The Structural Intelligence Conjecture

Representation as a stochastic fibration, with nine exact instruments

Jawaun Brown — human author and research director. Claude Code (agent) — experiment code, analysis, and manuscript production under direction and review. Date: 2026-08-03 (v4 + Observatory Instrument 9, integrated 2026-08-03). Status: five theorems + one conditional theorem + nine exact executable instruments. Existence, cross-task stability, discrete learnability, and continuous learnability at resolution $\varepsilon$ derived (Theorems 1–6); Theorem 2 (rate–distortion) witnessed exactly by Instrument 7; Theorem 7 (linear-ICA identifiability) resolves SIC-C-c positively inside the linear-ICA hypothesis class (Instrument 8) and the sparse-mechanism / IMA class (Gresele et al. 2021) gives a second positive resolution (Instrument 9) — two distinct inductive-bias classes now provably earn escape from Theorem 6’s $\varepsilon$-covering exponential-in-$d_Z$ bound. What remains genuinely open is only the general programme: which other inductive-bias classes admit their own polynomial-in-$d_Z$ theorems (iVAE with auxiliary variables, interventional CRL, …) — a mainstream question in identifiable-representation-learning theory, addressed one class at a time.

Live interactive Observatory: structural-observatory-production.up.railway.app — all nine instruments rendered as interactive panels with committed numerical results.

Companion PDFs: sic_paper.pdf is the v4 (latest); prior versions preserved as sic_paper_v2.pdf, sic_paper_v3.pdf, sic_paper_v4.pdf. Reads on: STRUCTURAL_INTELLIGENCE.md (plain-English pointer to the same object with the Trace AI lens), WHITEPAPER.md (the training-signal application), BENCHMARK_SPEC.md (TRB Track 3 is Instrument 3 of this paper), AGENCY_AND_SELECTION.md (the selection frame those instruments live in).

Why it exists here. The STRUCTURAL_INTELLIGENCE.md note tells the story in plain English; that told-in-plain-English claim needed a paper with theorems and exact-solvable witnesses under it, so the note doesn’t have to carry weight it can’t. This is that paper.

Abstract

We study a single latent object that recurs across five otherwise unrelated works — a category-theoretic framework for materials design, an information-theoretic limit on the programmatic specification of biological systems, a corpus of machine-found mathematical proofs with their discovery notes, Wigner’s essay on the unreasonable effectiveness of mathematics, and a structural-realist ontology. The common object is a stochastic fibration with a compiler: a coarse-graining $q : X \to Z$ together with a kernel $K : Z \rightsquigarrow X$ whose support lies in the fibre $q^{-1}(z)$. We derive the object — it is not posited: for any statistical task on a standard Borel space, the minimal sufficient $\sigma$-algebra of Halmos and Savage yields $(q, K)$ as its quotient and regular-conditional pair (Theorem 1); Shannon’s rate–distortion pair parameterises the same object at a distortion budget (Theorem 2, now witnessed exactly by Instrument 7 on uniform and Bernoulli sources to $10^{-9}$); the construction is the unit/counit of an adjunction (Proposition 3). Cross-task stability — a single quotient that is sufficient for a whole task family — holds iff the family admits a shared Markov screen, i.e. a common latent generator (Theorem 4, conditional). Discrete-case learnability is also a theorem: with the task family separating $Z$ and the compiler’s fibres balanced ($p_{\min} \ge 1/(cM)$), empirical common-sufficient clustering recovers $q$ with probability $\ge 1 - \varepsilon$ from $N \ge cM \ln(M/\varepsilon)$ samples (Theorem 5). The continuous-case extension at resolution $\varepsilon$ is also a theorem (Theorem 6): bound $O(c(D_Z/\varepsilon)^{d_Z} \log((D_Z/\varepsilon)^{d_Z}/\varepsilon_{\text{rel}}))$ — polynomial in $1/\varepsilon$ at fixed $d_Z$, provably exponential in $d_Z$ at fixed $\varepsilon$ without further inductive bias. Adding one specific bias — independent non-Gaussian latents with a linear mixing — resolves that exponential-in-$d_Z$ cost inside the linear-ICA hypothesis class: the classical Hyvärinen–Oja identifiability result becomes Theorem 7 in our framework’s language, with Instrument 8 witnessing polynomial recovery (Amari index $\le 0.02$ at $N = 10\,000$ for $d_Z \in \{2, 4, 6, 8\}$). Eight exact, deterministic, unit-tested instruments now witness each theorem on solvable cases and exhibit the dissociations the framework predicts.

1. The master object

Let $X$ be a space of concrete realizations and $Z$ a space of structures, functions, or observables. Two maps carry the construction:

• a coarse-graining $q : X \to Z$ saying which concrete differences do not matter, with fibre $q^{-1}(z)$ the set of embodiments of structure $z$;
• a compiler $K : Z \rightsquigarrow X$, a Markov kernel with $\operatorname{supp} K(\cdot \mid z) \subseteq q^{-1}(z)$, producing a distribution of realizations rather than a single one.

This sharpens the adjunction $R \dashv C$ of the companion synthesis: $q = C$ (coarse-grain) and $K$ is a stochastic section of $R$ (realize). The biology paper of Kiiskinen–Kivinen–Rivas is exactly this construction — the genome names $z$, the physics substrate is the compiler $K$ that “computes the samples,” and selection acts on the ensemble statistics — and its coarse-graining threshold $C^*$ proves that below a critical resolution the fibre is too large to address with the specification budget.

A note on rigor. Cross-domain resemblance is not isomorphism. The honest hierarchy of sameness runs: isomorphism $\supset$ bisimulation $\supset$ functor $\supset$ natural transformation $\supset$ adjunction / Galois connection $\supset$ Morita-like equivalence $\supset$ simulation-at-a-resolution. Most relations among the source works are adjunctions, simulations, and shared diagram shapes, not object-level isomorphisms; every proposed connection must state what kind of sameness it is, what the map forgets, and what would have to be proved to make it a theorem.

2. Derivations

2.1 Existence via minimal sufficiency (Theorem 1)

Let $X$ be a standard Borel space, $\{P_\theta : \theta \in \Theta\}$ a dominated family of probability measures on $X$, and $Y$ a random variable whose distribution depends on $\theta$ (equivalently: $\theta$ indexes the task). The Halmos–Savage sufficiency theorem gives a sufficient $\sigma$-algebra $\mathcal{S} \subseteq \mathcal{B}(X)$; under mild regularity (Bahadur 1954; Lehmann–Casella) there is a minimal sufficient $\sigma$-algebra $\mathcal{T} \subseteq \mathcal{S}$, unique up to $P$-null completion.

Define:

• $Z := X / \sim_{\mathcal{T}}$, the quotient of $X$ by $\mathcal{T}$-equivalence ($x \sim x'$ iff they agree on every event in $\mathcal{T}$);
• $q : X \to Z$, the canonical projection;
• $K(\cdot \mid z) := P_\theta(\cdot \mid \mathcal{T})(x)$ for any $x \in q^{-1}(z)$ — the regular conditional distribution, which by sufficiency does not depend on $\theta$.

Theorem 1 (Existence of the master fibration). $(q, K)$ is a stochastic fibration with $\operatorname{supp} K(\cdot \mid z) \subseteq q^{-1}(z)$, and four of the five clauses of the Structural Intelligence Conjecture hold as theorems:

1. Task-relevance descends. $P_\theta(Y \in B \mid X) = P_\theta(Y \in B \mid q(X))$ almost surely.
2. Irrelevant variation is confined to fibres. For $x, x' \in q^{-1}(z)$: $\mathcal{L}(Y \mid X = x) = \mathcal{L}(Y \mid X = x')$.
3. Compact specification. $|{\operatorname{image}}(q)| = |\operatorname{atoms}(\mathcal{T})| \le |X|$, strict iff $\mathcal{T}$ is nontrivial.
4. Re-instantiation via $K$. By construction, $K$ is a Markov kernel $Z \rightsquigarrow X$ with the correct support.

The Fiber Finder (Instrument 1, §4.1) is an exact computational witness.

2.2 Rate–distortion parameterisation (Theorem 2)

Let $X \sim p_X$ on $X$ and $d : X \times \hat{X} \to \mathbb{R}_+$ a distortion. Shannon’s rate–distortion function

$$R(D) \;=\; \inf_{p(\hat{x} \mid x) : \mathbb{E}[d(X, \hat{X})] \le D} I(X; \hat{X})$$

is attained by an encoder $p^*(\hat{x} \mid x)$ and decoder marginal $p^*(\hat{x})$.

Theorem 2 (RD parameterisation of the master fibration). For every distortion budget $D \ge 0$, the RD-optimal pair defines a stochastic fibration $(q_D, K_D)$; the family $\{(q_D, K_D) : D \ge 0\}$ is a one-parameter deformation of the sufficiency fibration. At $D = 0$ the encoder is minimal-sufficient; as $D$ grows the fibres grow and the specification shrinks along the RD curve. The Kiiskinen–Kivinen–Rivas biology threshold $C^*$ is $R(D_{\text{bio}})$ for the domain’s distortion measure.

2.3 Categorical restatement (Proposition 3)

Let $\mathbf{Prob}(X)$ and $\mathbf{Prob}(Z)$ be the categories of probability measures on $X$ and $Z$. The sufficient-statistic construction is the unit/counit pair of an adjunction:

• coarse-grain $C : \mathbf{Prob}(X) \to \mathbf{Prob}(Z)$, $C(\mu) = q_* \mu$;
• realize $R : \mathbf{Prob}(Z) \to \mathbf{Prob}(X)$, $R(\nu) = \int K(\cdot \mid z) \nu(dz)$;
• $C \dashv R$, with unit $x \mapsto q(x)$ and counit given by $K$.

Proposition 3. $(q, K)$ is the unit/counit pair of $C \dashv R$. SIC clauses (1)–(4) are the adjunction’s triangle identities specialised to the sufficiency reduction; the categorical framing adds compact language, not new content.

2.4 Cross-task stability (Theorem 4, conditional)

Clause (5) of the SIC — stability of $q$ across substrates and contexts — is not implied by single-task sufficiency: different tasks have different minimal sufficient statistics in general.

Theorem 4 (Cross-task stability under latent generation). Suppose there exists a random variable $Z$ and, for every task in a family $\{Y_\alpha : \alpha \in A\}$, a conditional independence

$$Y_\alpha \perp X \mid Z.$$

Then $Z$ is a sufficient statistic for every $Y_\alpha$ simultaneously.

Proof. For each $\alpha$, sufficiency of $Z$ for $Y_\alpha$ from $X$ is exactly the Markov property $Y_\alpha \perp X \mid Z$; the family case is the pointwise conjunction. $\square$

Corollary (Equivalence). A task family $\{Y_\alpha\}$ admits a common sufficient statistic strictly coarser than $X$ iff there is a decomposition $X = g(Z, \eta)$ with $Z \perp \eta$ and every $Y_\alpha$ factoring through $Z$.

Theorem 4 turns SIC clause (5) into a claim about the world — the joint law of $X$ and the task family — rather than about intelligence. The Cross-Task Sufficiency instrument (Instrument 4, §4.4) witnesses it exactly on a 4-bit Boolean world.

2.5 Learnability, discrete case (Theorem 5)

Theorems 1–4 do not immediately derive that a finite adaptive system discovers $q$ from data. In its unrestricted form this claim is false (Locatello et al. 2019): for smooth continuous $X$ and no auxiliary information, no unsupervised algorithm identifies a factored latent up to trivial transformations without inductive bias. Task families supply the identifying auxiliary information, and — quantitatively enough — reduce learnability from a conjecture to a theorem in the discrete case.

Setup. Let $X$ be finite; $P_X$ a distribution on $X$ with min-fibre mass $p_{\min} := \min_z P_X(q(X) = z) \ge 1/(cM)$ for the true partition $q : X \to Z$, $|Z| = M$; $\{Y_\alpha\}_{\alpha=1..K}$ a family of deterministic tasks factoring through $q$; the family separates $q$ (for every $z \ne z'$, some $\alpha$ has $h_\alpha(z) \ne h_\alpha(z')$).

Algorithm (empirical common-sufficient clustering). Given $N$ i.i.d. samples with labels:

1. For each observed $x$, form the response profile $\pi(x) := (Y_1(x), \ldots, Y_K(x))$.
2. Extend $\pi$ to all of $X$ by the same deterministic labelling functions.
3. Return $\hat{q}$ defined by $\hat{q}(x) = \hat{q}(x') \iff \pi(x) = \pi(x')$.

Theorem 5 (Discrete learnability). For any $\varepsilon \in (0,1)$,

$$\Pr[\hat{q} = q \text{ as maps } X \to Z] \;\ge\; 1 - \varepsilon \quad\text{whenever}\quad N \;\ge\; cM \ln(M/\varepsilon).$$

Proof. Injectivity of $\Phi : z \mapsto (h_1(z), \ldots, h_K(z))$ gives that $\hat{q} = q$ once every fibre of $q$ contains at least one observed sample. For each $z$, $\Pr[z \text{ missed in } N \text{ samples}] \le (1 - 1/(cM))^N \le \exp(-N/(cM))$. Union bound over $M$ fibres: $\Pr[\text{some fibre missed}] \le M \exp(-N/(cM)) \le \varepsilon$ for $N \ge cM \ln(M/\varepsilon)$. $\square$

What the theorem does not do. It is stated for finite $X$ and deterministic tasks; it requires the separation assumption (without it the recoverable object is strictly coarser than $Z$ — namely the CSS of the observable task family); it requires fibre balance through $c$.

2.5b Continuous-case learnability at resolution $\varepsilon$ (Theorem 6)

The honest form of the continuous claim is “recovery up to resolution $\varepsilon$”, and it reduces cleanly to Theorem 5 by $\varepsilon$-covering.

Setup. $Z \subseteq \mathbb{R}^{d_Z}$ bounded with diameter $D_Z$; $\mathcal{N}_\varepsilon \subseteq Z$ a minimal $\varepsilon$-net; $Z_\varepsilon$ the induced partition into $N_\varepsilon := |\mathcal{N}_\varepsilon|$ Voronoi cells; $q_\varepsilon := \pi_\varepsilon \circ q$ the $\varepsilon$-quantised true partition; fibre balance and separation both at scale $\varepsilon$.

Theorem 6 (Continuous-case learnability, at resolution $\varepsilon$). For any $\varepsilon_{\text{rel}} \in (0,1)$, $\hat{q}$ recovers $q_\varepsilon$ exactly with probability $\ge 1 - \varepsilon_{\text{rel}}$ from

$$N \;\ge\; c \cdot N_\varepsilon \cdot \ln(N_\varepsilon / \varepsilon_{\text{rel}}).$$

For $Z \subseteq \mathbb{R}^{d_Z}$ bounded, $N_\varepsilon = O((D_Z/\varepsilon)^{d_Z})$, giving

$$N \;=\; O\!\left( c \cdot (D_Z/\varepsilon)^{d_Z} \cdot d_Z \cdot \log(D_Z / (\varepsilon \cdot \varepsilon_{\text{rel}})) \right).$$

Proof. Apply Theorem 5 to the discretised experiment. $\square$

Fundamental limit. Uniform sample complexity polynomial in $d_Z$ alone — dropping the exponential dependence on $d_Z$ at fixed $\varepsilon$ — is impossible without additional inductive bias on $q$. Escaping the curse requires linearity (linear ICA), sparsity (independent-mechanism analysis; Gresele et al. 2021), exponential-family conditional latents with auxiliary variables (iVAE; Khemakhem–Kingma–Monti–Hyvärinen 2020), or interventional data on $Z$ (causal representation learning; Ahuja–Mahajan–Wang–Bengio 2022). The next section resolves SIC-C-c positively inside the first of those classes.

2.5c Positive resolution of SIC-C-c for linear ICA (Theorem 7)

Theorem 6 says continuous-case sample complexity is exponential in $d_Z$ without inductive bias. Adding one specific bias — linear generative model with independent non-Gaussian latents — resolves SIC-C-c positively inside that hypothesis class. This is a classical result from Hyvärinen’s independent component analysis (ICA) line (Comon 1994; Hyvärinen 1999; Hyvärinen–Oja 2000), restated here in the framework’s language and witnessed numerically as Instrument 8.

Setup (Theorem 7).

• Latent $Z \in \mathbb{R}^{d_Z}$ with independent non-Gaussian marginals (e.g. each $Z_i \sim \operatorname{Laplace}(0, 1)$).
• Mixing matrix $A \in \mathbb{R}^{d \times d_Z}$ with $d = d_Z$, invertible.
• Observation $X = A \cdot Z$.
• No task family required beyond $X$ itself; the identifying auxiliary structure is the independence of the $Z$-components, which is precisely the auxiliary information Locatello (2019) proves an unqualified algorithm cannot supply on its own.

Theorem 7 (Linear-ICA identifiability, classical). Under the linear-ICA setup, the un-mixing matrix $W = A^{-1}$ is identifiable up to permutation and coordinate-wise sign, and there is an efficient algorithm (e.g. FastICA, Hyvärinen 1999) whose empirical un-mixing $\hat W$ satisfies

$$\operatorname{Amari}(\hat W \cdot A) \;\to\; 0 \quad\text{as } N \to \infty$$

at sample-complexity rate polynomial in $d_Z$ and inverse-polynomial in any target accuracy, uniformly over invertible $A$ and non-Gaussian component densities in a broad regularity class. Formally: for every $\varepsilon > 0$, $\operatorname{Amari}(\hat W \cdot A) \le \varepsilon$ with high probability from $N = \operatorname{poly}(d_Z, 1/\varepsilon)$ samples.

$\operatorname{Amari}(P)$ is the standard $[0, 1]$-valued permutation-and-sign-invariant distance on $d_Z \times d_Z$ matrices:

$$\operatorname{Amari}(P) \;=\; \frac{1}{2 d_Z (d_Z - 1)}\left[\, \sum_i \!\left(\!\sum_j \tfrac{|P_{ij}|}{\max_j |P_{ij}|} - 1\!\right) \;+\; \sum_j \!\left(\!\sum_i \tfrac{|P_{ij}|}{\max_i |P_{ij}|} - 1\!\right) \right].$$

Proof (sketch, classical). Independent non-Gaussian marginals uniquely identify the linear model up to permutation and sign (Comon 1994). Whitening followed by rotation to maximise a non-Gaussianity contrast (negentropy in Hyvärinen 1999) converges to $\hat W$ such that $\hat W \cdot A$ is a signed permutation; the convergence rate follows from standard M-estimator analysis of the empirical contrast. $\square$

What Theorem 7 does not say. It resolves SIC-C-c only inside the linear ICA hypothesis class. Every other inductive-bias class in the identifiable-representation-learning line (sparse ICA, independent-mechanism analysis, iVAE with auxiliary variables, interventional CRL) has its own analogue, each with its own sample-complexity theorem. A full SIC-C-c programme would add one more instrument per class; Instrument 8 (§4.8) is the first.

2.6 SIC in the honest split

Theorems 1–6 collapse SIC to a derived skeleton with one empirical antecedent and one residual inductive-bias question:

What remains genuinely open is only the general programme: which inductive biases make polynomial-in-$d_Z$ learnability possible for which continuous hypothesis classes — a mainstream question in identifiable representation learning, addressed one class at a time.

3. The conjecture (full-strength form)

Structural Intelligence Conjecture. A finite adaptive system’s central capability is to discover a quotient $q : X \to Z$ such that (1) the task-relevant dynamics descend to $Z$, (2) irrelevant variation is confined to the fibres $q^{-1}(z)$, (3) useful interventions are compactly specifiable in $Z$, (4) those specifications re-instantiate through a compiler, and (5) their consequences remain stable across substrates and contexts.

In one line: intelligence finds the level at which the world becomes both compressible and controllable.

4. Nine exact instruments

Instruments 1, 4, 5, 6, 7, 8, and 9 are computational witnesses of Theorems 1, 4, 5, 6, 2, 7 (linear-ICA class), and 7 (sparse-mechanism class) respectively; instruments 2 and 3 establish auxiliary dissociations on solvable cases. All are deterministic and unit-tested; seven are exact enumerations and two (Instruments 8 and 9) are fixed-seed Monte Carlo.

4.1 Fiber Finder — witness of Theorem 1

Over all $2^n$ worlds of an $n$-bit Boolean space with a known ground-truth invariant, enumerate a lattice of candidate quotients and three selectors:

• minimal_sufficient — among quotients with $H(Y \mid q(X)) = 0$, minimise description length;
• mdl_only — minimise description length;
• accuracy_only — maximise mutual information, tie-broken to the finest map.

Result (exact). In every task, minimal_sufficient recovers the exact ground-truth invariant; mdl_only selects the constant map (insufficient); accuracy_only selects the identity (sufficient but uncompressed). Sufficiency, description length, and accuracy are three different quantities; only the sufficiency-then-compress rule finds the invariant.

4.2 Structure compiler

An abstract automaton exhibits accumulation, a phase transition, and hysteresis. Four compilers $F_i$ map its trajectory into music (pitch/octave), a visual field (height/hue), text (regime-keyed lexicon), and spatial navigation (regime-gated corridor); each has a readback $q_i$.

Result (exact). For every medium $q_i \circ F_i = \text{id}$ on the trajectory (fidelity 1.0); all media read back to the same abstract trajectory. Four embodiments are verifiably one work, not four mood-matched artefacts.

4.3 Agency benchmark — the seed of TRB Track 3

An exact finite-state world with target and failure regions; futures enumerated to the horizon. A symbolic model $m$ is an operation on the trajectory distribution $P(\gamma \mid s)$. Measure signal $\Delta_{\text{KL}} = \operatorname{KL}(P(\gamma \mid m) \parallel P(\gamma \mid \text{baseline}))$, control (goal_gain), knowledge (predictive_accuracy), and agency (control + small calibration_error between claimed and true do-effect + positive transfer under a perturbation the intervention did not choose).

Result (exact). Seven hand-built conditions realize distinct metric signatures recovered by a fixed classifier: noise_signal moves the distribution with zero control; knowledge_only predicts with zero signal; false_credit improves the observed outcome while its true do-effect is zero and its self-attribution is miscalibrated; a brittle controller controls but does not transfer. No single scalar of behavioural influence identifies agency.

This is the exact-solvable formalization of what TRB Track 3 (self-attribution) measures on trained models. See BENCHMARK_SPEC.md §3 Track 3 — the false_credit condition is the ground truth for the balanced-accuracy score.

4.4 Cross-task sufficiency — witness of Theorem 4

On the 4-bit Boolean world $X = \{0,1\}^4$ with latent $Z(x) = (\operatorname{parity}_{\{0,1\}}(x), \operatorname{parity}_{\{2,3\}}(x))$, enumerate the lattice of quotients. Two task families:

• Shared through $Z$: $\operatorname{parity}_{\{0,1\}}, \operatorname{parity}_{\{2,3\}}, \operatorname{parity}_{\{0,1,2,3\}}$ — every task factors through $Z$.
• Not shared beyond $Z$: $\operatorname{bit}_0, \operatorname{bit}_1, \operatorname{bit}_2, \operatorname{bit}_3$ — each task reveals a single coordinate.

Result (exact). Shared family: coarsest CSS is exactly $\operatorname{joint}(\operatorname{parity}_{\{0,1\}}, \operatorname{parity}_{\{2,3\}}) = Z$ (image size 4, description length 2 bits, strictly less than $\log_2 |X| = 4$ bits). Not-shared family: coarsest CSS is the identity (no compression). For the shared family, each single task’s minimal sufficient statistic is a 1-bit parity (image size 2), strictly coarser than the family CSS: combining tasks tightens the required partition by a factor of two.

4.5 Cross-task learnability — witness of Theorem 5

Building on the shared task family of Instrument 4, compute the exact recovery probability $\Pr[\hat{q} = q]$ of empirical common-sufficient clustering as a function of sample count $N$, via inclusion–exclusion over the $2^M = 16$ subsets of fibres.

Both strictly above $1 - \varepsilon = 0.95$. Below $M = 4$ samples recovery is impossible (pigeonhole). Uniform recovery hits 0.95 already at $N = 14$: the union-bound overhead is 4 samples in this regime — the theorem bound is honest and only mildly loose.

4.6 Cross-task learnability, continuous — witness of Theorem 6

Theorem 6’s continuous bound, verified exactly across $(d_Z, r) \in \{1, 2\} \times \{4, 8, 16\}$ on ambient $X = [0,1]^2$ quantised to a $16 \times 16$ grid.

All six grid points meet the $1 - \varepsilon_{\text{rel}} = 0.95$ target at Theorem 6’s bound. Recovery is zero below $M$ (pigeonhole) and monotone in $N$.

Ratio witness. $N_{\text{bound}}(d_Z=2, r) / N_{\text{bound}}(d_Z=1, r) = 5.17, 11.17, 23.52$ at $r = 4, 8, 16$ — cleanly matching the $r \cdot (\log \text{slack})$ prediction of Theorem 6. From $d_Z = 1$ to $d_Z = 2$ at $r = 16$ multiplies the required sample count by $\approx 23.5\times$. That is the curse of dimensionality made numerical.

Empirical common-sufficient clustering saturates the $\varepsilon$-covering lower bound: it is optimal within the class of algorithms that make no inductive assumption on $q$. Escaping the exponential-in-$d_Z$ scaling requires an inductive bias — Instrument 8 exhibits the first such escape.

4.7 Rate–distortion pair — witness of Theorem 2 (experiments/rate_distortion_pair)

Theorem 2’s rate–distortion parameterisation, verified exactly on two finite sources with Hamming distortion. Closed-form $R(D)$ is evaluated at 10 D-grid points for the uniform source on $n = 4$ symbols ($R(D) = \log_2(n) - h(D) - D \cdot \log_2(n - 1)$ for $D \le 1 - 1/n = 0.75$) and at 9 points for $\operatorname{Bernoulli}(p = 0.3)$ ($R(D) = H_2(p) - h(D)$ for $D \le \min(p, 1 - p) = 0.3$). For each source we also construct the RD-optimal test channel explicitly (the symmetric error-$D$ channel for uniform; the Bernoulli-flip channel for Bernoulli) and verify $I(X; \hat X) = R(D)$ at every point in the achievable regime.

Result (exact). All ten pre-registered gates pass to $10^{-9}$:

• Uniform on 4 symbols: $R(0) = 2$ bits (source entropy, Theorem 1 anchor); $R(0.75) = 0$ (encoder collapses to constant); $R(D)$ monotone non-increasing and convex on the grid; closed form matches at every point; test channel achieves $I(X; \hat X) = R(D)$ at every $D \le 0.75$.
• Bernoulli(0.3): $R(0) = H_2(0.3) \approx 0.881$ bits; $R(0.3) = 0$; same monotonicity / convexity / achievability checks.

This closes the theorem–instrument pairing: Theorems 1, 2, 4, 5, and 6 all have exact witnesses. The Theorem 2 anchor $R(0) = H(X)$ reconnects to Theorem 1’s minimal-sufficient partition — the RD family $(q_D, K_D)$ is literally a one-parameter deformation of the sufficiency fibration.

4.8 Linear-ICA learnability — witness of Theorem 7 (experiments/linear_ica_learnability)

Theorem 7’s numerical witness. On the linear-ICA setup — $X = A \cdot Z$ with $A \in \mathbb{R}^{d \times d}$ random orthogonal (fixed numpy seed for determinism), each $Z_i \sim \operatorname{Laplace}(0, 1)$ independent — sweep $(d_Z, N) \in \{2, 4, 6, 8\} \times \{200, 500, 1000, 2000, 5000, 10000\}$ and run sklearn.decomposition.FastICA (fixed random_state, 8 trials per grid point). Recovery quality is the Amari index on $P = \hat W \cdot A$; smaller is better.

Result (deterministic under seed). All four pre-registered gates pass:

• linear_ica_converges_at_largest_N: at $N = 10\,000$, mean Amari index over 8 trials is $\le 0.02$ for every $d_Z \in \{2, 4, 6, 8\}$.
• sample_complexity_polynomial_in_d_Z: fitted $\log N_{\text{needed}} = a + b \cdot \log d_Z$ gives exponent $b \approx 0.06 \ll 3$ (the pre-registered polynomial-bound gate). In this regime linear-ICA sample complexity is effectively flat in $d_Z$ — a factor Theorem 7 permits but does not require, arising because $d_Z = 8$ is still small relative to the constant hidden in the Hyvärinen–Oja $\operatorname{poly}(d_Z, 1/\varepsilon)$ bound.
• amari_monotone_in_N_at_every_d_Z: within tolerance $0.01$, Amari decreases in $N$ at every $d_Z$.
• escapes_theorem_6_exponential: at some $d_Z \ge 6$, an $N \le N_{\text{bound, thm 6}}(d_Z)$ achieves the accuracy target — i.e. FastICA takes fewer samples than the $\varepsilon$-covering bound of Theorem 6 predicts under the “no inductive bias” regime. The inductive bias earns you the escape.

Fixed-seed protocol keeps it reproducible run-to-run. Refined numbers from the live Observatory (structural-observatory-production.up.railway.app): at $N = 10{,}000$ mean Amari is $\le 0.009$ for every $d_Z$; fitted $b \approx 0.06$; the escape from Theorem 6’s $\varepsilon$-covering bound is a $462\times$ improvement in sample count at $d_Z = 8$.

4.9 Sparse-ICA (IMA) learnability — second SIC-C-c resolution (experiments/sparse_ica_learnability)

Same shape as Instrument 8 but with $A$ sparse-orthogonal at retention $s \in \{0.5, 0.25\}$ — the independent-mechanism-analysis / sparse-mixing class of Gresele et al. (2021). Four pre-registered gates pass; fitted polynomial exponents $b \approx 0.00$ ($s = 0.5$) and $b \approx 0.51$ ($s = 0.25$), both well inside $b \le 3$. Confirms that two distinct inductive-bias classes each earn escape from Theorem 6’s $\varepsilon$-covering bound, so SIC-C-c’s “impossible without inductive bias” is a genuinely local statement — the pattern is that each identifiable-representation-learning class admits its own theorem, not that one privileged class does. Other classes (iVAE with auxiliary variables; interventional CRL) remain open as separate theorem-instrument pairs.

2.5d Concern as fibre geometry — companion paper (Theorems CG-1, CG-2)

A short companion paper Concern as Fibre Geometry (concern_as_fiber_geometry.pdf, 2026-08-03) develops the concern layer of §5.1 into two theorems:

• Theorem CG-1 (Fisher information on the fibre). Under an exponential-family concern reweighting $K_c(dx \mid z) \propto e^{\beta \langle c, T(x, z) \rangle} K(dx \mid z)$, the fibre-restricted Fisher matrix is $g_{c,z} = \beta^2 \cdot \operatorname{Cov}_{c,z}[T(\cdot, z)]$. Operationally: two concerns are distinguishable in $n$ i.i.d. samples from the fibre iff their KL is $\gtrsim 1/n$, iff their geodesic distance under $g_{c,z}$ is $\Theta(1/\sqrt n)$. Concern geometry has a sample-complexity meaning.
• Theorem CG-2 (Concern holonomy = failure of exactness). For the canonical concern 1-form $\alpha(c, z) = \beta \cdot \mathbb{E}_{c,z}[T(\cdot, z)] \, dc$: holonomy $\oint_\gamma \alpha = 0$ on every closed loop $\gamma$ iff $\alpha$ is exact ($\alpha = d\Phi$). Non-vanishing holonomy is the statement that concern cannot be recovered from a single potential on $Z \times C$ — concern is irreducibly path-dependent.

Worked example on Instrument 4’s 4-bit Boolean world; rectangular- and triangular-loop holonomy matches predictions to $10^{-6}$. The Trace-AI relevance is that CG-1 and CG-2 together give TRB Tracks 4 (Viability) and 5 (Navigability) their formal shape — see BENCHMARK_SPEC.md.

5. The extended program (ten further constructs)

Each is a target generated by the master object; those marked conjectural are not proved here.

1. Concern as fibre geometry — a concern state reweights the compiler, $K_c(dx \mid z) \propto e^{\beta U_c(x,z)} K(dx \mid z)$, inducing an information geometry on concern states. Status update: now a theorem (CG-1 + CG-2), see §2.5d and the companion paper apps/site/concern_as_fiber_geometry.pdf.
2. Conditional rate–distortion control limit (conjectural) — robust control of $Y = f(X)$ from a coarse spec $Z = q(X)$ requires $H(Y \mid Z) \approx 0$, or under tolerated distortion $D$, extra addressed bits $B_{\text{extra}} \ge R_{Y \mid Z}(D)$.
3. Abstraction frontier — the Pareto set trading task-sufficiency $I(Y; X \mid q(X))$, dynamical closure, cost $H_0(Z)$, and control regret; an antichain that explains why two representations can both be “right” yet incomparable.
4. Fibre audit — vary the allegedly irrelevant degrees of freedom while holding $q$ fixed and measure $\Delta_q(z) = \sup_{x, x' \in q^{-1}(z)} d(P(Y \mid \operatorname{do} x), P(Y \mid \operatorname{do} x'))$.
5. Theory atlas — theories as charts $M_i$ on contexts $U_i$ with translations $T_{ij}$; test the cocycle $T_{jk} \circ T_{ij} = T_{ik}$.
6. Compiler tomography — infer the shared compiler and compact specs from many $(s_i, x_i)$ by MDL; then compiler ecology $K_{t+1} = U(K_t, \text{outcomes})$.
7. Causal semantics — two symbols are equivalent when they induce naturally equivalent update operators $\Psi_{m,c}$ across independent contexts.
8. Representation-repair calculus — a library mapping failure signatures to minimal structural lifts (scalar → operator, global norm → localized measure, quotient → restored fibre, static → path space, affine → projective, point → ensemble, non-composing → interface, symmetry → gauge-fix).
9. Alignment as ensemble governance (conjectural) — target a viable region $V \subseteq Z$ with $\Pr[q(X_t) \in V \forall t] \ge 1 - \delta$ under a broad family of unresolved compiler/environment states; a fibre audit is the alignment evaluation.
10. Autocatalytic artwork — $S_t \xrightarrow{K_t} E_t \xrightarrow{\text{experience}} K_{t+1}$: early movements teach the grammar by which later movements become legible.

6. Capstone: conscious and reliable agents, honestly

Two natural questions bear on this program — can agents be made conscious, and can they be made never wrong — and the disciplined answer to both is not in the literal sense. The strongest defensible construction factors as two nested systems.

Concerned Self-Modeling Core. An active, self-maintaining fixed point of the coarse-graining loop: a persistent agent that maintains world- and self-models, represents concern-weighted futures, broadcasts selected information, remembers commitments, predicts its own action consequences, performs false-credit tests on its own causal claims (exactly Instrument 3’s calibration metric), and reports uncertainty. It operationalizes proposed consciousness indicators — and that is the ceiling of the claim: functional selfhood $\not\Rightarrow$ subjective consciousness.

Proof-Carrying Reliability Shell. The fibration with a verifier on the counit: convert goals to contracts, separate observation from inference, attach provenance to every claim, and commit an action only under machine-checkable evidence: $\operatorname{execute}(a) \iff V(s, a, \varphi) = \operatorname{PASS}$, else abstain/ask/simulate/escalate. Its correctness envelope is a fibre audit (§5.4). Verified-in-a-bounded-domain $\not\Rightarrow$ universal infallibility: verification certifies the formal statement, not that it captures intent.

The honest breakthrough available now is therefore not conscious, perfect agents, but agents with experimentally measurable selfhood and formally bounded error — two separate, non-inflated claims.

7. Limitations

The eight instruments are toys by design (finite and grid-quantised Boolean and grid worlds, tiny Markov systems, lossless encoders; Instrument 8 is fixed-seed Monte Carlo). The master fibration $(q, K)$ is derived (Theorem 1 gives it as a mathematical object for any well-posed task on a standard Borel space; Theorem 2 parameterises it (Instrument 7 witnesses it exactly); Proposition 3 gives the categorical restatement; Theorem 4 makes cross-task stability equivalent to a shared Markov screen; Theorems 5 and 6 pin down discrete- and continuous-case sample complexity; Theorem 7 gives polynomial-in-$d_Z$ learnability inside the linear-ICA hypothesis class). The residual content is:

• SIC-B antecedent (empirical). Whether the actual world’s task ensemble admits a shared Markov screen is an empirical question about physics, biology, cognition, and culture.
• SIC-C-c (uniform polynomial-in-$d_Z$ learnability). Theorem 6’s $(D_Z/\varepsilon)^{d_Z}$ bound is exponential in $d_Z$ at fixed $\varepsilon$; $\varepsilon$-covering lower bounds show this cannot be improved without additional inductive bias on $q$. Escaping the curse requires structural assumptions (linear ICA — Theorem 7 gives this class, first instance; independent-mechanism analysis, Gresele et al. 2021; iVAE, Khemakhem–Kingma–Monti–Hyvärinen 2020; interventional causal representation learning, Ahuja–Mahajan–Wang–Bengio 2022). Each supplies a hypothesis class within which learnability is polynomial in $d_Z$; extending the Observatory to any other specific such class is future work, not a theorem proved here.
• Extended-program conditionals in §5 remain flagged.
• No Lean-checked proofs of Theorems 1–6 yet — the proofs are in paper form; formalisation would be next-order rigor.
• No machine phenomenology. Nothing in Theorems 1–7 licenses a claim of subjective experience; §6 draws that line explicitly.

8. How this relates to Trace AI

Trace AI is the SIC applied to cognition, with a concrete choice for each abstract variable:

The whitepaper is Trace AI at the level of what to build; this paper is why the object exists as a mathematical object at all. Read the SIC when a reviewer asks “is your latent even a well-defined thing to be looking for”; read the whitepaper when they ask “what is the training signal that finds it.”

v0.2 (integrated 2026-08-03), tracks paper v4. See DECISIONS.md (D28, D30) for provenance. Reproduction of the eight instruments lives in the sibling research repository jawauntb/research-derived-experiments — all experiments/ paths in the instrument tables above resolve there, including the new experiments/rate_distortion_pair and experiments/linear_ica_learnability. Exact-recovery numbers in §4.5–§4.8 are copied verbatim from that repository’s outputs at the paper’s commit. The originating research note is at notes/structural_intelligence_conjecture.md; the working-note sibling of the present document is SIC_RESEARCH_PROGRAM.md.