Discovery, Persistence, and Cumulative Organization: Viable substructures, sharp assembly thresholds, and resource limits

When can a process find viable combinations of components, extend them, and retain successive gains? This mathematical working paper separates the existence of stable substructures, their discovery, and their persistence. It provides quantitative sufficient conditions for viable growth from a reservoir of fixed interactions, together with exact retention criteria for finite Markov search models. Organization means the number of retained active components; functional or biological complexity requires a separate definition. Large size alone does not ensure cumulative organization. Ramsey theory guarantees homogeneous substructures, but an additional interaction bound is needed to force a favorable type. Pairwise stability also does not imply collective stability. Even a process that reaches every finite assembly size may repeatedly return to zero. The results identify the assumptions that bridge these distinct obstacles. Principal results Stable substructures from bounded interaction. Components inherit finite principal matrices A = −D + K + E, where D has diagonal entries at least d* > 0, K is skew-symmetric, E is symmetric with zero diagonal, and every absolute row sum of E is at most L. A selected subsystem with symmetric row load at most ρ < d* contracts at rate at least d* − ρ. This certificate is inherited by subsets and preserved by admissible additions and deletions. Ramsey and weighted partition bounds. For a target size k ≥ 2 and L > 0, set s = 1 + ⌈L(k − 1)/ρ⌉. Any pool of at least R(k, s) components contains a certified k-component subsystem; the standard binomial Ramsey bound gives an explicit sufficient size. A stronger linear bound follows from the classical weighted improper-coloring theorem of Araújo and coauthors: an M-component pool can be partitioned into q classes with internal row loads at most L/q, and some class has at least ⌈M/q⌉ components. These existence results do not alone extend a prescribed assembly. Viable extension without resetting the retained state. Starting from the empty assembly, a selection rule uses pairwise disjoint fresh candidate pools and never reintroduces discarded or deleted components. It controls the lifetime interaction accumulated by each retained component. For 0 < θ, η < 1 with Lθ(2 − θ) < d*, a uniform proposal from a pool of size Mₙ = max{1, ⌈n/[θ(1 − η)]⌉} is accepted with conditional probability at least η, while preserving contraction. Pool choices and deletion choices may depend on the complete past. Quantitative retention under loss. With proposals at Poisson rate r₀ and independent loss events at total rate δ, each removing one retained component, r₀η > δ guarantees Nₜ → ∞ almost surely and lim inf Nₜ/t ≥ r₀η − δ. The expected time to reach size k is at most k/(r₀η − δ), with a sharper finite-sum bound in the paper. Expected candidate exposure up to that time is O(k²). These are guarantees for the specified selection protocol, not universal growth laws. Positive material yield by batch selection. A weighted partition and finite linear program select compatible batches while preserving the old assembly. With q = ⌈2/θ⌉ and a sufficiently large fresh pool relative to the current size, at least the fraction θ/(2q) of its candidates can be retained. When every selected batch is inserted in full and there are no losses, the same fraction bounds cumulative material yield from below. The theorem does not assign a physical running time to batch selection. Finite discovery and repair. A killed Markov-chain resolvent gives exact success probabilities, search durations, occupation-weighted rates, and deadline bounds. Reversible search with stationary entry and constant loss admits a semidefinite success certificate and conductance sensitivity formulas. A birth–death scale series distinguishes lasting accumulation from infinitely many returns to zero. Exact passage times exhibit linear, quadratic, and exponential regimes. Sharp rate-budget thresholds. In the two-state search model, exploration rate c, discovery rate κ, and loss rate δ satisfy c + κ ≤ B. Lasting accumulation is possible exactly when B > 4δ. The maximum sustained speed is 2(B − 4δ)/9, attained at c = (B − δ)/3 and κ = (2B + δ)/3. Varying budgets have an exact product-series criterion. In the stated family δₙ = B/4 − a/(n + n₀), accumulation occurs exactly when a > 9B/32, despite a favorable local margin at every level; the optimal expected passage time grows quadratically. Viability, memory, and maintenance limits. Common quadratic and Volterra Lyapunov functions certify linear and positive-population assemblies during change. History-uniform comparisons and finite interface kernels treat dependent stages and irreversible loss. For any stable linear system under isotropic noise, the inverse-covariance trace identity yields a sharp mean-variance bound and a dimension ceiling at fixed total damping. Per-mode precision is distinguished from whole-assembly retention. Methods and evidence Complete analytic proofs combine Ramsey theory, weighted coloring, lifetime interaction accounting, linear programming, Poisson coupling, finite-state Green operators, birth–death scale functions, semidefinite inequalities, renewal arguments, and Lyapunov identities. Seventy-seven exact symbolic and rational checks verify finite formulas and examples; they do not substitute for the infinite-time proofs. Established combinatorial, probability, and covariance results are attributed explicitly. Limitations and open questions The conclusions depend on their specified interaction, isolation, sampling, loss, reset or comparison, and nonexplosion assumptions. Continuing growth requires fresh components and maintenance resources; the proposal rate must account for inspection and selection time. Rate and damping budgets are not automatically thermodynamic power budgets. The models are not empirically calibrated and do not establish inevitable complexity growth or explain fundamental physical constants. Open questions include sharp repair criteria with persistent internal memory; thermodynamically derived transition and maintenance costs; viable extension with reused pools, partial interaction information, or limited inspection time; loss hazards derived from collective noise-driven failure; and rigorous links between retained component count and an independently specified functional quantity. Included materials The project contains the manuscript PDF, editable LaTeX and BibTeX source, a reproducible figure, verification code and results, a separate correction and scope report, and Zenodo description and keyword files.