# Persistence and Primitive Closure in Finite Computational Systems ## Canonical theory statement This program poses separately operationalized questions about preservation of joint-state dependence through lossy transformations, primitive temporal closure under local compatibility rules, and the predictive value of carried state under resource constraints. Tests 01 and 02 use exact measurement or enumeration; Test 03 is a prespecified empirical pilot. Whether these properties jointly favor reusable simulation architectures remains an open hypothesis. ## Originating conjecture A resource-constrained persistent simulation may favor recurrent, reusable, or closed computational structures. This program began with a speculative architectural question: if a persistent world were implemented on a finite computational substrate, would resource constraints favor compact spatial organization, recurrent state reuse, controlled resets, or closed computational histories? These possibilities motivate the research; they are not assumptions or results of the completed experiments. ## How this program is organized The program decomposes a broad simulation-inspired architecture into independently measurable properties that can fail separately. It combines exact reset-channel measurements, a bounded exhaustive primitive-cycle census, and a prespecified approximately parameter-matched recurrence pilot without claiming that any one property implies another. This describes the present program without characterizing prior literature; comparative novelty or priority claims require a dedicated literature review. ## Toroidal cyclic architecture conjecture - Spatial compactness: a 2- or 3-torus supplies periodic spatial boundary conditions. It does not create temporal recurrence or reduce computation by itself. - Recurrent state: carried state may improve prediction under resource constraints. The direct pilot remains non-evidence until its mandatory gate and reserved-seed execution pass. - Reset fidelity: a specified lossy reset may preserve selected joint organization. Test 01 validates the measurement pipeline but does not establish a confirmatory effect. - Primitive closure: C_T = 1 means exactly one primitive period-T cycle modulo temporal rotation. A torus or one closed history does not imply C_T = 1. - Entropy boundary: memory reuse does not eliminate thermodynamic cost, and expansion or contraction does not automatically erase entropy. ## The three operational studies 1. Coupling/reset persistence asks whether interaction-dependent joint-state organization remains measurable after a controlled, entropy-shedding reset under frozen policy weights. Its primary operational contrast is P_rel = I(S_end; S_seed) - sum_i I(s_i,end; s_i,seed). 2. Closure/cycle uniqueness asks whether a finite local compatibility relation permits exactly one primitive period-T cycle after shorter-period repetitions and temporal rotations are removed. W_T = trace(M^T), P_T is the Mobius-inverted count of rooted histories with minimal period T, and C_T = P_T/T. The uniqueness condition is C_T = 1. 3. The pending recurrence pilot asks whether a carried-state RNN provides a late-horizon predictive advantage over an approximately parameter-matched fixed-window MLP in persistent-memory worlds. Parameter counts are approximately matched; stored context, recurrent activations, and compute are reported separately. No total-resource matching claim is made. ## Required interpretation The tests share a theoretical motivation but are not mathematically or causally equated. The evidence does not show that persistence causes closure, closure causes coupling, or P_rel and C_T measure the same property. ## Evidence status - Test 01 is a completed method-validation and threshold-calibration pilot. It does not establish an independent confirmatory coupling effect. The future 99th-percentile paired sign-flip threshold is 0.200407323 nats. - Test 02 exhaustively enumerates all 66,064 labeled binary relations for state sizes 2, 3, and 4 at periods 1 through 8. Its results are exact within those bounds. - Test 03 is pending/non-evidence. The implementation has not passed the mandatory independent code gate; no confirmatory manifest was created and reserved seeds were not accessed. It requires independent code approval, a frozen SHA256 manifest, and one reserved-seed execution. ## Supported - The Test 01 exact measurement implementation passes its stated validation gates. - Entropy shed is positive at all five tested reset scales. - Partner-only ablation changes frozen coupled behavior. - Unique primitive cycles exist within the completely enumerated Test 02 domain. This is a bounded descriptive result, not by itself discriminating evidence for the broader theory. - The cyclic-SCC conjecture is not a general characterization; its direction is supported in 12 of 24 cells. ## Not claimed - Physical time loops, cosmological implications, backwards causality, consciousness, or subjective continuity. - A confirmatory coupled-versus-isolated effect. - Generalization to larger or continuous systems. - A causal or mathematical equivalence between the two tests. - A demonstrated resource advantage for recurrent models or computational advantage from toroidal topology. - An inference from spatial periodicity to temporal recurrence or C_T = 1. - Entropy removal by contraction or avoidance of thermodynamic cost through memory reuse. ## Canonical machine-readable resource - theory.json: structured theory, tests, evidence status, and claim boundaries. ## Human-readable resource - The website root contains the theory, protocol-to-evidence mapping, methods, results, deviations, reproducibility inventory, and claims ledger. - The Primitive Closure Laboratory on the website runs exact small-system Test 02 calculations in the browser. It reports W_T, P_T, C_T, classification, and a representative primitive cycle in both visible prose and machine-readable JSON.