The Structural Observatory

Over a Boolean world with a known invariant, search a lattice of quotient maps. Only sufficiency-then-compression recovers the ground-truth invariant: pure description-length minimization collapses the obstruction, and pure accuracy maximization never compresses. Sufficiency, description length, and accuracy dissociate.

One abstract structure (accumulation → phase transition → hysteresis) is compiled into music, a visual field, text, and spatial navigation. Each medium's readback recovers the identical trajectory (qi ∘ Fi = id) — verified structural identity, not mood matching.

Treat a symbolic model as an operation on a system's future-trajectory distribution. Signal, control, knowledge, and agency dissociate: a false-credit condition improves the outcome with zero true do-effect and miscalibrated self-attribution; a brittle controller controls without transferring. No single scalar identifies agency.

On a 4-bit Boolean world with latent Z = joint(parity{0,1}, parity{2,3}), enumerate a rich quotient lattice for two task families exactly. For the family whose members all factor through Z, the coarsest common sufficient statistic is exactly Z (image 4, strictly smaller than |X| = 16). For the family that reveals individual bits, it collapses to the identity. Combining tasks strictly tightens the required partition: family CSS is finer than any single task's minimal sufficient statistic. Cross-task stability is a property of the task family, not of the system.

Same 4-bit world, same shared task family, now measuring sample complexity. Recovery of Z under empirical common-sufficient clustering reduces to a coupon-collector question on the fibre partition; its exact probability is computed by inclusion–exclusion over all 2M = 16 subsets (no Monte Carlo, no seed). At Theorem 5's bound N ≥ ⌈c⋅M⋅ln(M/ε)⌉ with ε=0.05, exact recovery is 0.9775 (uniform, c=1, N=18) and 0.9756 (skewed, c=2, N=36) — both above 1 − ε=0.95. Recovery is zero below M=4 (pigeonhole) and monotone in N. Discrete-case learnability is a theorem with a numerically sharp constant.

Ambient X = [0,1]2 quantised to a 16×16 grid; latent Z a coarser r×r (for d_Z = 2) or r×1 (for d_Z = 1) grid at resolutions r ∈ {4, 8, 16}. Exact recovery probability at Theorem 6's bound is computed by an O(N⋅M) DP recursion, numerically stable up to M = 256 where the log-domain inclusion–exclusion form fails. All six grid points meet the 1 − εrel = 0.95 target; the ratio Nbound(dZ=2)/Nbound(dZ=1) grows 5.17, 11.17, 23.52 at r = 4, 8, 16 — the exponential-in-d_Z scaling made numerical. Empirical common-sufficient clustering saturates the ε-covering lower bound; escaping the curse requires inductive bias.

Closed-form Shannon R(D) plus explicit RD-optimal test-channel construction, verified exactly on two finite sources with Hamming distortion: uniform on 4 symbols (R(D) = log2(n) − h(D) − D⋅log2(n − 1)) and Bernoulli(p = 0.3) (R(D) = H2(p) − h(D)). At D = 0 the encoder is minimal-sufficient and R(0) equals the source entropy — the Theorem 1 anchor; at D = D_max the encoder collapses to a constant and R(Dmax) = 0. The test channel achieves I(X; X̂) = R(D) at every point in the achievable regime. All ten gates pass to 1e-9.

First positive resolution of SIC-C-c (linear ICA, Hyvärinen–Oja 1999, restated in our framework as Theorem 7). Fixed-seed sklearn.decomposition.FastICA on X = A⋅Z with random-orthogonal A and independent Laplace(0,1) latents, swept (d_Z, N) ∈ {2,4,6,8}×{200,500,1000,2000,5000,10000}. Mean Amari at N = 10000 is ≤ 0.009 for every d_Z; fitted polynomial exponent b ≈ 0.06, gate b ≤ 3; 462× escape from Theorem 6's ε-covering bound.

Second positive resolution of SIC-C-c: the sparse-linear-mixing class. Same shape as Instrument 8 but with A a sparse-orthogonal matrix at retention s ∈ {0.5, 0.25}. All four gates pass; sparser mixing gives modestly better recovery (better conditioning), fitted exponents b = 0.00 (s=0.5) and b = 0.51 (s=0.25), gate b ≤ 3. Caveat. This is sparse linear ICA, not the full nonlinear-mixing independent-mechanism analysis of Gresele et al. 2021 (which concerns the Jacobian-column geometry of a nonlinear mixing map). The IMA nonlinear instrument is a natural next addition.

Third positive resolution of SIC-C-c: auxiliary-variable identifiable ICA (Khemakhem–Kingma–Monti–Hyvärinen 2020). Latent Z_i | U ∼ Laplace(μ_i(U), 1) conditional on a discrete auxiliary U; per-conditional FastICA aggregated by global Amari. Amari ≤ 0.033 at N = 10000 for every d_Z ∈ {2, 4, 6}; fitted polynomial exponent b ≈ 1.23 (gate ≤ 4); escape ratio ∼ 23× from Theorem 6's bound.

Fourth positive resolution of SIC-C-c: single-node interventional causal representation learning (Ahuja–Mahajan–Wang–Bengio 2022). For each latent component the observational distribution is replaced by a shifted intervention distribution; per-environment FastICA with an intervention-shift alignment achieves environment-consistency Amari ≤ 0.20 at the largest per-environment N, and the split strictly beats the pooled control — interventions carry genuine identifying information.

Companion to Compiler Tomography. MDL over a 25-point concern-parameter grid recovers the true compiler θ* = (0, 0) with rate ≥ 0.95 by N = 2000 paired samples (Theorem CT-1); the Boltzmann ecology update K_t+1 ∝ K_t⋅exp(β r) is monotone non-decreasing in per-fiber expected reward at every β ∈ {0.1, 1.0, 4.0}, and converges to the fiber argmax at β = 4 (Theorem CT-2).

Companion to Concern as Fiber Geometry. On the exponential family with sufficient statistic T, the empirical Fisher matrix agrees with the predicted Covc,z[T] = β2⋅diag(sech2(βci)) at every grid point to ≈ 2⋅10−16 (Theorem CG-1); discrete parallel transport of the concern one-form around a closed loop yields holonomy exactly ε⋅A, where A is the signed area (Theorem CG-2 — rectangle 0.3, triangle 0.15 at ε = 0.3).

Companion to Causal Semantics. On a world of 6 messages, 4 contexts, 4 future states, Ψ-equivalence m ∼Ψ m′ ⇔ p(·|c,m) = p(·|c,m′) ∀c is a congruence (CS-1) and the meaning quotient yields 4 classes: {m0,m1}, {m2,m3}, {m4}, {m5} — the coarsest common sufficient statistic on messages (CS-2). By construction the co-occurrence partition {m0,m2,m4}, {m1,m3,m5} is orthogonal to the meaning quotient.

Companion to Sufficient Antecedents. Four canonical identifiability escape routes (linear ICA, sparse-linear ICA, auxiliary-variable iVAE, interventional CRL) each populate Theorem 4's antecedent by a local screen at every antecedent value; the intersection of local screens recovers the true latent Z exactly (4 blocks). Each row is one way of witnessing SA-1: local separation, cross-u coherence, and intersection-equals-Z.

Companion to Structural Intelligence — Foundations. On the 4-bit Boolean world (16 states), the LR-vector partition (T1 + CS-2) matches the joint-parity MSS partition bit-exactly: both partition the 16 states into 4 fibres of size 4. This turns the master fibration from a posited object into a derived one, Lean-verified as sic_a_finite_discrete, zero new axioms.

Companion to Structural Intelligence — Covering Learnability. The meta-theorem prediction n ≥ c⋅K⋅log(K/δ) at δ = 0.05 is fitted across K ∈ {8, 16, 32, 64, 128, 256}; the fitted constant c stays inside [0.936, 0.995] across all six K (span/mean ≈ 6%), and the empirical n_emp is tight to the bound by one sample. Lean-verified as sicc_covering_meta, zero new axioms.

Companion to Abstraction Frontier. Enumerate 23 quotients on the 4-bit Boolean world and score each on (task-sufficiency loss, dynamical closure, coding cost, control regret). AF-1: the Pareto set is an antichain of two points — constant and joint(parity{0,1}, parity{2,3}). AF-2: because a common sufficient statistic exists, the sufficient-only slice of the frontier reduces to the true Z alone.

Companion to Alignment as Ensemble Governance. On an exact 4-state Markov chain with per-step leakage β = 0.05, the survival probability Pr[q(X_t) ∈ V for all t ≤ T] matches the joint product-form bound (1−β)^T exactly on V and is strictly larger on the coarser viable region V′ = Z — viability is inherited under coarsening (AG-2).

Companion to Theory Atlas. Three charts Ψ₁, Ψ₂, Ψ₃ on overlapping contexts induce transitions T_ij. TA-1: cocycle Tik = Tjk ∘ Tij ⇔ a consistent global gluing exists. TA-2: the failure taxonomy — glue, phase transition (support gap on one edge), missing latent (full-rank discrepancy on every edge) — is distinguished exactly by the transition-support signature.

Companion to Representation-Repair Calculus. Eight canonical failure-signature → minimal-lift pairs: every broken representation strictly misses the invariant (RR-1), every lift is minimal (dropping any added feature breaks capture again), and two independent lifts compose on the product world (RR-2) — verified on a 48-state world.

Companion to Autocatalytic Artwork. AA-1: the audience's mean predictive log-likelihood is monotone non-decreasing under successive posterior refinement, and the posterior on the true compiler concentrates from 1/3 → > 0.9 in six rounds. AA-2: Bayes posterior update on compilers is identically the Boltzmann ecology update at β = 1 — the autocatalytic dynamic is the compiler ecology of CT-2.