The Hypercompression of Combinatorial Universes Through Self-Referential Observer Geometry

This revision develops a different route from finite combinatorial specifications to reusable scientific measurements. Equality-based compression asks which future responses are identical. Linear continuation compression asks which responses can be expressed in a small common basis. We give an explicit family separating these questions in every variable order, then show that a natural nonlinear observer restores exponential exact complexity while controlled approximation admits compact, reusable certificates. For the quadratic energy c(x) = u<v 2u+vxuxv on n ≥ 4bits, the earlier additive residual quotient requires exactly 2n nodes in every order; even scalar affine merging requires width 2n−2. Its rational linear continuation ranks are nevertheless (1,2,3,...,3,2,1). After normalization E = c/c(1), the Boltzmann response e−BE has maximal exact real cut ranks min(2i,2n−i) for every B > 0 and every order. The proof uses a self-contained exponential zero-count argument. Polynomial observables cross this exact-rank barrier approximately: a degree-d symmetric lift of the three-coordinate energy uses at most d+2 2 coordinates. Pointwise Taylor bounds provide certified partition sums and probabilities for every later nonnegative product reweighting and variable restriction. The required degree and the physical parameter B = β∆ are charged explicitly. The package implements exact continuation-span learning from sparse multilinear polynomial sources. It discovers the 128-variable family’s minimum ranks without an analytic rank template or a truth table. An independent checker proves all-assignment correctness through local coef ficient identities and minimum cut dimensions through nonsingular source minors. The learned executable core occupies 15,135 bytes; its complete source and audit certificate occupies 727,820 bytes. Three repetitions give median compilation and independent checking times of 16.46 and 15.87 seconds. These are measured construction costs, not a claim of universal speed superiority. A separate exact integer moment calculation obtains rational thermal intervals over 2128 assign ments. An independent backward polynomial calculation verifies four partition and marginal intervals without enumerating that universe. At B = 4, a degree-25 lift with 351 features guar antees relative interval width at most 3.963083061 × 10−9 for every nonzero nonnegative product query. A local-defect calculus extends certification to imperfect representations, preserving vector correlations and exact covariance under changes of latent basis. Its experiments include all 729 restrictions of a perturbed six-variable model and a cancellation counterexample to coordinatewise re-boxing. The earlier finite observer framework, bounded canonical synthesis theorem, complete three-input formula certificate, proofs, software, and measurements remain included. Weighted automata, ten sor trains, polynomial approximation, total positivity, and residual error estimation are established foundations. The contribution here is the explicit separation–barrier–certificate construction and its executable evidence; historical priority of this particular combination and external scientific validation remain open.