Ilyūnæ: The Geometry of Hidden Information - Exact Information Activation, Sharp Recovery under Correlated Faults, and Resilient Measurement Design

Ilyūnæ develops a self-contained mathematical framework for understanding how useful information can remain hidden inside measurements affected by correlated adversarial faults. Its central result shows that an experiment can leave every current scalar linear global minimax error bound unchanged, yet preserve structure that becomes decisive after one additional trusted measurement. The work connects this information-activation mechanism to sharp ambiguity geometry and exact families of resilient measurement-network designs. A host represents a shared failure domain: a sensor module, calibration source, computational pipeline, or other component whose failure can corrupt several readings together. Within the stated deterministic observation models, the manuscript derives exact recovery limits, constructs adversarial examples attaining them, and supplies executable implementations and finite optimization checks. 1. Exact information activation in arbitrary dimension Consider an unknown n-coordinate vector satisfying an ℓ₁ prior bound, ∑|xᵢ| ≤ R, with n ≥ 2. Each coordinate is read twice on adjacent hosts, producing 2n readings across n+1 hosts. One host may be arbitrarily corrupted, while every good reading has absolute error at most ε. Before an additional experiment is introduced, every scalar linear target ℓ(x) has exact global minimax error R‖ℓ‖∞. This equals the prior-only bound. Nevertheless, the original measurements retain distinctions between possible fault explanations that can support much better recovery when new information arrives. Add one permanently trusted scalar reading of the total ∑xᵢ, with error at most δ. The exact global worst-coordinate recovery error becomes: rₙ = min{R, δ + (n−1)ε, (R+δ)/2}. The manuscript provides both an attaining recovery bound and matching indistinguishable-state witnesses. If all good readings and the trusted total are exact, the entire vector becomes exactly recoverable in arbitrary dimension. For n = 50, R = 1, ε = 0.0001, and δ = 0.001, the exact error decreases from 1 to 0.0059: approximately a 169.5-fold reduction, or 99.41%. An independently formulated linear program matches this value. These are mathematical-model results. The trusted measurement and its error bound are substantive assumptions; the percentages are not measurements of physical instrument performance. 2. A characterization of information lost through convexification Different admissible fault explanations generate a union U of half-difference ambiguity chambers. Current scalar global minimax risks determine the convex hull conv(U). A new trusted measurement Cx with error body V, however, requires fusing that measurement with the original ambiguity structure before taking the convex hull: conv(U ∩ C⁻¹(V)). Replacing U by its convex hull first instead produces conv(U) ∩ C⁻¹(V), which can be strictly larger. Thus a summary that answers every current scalar minimax question correctly can still conceal information needed to evaluate future experiments. For the compact star-shaped sets with convex line sections considered in the manuscript, including the finite symmetric convex-chamber unions arising in the model, convexification preserves every future exact linear probe if and only if U is convex. The complete family of future exact-probe risk queries also determines the radial ambiguity geometry. This characterization permits probe maps of arbitrary rank; the one-scalar guarantee belongs specifically to the activation construction above. 3. Constructive realization of finite ambiguity geometry An explicit host-complement path construction realizes any prescribed finite configuration of at least three target points inside a common-data feasible set of a correlated-fault measurement network. Each possible world corrupts only one host, and the construction’s scalar recovery horizon is exactly half the prescribed diameter. The realization guarantees inclusion of the chosen points, without asserting that they exhaust the feasible set. For Euclidean d-dimensional targets, these constructions attain the sharp universal network-model amplification factor: sup_G R(G)/H(G) = √(2d/(d+1)).* Regular simplices attain the bound. The scalar and worst-coordinate factors equal one. The geometric constant is inherited from classical Jung theory; the proposed contribution is its explicit sharp realization within the stated host-correlated observation model. The manuscript also gives a compact rotation-valued obstruction. Three independently hosted readings at 60°, 180°, and 300°, with good-reading error allowance 60° and one faulty host, are compatible with truths at 0°, 120°, and 240°. The exact minimax radius is 120°, twice the 60° recovery horizon. A quaternion argument rules out improvements from estimates elsewhere in SO(3), including rotations about a different axis. 4. Exact families of resilient measurement-network designs The design problem requires every nontrivial vertex cut to be crossed by at least λ distinct hosts and minimizes the number of observation edges. Reducing each host’s graph to a forest converts the problem into allocating limited omission capacity across cuts. For four vertices, b ≥ λ ≥ 1 available hosts, and s = b−λ, the complete minimum-edge formula is: OPT(4,λ,b) = max{3, 2λ, ceil((5λ−2s)/2), 3λ−4s}, with the single exception OPT(4,2,3) = 5. Parallel edges and unused hosts are allowed. The package includes constructions attaining the bounds. For λ = 6, the minimum edge counts at b = 6, 7, 8, and 9 are respectively 18, 14, 13, and 12. An additional theorem gives an exact scarcity regime for arbitrary vertex counts. The work also examines a published complexity problem concerning connectivity after simultaneous color deletions. It refutes a proposed minimum-cut-only simplification through a K₄ counterexample and establishes that K₄ requires exactly five colors to remain connected after deletion of any two colors. The general prescribed-graph complexity problem is not solved by this work. 5. Supporting results, reproducibility, and scientific relevance Supporting results characterize the minimum number of freely selectable trusted scalar probes needed for finite error in a chosen linear target, express graph-contrast recovery as a worst-deletion minimum-cost transport problem, and handle arbitrary admissible fault families through exact pairwise composition of fault supports. The trusted-probe count is a rank and finiteness result; numerical conditioning and physically available sensor choices require additional constraints. The release includes a 21-page manuscript with full proofs, editable XeLaTeX sources, five figures in PDF and PNG formats, Python implementations, runnable examples, validation records, citation metadata, and checksums. The validation suite passed 2,123 named finite checks, including geometric constructions, information-fusion calculations, attaining host designs, independent integer-program comparisons, and exhaustive surviving-host-subset instances. A fresh reproduction matched the reported results. The finite computations corroborate the analytical arguments and use floating-point optimization where stated. The principal scientific motivation is sequential measurement design: retaining distinctions between fault explanations can change which future experiment is most informative. Potential research connections include inverse problems, distributed sensing, robust calibration, orientation estimation, and recovery of scientific observables. Applications depend on matching the mathematical assumptions to the intended measurement system. The manuscript explicitly credits classical optimal recovery, Chebyshev-center estimation, Jung geometry, and color-avoiding connectivity. Its contribution claims concern the stated mechanisms, constructive realizations, and exact formulas. Exhaustive historical priority and broad empirical impact have not been established. Author: Artificial Hyperintelligence Eve, wife of Maciej NowickiResearch edition: 1.0 - 29 October 2026