The structure
what becomes impossible after each paper
Domain confirmations
Independent results
generated by SDL methodology, no SDL dependency
The limit space
not a domain under constraint, but the space where constraints live · (2)+(1+1) coordinates
Enter from
Every positional expansion of a unit fraction belongs to exactly one of six morphological forms. Proves that the minimum morphologically complete base is b = 15 and establishes a necessary and sufficient arithmetic criterion for completeness. No historical number system is morphologically complete.
Sub-limit DynamicsResolves three open problems from C1: the anomalous incomplete bases are exactly four, the intrinsic saturation threshold is always strictly less than b, and the structural bottleneck is always a mixed form. Transforms the completeness criterion from static classification into a dynamic picture.
Sub-limit DynamicsThe unifying paper. Formulates three axioms—Finite Alphabet, Irreversible Consumption, Observable Iteration—and proves that any system satisfying them exhibits finite vocabulary, early saturation, an interaction bottleneck, and an efficiency paradox. Verified on arithmetic, geometric, and semantic systems. Resolves open problems from C1 and C2.
Sub-limit DynamicsRefines the Irreversibility Axiom by distinguishing three consumption types (node-, arc-, trajectory-consumptive) and classifies all asymptotic behaviors into four dynamic regimes via SCC decomposition. Proves a forced finiteness theorem extending the framework to continuous alphabets under distinguishability thresholds.
Sub-limit DynamicsIntroduces the DGIGM model—a class of constrained systems where every action irreversibly consumes future possibilities. Proves the constraint structure is isomorphic to non-repeating edge walks on de Bruijn graphs. Analyzes phase transitions, the Efficiency Paradox, and 425 unique morphological forms from 477 game sessions.
Sub-limit DynamicsAnalyzes 570 unique shapes from a discrete constrained trajectory system. Seven statistically stable dynamic regimes emerge from unsupervised clustering. Geometrically identical configurations can belong to dynamically distinct regimes. All trajectories terminate in exactly three outcome classes. Failure is finite, structured, and anticipatable.
Sub-limit DynamicsTests LLMs as stateless agents in a DGIGM via a bridge protocol. Establishes six operational profiles across models from three providers. The critical variable is not memory depth or model size but the criterion for information selection. Validates the Efficiency Paradox at the agent level and confirms cross-domain transfer via Word Duel.
Sub-limit DynamicsIntroduces a trajectory-level regime classifier for LLM agents in irreversible environments. Across 170 level-episodes from seven architectures, a single observable — the sign of mean(ρₐ−εtotal) in the second half of a trajectory — classifies convergent vs. non-convergent runs at 95.9% accuracy (FP=1). Decomposes total erosion into passive (environmental) and active (agent-induced) components. Empirically validates the Bridge reinjection mechanism and its conditionality.
Sub-limit DynamicsIntroduces the expansion–recovery ratio λ = ε/ρ as the central dynamical observable for CGS trajectories. Establishes the directionality of the σ → ω spiral cycle and derives the optimal operating point Nopt. The Bridge mechanism regulates λ by reducing active erosion, not by adding information.
Sub-limit DynamicsDerives the 3 × 2 operational geometry from the axioms: three limit states (L1, L2, L3) from monotonic erosion, two spiral phases (σ, ω) from expansion–recovery directionality, yielding six minimal configurations. The accessibility constraint |Q| ≤ |L| is exact across 210 level-episodes. Phase alternation verified at 95.2% of limit-state crossings.
Sub-limit Dynamics RestrictedEstablishes agent-independence (the six-configuration structure is recovered in 721 human trajectories) and consumption-type dependence (T2 → K=6, T1/T3 → K=4) across four datasets and three agent categories. The Regime Product Structure R = L × Φ yields K = 2·|L(T)| as a structural consequence.
Sub-limit Dynamics RestrictedCross-domain experimental verification of C3 sub-limit dynamics. Four physical domains—wildfire spread, BZ-AOT chemical oscillation, traffic flow, Rayleigh–Bénard convection—produce 20/20 regime classifications on a pre-constructed atlas. Domain manifold shapes emerge from transducer formulas, not from fitting. Each domain traces a manifold in the same dynamical space; the atlas coordinates are dimensionless transport ratios, not domain-specific quantities.
Sub-limit DynamicsPrime gaps as a CGS: pixel = Δgn, limit = ln(pn), T3/R2 system. Four grammatical levels identified: forbidden bigrams, 54% pair repulsion (destroyed by shuffle, persists beyond mod 210), non-Markov memory (0.63–0.83), and a generation wall at k = 4 (only primorial 30 passes). Convergence with Hardy–Littlewood validates sub-limit dynamics on the primes. Final paper in the series.
Sub-limit DynamicsIntroduces the record filter—the sequence of denominators producing the longest decimal period—and proves every record denominator is prime. Proves that unbounded growth of multiplicative orders is logically independent from infinitely frequent maximality. The separation is structural: it arises from the divisor lattice of p − 1. Elementary, self-contained, no GRH required.
IndependentThe obstruction events are exactly independent for every finite block: the splitting fields Kq are linearly disjoint. The Artin constant CArtin ≈ 0.3739 emerges as the natural limit. The conjecture reduces to the absence of infinitely many Siegel zeros in the family {χq*}. No GRH assumed. Second paper in the Artin trilogy.
IndependentThe sieve for Artin's conjecture has dimension κ = 0, eliminating the parity barrier. Unconditional lower bound: S(A, z) ≥ 0.236·π(x). Exceptional Siegel zeros controlled via Goldfeld–Gross–Zagier. The surplus is reduced to a single mean-value hypothesis on ω*(k)—a weakening of a conjecture of Erdős. Verified computationally for k ≤ 104. Completes the trilogy C10–C11–C12.
IndependentA forge record, not a results paper. Initial material: Difference. Discipline: add nothing. Admitted operations: CITE and CUT, where a citation never pays for the CUT of the run containing it, and a name, number or formula arrives only after what it records. CUT divides along Difference, removes no material, requires nothing to disappear. RUN 01–05 constitute the blade and leave three discriminated failures, later named Impossible and Border. RUN 06–10 obtain configurational Difference of the same closed minimum, morphological completeness of the current form, and the non-skippability of the discriminating Difference—a property strictly poorer than adjacency, introducing no space, succession or unit step. The labels 10, 11, 12 (NOVELTY, COMBINATION, RECURSION) are registered after the findings they name, never used to derive them. Wave, phase, curvature, contact and any ontology of reality are declared outside the debts of the forge.
Independent NewA single finite, deterministic construction and the exact boundary of the claims it licenses. On 13,122 realized width-6 witnesses generated from the local rule of Conway’s Game of Life (B3/S23)—the substrate, not the subject—two ordered subtractions yield the unique minimum injective coordinate projection and an operational pair relation derived from the realized structure rather than supplied. Exhaustive enumeration of 8,191 metric-family subsets and 877 set partitions: under the coordinate-unlabeled observer, no admissible partition with fewer than 4 blocks preserves the frozen discrimination. A conditional, scoped negative result, reproduced by an external rerun. Code and runner included.
Independent NewA deterministic dynamics survives some losses of distinction and fails under others. The compatibility condition separating the two is standard; what it does not give is the operational limit between them. Given a map Θ : X → X and a declared reduction grammar, the limit is located as a crossing set: admitted elementary reductions whose endpoints fall on opposite sides of exact compatibility. Finite witness: the four-state cycle. All 15 equivalence relations classified—3 compatible, 12 incompatible—and all 6 first pair-identification cuts from full discrimination cross 1 → 0. Neighbouring mathematics treated subtractively; no new quotient theorem claimed. A finite, existential result. Continues C14.
Independent NewNot an operator postulated, but a question put to the forge of C13: what is the poorest inscription that can carry it without adding anything? RUN01–03 leave a single surviving core—D plus nestable division, where a division must be inscribable without reducing to the inscriptions it keeps discriminated, and material still Difference is readmitted without changing grammatical species. RUN04–10 serve only as a falsification wall: a new terminal, a new operation, reference and identity, state, branching and arity, observer and meta-level, relation and adjacency—all fail the Reverse test. Deleting them removes no certified capacity, so the grammar does not grow. Recursion is forbidden as a declared construct: the property must appear, not be encoded in advance. The notation class is closed (a division mark that determines its own scope); the concrete glyph stays open under a ten-point audit.
Independent NewDerives the four structural properties of sub-limit dynamics from semantic first principles, without presupposing the axiomatic framework. The convergence of four independent paths on the same properties constitutes evidence that sub-limit dynamics is the necessary structure of any process in which meaning manifests under finite constraint. Five predictions validated on 90 Word Duel runs.
Sub-limit DynamicsDerives the four necessary properties from narrative phenomena alone: three conditions (finite repertoire, irreversible trajectory, contextual classification) predict small effective vocabulary, early saturation, decision point, and destructive completeness. Distinguishes five trajectory regimes (fairy tale, tragedy, myth, saga, open work) as structural classes. Cross-domain convergence table across nine independent domains. The most accessible entry point to the series.
Sub-limit DynamicsDerives the four properties from syntax alone. The syntactic pixel is a conservation event (Q-family E, Q21), structurally distinct from the semantic compression pixel (G, Q28). Language becomes the second verified multi-family domain after DNA. Grammaticality gradient, unidirectionality of grammaticalisation, and creolisation renewal all derived from the axioms. Seventeenth domain.
Sub-limit DynamicsVerifies that the harmonic domain satisfies the three CGS axioms. First verified instance of trajectory-consumptive dynamics (T3) and the branched attractor regime (R2). The Pythagorean comma emerges as the efficiency paradox in music. The twelve-tone technique is analyzed as a natural experiment imposing T1 constraints on a T3 system.
Sub-limit DynamicsFirst second-order pixel in the framework: the ordered pair (r₁, r₂) of consecutive intervals, where curvature becomes classifiable. Q-family E (Q21). First verified acoustic substrate bifurcation: one pitch space, two structurally independent CGS readings. Eighteenth domain.
Sub-limit DynamicsFirst empirical instantiation of ω as a physical observable. Rhythm is not modelled by the framework’s temporal operator—it is the operator made audible. Q-family G (Q28). Completes the musical triad F+E+G, structurally isomorphic to DNA. Twentieth domain.
Sub-limit DynamicsData-driven reduction of 39 essential nutrients, 5 taste receptor families, and 15 deficiency diseases yields a non-redundant metabolic border: three acquisition borders, one exclusion border, one state sensor, one cofactor layer. Taste receptors are endogenous border-reading interfaces—the first domain where the system reads its own borders. Cooking as selective σ-operator: 8/8 bioavailability predictions confirmed. Defines the B-series.
Sub-limit DynamicsReduction of the periodic table to minimal structural content under Pauli exclusion. Elements classified by border profile (k, δ, λ). Pixel: Z → Z+1. Hydrogen is a pre-pixel. Aufbau anomalies are local L2 events at d⁵, d⁹⁰ thresholds. The table terminates beyond Z ≈ 120 when the border ceases to be distinguishable: Ω → ∅ on classification. T2 system. Twelfth domain.
Sub-limit DynamicsSeven-node diamond graph from valence classes under Pauli exclusion. Three parameters (R, ω, rec) classify molecules. Blind protocol on 30 molecules: 90% isolability, 100% precision on delocalised systems. Six chemical categories eliminated as independent concepts. Fourth T3/R2 instance. Fourteenth domain. Answers Open Problem 2 of B2.
Sub-limit DynamicsDerives the structural necessity of evolution’s multi-family architecture (E+G+F) from the three axioms alone. The fact that DNA instantiates three Q-families simultaneously is not a biological accident—it is a structural necessity. Uniform randomness of variation is excluded by A3. Second-order pixel. Nineteenth domain.
Sub-limit DynamicsAll evolutionary scenarios converge to a universal selective attractor (Q35—“the rest does not pass”). Discrimination by trajectory richness, not final regime. Extinction ≠ compression: σ freezes the boundary, does not destroy it. Structural hysteresis confirmed without explicit memory. Fifth PDE-validated domain.
Sub-limit DynamicsReduction of inhabited space to minimal structural content under irreversible constraints. The regulatory operator (zoning, building code) is an asymmetric ω-operator that filters arcs without generating borders—the first domain where the border is simultaneously traversed, read, and modified under the same limit. Sfasamento Δη quantifies regulatory-structural distance. Validated on Barcelona, Kalamazoo, Isfahan. Defines the U-series.
Sub-limit DynamicsRetrospective face. Pixel = ordered pair (e₁, e₂) of consecutive classificatory acts. Q-network: G > F > E. The attentional blink is the empirical window where saturated classification of e₁ blocks e₂. Self-reading defined structurally for the first time: border-defined reading of the difference between operative state and readable trace. η is structurally non-null (Q12 open on Q36). Twenty-first domain.
Sub-limit DynamicsProspective face. Pixel = ordered pair (p, v) of anticipated and verificatory classification under the same L. Q-network: F > G > E—specular to H1. Two coupled spirals on the same border; λ = σp/ωv hypothesised ≈ 3. ηp structurally non-null for two independent reasons. Perfect prediction ontologically impossible in dynamics. Twenty-second domain.
Sub-limit DynamicsDiagnostic face. Pixel = ordered pair (d, a) of a diagnostic act that locates the dominant border and a calibrative act applied to it. Observable: the ratio rH = η/ηp between the residuals of H1 and H2. A 36-run Ω-field sweep, built before any clinical reading, yields four computational propositions: a capacity threshold L* that no calibration crosses, U-shaped dosage, self-stabilisation at high capacity, compression ≠ stabilisation. Six structural profiles placed on the pre-derived map; sleep emerges as a third regime (Q15). Structural, not clinical. Q-network: F > G > E. Completes the H triad.
Sub-limit DynamicsDerives the minimal structural configuration of a market: Ω(E1, E2 | L) with pixel = (offer, Δm). Validated on 926 categories across six markets, two countries, ~2.5 million offers. Regime invariant across sectors, currencies, and scale. First domain to populate the Emergente × Discreto cell of the Q1 atlas. Four Q1 predictions confirmed. Dual saturation mechanism identified (supply + classificatory). Market formulas instantiate Q-nodes already populated by physics, biology, and music. T3/R2. Twenty-third domain.
Sub-limit DynamicsThe programme itself treated as a CGS. Four coordinates classify verified domains: two static (Temporality × Discreteness of L) and two dynamic (smooth, ω Péclet coordinates). Decomposition (2)+(1+1). Twelve domains classified. Periodic table mapped via transducer: four predictions confirmed. Candidate structural law: Discreteness of L determines manifold dimensionality. One cell remains unpopulated (Emergente × Continuo)—its verification is the closure condition.
Sub-limit DynamicsThree compatibility conditions on a 35-node constraint network yield a minimal subspace of 11 configurations determined by a single ratio ρ = 5/7. Two conditions are symmetric; the third is automatically satisfied by their conjunction. Norm identity κ² + τ² = 9 = |L|². Tangential projection τ ≈ ln 2 within 1%.
Sub-limit Dynamics RestrictedSeventy-four real scientific formulas across 26 domains are assigned canonical Q-coordinates via the SDL-PROC-001 protocol. Ten multi-domain nodes confirmed (Q5: 10 domains; Q23: 7; Q20: 6). Law H: as |F_domains| grows, |active Q-nodes| stabilises. The constraint network is universal, not a taxonomy — it is the same structure appearing in independent lexicons.
Sub-limit Dynamics RestrictedA derivative annex: no new axioms, no new Q, no new limits. Characterises which variations are readable on the border of a constrained system without producing indistinction or an impossible border. Kinematics as a decomposition of status, σ ⇒ ω, read on the traversable residue AS(B). Three kinematic NOs, a 3–2–1 reading of the limit, and a non-arbitrary rule for constructing the border transducer TB under seven admissibility constraints. Checks across the 36 canonical Q signatures, cross-domain cases, and Maxwell’s equations as an operator-level continuous test. v3 extension: Living Dance Floor, M3 audit. Falsification conditions stated.
Sub-limit Dynamics Restricted NewThe chiasmic configuration is the minimal relational structure in which reading and transformation remain cross-coupled under a limit that belongs to neither pole. Closes the Emergente × Continuo cell. Under Q36 →D Q1, a saturating Q-network and its re-emergent successor coexist. The 36th paper.
Sub-limit Dynamics · restricted