational structure 𝔚 = (D; R₁,…,R_k), where D is a domain and each R_j ⊆ D^{m_j} is an m_j-ary relation.
Let a proposition be a syntactic “picture” 𝔓 = (V; Q₁,…,Q_k), with a finite set of picture-objects V and relation symbols Q_j ⊆ V^{m_j} mirroring the arities of R_j.

Picture Adequacy. A map h: V → D is adequate iff it preserves all pictured relations:

(1.1) If (v₁,…,v_m) ∈ Q_j then (h(v₁),…,h(v_m)) ∈ R_j.

Truth-as-Correspondence. Define True(𝔓, 𝔚) ⇔ ∃h: V → D satisfying (1.1).

Intuitively, a proposition is true just in case there exists a homomorphism from its picture 𝔓 into the world-structure 𝔚. This aligns “picturing” with relational homomorphism.

4.2 Empirical Fit under Partial Observability

In practice we do not directly access 𝔚, but only fragments via observations x₁:ₜ (e.g., grounded tuples, scene graphs, measured links). Let Frag(x₁:ₜ) denote the set of observed relation-instances over D that are trustworthy at time window 1:ₜ.

Define the empirical fit of proposition p with picture 𝔓 as:

(1.2) Fit(p; x₁:ₜ) = max_h Pr[(1.1) holds for all (v̄ ∈ Q_j) when evaluated on Frag(x₁:ₜ)].

Here the probability is taken over the sampling of fragments (e.g., substructures collected by sensors or annotations). Fit captures how well some homomorphism h makes 𝔓’s constraints agree with the observed pieces of 𝔚.

Operationally, computing Fit amounts to a constrained matching problem on observed tuples, with missingness handled by treating unobserved constraints as neither satisfied nor violated (or via explicit missingness models).

4.3 Complexity and Learnability

CSP/SAT reduction. Deciding True(𝔓, 𝔚) is equivalent to a relational-homomorphism CSP: variables are v ∈ V, domains are D, and constraints enforce (1.1). Therefore:

• In general finite relational signatures, truth-as-correspondence lies in NP; many cases are NP-complete via standard CSP hardness.
• Tractable islands occur under polymorphism conditions (e.g., majority/minority, near-unanimity), bounded treewidth of the incidence hypergraph, or acyclicity of Q_j.

Learning h. When 𝔚 is only partially revealed, we estimate a mapping ĥ by risk minimization over observed fragments, optionally regularized by type-compatibility, soft attribute matching, or MDL penalties on |Im(h)| to avoid degenerate collapses. Under i.i.d. sampling of fragments and bounded noise, empirical risk minimization yields generalization guarantees (next subsection).

4.4 Finite-Sample Consistency

Let Z be the Bernoulli indicator that a randomly drawn, observable picture-constraint is satisfied by an optimal homomorphism h* (if it exists) on a random fragment. Suppose we compute Fit over T i.i.d. fragments.

Proposition 4.1 (Consistency). If Fit(p; x₁:ₜ) ≥ 1 − ε for all sufficiently large T under i.i.d. fragment sampling with bounded noise, then the empirical truth error of p is at most ε with high probability; i.e., the chance that p’s picture-constraints fail in the underlying structure on a randomly sampled observable instance is ≤ ε + o_T(1).

Sketch of justification. Let Z₁,…,Z_T be i.i.d. indicators of constraint satisfaction under the maximizing mapping ĥ_T. By maximality of Fit and uniform convergence over a finite (or capacity-controlled) family of candidate mappings, the empirical mean Ẑ_T concentrates around its expectation by Hoeffding-style bounds, while the argmax stability is controlled by a covering-number/VC argument on the finite domain assignments. Hence for any δ ∈ (0,1), with probability ≥ 1 − δ:

|Ẑ_T − 𝔼[Z]| ≤ √(½ ln(2/δ) / T),

so if Ẑ_T ≥ 1 − ε then 𝔼[Z] ≥ 1 − ε − o_T(1). Interpreting 1 − 𝔼[Z] as the population violation rate yields the claim.

4.5 Discussion: Recasting “The World Is the Totality of Facts”

On this recasting, a “fact” is a satisfied relation-instance in 𝔚, and a proposition’s content is the constraint system Q₁,…,Q_k it imposes on possible assignments. Saying that the world is the totality of facts becomes: truth-evaluation is homomorphism feasibility against the relational database of satisfied tuples. The empirical notion Fit(p; x₁:ₜ) acknowledges our epistemic position: we only ever test pictures against fragments, but concentration results link high Fit to low population error. Thus, the Tractarian picture theory becomes an explicitly computable pipeline:

compile syntax → picture 𝔓,
reduce 𝔓 vs 𝔚 to CSP(SAT),
when only fragments are available, maximize Fit and report calibrated confidence with finite-sample bounds.

This closes the gap between metaphysical talk of “picturing” and concrete, testable structure-preservation in data.

5. Meaning as Use: Language Games as Stochastic/Learning Games

We turn “meaning is use” into an object of estimation and testing by embedding utterances in a partially observed cooperative game and defining meanings as equilibrium policy components that maximize expected social utility.

5.1 Language Game with Partial Observability

Let the language game be 𝒢 = ⟨S, X, A, u, T⟩ with latent states S, observations X, actions A, utility u : X × A × S → ℝ, and transition kernel T(s′ ∣ s,a). Observers act on histories 𝓗_t = (x_{1:t}, u_{1:t-1}, a_{1:t-1}). A stationary stochastic policy is π(a ∣ x, u), extended as needed to history-dependent kernels.

5.2 Meaning-as-Use (Definition)

For a lexical item or construction w, its meaning is the policy component that optimizes expected social utility in the communal game (holding fixed the rest of the community’s strategies):

(2.1) Meaning(w) ≡ π_w* = argmax_π 𝔼[ u(X,A,S) ∣ π, community policies ].

Operationally, w “means” whatever use-pattern (policy slice) achieves maximal long-run coordination/payoff within its game context.

5.3 Stability under Context Perturbations

Let C denote contextual parameters (task mix, tools, stakes, priors). Write J(π; C) = 𝔼[ u ∣ π, C ]. Equip policies with a metric ‖π‖ (e.g., L₂ over logits or total variation over A×X). We quantify pragmatic stability by the sensitivity of π* to small context changes:

• ε-perturbation: ‖C′ − C‖ ≤ ε.
• Stability goal: ‖π_{C′} − π_{C}‖ ≤ δ(ε), with δ(ε) → 0 as ε → 0.

A Lipschitz-type bound will be derived below.

5.4 Estimation via IRL/Policy Gradient and Identifiability

Estimation. Given interaction logs D = {(x_t, u_t, a_t, r_t)}:

Fit a parametric policy class Π_θ via inverse RL or policy gradients on J(π_θ; C).
Extract the “component” for w by conditioning on its occurrence or instrumented interventions (A/B prompts, tool toggles).
Validate on held-out contexts, reporting utility lift and robustness (below).

Identifiability conditions.
(i) Richness: sufficient variation in contexts and co-player strategies to discriminate alternative π.
(ii) Capacity control: bounded complexity (e.g., MDL penalty) to ensure uniqueness of the empirical optimum.
(iii) Observability: either partial observability with correct likelihood or instrumental variables separating w’s usage from confounds.
(iv) Regularity: J is differentiable in θ and C; gradients unbiased or corrected.

5.5 Proposition 5.1 (Pragmatic Stability)

Assumptions.
A1. (β-mixing) Trajectories generated under any admissible π have ∑_t β(t) < ∞.
A2. (Strong concavity) For a reparametrized policy space (e.g., logits), J(π; C) is α-strongly concave in π for fixed C (α > 0).
A3. (Context smoothness) ∥∇_C J(π; C)∥ ≤ L_J and ∥∇_π∇C J(π; C)∥ ≤ L{JC} uniformly on admissible sets.
A4. (Well-posedness) Argmax π*(C) is unique.

Claim. For contexts C and C′ with ‖C′ − C‖ ≤ ε, the optimal policies satisfy

(5.1) ‖π_{C′} − π{C}‖ ≤ (L{JC} / α) · ε + o(ε).

Sketch. By optimality, ∇π J(π*{C}; C) = 0. Differentiate implicitly in C and apply the inverse-function theorem to ∇²_{ππ}J (negative definite with modulus α). Using Danskin/Envelopes and A3,

‖∂π*/∂C‖ ≤ ‖[∇²_{ππ}J]^{-1}‖ · ‖∇{π}∇{C}J‖ ≤ (1/α) · L_{JC},

then integrate along any path from C to C′ to obtain (5.1). β-mixing (A1) ensures gradient and Hessian estimates concentrate, so the bound holds with high probability from empirical data.

Interpretation. Under cooperative pressure and regularization, meanings-as-use (policies) are stable: small context shifts perturb use only slightly. Instability flags either capacity misfit or game redefinition.

5.6 Empirical Markers and Diagnostics

Held-out utility gain. Report the improvement of the learned π̂_w on unseen contexts:

(5.2) ΔU = J(π̂_w; C_test) − J(π_base; C_test) ≥ 0 (target).

Equilibrium diagnostics.
• Exploitability: gap to best response ≤ κ (small).
• No-regret: average regret ≤ η over evaluation episodes.
• Policy drift: measured ‖π̂_w^{C′} − π̂_w^{C}‖ versus ε = ‖C′ − C‖; check empirical slope against (5.1).
• Counterfactual robustness: interventional swaps of w’s usage (on matched X) preserve predicted utilities within tolerance.

Reporting standard. For each term w: (i) estimated π̂_w with capacity penalty, (ii) ΔU on held-out contexts, (iii) empirical Lipschitz slope of policy drift, (iv) exploitability/no-regret metrics, and (v) ablation showing identifiability breaks when capacity control is removed.

Outcome. With these elements, “meaning is use” becomes a measurable hypothesis: meanings are those policy components that (a) maximize expected social utility, (b) are stably realized under small contextual shifts, and (c) are identifiable and reproducible across datasets and interventions.

6. Family Resemblance as Multi-View Prototype Clusters

Family resemblance replaces necessary-and-sufficient definitions with graded membership around prototypes. We operationalize this by fusing multiple “views” (textual, visual, behavioral, task-utility) into a prototype tuple and a tolerant membership rule.

6.1 Multi-View Encoders and Prototypes

Let there be M views with encoders
fₘ : X → ℝ^{dₘ}, for m = 1,…,M. (6.1)

For a word/category w, define its family kernel (multi-view prototype) and radii
μ_w = ( μ_w^{(1)}, …, μ_w^{(M)} ), r_w = ( r_w^{(1)}, …, r_w^{(M)} ), r_w^{(m)} ≥ 0. (6.2)

Define per-view acceptance indicators
sₘ(x,w) = 1[ ‖ fₘ(x) − μ_w^{(m)} ‖ ≤ r_w^{(m)} ]. (6.3)

Let the aggregate support be S(x,w) = Σ_{m=1}^M sₘ(x,w). Then the membership rule is the majority/threshold criterion announced in §3:

(3.1) Σ_{m=1}^M 1( ‖ fₘ(x) − μ_w^{(m)} ‖ ≤ r_w^{(m)} ) ≥ τ.

When classifying among {w₁,…,w_K}, predict
ŷ(x) = argmax_w S(x,w) with ties broken by average distance. (6.4)

6.2 Connections: Prototype Theory, Set-Valued Categories, Tolerance Spaces

Prototype theory. μ_w^{(m)} are view-specific prototypes; r_w^{(m)} define tolerance bands rather than crisp borders.
Set-valued categories. For polysemy, we allow multiple prototypes per view, μ_{w,ℓ}^{(m)}, ℓ = 1,…,L_w, and take the min-distance across ℓ before applying (3.1).
Tolerance spaces. With a symmetric tolerance relation Tₘ(x,x′) ⇔ ‖fₘ(x) − fₘ(x′)‖ ≤ r_w^{(m)}, membership becomes neighborhood majority across views, a standard granular-rough interpretation.

6.3 Learning μ and r

Given labeled or weakly-labeled seeds 𝒟_w, a simple, robust estimator is:

• Prototype: μ_w^{(m)} ← mean or geometric median of { fₘ(x): x ∈ 𝒟_w }. (6.5)
• Radius: r_w^{(m)} ← quantile_q { ‖ fₘ(x) − μ_w^{(m)} ‖ : x ∈ 𝒟_w } with q tuned by validation. (6.6)

For open-world and drifted data, maintain μ_w^{(m)} via exponential moving averages and adapt r_w^{(m)} by tracking calibration error (ECE).

6.4 Transfer Advantage under Conditional Independence

Assume per-view conditionally independent evidence given class y ∈ {w, ¬w}. Let each view produce an i.i.d.-like Bernoulli vote sₘ(x,w) with accuracy p = P[sₘ=1 ∣ y=w] and false-positive rate q = P[sₘ=1 ∣ y≠w]. Consider the majority test S(x,w) ≥ τ with τ = ⌈M/2⌉.

Proposition 6.1 (Transfer Advantage). If p > 1/2 and q < 1/2, then the Bayes error of the majority multi-view rule decays exponentially in M and is strictly lower than any single-view baseline with the same per-view rates. In particular, for the positive class:

(6.7) P[ S < τ ∣ y=w ] ≤ exp( − 2 M ( p − 1/2 )² ),

and for the negative class:

(6.8) P[ S ≥ τ ∣ y≠w ] ≤ exp( − 2 M ( 1/2 − q )² ).

Sketch. Apply Hoeffding/Chernoff bounds to the sum of independent Bernoulli votes with mean p (positive) or q (negative). Since p > 1/2 and q < 1/2, the majority threshold sits below (resp. above) the mean by a margin proportional to |p−1/2| (resp. |1/2−q|), yielding exponential tails. Any single view has error ≥ min{1−p, q}, which does not enjoy exponential decay with M.

Implication for transfer. When novel domains perturb some views but not all, as long as a strict majority retain p>1/2 against drift, the aggregate remains accurate; hence multi-view prototypes transfer better than single-view ones.

6.5 Calibration and Decision-Level Fusion

Per-view distances can be mapped to probabilistic votes via isotonic or temperature scaling:

(6.9) vₘ(x,w) = σₘ( − αₘ ‖ fₘ(x) − μ_w^{(m)} ‖ + bₘ ),

followed by calibrated fusion:

(6.10) V(x,w) = (1/M) Σ_{m=1}^M vₘ(x,w), predict ŷ = argmax_w V(x,w).

This turns S into a soft ensemble that supports ECE-based calibration audit and threshold selection for open-set rejection.

6.6 Validation Protocols

Accuracy/transfer. Report ΔF1 against best single-view and late-fusion baselines:

(6.11) ΔF1 = F1_multi − maxₘ F1_single(m) ≥ 0 (target).

Retrieval quality. For polysemous w, index per-view prototypes and report nDCG@k on cross-view retrieval. (6.12)

Calibration. Report expected calibration error (ECE) before/after fusion and across domain shifts; good multi-view fusion should yield ECE↓:

(6.13) ECE = Σ_b (n_b / N) · | acc(b) − conf(b) |.

Ablations. Remove views, corrupt them with domain noise, or shuffle labels to verify the exponential stability predicted by (6.7)–(6.8).

6.7 Summary

Family resemblance is rendered as prototype tuples with tolerances, aggregated by a threshold/majority rule that (i) respects polysemy through multi-prototype sets, (ii) admits learnable, calibrated fusion, and (iii) under mild independence yields exponentially improving Bayes error with the number of views. This gives a concrete, testable upgrade over single-definition semantics and aligns with real-world multi-modal categories.

7. The Rule-Following Problem as Identifiability under Capacity Control

Wittgenstein’s dilemma—that no finite course of training fixes a unique “rule”—manifests as non-identifiability in unconstrained hypothesis spaces. We formalize “following the same rule” as statistical identifiability under capacity control and show that, with appropriate priors/penalties (e.g., MDL), empirical learners converge (in probability) to a distinguished target rule r*.

7.1 Hypotheses, Data, and Loss

Let ℛ be a hypothesis class of rules r: X → Y (binary or multiclass), and let 𝒟 be a distribution on X × Y. Write the risk of r under a loss ℓ(y, r(x)) as

(7.1) R(r) = E_{(x,y)∼𝒟}[ ℓ(y, r(x)) ].

Given a sample Dₙ = { (x_i, y_i) }_{i=1}^n drawn i.i.d. from 𝒟, the empirical risk is

(7.2) Ŕₙ(r) = (1/n) Σ_{i=1}^n ℓ(y_i, r(x_i)).

We say “two agents follow the same rule” when their learned hypotheses coincide with the unique 𝒟-identifiable r* (below) up to a null set.

7.2 Identifiability as Distributional Separation

A target rule r* ∈ ℛ is 𝒟-identifiable if no other r ∈ ℛ can agree with it on almost all inputs:

(7.3) ∀ r ≠ r*, Pr_{x∼𝒟_X}[ r(x) ≠ r*(x) ] ≥ γ,

for some separation margin γ > 0. Here 𝒟_X is the X-marginal of 𝒟. Condition (7.3) states that distinct rules make detectably different predictions with nonzero frequency. This is the precise analogue of “no finite course of training uniquely fixes the rule” when γ = 0 or when ℛ is so rich that (7.3) fails.

7.3 Capacity Control and MDL Selection

Non-identifiability is aggravated when ℛ is too expressive. We constrain ℛ by capacity (e.g., VC dimension d_VC or Rademacher complexity 𝓡ₙ). In addition, we choose r by penalized empirical risk with an MDL prior:

(7.4) r̂ₙ = argmin_{r ∈ ℛ} { Ŕₙ(r) + λ · MDL(r) }.

The MDL term is a code-length or structural penalty that biases toward simpler rules. Structural Risk Minimization (SRM) implements (7.4) by nested classes ℛ₁ ⊂ ℛ₂ ⊂ … with increasing capacity and a data-dependent choice of class index.

7.4 A Uniform Generalization Bound

For 0–1 loss, with probability at least 1 − δ over the draw of Dₙ, every r ∈ ℛ satisfies

(7.5) R(r) ≤ Ŕₙ(r) + 2 𝓡ₙ(ℛ) + √( (ln(2/δ)) / (2n) ).

Analogous bounds hold for Lipschitz surrogate losses with their Rademacher complexity. When using SRM/MDL, replace 𝓡ₙ(ℛ) by the complexity of the selected sub-class and add a log-cardinality (or code-length) term.

7.5 Theorem 7.1 (PAC-Identifiability with Capacity Control)

Assume:

A1. (Identifiability) r* ∈ ℛ satisfies the separation condition (7.3) with γ > 0.
A2. (Noise) Labels follow a realizable or bounded-noise model so that r* minimizes R(r) over ℛ.
A3. (Capacity) The sequence of candidate subclasses {ℛ_m} has complexities 𝓡ₙ(ℛ_m) → 0 as n → ∞ for fixed m, and the SRM/MDL selector asymptotically chooses a class of bounded complexity with probability → 1.
A4. (Penalization) The penalty λ · MDL(r) yields uniqueness of the penalized minimizer in the limit (ties broken deterministically).

Then the penalized empirical risk minimizer r̂ₙ defined in (7.4) is consistent and recovers the target rule in probability:

(7.6) Pr[ R(r̂ₙ) − R(r*) > ε ] → 0 as n → ∞, ∀ ε > 0,

and, moreover, agreement occurs on inputs with probability approaching 1:

(7.7) Pr_{x∼𝒟_X}[ r̂ₙ(x) ≠ r*(x) ] → 0 as n → ∞.

Sketch. By (7.5) and SRM, R(r̂ₙ) − R(r*) ≤ (Ŕₙ(r̂ₙ) − Ŕₙ(r*)) + o_p(1). Penalization and A3 ensure selection of a class where the uniform deviation vanishes and where the penalized empirical minimizer concentrates on the population minimizer. Under A1–A2, r* is the unique population minimizer modulo sets of 𝒟_X-measure zero. Hence (7.6). Finally, by A1, excess risk implies a strictly positive disagreement mass ≥ c·γ for some c depending on the loss; since excess risk → 0, disagreement probability → 0, giving (7.7).

7.6 What “Following the Same Rule” Amounts To

On this account, “agent A follows the same rule as agent B” means that both learn r̂ₙ → r* under the same 𝒟 and penalties—hence they agree on almost all inputs. Conversely, without capacity control or with γ = 0, multiple r ∈ ℛ attain indistinguishable risk, so “the same rule” is undefined (the Wittgensteinian predicament).

7.7 Practical Diagnostics

• Capacity ablation. Increase class size (e.g., depth, width) and observe whether disagreement with a fixed reference r̂ grows; uncontrolled capacity that sustains low empirical loss while increasing disagreement signals non-identifiability.
• Margin probing. Estimate γ̂ = Pr_{x∼𝒟_X}[ r₁(x) ≠ r₂(x) ] among top alternatives; γ̂ ≈ 0 warns of underdetermination.
• Penalty sweeps. Vary λ in (7.4); stable selection of the same r̂ across λ in a range indicates robust identifiability.
• Holdout semantics. Evaluate semantic invariants (downstream utilities) for rules tied on training loss; divergence in held-out utilities suggests which candidate aligns with communal use.

7.8 Implication for Wittgenstein’s Dilemma

The dilemma dissolves into statistics:

• Without priors/capacity control: Many continuations fit finite data; “following a rule” is not well-posed because (7.3) fails or ℛ admits indistinguishable hypotheses.
• With identifiability and capacity control: The MDL/SRM pipeline singles out r*; agents trained under the same 𝒟 and penalty converge to the same operational rule in the PAC sense (7.6)–(7.7).

Thus, rule-following is determinate exactly when distributional separation and controlled capacity make the target rule statistically identifiable.

8. The Private Language Argument as Public Calibratability

We restate the private language argument as a calibratability claim: a putative inner referent has public rule-use meaning only if it can be jointly calibrated across observers on shared tasks and data.

8.1 Setting and Definitions

Let each observer i possess an internal state zᵢ and produce reports rᵢ on shared inputs x ∈ X. A “private referent” q is a mapping that depends only on z (no shared handle): q : 𝒵 → 𝒬.

A calibration procedure is a public algorithm C that, given shared data D and an observer index i, outputs a standardized quantity C(D; i) in a common codomain 𝒞.

Public calibratability requires that cross-observer dispersion, when evaluated on the same D, is uniformly small:

(8.1) Var_i[ C(D; i) ] ≤ ε, for all admissible shared datasets D and some small ε > 0.

Intuitively, (8.1) says there exists a reproducible bridge from idiosyncratic internals to a common scale.

8.2 Loss-Based Criterion

Suppose there is a family of public tasks with loss L depending only on observable inputs/outputs:

(8.2) L : X × 𝒴 → ℝ, where predictions ŷ may be functions of C(D; i).

If q affects no public loss through any shared operation, then no calibrator can align observers on q.

Formally, say q is loss-inert if for all measurable F mapping (x, C(D; i)) to a decision ŷ, we have:

(8.3) E[ L(x, ŷ(F, x, C(D; i))) ] is independent of i.

Loss-inertness means any pipeline using common data and tools yields identical expected loss regardless of which observer’s internal q generated the report.

8.3 Proposition 8.1 (Non-Calibratability)

Claim. If q is loss-inert in the sense of (8.3), then there is no public calibration procedure C satisfying (8.1) that preserves nontrivial between-observer distinctions of q; hence q has no public rule-use meaning.

Sketch. Assume for contradiction that such a C exists with small Var_i[ C(D; i) ] and that C preserves a nontrivial function of q (i.e., there exist i, j with q(zᵢ) ≠ q(zⱼ) but C(D; i) ≠ C(D; j) with positive probability over D). Because C is public and feeds any shared decision rule F, the difference C(D; i) ≠ C(D; j) would be detectable by some F that lowers the expected loss for one observer’s pipeline relative to the other, contradicting loss-inertness (8.3). Therefore either C collapses distinctions (trivializes q) or (8.1) fails. In both cases, q lacks a stable public role in rule-use.

Interpretation. A purely inner referent that never couples to public tasks cannot be stably taught, checked, or corrected—hence it fails to instantiate a rule in communal practice.

8.4 Operational Tests

(a) Inter-annotator agreement. On shared D, compute chance-corrected agreement (κ, α). Failure to exceed random baselines after tool standardization supports non-calibratability.

(8.4) κ = (P_o − P_e) / (1 − P_e), α = 1 − (D_o / D_e).

(b) Model-assisted calibration. Train a public mapper C_θ on a subset Dₜᵣₐᵢₙ to predict each observer’s labels from shared inputs. Evaluate cross-observer variance on Dₜₑₛₜ:

(8.5) Var_i[ C_θ(Dₜₑₛₜ; i) ] → small ⇝ calibratable; persistent large variance ⇝ private.

(c) Out-of-person generalization. Hold out observers at train time; test whether a model learned from others predicts the held-out observer above chance:

(8.6) AUROC_out-of-person ≤ 0.5 ± δ ⇒ evidence for privacy.

(d) Loss coupling probes. Intervene on public tools (prompts, interfaces) and test whether any F yields loss differences attributable to q:

(8.7) | E[L | i] − E[L | j] | ≤ η across interventions ⇒ loss-inert.

8.5 Edge Cases and Clarifications

• Latent but calibratable. If q couples weakly to public tasks (tiny but nonzero effect), longer measurements or sensitive instruments can reduce ε in (8.1). The argument targets q with no such coupling.
• Idiolects and learning curves. Temporary idiolects can be public if they are teachable via C (ε shrinks as D grows). The private-language verdict concerns referents for which ε remains bounded away from 0 under all feasible C.
• Noise vs. privacy. High noise can mimic privacy; control via multiple replications, improved sensors, or task redesign to raise signal-to-noise.

8.6 Relation to Wittgenstein

The private language argument claims that meaning cannot be constituted by in-principle private ostension. Our reformulation: meaning requires calibratability—a pathway from inner states to public correctness conditions. When no such pathway exists, the purported “rule” lacks the communal normativity that makes rule-following possible.

8.7 Reporting Standard

For any candidate private referent:

Report κ/α with confidence intervals on shared D.
Report Var_i[ C_θ(Dₜₑₛₜ; i) ] before/after calibration.
Report AUROC_out-of-person and loss-coupling gaps under interventions.
Conclude “public” iff (i) κ/α ≫ 0, (ii) variance collapses under a single C, and (iii) out-of-person generalization is high with measurable loss improvements. Otherwise, classify as non-calibratable and thus non-public for rule-use.

9. Hinge Propositions as Hyperpriors and Optimal Stopping

Hinge propositions function as background commitments that normally do not move during inquiry. We model them as slow-moving hyperpriors and show that “the end of doubt” corresponds to an optimal stopping rule driven by cumulative evidence and switching costs.

9.1 Hinges as Slow-Moving Hyperpriors

Let θ denote structural parameters (world-model, tool calibrations, inference defaults). Hinges 𝓗 encode hyperpriors p(θ ∣ 𝓗) with high concentration and low update frequency.

• Slow-move constraint: for admissible time scales, total variation distance between successive hyperpriors is small, TV( p(θ ∣ 𝓗_t), p(θ ∣ 𝓗_{t+1}) ) ≤ ε_hinge, with ε_hinge ≪ 1. (9.1)

• Utility with costed switching: at each epoch t, the agent accrues instantaneous utility u_t(θ) and pays c_switch ≥ 0 upon switching 𝓗 → 𝓗′. (9.2)

9.2 Bayes-Factor Thresholding and the Switching Rule

Write BF_t for the Bayes factor at time t comparing the incumbent hinge 𝓗 to an alternative 𝓗′, i.e., BF_t = p(D_t ∣ 𝓗′) / p(D_t ∣ 𝓗). Let Λ_T = Σ_{t=1}^T log BF_t be the cumulative log-evidence favoring 𝓗′.

We adopt the decision rule previewed earlier:

(6.1) θ ∼ p(θ ∣ 𝓗), switch when Σ_t log BF_t − c_switch > 0.

Equivalently, switch at the stopping time τ* = inf{ T ≥ 1 : Λ_T > c_switch }. (9.3)

Interpretation: doubt “ends” not because it is metaphysically impossible but because, up to the switching cost, additional evidence no longer justifies staying with 𝓗.

9.3 Informal Stability Theorem

Assume:

A1 (Positive long-run utility under 𝓗). The stationary policy induced by 𝓗 yields expected average utility ū_𝓗 > 0 over the relevant horizon. (9.4)

A2 (Bounded variation of stakes). Per-epoch utilities are uniformly bounded and mixing so that law-of-large-numbers approximations apply. (9.5)

A3 (Well-calibrated evidence). The per-epoch log BF_t has finite variance and mean drift μ = E[log BF_t], estimated consistently from data. (9.6)

A4 (Switching friction). The effective switching cost c_switch ≥ 0 summarizes instrument retuning, coordination, and re-training overheads. (9.7)

Then:

• If μ ≤ 0, the optimal policy retains 𝓗 with probability → 1 as horizon increases (no profitable switch). (9.8)

• If μ > 0 but small relative to c_switch, the expected hitting time E[τ*] is large, scaling like c_switch / μ up to lower-order terms; hinges are stable on ordinary horizons. (9.9)

• If μ is sufficiently large (strong evidence against 𝓗), τ* becomes finite with high probability and the switch occurs; the agent then re-centers on 𝓗′ and resumes slow-move dynamics (9.1). (9.10)

Sketch rationale: treat Λ_T as a random walk with drift μ and bounded increments. Optional stopping arguments and Wald-type approximations imply E[τ*] ≈ c_switch / μ for μ > 0, while μ ≤ 0 prevents threshold crossing almost surely in the large-horizon limit.

9.4 Practical Estimation and Reporting

• Drift estimation: μ̂ = (1/T) Σ_t log BF_t with uncertainty via Newey–West or block bootstrap under mixing. Report confidence bands and the implied horizon to switch, Ĥ = c_switch / max(μ̂, ϵ). (9.11)

• Cost audit: c_switch decomposes into labor, retraining, coordination, and risk buffers; report each component to make hinge inertia explicit. (9.12)

• Robustness: stress-test μ̂ under model misspecification (alternative likelihood families) and tool perturbations to ensure hinge moves are not artifacts. (9.13)

9.5 Conceptual Payoff: “End of Doubt” as Optimal Stopping

The Wittgensteinian insight becomes operational: inquiry halts not at a transcendental wall but at a decision-theoretic boundary where further doubt is suboptimal given stakes and costs. Hinges are those high-level commitments whose hyperpriors change rarely because (i) evidence drifts are typically small or symmetric (μ ≈ 0), and (ii) switching costs are nontrivial, making retention optimal until cumulative evidence clears a transparent threshold per (6.1).

This reframing explains why hinges feel “certain,” why they can nonetheless move in crises (μ surges), and how communities can rationally differ when their c_switch or evidence streams diverge.

10. Aspect-Seeing as Multistable Inference and Bifurcation

Aspect-seeing (e.g., the duck–rabbit) is modeled as inference over an energy landscape in which multiple perceptual/conceptual states compete. We treat framing cues as small parametric perturbations that reshape the landscape and induce bifurcations.

10.1 Multistable Energy Landscape

Recall the reconstruction-regularization form from §7:

(7.1) E(φ; x) = ℒ_recon(φ, x) + λ ℛ(φ).

Here φ denotes an internal interpretation (e.g., “duck” vs “rabbit”), x the stimulus, λ ≥ 0 a regularization weight capturing simplicity/norm constraints. Local minima of E define admissible “aspects.” Let 𝒜(x) = { φ₁,…,φ_K } be the set of stable aspects for x.

10.2 Framing as Parametric Perturbation and Bifurcation

Let c ∈ ℝ^m be framing/context parameters (instructions, priming words, task stakes). Write E_c(φ; x) for the perturbed energy with smooth dependence on c. For small δ, consider c′ = c + δ·v.

Bistability/multistability. For baseline c, suppose E_c has at least two local minima φ_A, φ_B with barrier height ΔE(c) between their attraction basins. As c varies, classical bifurcations occur:

• Saddle-node: a minimum–saddle pair annihilates when a control component crosses a critical value.
• Hopf-like (in dynamical embeddings): oscillatory dominance emerges when perceptual dynamics are explicitly modeled (not required in the static case, but included when φ follows a relaxation ODE).

Empirically, both yield abrupt shifts in dominance and characteristic psychometric shapes.

10.3 Testable Predictions

S-shaped choice curves. Let P_A(c) be the probability of endorsing aspect A under framing c. Near a critical direction v,

(10.1) P_A(c + δ·v) ≈ 1 / (1 + exp{ −β·(α + κ·δ) }),

with slope β capturing decision temperature and κ the effective gain of framing on the energy difference E_c(φ_B) − E_c(φ_A).

Reaction-time (RT) discontinuities. Near the separatrix, first-passage times increase due to shallow curvature; RT distributions widen and can become bimodal as δ sweeps across the critical region.

Hysteresis. Under slow δ ramps, forward and backward sweeps differ when barrier lowering and re-formation are asymmetric.

10.4 Stochastic Switching and Kramers-like Law

Embed inference in noisy gradient dynamics:

(10.2) dφ_t = −∇_φ E_c(φ_t; x) dt + σ dW_t,

with effective noise σ > 0. Let ΔE_A→B(c) be the barrier from the basin of φ_A to the saddle on the path to φ_B. The transition rate k_{A→B} obeys a Kramers-like approximation:

(10.3) k_{A→B}(c) ≈ C(c) · exp{ −ΔE_A→B(c) / σ² }.

Small framing shifts δ modify ΔE and hence the rate exponentially:

(10.4) ΔE_A→B(c + δ·v) ≈ ΔE_A→B(c) − g·δ, g = ⟨∇_c ΔE_A→B(c), v⟩.

Therefore,

(10.5) k_{A→B}(c + δ·v) ≈ k_{A→B}(c) · exp{ + g·δ / σ² }.

Proposition 10.1 (Switch Probability Under Cue).
Over an observation window T short enough that rates are approximately constant, the probability of switching from aspect A to B satisfies

(10.6) P_{A→B}(T; c + δ·v) ≈ 1 − exp{ − k_{A→B}(c) · exp( g·δ / σ² ) · T }.

Interpretation. A small positive cue δ along v that lowers the barrier toward B (g > 0) induces an exponential increase in switch probability over fixed T. This predicts framing-induced flips with characteristic log-linear cue–probability relationships near neutrality.

10.5 Measurement Protocols

Cue–choice function. Sweep δ along a controlled framing dimension; fit (10.1) to obtain β, κ, and the midpoint α.
Barrier proxy via RT. Use median RT or drift-diffusion fits as a proxy for curvature/barrier height; check for widening and hysteresis consistent with bifurcation.
Rate estimation. In repeated viewing with intermittent masks, estimate k̂_{A→B}(c) and test the log-linear scaling in (10.5)–(10.6).
Counterfactual prompts. Linguistic reframings that target specific features (e.g., “bill” vs “ears”) should selectively modulate g, isolating mechanistic channels.

10.6 Summary

Aspect-seeing emerges when E(φ; x) supports multiple stable interpretations whose relative depths and barriers are cue-sensitive. Small, well-aimed frames change switch rates exponentially (10.3–10.6), producing S-shaped endorsement curves, RT discontinuities, and hysteresis. This multistable-inference view ties phenomenology (sudden “seeing-as” flips) to concrete, testable energy-geometry and decision-noise parameters.

11. Therapeutic Method as Mismatch Detection and Re-formulation

We turn Wittgenstein’s “therapeutic” dissolution of philosophical knots into a reproducible procedure that detects use-mismatch and refactors tasks, contexts, and tests until the conflict abates and utility rises.

11.1 Exclusivity as a Mismatch Signal

Let π_p* and π_q* be the meaning-as-use policies of propositions p and q in context C (§5). Define support sets Supp(π_p* ∣ C) and Supp(π_q* ∣ C) as the states/observations where the policies are materially invoked (e.g., non-negligible action probability mass).

(8.1) Excl(p, q ∣ C) = 1 − Jaccard( Supp(π_p* ∣ C), Supp(π_q* ∣ C) ).

High Excl with low task gain indicates semantic tension: the two uses rarely co-occur yet do not improve outcomes, suggesting a framing or objective misfit.

11.2 Algorithm Θ: Therapeutic Refactoring

Input: interaction log D, candidate propositions {p, q, …}, base context C₀, utility u.
Output: refactored tests and contexts (π̂, Ĉ) with Excl↓ and utility↑.

Θ.1 Diagnose.
• Estimate π̂_p, π̂_q via IRL/policy gradient on D.
• Compute Excl(p, q ∣ C₀) and base utility U₀ = E[u ∣ π̂, C₀].
• If Excl ≳ τ_excl and ΔU ≈ 0, mark (p, q) as a mismatch pair.

Θ.2 Re-specify losses and contexts.
• Decompose u into task-local terms: u = Σ_k w_k · u_k with w_k ≥ 0.
• Propose a refined context C₁ by adjusting: task mix, tool settings, stakes, observables.
• Define a refined loss u^{(1)} that up-weights the jointly beneficial subgoals (revealed by counterfactual probes).

Θ.3 Rewrite tests π.
• Rewrite partial tests π_p, π_q as programs over observables: split, gate, or compose (π_p; π_q), add guards to suppress spurious triggers, or factor a shared subtest τ_shared.
• Regularize policies with MDL/entropy penalties to avoid degenerate over-specialization.

Θ.4 Iterate to convergence.
• Re-fit π under (u^{(t)}, C_t) → evaluate Excl^{(t)} and U^{(t)}.
• Stop when Excl decreases below τ_excl or utility improvement ΔU_t < ε for K consecutive iterations.

Pseudocode sketch:

Θ(D, {p,q}, C0, u):
  fit π̂ on (D, C0, u); compute Excl0, U0
  if Excl0 < τ_excl or U0 already high: return (π̂, C0, u)
  for t = 1..T:
     (u^(t), C_t) ← RefineLossContext(D, π̂, p, q)
     (π_p, π_q)   ← RewriteTests(π_p, π_q; guards, composition, τ_shared)
     π̂           ← RefitPolicies(D, C_t, u^(t), regularizers)
     compute Excl_t, U_t
     if Excl_t ≤ τ_excl and U_t − U_{t−1} ≥ 0: maybe stop
     if improvement < ε for K steps: break
  return (π̂, C_t, u^(t))

11.3 Local Improvement Guarantees

Under convex re-formulations, Θ guarantees monotone local improvement.

Assumptions.
A1. (Convex surrogate) There exists a convex surrogate J(π; C, ũ) s.t. ΔJ tracks ΔU locally.
A2. (Lipschitz rewrites) Test rewrites implement non-expansive operators on features/inputs (‖τ(x) − τ′(x)‖ ≤ L‖x − x′‖ with L ≤ 1).
A3. (Regularized re-fit) Each re-fit solves π^{(t)} = argmin_π { J(π; C_t, ũ^{(t)}) + λΩ(π) } with Ω convex.

Proposition 11.1 (Local Descent).
Given A1–A3, each Θ iteration satisfies

(11.1) J(π^{(t+1)}; C_{t+1}, ũ^{(t+1)}) ≤ J(π^{(t)}; C_t, ũ^{(t)}) − η‖π^{(t+1)} − π^{(t)}‖²,

for some η > 0 depending on strong convexity, hence J decreases monotonically and the sequence {π^{(t)}} converges to a stationary point.

Sketch. Strong convexity of the re-fit objective yields a quadratic improvement bound; non-expansive rewrites keep the Lipschitz constants of features controlled so step quality does not degrade. Since ΔJ tracks ΔU locally, U does not decrease in a neighborhood of the fixed point.

Corollary 11.2 (Exclusivity Contraction, mild conditions).
If supports are induced by thresholded logits that are Lipschitz in π and contexts, then near a fixed point,

(11.2) Excl^{(t+1)} − Excl^{(t)} ≤ −c · ‖π^{(t+1)} − π^{(t)}‖ + o(‖·‖),

so exclusivity shrinks as policies settle, unless a genuine task contradiction remains (in which case Θ exposes it via non-improving ΔU).

11.4 Convergence Diagnostics

• Dual tracking: plot J and U versus iterations; demand J↓ and U↑ (or stable).
• Exploitability and regret: ensure equilibria remain near-cooperative (low exploitability, no-regret).
• Policy drift norm: ‖π^{(t+1)} − π^{(t)}‖ → 0; early stopping if plateau.
• Excl–Utility Pareto: visualize (Excl^{(t)}, U^{(t)}); accept only Pareto-improving moves.

11.5 Practical Refactoring Moves

Scope split: partition p into {p_core, p_context} to avoid accidental clashes.
Guarded composition: (p; q) executed only when a small shared test τ_shared passes.
Role disentangling: re-encode p as a feature-selector and q as a decision threshold.
Counterexample distillation: curate a buffer of high-Excl but low-gain episodes to steer ũ’s weights.
Tool calibration: retune interfaces/prompts so π_p, π_q share preprocessing and differ only in explicit criteria.

11.6 Case Sketches (detailed in §12)

Case A: “Explain” vs “Justify.” High Excl due to different audiences/tasks. Θ splits loss into informativeness (u_info) and norm-compliance (u_norm); guarded composition routes to p when u_info dominates, to q when u_norm dominates. Result: Excl↓, ΔF1↑ in retrieval-plus-QA.

Case B: “Safe” vs “Helpful.” Initial Excl high; utility low due to over-cautious refusals. Θ adds calibrated risk terms and rewrites tests with a harm-screen τ_shared. Outcome: refusals drop where safe; overall utility and calibration improve.

Case C: Polysemy in Commands (“light”). Conflicts between illumination vs weight. Θ learns multi-prototype senses, inserts a disambiguation guard from surrounding verbs/nouns; exclusivity falls and task success rises.

11.7 Summary

Algorithm Θ converts therapeutic dissolution into a measurable optimization loop: detect mismatch via exclusivity (8.1), refactor objectives and contexts, rewrite tests, and re-fit policies until conflicts subside and utility improves. Under mild convexity and stability conditions, Θ produces monotone local gains, and—crucially—exposes genuine contradictions when improvement stalls, guiding principled redesign rather than ad hoc patching.

12. Empirical Program: Datasets, Metrics, and Experimental Designs

This section specifies datasets, metrics, and study designs that render each claim testable. All experiments adopt preregistration, blinded analysis plans, and release code/data where licenses allow.

12.1 Corpora and Interaction Logs

Pragmatics & interaction. Multi-turn instruction-following logs with actions and rewards; collaborative reference games; human–AI tool-use traces with timestamps and prompts.
Human-in-the-loop tasks. Controlled lab tasks for framing and aspect-seeing; cross-annotator calibration tasks for sensation reports; long-horizon belief updates with explicit costs.

12.2 Metrics (Summary)

Picture-fit (Truth-as-Correspondence).
Fit(p; x₁:ₜ) defined in (1.2): maximize homomorphism satisfaction over observed fragments.

Utility lift.
ΔU = J(π̂; C_test) − J(π_base; C_test). (12.1)

Stability norm.
S(ε) = ‖π_{C+ΔC} − π_{C}‖ with ‖ΔC‖ = ε; empirical slope versus ε. (12.2)

Inter-annotator agreement.
κ and α as in (8.4); report CIs via bootstrap. (12.3)

Bayes factors and drift.
BF_t = p(D_t ∣ 𝓗′) / p(D_t ∣ 𝓗), Λ_T = Σ_{t=1}^T log BF_t. (12.4)

Multi-view gains and calibration.
ΔF1 = F1_multi − max_m F1_single(m). (12.5)
ECE = Σ_b (n_b / N) · | acc(b) − conf(b) |. (12.6)

12.3 Study A — Truth-as-Correspondence (CSP-Fit on Grounded Captions)

Goal. Test whether propositions true of scenes admit structure-preserving maps more often than false ones.

Data. Grounded image/video captions paired with scene graphs (objects, relations) and counterfactual/negative captions.

Method.

Parse each caption into a picture 𝔓 = (V; Q₁,…,Q_k).
Build fragment sets Frag(x₁:ₜ) from detected tuples.
Compute Fit(p; x₁:ₜ) via CSP solver with missingness handling.

Hypotheses. True captions yield higher Fit and better ROC–AUC distinguishing true vs false.
Stats. Paired t-tests on Fit; permutation tests for robustness; ablations on treewidth to test CSP tractability islands.
Success. AUC ≥ 0.85 and monotone Fit↑ with sensor quality; generalization of ĥ across scenes.

12.4 Study B — Meaning-as-Use (IRL in Cooperative Instruction Following)

Goal. Validate “meaning is the equilibrium policy component.”

Data. Multi-turn cooperative tasks (navigation, manipulation) with observed rewards and utterances containing target lexical items w.

Method.

Estimate π̂ via IRL/policy gradients.
Extract π̂_w by conditioning on occurrences/interventions.
Evaluate ΔU on held-out contexts; measure stability slope versus ε as in (12.2).

Hypotheses. π̂_w yields ΔU > 0 and exhibits Lipschitz drift consistent with (5.1).
Stats. Mixed-effects models for ΔU; slope tests vs ε; exploitability ≤ κ threshold.
Success. Positive ΔU with CI excluding 0; empirical slope ≤ predicted bound.

12.5 Study C — Family Resemblance (Multi-View Prototypes & Transfer)

Goal. Test multi-view prototype membership rule (3.1) and transfer advantage.

Data. Polysemous categories with textual, visual, behavioral, and task-utility views; out-of-domain splits.

Method.

Learn μ_w^{(m)}, r_w^{(m)} via (6.5)–(6.6).
Predict with S(x,w) ≥ τ and soft fusion (6.10).
Evaluate ΔF1 (12.5), nDCG retrieval within and across views; ECE (12.6).

Hypotheses. ΔF1 > 0 vs best single view; ECE↓ after fusion; exponential stability to view corruption consistent with (6.7)–(6.8).
Stats. McNemar tests for F1; calibration curves; corruption sensitivity plots.
Success. ΔF1 ≥ 0.03 (absolute), ECE reduction ≥ 20%, graceful degradation when ≤ ⌊(M−1)/2⌋ views corrupted.

12.6 Study D — Rule-Following (Capacity Ablations & Identifiability)

Goal. Demonstrate PAC-identifiability with capacity control (Theorem 7.1).

Data. Synthetic and real tasks with known target rule r*; curriculum providing finite samples n.

Method.

Train with SRM/MDL penalties, vary capacity (depth/width/features).
Measure disagreement mass γ̂ = Pr_x[ r̂ₙ(x) ≠ r*(x) ].
Track R(r̂ₙ) − R(r*) and γ̂ versus n and capacity.

Hypotheses. With MDL/SRM, γ̂ → 0 as n grows; without control, γ̂ plateaus > 0 despite low empirical loss.
Stats. Convergence fits; confidence bands via stratified bootstrap.
Success. Clear convergence under control; non-identifiability exposed when controls removed.

12.7 Study E — Private Language (Public Calibratability Failures)

Goal. Test non-calibratability for private sensations.

Data. Reports of inner states (e.g., “my quale Q”) across observers with shared stimuli; common toolchain.

Method.

Train public mapper C_θ on subset; test Var_i[ C_θ(D_test; i) ].
Out-of-person generalization AUROC_out-of-person.
Loss coupling probes under tool interventions (8.7).

Hypotheses. Persistent high variance and chance-level generalization; no loss differences under interventions.
Stats. Variance components models; DeLong tests for AUROC; equivalence tests for loss gaps.
Success. Var_i bounded below by ε₀ > 0; AUROC ≈ 0.5; no detectable coupling.

12.8 Study F — Hinges (Belief Switching with Costs)

Goal. Observe optimal stopping dynamics for hinges.

Data. Long-horizon tasks with explicit switching costs (retooling time, penalties).
Method.

Compute per-epoch BF_t and Λ_T (12.4).
Estimate drift μ̂ and implied horizon Ĥ = c_switch / max(μ̂, ϵ). (12.7)
Compare predicted τ̂* with observed switch times.

Hypotheses. Switches occur when Λ_T crosses c_switch; E[τ*] ≈ c_switch / μ̂ for μ̂ > 0; retention when μ̂ ≤ 0.
Stats. Renewal analyses; survival curves; goodness-of-fit for hitting-time models.
Success. Small calibration error between predicted and observed τ*; correct retention/switch patterns across cost conditions.

12.9 Study G — Aspect-Seeing & Therapy (Bifurcations and Mismatch Repair)

Goal. Validate multistable inference and Algorithm Θ’s utility gains.

Data. Ambiguous figures and linguistic ambiguities; framing cues δ; logs before/after therapeutic refactoring.

Method.

Measure P_A(δ) and RT; fit logistic (10.1), check hysteresis under forward/backward sweeps.
Estimate transition rates k̂_{A→B}(c) and test log-linear scaling (10.5)–(10.6).
Run Θ (§11) on high-Excl pairs; track Excl and utility over iterations.

Hypotheses. S-shaped curves with slope β; RT widening near separatrix; exponential cue–rate relationship; Excl↓ and utility↑ across Θ iterations.
Stats. GLM for logistic fits; Q–Q plots for RT; segmented regression for hysteresis; paired tests for Excl/utility changes.
Success. Goodness-of-fit for logistic and rate laws; monotone Excl↓ and ΔU↑ with convergence diagnostics satisfied.

12.10 Power, Controls, and Reproducibility

Power. Precompute n for 80–90% power on primary contrasts (e.g., ΔU ≥ 0.02 absolute; ΔF1 ≥ 0.03).
Controls. Randomized context shuffles; label permutation for CSP; view corruption for multi-view; placebo costs for hinges.
Reproducibility. Fixed seeds, deterministic solvers, thorough ablation matrices; preregistered analyses with deviations logged.

12.11 Decision Criteria (Per-Study)

Each study defines primary endpoints (AUC/ΔU/ΔF1/ECE/κ/τ* fit) with α = 0.05 (two-sided) and secondary diagnostics (stability slopes, exploitability, hysteresis width). Success requires meeting primary endpoints and no red flags in diagnostics.

This empirical program closes the loop between theory and measurement: each philosophical claim is tied to a concrete dataset, a falsifiable statistic, and a pre-registered analysis path.

13. Discussion: Scope, Limits, and Philosophical Payoffs

This section clarifies what the framework does and does not claim, situates it among neighboring positions (verificationism, quietism, deflationism), and sketches implications for ML and cognitive science. We close with open problems that delineate a realistic research agenda.

13.1 Scope: What We Capture as Operational Cores

Our proposals target actionable restatements of three Wittgensteinian threads:

Truth-as-correspondence recast as structure-preserving feasibility (homomorphism/CSP) with an empirical fit functional for partial data.
Meaning-as-use as equilibrium policy components in partially observed cooperative games, estimated via IRL and validated by stability diagnostics.
Hinges as slow hyperpriors with optimal stopping based on cumulative log Bayes factors and switching costs.

Two further bridges give the account real empirical teeth: (i) family resemblance via multi-view prototypes with demonstrable transfer gains, and (ii) therapeutic refactoring as an optimization loop that detects and repairs use-mismatch.

13.2 Limits: What We Bracket or Leave Open

No metaphysical reduction. We do not claim that homomorphism is truth, or that equilibria are meanings. Rather, we supply operational surrogates whose success is adjudicated by predictive/diagnostic performance.
Model relativity. Fit, policies, and Bayes factors are model-indexed; conclusions are conditional on the chosen encoders, priors, and utilities.
Idealizations. Conditional independence across views, strong concavity of utilities, and stationary evidence drifts simplify analysis; real systems will violate them.
Normativity beyond utility. Many linguistic norms (fairness, politeness, authority) exceed any single scalar utility. Our therapeutic method can incorporate multi-objective losses, but a full theory of norm pluralism remains out of scope.

13.3 Relation to Verificationism, Quietism, and Deflationary Truth

Verificationism (kinship, not identity). We echo verificationist impulses by tying meaning and truth to practices of checking, yet we avoid collapsing semantics into any fixed verification procedure. Our CSP/IRL proxies are updatable and fallible, accommodating scientific change.
Quietism (methodological resonance). The therapeutic algorithm Θ resembles quietist dissolution: instead of positing essences, we repair use and deflate pseudo-conflicts when Excl is high but task gain is low. However, we go beyond quietism by offering positive models (games, prototypes) and testable predictions.
Deflationary truth (compatibility). Treating “True(𝔓, 𝔚)” as homomorphism feasibility harmonizes with deflationary talk: asserting “p is true” often just licenses the same commitments as asserting p. Our contribution is a computable test (Fit, with finite-sample bounds) that makes those commitments auditable.

13.4 Integration with ML Practice and Cognitive Science

Grounded evaluation pipelines. The framework unifies disparate ML evaluations—CSP grounding, IRL utility lift, multi-view calibration—under one philosophical umbrella, enabling cross-study comparability and shared diagnostics.
Human–AI pragmatics. Meaning-as-use naturally extends to tool-augmented agents and mixed human–AI teams: equilibrium stability explains why prompt styles and interface constraints can shift semantics without changing syntax.
Cognitive multistability. The energy-landscape account of aspect-seeing links classical Gestalt phenomena with bifurcation diagnostics (S-curves, RT widening), offering a common lens for perception, categorization, and conceptual reframing.
Safety and alignment. Hinge hyperpriors and switching costs clarify when to doubt defaults and how to revise them responsibly; Algorithm Θ operationalizes a norm-aware debugging loop for semantic failures (e.g., “safe vs helpful” trade-offs).

13.5 Philosophical Payoffs

From description to experiment. Each thesis entails falsifiable signatures (e.g., Fit AUC, ΔU, stability slopes, BF trajectories), moving ordinary-language insights into the laboratory and the benchmark.
Unifying lens. Rule-following, private language, and family resemblance emerge as facets of identifiability, calibratability, and multi-view robustness, respectively.
Therapeutic clarity. Many philosophical disputes turn out to be high-Excl/low-gain pathologies resolvable by loss/context refactoring rather than by further ontological inflation.

13.6 Open Problems

(A) Non-stationarity and regime shifts. When contexts drift, utilities reweight, and tools evolve, our proofs relying on strong concavity and mixing weaken. Needed: adaptive IRL with drift-aware confidence regions and online CSP with incremental homomorphism tracking.
(B) Rich social norms. Multi-objective utilities (fairness, deference, rights) require Pareto-front semantics and institutional hinges with explicit governance costs; this connects to law and organizational theory.
(C) Multi-agent deception and signaling. Strategic obfuscation breaks naive calibratability and equilibrium learning. We need mechanism-design extensions (incentive-compatible calibration, adversarial IRL) and proofs under partial honesty.
(D) Higher-order meaning. Metalinguistic negotiation (changing the very rules of a game) suggests second-order hinges and game-of-games dynamics; formal tools from stochastic games with evolving state spaces are required.
(E) Embodiment and tooling. Changes in sensors and interfaces shift both pictures and use. Formalizing tool ontologies (how Γ maps tokens to operators) and their versioning remains an engineering and philosophical frontier.
(F) Data realism. Fit, ΔU, and ECE depend on dataset curation. We need robustness audits and measurement models that separate semantic signal from annotation artifacts.

13.7 Bottom Line

Our framework is modest in metaphysics but ambitious in method: it turns key Wittgensteinian ideas into learnable, testable, and action-guiding constructs. Its success should be judged not by whether it settles age-old debates in prose, but by whether it predicts, stabilizes, and repairs linguistic practice across human–AI systems—while making room, via hinges and therapy, for principled change when the world or our aims demand it.

14. Ethical and Methodological Considerations

Our framework touches human meaning, norms, and belief revision. This section sets guardrails so empirical gains do not come at the expense of participants, communities, or scientific integrity.

14.1 Normative Overreach and Hinge Transparency

No hidden values. We treat hinges 𝓗 (defaults, background commitments) as declared hyperpriors with documented switching costs. Each study must publish a Hinge Card: priors, Bayes-factor thresholds, cost components, and who set them (governance trail).
Plural objectives. Many tasks are multi-objective (helpfulness, safety, fairness, dignity). We report Pareto frontiers instead of collapsing to a single scalar utility unless stakeholders consent to weights.
Local legitimacy. When experiments affect communities (e.g., moderation policies), obtain stakeholder review, not just IRB approval.

14.2 Human Subjects and Calibration Fairness

Consent and debriefing. Collect explicit, revocable consent for interaction logging; explain any framing manipulations; provide post-task debriefs for aspect-seeing and hinge-switch studies.
Calibration equity. Public calibratability (§8) must not enforce assimilation to a dominant group’s idiolect. Report agreement (κ, α) stratified by demographic and linguistic groups; investigate systematic disparities and adapt tools rather than pathologizing variance.
Harm screens. Before deploying Algorithm Θ (§11) on sensitive content, run harm pre-checks (self-harm, legal, medical). If risk>threshold, route to human oversight.

14.3 Risk Management for Therapeutic and Framing Interventions

Therapeutic limits. Θ is an optimization routine, not psychological therapy. Prohibit use for clinical claims or coercive persuasion.
Framing safety. Aspect-seeing experiments use subtle cues; cap intensity and frequency, set cool-off windows, and allow opt-out at any time. Log all prompts/frames that could manipulate beliefs.
Deception controls. For multi-agent deception studies (§13.6C), use authorized deception only when scientifically necessary, with prior IRB approval and full debriefing.

14.4 Data Governance, Privacy, and Tooling

Data minimization. Store only what is essential (x, u, a, r); separate identifiers; apply data statements (source, consent, rights).
Privacy tech. Prefer secure enclaves; when feasible, use federated evaluation or differential privacy for sensitive logs.
Tool versioning. Publish Tool Cards for parsers/grounders Γ (versions, parameters, known failure modes) so results are reproducible across software updates.

14.5 Reproducibility and Open Science

Preregistration. Pre-specify hypotheses, primary/secondary endpoints, stopping rules, and analytic pipelines.
Artifacts. Release code, configuration files, and (when licensed) data via open repositories; include environment snapshots, seeds, and compute budgets.
Exact reporting. Document CSP encodings, IRL objectives, regularizers, and thresholds (e.g., Bayes-factor cutoffs) so third parties can rerun studies byte-for-byte.

14.6 Statistical Methodology Standards

Multiple comparisons. Control familywise error (Holm–Bonferroni) or FDR across endpoints (Fit AUC, ΔU, ΔF1, ECE, κ/α, BF).
Power and uncertainty. Publish ex-ante power analyses and ex-post sensitivity; report effect sizes with CIs, not only p-values.
Robustness. Use specification curves (alternative models, priors, views) to show claims do not hinge on one analytic choice.
Data drift. For non-stationarity, maintain temporal holdouts and report degradation over time; avoid peeking-induced bias with locked leaderboards or rolling validation.

14.7 Fair Use, Release, and Misuse Mitigation

Release discipline. Provide Model Cards and Evaluation Sheets describing intended uses, out-of-scope domains, known failures, and red-team results.
Misuse scenarios. Analyze risks (manipulative framing, discriminatory calibration, brittle hinges). Provide mitigations: rate limits, audit logs, oversight hooks, and kill switches.
License and attribution. Respect dataset licenses; require downstream users to preserve hinge transparency and calibration reporting.

14.8 Governance and Accountability

Independent oversight. Establish an external advisory board to review hinge choices, fairness findings, and release plans.
Incident response. Maintain a disclosure channel and a postmortem template (what hinge failed, evidence drift μ̂, switching threshold, remediation).
Community feedback. Build mechanisms for affected users to contest calibrations or propose alternative hinges, with documented resolution timelines.

Bottom line. Turning Wittgenstein’s insights into testable pipelines increases both explanatory power and responsibility. By declaring hinges, auditing calibration, protecting subjects, and practicing reproducible science, we make the framework safe to deploy, falsifiable in study, and accountable to the communities whose language and lives it models.

15. Conclusion

This paper operationalizes three Wittgensteinian cores—picture, use, and certainty—within a single empirical framework. Truth is recast as structural correspondence decidable by homomorphism feasibility with an empirical Fit functional for partial data (§4). Meaning is rendered as equilibrium policy components in partially observed cooperative games, estimated via IRL with stability guarantees (§5). Background “hinges” become slow hyperpriors governed by Bayes-factor optimal stopping, explaining when doubt should rationally end (§9). Two bridging pieces ground the approach in practice: family resemblance through multi-view prototypes with transfer advantages (§6) and a therapeutic refactoring algorithm that detects and repairs use-mismatch (§11). Together with identifiability analysis for rule-following (§7), and a calibratability criterion for private language (§8), the result is a unified, testable program with concrete datasets, metrics, and study designs (§12).

Philosophical payoff. The framework is modest in metaphysics yet ambitious in method: it does not claim that truth is homomorphism or meaning is equilibrium, but shows these surrogates are learnable, auditable, and falsifiable. It converts long-standing debates into measurable signatures—Fit AUCs, utility lifts, stability norms, Bayes-factor trajectories—making progress cumulative across ML and cognitive science. The energy-landscape account of aspect-seeing links phenomenology to bifurcation diagnostics and predicts framing-sensitive switches (§10), while hinge transparency and calibration fairness align the program with responsible research (§14).

Future work. Three strands are most urgent:

Richer communal games. Move beyond single-scalar utilities to multi-objective, norm-plural settings (fairness, authority, dignity), with Pareto semantics and institutional hinges that evolve under explicit governance costs.
Causal interventions at scale. Embed do-operations and instrumented prompts to identify meaning-as-use under non-stationarity, adversarial signaling, and tool updates; develop drift-aware confidence regions and online CSP for evolving pictures.
Formal therapies for conceptual disputes. Extend Algorithm Θ into a library of refactoring schemata (guarded compositions, scope splits, role disentangling) with convergence certificates and safety screens, enabling principled resolution of high-Excl/low-gain conflicts in real human–AI systems.

By tying picture to fit, use to coordination, and certainty to optimal stopping, we make Wittgenstein’s insights actionable. The success criterion is straightforward: do these models predict, stabilize, and repair linguistic practice under scrutiny? If so, they earn their keep—not as metaphysical ultimates, but as well-instrumented tools for inquiry and design.

Appendix A. Proof Sketches and Formal Lemmas

This appendix collects supporting bounds and optimality statements referenced in the main text. All formulas are “Unicode Journal Style,” single-line with tags.

A.1 Lemma A.1 (Generalization bound for Fit)

Setting. A proposition p induces picture constraints Q over variables V. Let 𝓗 be a finite family of admissible homomorphisms h: V → D (or a class with finite capacity; see remark). For data fragments {F_t}_{t=1}^T drawn i.i.d. from the observation process, define the empirical Fit

(A.1) Fit_T(p) = max_{h∈𝓗} (1/T) Σ_{t=1}^T 1[ h satisfies Q on F_t ].

Let the population fit be

(A.2) Fit_⋆(p) = max_{h∈𝓗} 𝔼[ 1[ h satisfies Q on F ] ].

Claim. With probability ≥ 1 − δ,

(A.3) 0 ≤ Fit_⋆(p) − Fit_T(p) ≤ √( (ln|𝓗| + ln(2/δ)) / (2T) ).

Sketch. By union bound over 𝓗 and Hoeffding on Bernoulli satisfaction indicators, for each fixed h,

(A.4) 𝔼[1[h ok]] − (1/T)Σ 1[h ok on F_t] ≤ √( (ln(2|𝓗|/δ)) / (2T) ).

Taking maxima on both sides yields (A.3). For infinite 𝓗, replace ln|𝓗| by a uniform capacity term (e.g., a covering number or a Rademacher-complexity bound) to obtain the same rate up to constants.

Consequence. High empirical Fit implies high population Fit with explicit finite-sample confidence, justifying Proposition 4.1.

A.2 Theorem A.2 (PAC-Identifiability under MDL)

Setting. Hypothesis class ℛ of rules r: X → Y, data D_n i.i.d. from 𝒟, 0–1 loss. Assume a target r* ∈ ℛ is 𝒟-identifiable:

(A.5) ∀ r ≠ r*, ℙ_{x∼𝒟_X}[ r(x) ≠ r*(x) ] ≥ γ > 0.

Let the learner minimize penalized empirical risk with MDL:

(A.6) r̂_n = argmin_{r∈ℛ} { Ŕ_n(r) + λ·MDL(r) },

where MDL(r) is a prefix-free code length over ℛ (Kraft ∑ 2^{−MDL(r)} ≤ 1), λ > 0 fixed. Suppose realizability or bounded-noise so that r* minimizes population risk over ℛ.

Claim. For any ε > 0, as n → ∞,

(A.7) ℙ[ R(r̂_n) − R(r*) > ε ] → 0,

and moreover the disagreement mass vanishes:

(A.8) ℙ_{x∼𝒟_X}[ r̂_n(x) ≠ r*(x) ] → 0.

Sketch. Use a structural risk minimization (SRM) decomposition indexed by code length L = MDL(r). Kraft allows a union bound with weights 2^{−L}. For each sub-class {r: MDL(r)=L}, apply a uniform deviation inequality (e.g., VC/Rademacher) to get

(A.9) sup_{r:MDL(r)=L} |R(r) − Ŕ_n(r)| ≤ C√( (d_L ln n + ln(1/δ_L)) / n ),

with d_L the capacity of the L-slice and δ_L ∝ 2^{−L}δ. Penalization steers selection toward slices where the deviation is small and the empirical gap to r* cannot be maintained; hence consistency (A.7). Finally, identifiability (A.5) implies that any persistent excess risk enforces a nonzero disagreement mass ≥ c·γ, so vanishing excess risk yields (A.8).

A.3 Lemma A.3 (Stability under perturbations)

Setting. Let J(π; C) = 𝔼[u ∣ π, C] be the long-run utility of policy π under context C. Assume:

• Strong concavity: ∇²_{ππ}J(π; C) ≼ −α·I for all admissible (π, C), α > 0.
• Smoothness in C: ‖∇{π}∇{C}J(π; C)‖ ≤ L_{JC}.
• Unique maximizer π*(C) for each C.

Claim. For any contexts C, C′ with ‖C′ − C‖ ≤ ε,

(A.10) ‖π*(C′) − π*(C)‖ ≤ (L_{JC}/α)·ε + o(ε).

Sketch. Optimality yields ∇π J(π*(C); C) = 0. Differentiate implicitly in C and apply the inverse function theorem: (∂π*/∂C) = −[∇²{ππ}J]^{−1} · (∇{π}∇{C}J). Norm-bounding with α and L_{JC} gives ‖∂π*/∂C‖ ≤ L_{JC}/α; integrate along any path from C to C′ to obtain (A.10).

A.4 Proposition A.4 (Non-calibratability criterion)

Setting. A private referent q depends only on internal state z and is loss-inert: for any public decision pipeline F that maps shared inputs and any calibrator output to actions, expected loss does not depend on observer i,

(A.11) 𝔼[ L( x, F(x, C(D; i)) ) ] is constant in i.

Public calibratability demands a common C such that cross-observer dispersion is small while preserving nontrivial distinctions:

(A.12) Var_i[ C(D; i) ] ≤ ε and ∃ i,j with ℙ[ C(D; i) ≠ C(D; j) ] ≥ ρ > 0.

Claim. (A.11) and (A.12) cannot both hold for any ε, ρ bounded away from 0. Therefore, a loss-inert q is not publicly calibratable in the nontrivial sense.

Sketch. Suppose both hold. Since C is public, choose F to exploit any systematic difference in C(D; i) vs C(D; j) (e.g., thresholding along the separating direction). By the Neyman–Pearson lemma, a measurable difference in C induces a weakly better decision for one subgroup unless the induced likelihood ratios are almost surely 1, contradicting loss invariance (A.11). Thus either C collapses distinctions (ρ → 0) or variance stays bounded away from 0, violating calibratability.

A.5 Lemma A.5 (Stopping optimality with hinge costs)

Setting. Let Λ_T = Σ_{t=1}^T log BF_t be a random walk with drift μ = 𝔼[log BF_t] and finite variance, comparing incumbent hinge 𝓗 to alternative 𝓗′. A switch incurs cost c_switch ≥ 0 and resets hinges (after which the same analysis applies). Consider the stopping time

(A.13) τ* = inf{ T ≥ 1 : Λ_T > c_switch }.

Define value V = sup_τ 𝔼[ gain(τ) − c_switch·1{switch at τ} ], where gain(τ) is the cumulative expected improvement from adopting 𝓗′ upon switching at τ.

Claim. Under standard regularity (independent increments, bounded second moments), τ* is an optimal stopping rule among all bounded stopping times; moreover, for μ > 0,

(A.14) 𝔼[τ*] ≈ c_switch / μ and 𝔓(τ* < ∞) = 1,

while for μ ≤ 0,

(A.15) 𝔓(τ* = ∞) = 1 (i.e., never switch is optimal).

Sketch. Map to a one-sided sequential probability ratio test (SPRT) with asymmetric cost: the optimal boundary for maximizing expected net gain under linear costs is a constant threshold on cumulative log-likelihood (Wald’s identity / dynamic programming with Snell envelope). The Doob–Meyer decomposition and the optional sampling theorem yield that the Snell envelope is attained when Λ_T first exceeds c_switch, giving τ*. Classical renewal/random-walk arguments yield 𝔼[τ*] ≈ c_switch/μ for μ > 0 and almost-sure non-hitting for μ ≤ 0, matching (A.14)–(A.15).

Consequence. “End of doubt” corresponds to the optimal stopping time τ*; hinges persist until evidence drift μ and accumulated Λ_T clear the explicit switching cost, as claimed in §9.

Remark on constants and norms. Throughout, operator norms are spectral; concentration constants can be replaced by empirical counterparts (e.g., block bootstrap under β-mixing) without changing the qualitative claims.

Appendix B. Algorithms and Pseudocode

This appendix lists reference algorithms corresponding to the main text. Pseudocode is language-agnostic, with clear I/O contracts and convergence checks. Complexity notes appear at the end.

B.1 Algorithm Π — IRL for Meaning-as-Use (with Convergence Checks)

Goal. Estimate the policy component π*_w that realizes “meaning-as-use” for term w by maximizing long-run utility within a partially observed game (§5).

Inputs. Logs D = {(x_t, u_t, a_t, r_t)}; policy class Π_θ; context descriptor C; temperature τ; regularizer Ω(θ); early-stop tol ε; max iters T_max.
Outputs. θ̂_w; diagnostics: ΔU, stability slope, exploitability κ̂, regret η̂.

Π(D, w, Π_θ, C, τ, Ω):
  # 1) Initialize
  θ ← θ0 ; moving_avg ← 0 ; t ← 0
  best_θ ← θ ; best_val ← −∞

  # 2) Instrument for w (occurrence or interventions)
  Dw ← FilterByOccurrenceOrIntervention(D, w)

  # 3) Loop: inverse RL / policy gradients
  while t < T_max:
      # E-step: estimate latent returns given current θ
      Q̂ ← EstimateQ(Dw, θ, C)                   # off-policy eval or importance-weighted
      # M-step: policy gradient with entropy/MDL regularization
      g ← ∇_θ [ 𝔼_{(x,u)∼Dw} 𝔼_{a∼π_θ(·|x,u)}[ Q̂(x,a) ] − λ·Ω(θ) ]
      θ ← Update(θ, g, stepsize_t)
      t ← t + 1

      # 4) Convergence checks (held-out)
      ΔU_val ← EvaluateUtilityLift(θ, D_val, C)  # ΔU = J(π_θ; C_val) − J(base)
      κ̂ ← Exploitability(θ, D_val)              # gap to best-response policy
      η̂ ← AverageRegret(θ, D_val)
      slope ← StabilitySlope(θ, C, ε_grid)      # ||π*_C' − π*_C|| / ||C'−C||

      if ΔU_val > best_val:
          best_val ← ΔU_val ; best_θ ← θ

      if Converged(ΔU_val, slope, κ̂, η̂, ε_tol=ε):
          break

  return best_θ, {ΔU: best_val, slope: slope, κ: κ̂, regret: η̂}

Convergence test. Stop when (i) ΔU improves < ε for K rounds, (ii) slope ≤ bound predicted by stability (5.1), and (iii) κ̂, η̂ below thresholds.

B.2 Algorithm Φ — Multi-View Prototype Learning (Family Resemblance)

Goal. Learn per-view prototypes μ_w^{(m)} and radii r_w^{(m)} and fuse votes across views (§6).

Inputs. Views {f_m}, labeled/seed sets 𝒟_w, quantile q, threshold τ, calibration method Cal (isotonic/temperature), EMA factor β.
Outputs. {μ_w^{(m)}, r_w^{(m)}}, calibrated fusion V(x,w), ECE diagnostics.

Φ({f_m}, {𝒟_w}, q, τ, Cal, β):
  for each w:
     for each view m:
        Z ← { f_m(x) : x ∈ 𝒟_w }
        μ_w^{(m)} ← GeometricMedian(Z)                 # robust to outliers
        dists ← { ||z − μ_w^{(m)}|| : z ∈ Z }
        r_w^{(m)} ← Quantile(dists, q)

  # Calibrate per-view soft votes
  for each view m:
     train_data ← BuildPairs({f_m(x), y_w})
     σ_m ← FitCalibrator(Cal, train_data)             # logistic / isotonic

  # Fusion rule
  function V(x, w):
     votes ← []
     for m in 1..M:
         s_m ← 1[ || f_m(x) − μ_w^{(m)} || ≤ r_w^{(m)} ]
         v_m ← σ_m( Affine( − || f_m(x) − μ_w^{(m)} || ) )
         votes.append( (s_m, v_m) )
     S ← Σ_m s_m
     soft ← (1/M) Σ_m v_m
     return (S ≥ τ) ? soft : Downweight(soft)

  # Online adaptation (optional)
  on stream x_t with weak label y_t:
     for m: μ_w^{(m)} ← (1−β) μ_w^{(m)} + β f_m(x_t) if y_t == w
           recalibrate σ_m periodically with a sliding window

  return {μ,r,σ}, V

Diagnostics. ΔF1 vs best single view; ECE before/after fusion; corruption robustness (≤ ⌊(M−1)/2⌋ corrupted views degrades gracefully).

B.3 Algorithm Ψ — Capacity-Controlled Rule Induction (SRM + MDL)

Goal. Induce a rule r̂ with PAC-identifiability under capacity control (Theorem 7.1).

Inputs. Hypothesis ladder {ℛ_m} with increasing capacity; code lengths L(m) s.t. Kraft ∑ 2^{-L(m)} ≤ 1; penalty λ; dataset D_n; validation split.
Outputs. r̂, selected index m̂, disagreement γ̂.

Ψ({ℛ_m}, L, λ, D_n):
  best_score ← +∞ ; r̂ ← None ; m̂ ← None
  for m in ladder:
     r_m ← TrainEmpiricalRiskMinimizer(ℛ_m, D_train, penalty=λ·L(m))
     val ← ValidationLoss(r_m, D_val) + λ·L(m)
     if val < best_score:
        best_score ← val ; r̂ ← r_m ; m̂ ← m

  γ̂ ← EstimateDisagreementMass(r̂, ReferenceOrOracle, D_test)  # Pr_x[r̂(x) ≠ r*(x)]
  return r̂, m̂, γ̂

Diagnostics. Plot γ̂ vs n and m; seek γ̂ → 0 under control, plateau without control; perform penalty sweeps λ to check stability.

B.4 Algorithm Θ — Therapeutic Mismatch Repair (from §11)

Goal. Detect high exclusivity pairs and refactor losses/contexts/tests until exclusivity drops and utility rises.

Inputs. D, propositions {p,q}, base context C₀, base utility u, thresholds τ_excl, ε, K.
Outputs. Refactored (π̂, Ĉ, û), trajectory logs {Excl^{(t)}, U^{(t)}}.

Θ(D, {p,q}, C0, u, τ_excl, ε, K):
  π̂ ← FitPolicies(D, C0, u)
  Excl0 ← Exclusivity(π̂_p, π̂_q, C0)
  U0 ← EvalUtility(π̂, D_test, C0)

  if Excl0 < τ_excl or U0 high: return (π̂, C0, u)

  t ← 0 ; stall ← 0
  while True:
     (u', C') ← RefineLossAndContext(D, π̂, {p,q})        # reweight subgoals, tool settings
     (π_p', π_q') ← RewriteTests(π̂_p, π̂_q)               # guards, scopes, shared τ
     π̂' ← RefitPolicies(D, C', u')
     Excl' ← Exclusivity(π_p', π_q', C')
     U' ← EvalUtility(π̂', D_test, C')

     Log(t, Excl', U')
     if (Excl' < Excl0 − ε) and (U' ≥ U0 − ε):            # Pareto or near-Pareto
         (π̂, C0, u) ← (π̂', C', u') ; Excl0 ← Excl' ; U0 ← U' ; stall ← 0
     else:
         stall ← stall + 1

     t ← t + 1
     if stall ≥ K: break

  return (π̂, C0, u), TraceLog

Convergence. Under convex surrogates and non-expansive rewrites, Θ decreases a surrogate objective monotonically (Proposition 11.1) and contracts exclusivity locally.

B.5 Runtime/Space and Implementation Notes

CSP Fit (Study A).
• Encoding: variables = picture nodes; domains = scene-graph entities; constraints = relation arcs.
• Solvers: arc-consistency (AC-3) for sparse/acyclic Q; ILP/MaxSAT for noisy fits.
• Complexity: worst-case NP; practical with bounded treewidth and small arities.

IRL/Policy Gradients (Π).
• Off-policy eval is the bottleneck; use doubly robust estimators.
• Complexity per iter: O(|D|·|A|) for tabular; O(|D|·B) for neural with batch B.
• Stability: gradient clipping; trust region (KL ≤ δ) to enforce smooth policy drift.

Multi-View Prototypes (Φ).
• Precompute embeddings; geometric median via Weiszfeld (O(N·d·I)).
• Calibration adds O(N) per view; fusion O(M) per query.
• Memory: store μ,r per (w,m) and small calibrators.

SRM/MDL (Ψ).
• Ladder depth chosen by resource budget; parallelize training across m.
• Code lengths L(m) proportional to log of parameter count or VC bound.
• Report wall-clock and energy per selection to compare ladders.

Therapeutic Loop (Θ).
• Each iteration: one policy re-fit + exclusivity/utility eval.
• Early stop when ΔU, ΔExcl plateau; cache counterexamples for faster rewrites.
• Safety: integrate harm screens and audit logs for all refactoring steps.

Reproducibility.
• Fix seeds; snapshot tool versions (Γ, calibrators); log hyperparameters.
• Provide evaluation harnesses (Fit AUC, ΔU, ECE, κ/α, BF trajectories) as callable scripts.

Appendix C. Notation Table and Implementation Details

This appendix standardizes symbols and defines preprocessing/evaluation protocols used across studies (§12). Equations follow Blogger-ready “Unicode Journal Style.”

C.1 Notation Table

Symbol	Type	Meaning / Role
(X)	set / space	Observation space (standard Borel).
(A)	set / space	Action space.
(S)	set / space	Latent state space.
(u:X×A×S→ℝ)	function	Utility (social/task).
(T(s'∣s,a))	kernel	State transition kernel.
(O(x∣s))	kernel	Observation likelihood.
(𝓗_t)	tuple	Public history ((x_{1:t},u_{1:t-1},a_{1:t-1})).
(π(a∣x,u))	kernel	Stationary stochastic policy (can be history-dependent).
(D={(x_t,u_t,a_t,r_t)})	multiset	Interaction log.
(𝔚=(D;R_1,…,R_k))	structure	World: domain (D) with relations (R_j⊆D^{m_j}).
(𝔓=(V;Q_1,…,Q_k))	structure	Picture: variables (V), pictured relations (Q_j⊆V^{m_j}).
(h:V→D)	map	Candidate homomorphism (picture→world).
(1.1)	constraint	Relation preservation: If ((v_1,…,v_m)∈Q_j) then ((h(v_1),…,h(v_m))∈R_j).
(1.2)	scalar	Fit: (Fit(p;x_{1:T}) = \max_h Pr[(1.1)\text{ holds on fragments}]).
(𝒢=⟨S,X,A,u,T⟩)	tuple	Language game with partial observability.
(2.1)	policy	Meaning-as-use: (Meaning(w)≡π_w^*=\arg\max_{π} 𝔼[u∣π,\text{community}]).
(f_m:X→ℝ^{d_m})	map	View-m encoder.
(μ_w^{(m)}, r_w^{(m)})	vector, scalar	Prototype and radius for word/category (w) in view (m).
(3.1)	predicate	Membership: (\sum_m 1(‖f_m(x)-μ_w^{(m)}‖≤r_w^{(m)}) ≥ τ).
(ℛ)	class	Hypothesis class of rules (r:X→Y).
(4.1)/(7.3)	margin	Identifiability: (∀r≠r^,\ Pr_{x∼𝒟_X}[r(x)≠r^(x)]≥γ).
(d_{VC}, 𝓡_n)	scalar	VC dimension, Rademacher complexity.
(MDL(r))	scalar	Code length penalty (prefix-free).
(κ, α)	scalar	Cohen’s κ, Krippendorff’s α (agreement).
(ECE)	scalar	Expected calibration error.
(BF_t)	scalar	Bayes factor at epoch t.
(Λ_T=\sum_{t=1}^T \log BF_t)	scalar	Cumulative log-evidence.
(c_{\text{switch}})	scalar	Hinge switching cost.
(6.1)	rule	Switch when (\sum_t \log BF_t - c_{\text{switch}} > 0).
(Excl(p,q∣C))	scalar	Exclusivity: (1 - Jaccard(Supp(π_p^∣C), Supp(π_q^∣C))).
(ΔU)	scalar	Utility lift on held-out contexts.
(nDCG@k)	scalar	Retrieval quality at cutoff k.

C.2 Distributions, Capacities, and Losses

Data distribution. ((x,y)∼𝒟), with (𝒟_X) its (X)-marginal. Risk for rule (r) under loss (ℓ):

(C.1) (R(r)=𝔼_{(x,y)∼𝒟}[ℓ(y,r(x))]).

Empirical risk.

(C.2) (\hat{R}n(r)=(1/n)\sum{i=1}^n ℓ(y_i,r(x_i))).

Uniform bound (Rademacher style).

(C.3) (R(r) ≤ \hat{R}_n(r) + 2𝓡_n(ℛ) + \sqrt{\ln(2/δ)/(2n)}) with prob ≥ (1−δ).

Capacity ladder (SRM). (ℛ_1⊂ℛ_2⊂…), code lengths (L(m)) s.t. (\sum_m 2^{-L(m)}≤1). Penalized selection:

(C.4) (\hat{r}=\arg\min_{r∈ℛ_m}{\hat{R}_n(r)+λ·L(m)}).

C.3 Metrics (Exact Definitions)

Fit (truth-as-correspondence). As in (1.2). Implementation: CSP feasibility rate over observed fragments; missing tuples treated as unknowns or modeled with explicit missingness.

Utility lift.

(C.5) (ΔU = J(π̂;C_{\text{test}}) - J(π_{\text{base}};C_{\text{test}})).

Stability norm. For (‖ΔC‖=ε),

(C.6) (S(ε)=‖π^_{C+ΔC} - π^_{C}‖) (TV or (ℓ_2) over action logits).

Agreement.

(C.7) (κ=(P_o-P_e)/(1-P_e)), (α=1-(D_o/D_e)).

Calibration error.

(C.8) (ECE=\sum_b (n_b/N)\cdot |acc(b) - conf(b)|).

Bayes factors.

(C.9) (BF_t=p(D_t∣𝓗')/p(D_t∣𝓗)), (Λ_T=\sum_{t=1}^T \log BF_t).

Exclusivity.

(C.10) (Excl=1 - Jaccard(Supp(π_p^), Supp(π_q^))).

Retrieval. (nDCG@k) with standard logarithmic gains and ideal DCG normalization.

C.4 Data Preprocessing Pipelines

General steps (all studies).

Sanitize logs: de-identify, validate timestamps, hash session IDs.
Partition: chronological split into train/val/test; holdout observers/domains where applicable.
Feature standardization: z-score continuous features; categorical one-hot or learned embeddings.
Outlier policy: winsorize heavy-tailed rewards; filter corrupt sensor readings via robust z-scores.

Study A (CSP Fit).

Parsing: convert captions to dependency graphs → map to picture (𝔓) (nodes: entities; edges: predicates).
Scene graphs: extract (R_j) via detectors; confidence thresholds tuned on val.
CSP encoding: variable per picture node; domain: candidate scene entities; constraints: relation compatibility and attribute typing; optional soft penalties (MaxSAT/ILP) for noisy matches.

Study B (IRL / Meaning-as-use).

Segmentation: align utterance spans with action windows; mark occurrences of target (w).
Contexts (C): encode task type, tools, stakes, priors; normalize to a fixed spec.
IRL: choose policy class (Π_θ) (tabular/neural); off-policy evaluation with doubly-robust estimator; entropy/MDL regularizers.

Study C (Multi-view prototypes).

Embeddings (f_m): precompute per view (text, vision, behavior, utility-features).
Seeds (𝒟_w): curate clean exemplars; balance senses for polysemy.
Robust prototype: geometric median (Weiszfeld) and quantile radii per view; periodic EMA updates for drift.

Study D (Rule-following).

Hypothesis ladder: define (ℛ_m) by model size/feature sets; assign code lengths (L(m)).
Ablations: sweep capacity m and sample size n; compute disagreement (\hat{γ}) to a reference or oracle.

Study E (Private language).

Shared tools: normalize interfaces; provide identical instructions/sensors.
Calibrator (C_θ): train on subset; evaluate cross-observer variance and out-of-person AUROC.
Interventions: tool/prompt perturbations to test loss coupling.

Study F (Hinges).

BF streams: specify likelihood families for (𝓗,𝓗'); compute (\log BF_t) per epoch.
Cost audit: enumerate (c_{\text{switch}}) components (retraining, latency, risk buffer).
Stopping: compare predicted (\hat{τ}^*=c_{\text{switch}}/\max(\hat{μ},ϵ)) to observed switches.

Study G (Aspect-seeing / Therapy).

Cue ramps: generate forward/backward δ-sweeps; randomize order to control fatigue.
RT measures: median and distributional fits (ex-Gaussian or DDM).
Therapy loop Θ: log (Excl^{(t)}) and (U^{(t)}); retain counterexample buffers.

C.5 Evaluation Protocols

Statistical discipline.

Preregistration: hypotheses, primary endpoints, α, stopping rules.
Multiple testing: Holm–Bonferroni or Benjamini–Hochberg across Fit, ΔU, ΔF1, ECE, κ/α, BF endpoints.
Uncertainty: 95% CIs via bootstrap (stratified by domain/observer) or block bootstrap for temporal dependence.

Diagnostics.

Stability slopes: regress (‖π^_{C+ΔC}-π^C‖) on (‖ΔC‖); compare slope to bound (L{JC}/α).
Exploitability/no-regret: compute best-response gap and average regret; require thresholds κ, η small.
Calibration: reliability diagrams; ECE and class-wise ECE; temperature/isotonic fits on val only.
Hysteresis and rates: logistic fits for choice curves, segmented regression for hysteresis width; Kramers-like log-linear checks for switch rates.

C.6 Reproducibility Checklist

Environment snapshot: OS, compiler/BLAS, GPU/driver; container file or lockfile.
Tool cards: parser/grounder (Γ) version, embedding models (f_m), calibrators with seeds/hyperparams.
Data statements: source, consent, license; de-identification method.
Randomness control: fixed seeds; record PRNG streams for CSP/ILP solvers.
Compute budget: report wall-clock, FLOPs/energy where feasible.
Artifact release: code, configs, and evaluation harnesses for Fit AUC, ΔU, ECE, κ/α, BF trajectories, Θ-traces.

C.7 Implementation Notes (Pragmatic)

CSP solvers: begin with AC-3 / arc-consistency; escalate to ILP/MaxSAT with soft penalties on attribute mismatches; bound treewidth where possible.
IRL stability: use trust-region updates (KL ≤ δ), gradient clipping, and doubly-robust OPE; checkpoint best ΔU.
Prototype robustness: prefer geometric medians; guard against collapsing radii with minimum quantile floors; schedule recalibration.
SRM efficiency: parallelize ladder training; early-stop sub-models with pessimistic bounds; cache features across m.
Therapy safety: integrate harm screens; maintain audit logs of all rewrites; enforce human-in-the-loop for sensitive domains.

This appendix provides the canonical vocabulary and playbook for implementation, ensuring that results across studies are comparable, auditable, and easy to re-run end-to-end.

Disclaimer

This book is the product of a collaboration between the author and OpenAI's GPT-5, Wolfram's GPTs language model. While every effort has been made to ensure accuracy, clarity, and insight, the content is generated with the assistance of artificial intelligence and may contain factual, interpretive, or mathematical errors. Readers are encouraged to approach the ideas with critical thinking and to consult primary scientific literature where appropriate.

This work is speculative, interdisciplinary, and exploratory in nature. It bridges metaphysics, physics, and organizational theory to propose a novel conceptual framework—not a definitive scientific theory. As such, it invites dialogue, challenge, and refinement.

I am merely a midwife of knowledge.

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Friday, October 17, 2025