Emergent Persistent Structures from a Pre-Geometric Medium

CFT GENESIS Emergent Persistent Structures from a Pre-Geometric Medium Stefan D. Ungureanu Independent Researcher ORCID: 0009-0007-6242-6305 June 2026 Significance Statement Physics assumes space exists before anything happens in it. This work asks: what if it does not? Starting from four minimal axioms — abstract entities, complex states, dynamic connectivity, and a principle of least action — with no metric, no coordinates, and no conservation laws imposed, a computational system spontaneously produces structures that persist, reconstitute their identity from a collective field, and generate their own geometry. Twelve independent quantitative analyses across three dynamical regimes confirm four emergent properties. New analyses reveal that amplitude is conserved in non-local transfer between modes, that 70% of mode pairs maintain stable phase correlations, and that modes under perturbation reconstitute at 1.4× their original density. The result reframes a question for fundamental physics: if the absolute minimum produces identity, geometry, and emergent conservation, what in real physics is truly necessary? Abstract We present CFT Genesis (Lido) — a minimal computational model in which a complex field ψ ∈ ℂ and a connectivity matrix W co-evolve on a pre-geometric substrate: a collection of Degrees of Freedom (DoF) with no coordinates, no metric, and no pre-established geometry. Geometry is not an ingredient of the system — it is a product of its dynamics. Through 18 quantitative analyses on simulations of up to 100 million steps, across three distinct dynamical regimes (κ = 0.02, 0.073, 0.10), we report four robust emergent properties: (I) persistent modes arise spontaneously from maximal disorder and reach a comparable density peak, followed by a mature density dependent on κ; (II) they survive total destruction of the topological substrate; (III) their identity resides in field ψ, not in connectivity W; (IV) identity is not created — it is reconstituted from the collective field at each (re)appearance. Six additional analyses characterize the system at equilibrium: amplitude transfer between modes is nonlocal (BFS median = 5, r = 0.28, amplitude ratio = 1.001); 70.8% of mode pairs maintain stable phase correlations; global coherence is strongly coupled to field amplitude (r = 0.86); persistent modes occupy zones of elevated local tension (+42%); and under perturbation the system reconstitutes at 1.41× the baseline density. These results are reported at κ = 0.02; cross-κ verification is in progress. The central result is a field-functional identity criterion: a persistent mode is defined by its recurrent capacity to be selected by the collective field to carry a characteristic functional profile, regardless of its local topological state. Terms such as “identity,” “persistence,” and “reconstitution” are used throughout as operational labels defined by measurable criteria in §8 — not as ontological claims. Keywords: pre-geometric adaptive systems, co-evolutionary networks, emergent modes, field identity, emergent geometry, CFT Genesis §1 The Founding Question Physics — classical or quantum — is built on a tacit premise: space exists before anything happens in it. Even in the most abstract formulations, there is a background geometry. Whether it is a grid, a Hilbert space, or a Riemannian manifold — something is already there before the dynamics begin. Structures appear on the stage. Not together with the stage. The question from which this project started was different: what happens if there is no stage? If there is no space, no metric, no pre-established geometry — only internal states of abstract entities and relations that change between them? The answer was not obvious. The system could remain in disorder. It could collapse numerically. It could produce uniform and trivial structures. There was no a priori reason for anything interesting to appear. The exact question, written into the v4 code before the first run: Do structures appear that should not survive, but do? The answer was: yes. §2 Axiomatic Foundation CFT Genesis (Lido) is defined by four axioms. They are not metaphors — they are operative specifications of the simulator. Everything that follows in this paper derives from them and from nothing else. A0 — The Constitutive Medium Ω There exists an entity called the Constitutive Medium (Ω) — a collection of Degrees of Freedom (DoF). Ω has no shape, no volume, no location. DoFs are not nodes, not points, not particles. They have no coordinates (x, y, z). They have a single property: they can carry a state and they can be connected to one another. This is the essential difference from any preceding model: we do not start with a space in which we place objects. We start with pure relational entities, without a geometric substrate. A1 — State (Ψ) The state of each degree of freedom i is described by a complex phase variable: qi = Ai · eiφi. This is the sole internally measurable reality. There is no position (x, y, z). There is only the value of state q. Amplitude A and phase φ are everything there is about a DoF — and all that is needed. A2 — Connectivity (Topology) Degrees of freedom interact through a dynamic adjacency matrix Wij ≥ 0 — a variable that co-evolves with field ψ. If qi and qr oscillate coherently (similar phases), coupling Wij strengthens. If they are in tension (diverging phases), the coupling erodes. Fundamental consequence: geometry is the inverse of connectivity — dij ∝ 1/|Wij|. Two strongly connected DoFs are, by definition, 'close'. 3D space appears as a projection of this network, not the reverse. W is not a fixed substrate — it is a dynamic variable that generates the substrate. A3 — Dynamics (Principle of Least Action) The system minimizes an Action S = ∫ ℒ dt, with the Lagrangian: ℒ = Σi α|q̇i|² − Σ⟨ij⟩ βWij|qi − qr|² − Σi (λ/2)|qi|⁴ From Euler-Lagrange, the equation of motion for the field: α · ψ̈i = Σr Wij(ψr − ψi) − λ · |ψi|² · ψi Substrate dynamics: dWij = −γ · |ψi − ψr|² · Wij · dt + T · noise · dt The ratio κ = γ/T controls the balance between erosion (γ) and creation of new connections (T). It is the primary regime parameter: at small κ, the network is dense and the dynamics are slow; at large κ, the network fragments rapidly. The phase transition boundary lies in the interval κ ∈ (0.02, 0.073) — confirmed across three distinct regimes. What does not exist in the system: no global normalization, no externally imposed amplitude control, no pre-established geometry, no programmed conservation law, no objective function. Stabilization is exclusively emergent — derived from the nonlinear term λ|ψi|²ψi present in the Lagrangian. §3 From v4 to v9.0: The Path to Clean Physics The axioms are simple. The simulator had to become equally simple. The path from the first version (v4) to v9.0 is a story of elimination: version by version, we identified and removed mechanisms that did not derive from the axioms. At each elimination, the persistent structures survived. This is, in itself, a result: it demonstrates that they are not artefacts of control mechanisms — they are properties of the system axiomatized in §2. v4 and v5.1 (Feb–Mar 2026): First functional simulation, N = 16,384 DoFs, real field ψ ∈ ℝ, leapfrog integrator. The system contained amplitude clipping — a global mechanism that forcibly cut |ψ| if it exceeded a threshold. Stabilization was partly imposed, not fully emergent. Despite this constraint, persistent structures appeared: in a controlled run at T = 0.05, γ = 0.001 (seed = 42), 170 persistent modes were identified out of 207 anomalous nodes (N = 16,384, 1.6M steps). These modes exhibited internal class transitions (Hermit → Dipol → Complex, up to 13 transitions per node), non-zero co-appearance correlations between dying and appearing modes (82.1% of simultaneous out→in pairs had prior coappearance history), and ψ survival through topological isolation (79% of Hermit episodes were followed by reappearance with W reconstructed around the surviving ψ). These results, produced before any methodological cleaning, confirm that persistent structures are a property of the axiom set — not an artefact of v9.0. The amplitude clipping and the real-valued field are the reasons v4 data are treated as historical reference only (R02, R07); the qualitative phenomena are present in both. A note on random seed: all principal simulations in this project — from v4 through v9.0 and v11.0 — were run with a free, unfixed seed. This was a deliberate choice, consistent with the broader elimination philosophy: just as we removed amplitude clipping, vortex detection, and cache delays, we also removed seed fixation as a constraint on the initial state. Fixing the seed would have introduced a preferred initial condition — a form of hidden selection. The free seed means each run explores an independent stochastic realization of the same dynamics. The consequence is that cross-κ comparisons combine regime variation with independent realizations; quantifying the resulting statistical spread via multi-seed ensemble at κ_c is an immediate direction (see §13, L5). v6.1: The transition to ψ ∈ ℂ was necessary — phase φ is a real degree of freedom. But v6.1 introduced vortex detection on pseudo-triangles artificially constructed from a graph without geometry. Two major results proved to be artefacts of this construction and were eliminated. v7.0: Leapfrog did not conserve energy over the long run. Replaced by Velocity Verlet. v8.0: Correct Velocity Verlet, but Wij was updated in CPU cache only every 100,000 steps — local_tension and phase_lock were seeing a network with 10× delay relative to reality. v9.0 — the reference: Cache updated every 10,000 steps. No global mechanism. Velocity Verlet. N = 65,536 DoFs. Runs of 15–50 million steps at three values of κ. v9.0 runs the axioms, not an approximation of them. All principal analyses in this paper are based on v9.0. §4 What Exists: Population Overview (S02, S24, S06) The first question is the simplest: what structures appear? Analysis S02 (complete census, directly from pkl) catalogued every persistent mode in the simulation. Equilibrium Density At all three κ regimes, the density of simultaneously active persistent modes reaches a similar peak (~22– 29 modes per 1,000 DoFs) and declines toward a κ-dependent equilibrium density: κ Density @equilibrium Unique modes Regime 0.02 ~4.72/1K @35.8M steps 11,068 Dense, slow, heavy tail 0.073 ~0.56/1K @15M steps 6,530 Transitional 0.10 ~0.29/1K @15M steps 5,639 Rapidly extinctive Empirical formula from 3 points: density @15M ≈ 0.00257 × κ−2.056. A fit on 3 data points has zero residual degrees of freedom; the exponent −2.056 is a descriptive summary, not a confirmed scaling relation — validation at a fourth point (κ = 0.04, v11.0) is an immediate direction. The similar density peak at all κ (~22–29/1K) suggests a universal genesis mechanism. κ controls long-term survival, not birth. Fig. 1. Density evolution (S02, S22). All three κ reach universal genesis peak (~22–29/1K), then diverge: κ = 0.02 → 3.6/1K, κ = 0.073 → 0.56/1K, κ = 0.10 → 0.29/1K. Curves from census anchor points. At κ = 0.02 @50M steps (v10.g run): n_nodes_ever_active = 11,120, peak n_persistent = 1,688 @3.5M (rate 25.76/1K), final n_persistent = 222 (rate 3.39/1K). The degree p99 of the network declines from 2.82 over 0–50M to 0.03 over 40–50M, confirming the late-phase regime of sparse, stable topology. Who Becomes a Persistent Mode Of 65,536 DoFs, only 8.6–16.9% ever become persistent. S24 (persistent vs. non-persistent contrast, 3 κ) identified the separator: Property Persistent (p50) Non-persistent (p50) Ratio Amplitude |ψ| at appearance ~4.0 ~0.15 ~27× invariant cross-κ W-degree at appearance 1 66–100 Opposite regions The primary separator is field amplitude ψ — not connectivity, not local context. A DoF with amplitude ~27× above median becomes a persistent mode. The ~27× ratio is invariant across all 3 κ. Longevity (S06) S06 (span ~ amplitude) found no power law at any of the 3 κ. R² < 0.07 at all. The conclusion from historical versions (α = 0.342 from v6.1) is not confirmed on clean v9.0 physics. v9.0 data speak directly. §5 Emergent Geometry A2 states that geometry is the inverse of connectivity: dij ∝ 1/|Wij|. This was not a philosophical axiom — it was empirically verified by v9.0. DoFs in the code have computational indices 0 to N−1 — they are labels, not positions. There are no coordinates, no assumed distance. And yet S18 measures topological distances in W — the number of BFS steps between two DoFs. This is real geometry, produced by dynamics. S12/Comp B computes localization length ξ ≈ 2.3–2.7 hops on the fresh network. At maturity, the field no longer has an exponential localization profile — the geometry has reorganized. Global IPR grows monotonically at all 3 κ (ratio 2–3.8×): the field concentrates on an ever smaller fraction of DoFs. Persistent modes hold 6–16% of total energy while representing < 1% of DoFs. Geometry was not placed in the system — it appeared. §6 Founders and Early Structure (S26) At κ = 0.073 and 0.10, early founders (first 10 checkpoints) oscillate ~1.5× more and persist ~1.8× longer than the rest of the persistent population. At κ = 0.02, early founders have extremely high W-degree (~108) — the network was maximally dense at the time of their appearance. A single structural never_died mode was identified across the entire simulation: κ = 0.073, active from step 300,000 to 15 million — 98.7% of the simulation duration, without any disappearance. Methodological note (L3): kearly = 10 represents 2% of duration at κ = 0.02 but 6.7% at κ = 0.073 — early founder fractions are not directly comparable cross-κ without kearly = 27 on κ = 0.02. The conclusion about founders remains qualitative. §7 The Internal Time Scale: τ_amp (S11) The amplitude autocorrelation of a persistent mode is not exponentially decaying — it is oscillatory with slow damping. The system does not forget immediately. It oscillates. τamp = 213–244 time steps, with a quasi-periodic period of ~1,000–1,300 steps. Invariant across all 3 κ. This is the only internal time scale identified in the system — emergent from dynamics, not imposed as a parameter. It defines the rhythm at which a persistent mode lives and reconstitutes itself. §8 Where Identity Resides: An Investigation by Elimination A persistent mode disappears. The system continues. The same DoF reappears — active, with high amplitude, with a coherent functional profile. We call it the same mode. But what justifies this claim? The topological substrate has changed. Neighbors in W are different. The local context is new. The question is precise: where is the identity of a persistent mode stored between two appearances? There are three possible answers. The data eliminate them one by one. Step 1 — Identity is not in the substrate (S03) Jaccard(W_exit, W_entry): fraction of common neighbors between the moment of disappearance and the moment of reappearance. Result: Jaccard = 0 in 99–100% of episodes, at all 3 κ. Of 26,477 episodes at κ = 0.02, a single case with genuine overlap. S25 confirms: throughout the entire inactivity interval, the fraction of W-neighbors that survive is 0 in ~98–99% of cases. The stage does not exist when the actor is absent. S12 adds a spectral signature: the subgraph induced by simultaneously persistent modes is nearly completely fragmented — isolated nodes and a few dimers (IPRnt p50 = 0.500, invariant cross-κ). Persistent modes are not connected to each other. They coexist in isolation within the global network. Step 2 — Identity is not in the local scene (S18) BFS topological distance between pairs of simultaneously persistent modes, against a random baseline. Result: Δ median = +2.0 topological steps above random, invariant across all 3 κ. The persistent distribution is bimodal at d = 4/5; the random distribution is unimodal at d = 3. Fraction of persistent pairs with no topological path: ~57–61%, vs ~10–20% random. Persistent modes do not cluster in regions. They maintain greater distances from each other than the network requires. There is no persistent topological region in which identity is localized. Step 3 — Identity is not in individual memory (S08, S09) Profile space with 5 observables (amplitude, phase, phase_lock, local tension, local energy). Cosine similarity on two questions. S08: mode active now vs. its own past profile. Median: ~0.57. S09: mode at reappearance vs. its own profile before disappearance. Median: ~0.57. Same at all 3 κ. 0.57 is the reference value of an active mode against any other moment in its own existence. A mode does not resemble its own past any more than it resembles the current state of the field around it. Individual memory is absent. Step 4 — The data on collective field similarity (S10) Similarity between a newly appeared mode and the profile of donors that recently disappeared from the collective field. Result: median score = 0.86. In >99% of births, the score exceeds 0.7. Invariant across all 3 κ. The jump from 0.57 to 0.86 is the difference Δ ≈ 0.29 — robust, reproducible, unexplained by any parameter of the system. An active mode resembles a random donor from the collective field far more than its own past. The field produces the same functional profile regardless of which DoF becomes active. Step 5 — The field-functional identity criterion A persistent mode is defined by its recurrent capacity to be selected by the collective field to carry a characteristic functional profile, regardless of its local topological state. Explicitly, from data: Entry condition (S24): amplitude |ψ| at appearance ≈ 27× above non-persistent. Invariant cross-κ. Maintenance condition (S11): the field self-organizes on scale τamp ~213–244 steps; modes live and reproduce this scale at each appearance. Separation condition (S18): modes maintain topological distance from each other greater than the network requires (+2.0 steps above random, invariant cross-κ). Reconstitution (S10): the functional profile at each (re)appearance is determined by the collective field, not by the mode's history. What remains between appearances: the index (a label in the data structure) and the capacity of the site to be reactivated — not a stored state. Identity is reconstituted from the field, not conserved in the mode. This is an ontological shift from any adaptive network model [4]: identity is not a property of an agent — it is a property of the relation between a site and the collective field. Fig. 2. Identity investigation (S03→S10), κ = 0.02. Four sequential hypotheses eliminated by data. S03: Jaccard = 0 in 88% — substrate eliminated. S08/S09: individual similarity ≈ baseline (0.57) — individual memory absent. S10: collective field similarity = 0.859 (99.8% > 0.7). Δ = +0.29 identifies the collective field as the source of identity §9 Equilibrium Structure: Transfer, Energy, and Phase (S19, S20, S21, S_PHASE) Six analyses characterize the system at equilibrium at κ = 0.02 @50M. Cross-κ verification is in progress. Amplitude Transfer (S19) Pairwise correlation between amplitude of dying modes and amplitude of modes appearing within a temporal window. n_deaths = 71,949; n_births = 72,171. Pearson r(amp_dead, amp_born): Window r n pairs 1 0.2832 22.1M 2 0.2821 33.1M 5 0.2793 66.1M 10 0.2755 120.7M r is stable at ~0.28 and declines slowly with window — a real signal, not noise. Ratio amp_born/amp_dead median = 1.001: amplitude is conserved in transfer. BFS distance between transfer pairs: median = 5 hops, frac_disconnected = 0.556. Transfer is predominantly non-local — 55.6% of pairs have no topological path at the moment of transfer. The near-exact conservation (ratio 1.001) is a result whose mechanism is not yet understood: whether it reflects an emergent conservation law, a structural consequence of the Lagrangian, or a property of the amplitude selection threshold remains an open question. Cross-κ verification will determine whether it is a general property of the system. Fig. 3. Amplitude transfer (S19), κ = 0.02. Left: r(amp_dead, amp_born) stable at ~0.28 across windows. Center: amplitude ratio median ≈ 1.001 — amplitude conserved. Right: BFS distribution for transfer pairs — median 5 hops, frac_disconnected = 55.6%. n_deaths = 71,949. Cross-κ pending. Energy Landscape (S20) E_total = Σ|ψi|² evolution and Pearson correlations with system observables. E_total is dominated by the field background (r = 0.997 with E_background). Persistent modes hold 6.4% (p50) of total energy while representing < 1% of DoFs. When E_total increases globally, n_persistent and nnz decrease (r = −0.684 and r = −0.946 respectively). The field background and persistent structures are in inverse equilibrium. Phase Correlations (S21 and S_PHASE) |Δφ| distribution peaks at 90.9° (131.5M observations) — modes active simultaneously tend toward quadrature. 70.8% of the 15,070,992 unique pairs have phase correlation r > 0.5 over multiple flushes (stable). BFS distance for stable pairs: median = 5 hops, frac_disconnected = 0.590 — phase-correlated pairs are predominantly topologically disconnected. Global coherence (S_PHASE, 5,002 flushes): mean = 0.455, std = 0.198. r(coherence, amp_mean) = +0.863 — when the field is more intense, global phase coherence is higher. r(coherence, nnz) = −0.831 — when the network is denser, coherence is lower. Local_tension differentiates persistent from non-persistent modes: persistent modes show +42% higher tension at p50 (13.57 vs 9.55). local_entropy and local_w_flux do not differentiate — topological observables are independent of field amplitude (r ≈ 0). Fig. 4. Phase correlations (S21), κ = 0.02, 916,765,970 pair-observations. Left: |Δφ| distribution — peak at 90.9° (quadrature). Right: fraction of pairs with stable phase correlation by threshold — 70.8% stable at r > 0.5, BFS median 5 hops, frac_disconnected = 59%. Cross-κ pending. The BFS distance of 4–6 hops appears in three independent analyses (S19 transfer, S21 phase-correlated pairs, S18 proximity) — a structural signature of the system. §10 Perturbation Response (S27) Experiment E1: shock at 50M steps, κ = 0.02 → 0.10. n_before = 237 persistent modes at the moment of shock. Time post-shock Survivors New modes Total active +8K steps 178 (0.751) 0 178 +100K 125 (0.527) 118 243 +1M 100 (0.422) 205 305 +3.2M 80 (0.338) 214 294 New equilibrium density: 334.6 modes (ratio 1.41× relative to baseline 237). The system produces more persistent modes at κ = 0.10 than at κ = 0.02 — consistent with S02 which shows similar peak densities across κ. amp_surviving p50: stable at ~5.59 throughout — survivors do not weaken under the new regime. nnz declines monotonically from 5.02M to 4.65M. Fig. 5. Perturbation response E1 (S27): κ 0.02→0.10 at step 50M. Left: survival curve — long-term ~33.8% of pre-shock modes persist. Right: mode counts — new equilibrium 334.6 modes (×1.41 baseline). Survivor amplitude stable throughout (p50 = 5.59). §11 Phase Diagram and Dynamical Regimes (S22) κ = γ/T controls the ratio between connection erosion and creation. Three values of κ define a preliminary phase diagram. The transition boundary lies in the interval κ ∈ (0.02, 0.073) — density factor between the two endpoints is ~14× at 15M steps. Power-law fit on 3 points (S22): density @15M ≈ 0.00258 × κ−2.055, Pearson r = −0.999983 (3 points, zero residual degrees of freedom). We observe a steep decrease consistent with a power law with exponent around −2, but this cannot be treated as a confirmed scaling relation — confirmation requires at least a fourth data point (κ = 0.04, v11.0, in progress). The exponent −2.056 is a descriptive summary only. At κ = 0.02, the total number of connections (nnz in W) decreases monotonically over 50M steps — from ~10M at start to ~5.7M at maturity. At κ = 0.073 and 0.10 the behavior is different, consistent with the phase transition hypothesis. Fig. 6. Phase diagram (S22). Log-log plot: persistent mode density vs. κ. Filled circles: density @15M steps. Open circles: equilibrium density. Triangles: peak (~28/1K, universal). Dashed line: power-law fit ρ ≈ 0.00257 × κ^(−2.056), Pearson r = −0.99998. Shading: transition κ_c ∈ (0.02, 0.073). §12 What This Means CFT Genesis was not built to prove something. It was built to be minimal — four axioms, no metric, no global control mechanisms — and left to run. The central property — identity resides in the field, not in the substrate — is not a tested hypothesis. It is a conclusion derived from data: Jaccard = 0 (S03), 100% reorganization across gap (S25), individual memory absent (S08, S09), profile imposed by collective field (S10). All at all 3 κ. The equilibrium analyses deepen this picture: amplitude transfer between modes is non-local, predominantly between topologically disconnected sites (BFS median = 5, frac_disconnected = 0.556), with amplitude conserved in the process. Phase correlations are stable in 70.8% of pairs, again predominantly non-local. Global coherence is strongly coupled to field intensity (r = 0.86), not to network density. Modes occupy zones of elevated local tension (+42%) — they live in the structurally most active regions of the pre-geometric medium. More importantly: geometry was not placed in the system — it appeared. The topological distances measured by S18, the spectral structure from S12, the localization length from S12/Comp B — all are properties of a space that the system built for itself, from axioms that contain no metric [8]. This raises a question for fundamental physics: if the absolute minimum — state + relation + principle of least action — produces identity, geometry, and emergent conservation, what in real physics is truly necessary and what is ornament added on top? Lido does not answer this question. But it formulates it more clearly than anything before. §13 Limitations and Future Directions Documented Limitations L1 — Profile score calibration: S08/S09/S10 use equal weights for the 5 observables. Δ = 0.29 is robust but the absolute magnitude depends on this choice. L2 — S08 vs S09 indistinguishable: with current tools, we cannot determine whether the median ~0.57 from S08 and S09 reflects the same phenomenon or two different mechanisms at the same scale. L3 — S26 kearly cross-κ sensitivity: kearly = 10 represents 2% of duration at κ = 0.02 but 6.7% at κ = 0.073. Early founder fractions are not directly comparable cross-κ without kearly = 27 on κ = 0.02. L4 — Dimer structure: the absence of trimers follows probabilistically from connection density (nnz/N² ≈ 0.0013) and simultaneous population size (~237 modes at κ = 0.02): expected number of trimers ≈ 0.005. Quantitative cross-κ verification pending. L5 — Free random seed (deliberate choice): all principal simulations use a free, unfixed seed — a design decision consistent with the elimination philosophy of the project (see §3). Seed fixation would introduce a preferred initial condition; the free seed ensures each run is an independent stochastic realization. The consequence is that cross-κ results combine regime variation with independent realizations. Quantifying the resulting statistical spread requires a multi-seed ensemble: minimum 3 independent realizations at κ_c = 0.04 (v11.0, Programul κ). This is an immediate direction. L6 — S19–S21, S27, S_ENTROPY, S_PHASE cross-κ pending: all equilibrium and perturbation analyses in §9–10 are currently at κ = 0.02 only. Cross-κ verification at κ = 0.073 and κ = 0.10 is in progress. Results are reported as observed at κ = 0.02, not as general system properties, until confirmed cross-κ (R18). L7 — Finite-size effects: all simulations use N = 65,536 DoFs. Whether the observed phenomena — persistent mode density, identity criterion, phase correlations — scale to N ≫ 10⁶ is an open question. No finite-size scaling analysis has been performed; this is a direction for future work. The current results are reported as properties of the N = 65,536 system, not as thermodynamic-limit claims. [A17] — κ = 0.063 @50M artefact: v10.g resume from v9.2 checkpoint @20M produced 1 persistent mode at 50M (vs 46 at 20M). Possible incompatibility at junction. κ = 0.063 data are certified only up to 20M (v9.2 pure). Phase diagram fourth point pending re-run. Immediate Directions S19–S21, S_ENTROPY, S_PHASE at κ = 0.073 and κ = 0.10 — required for R18 compliance on all §9 results. Each Zenodo deposition in this series (v4, v6, v9, v11) serves as a fixed reference point for the project’s evolution. The present document (v9.0 results) is the first complete description of the system in its cleaned form; subsequent depositions will add cross-κ verification, multi-seed ensembles, and finite-size scaling. Program κ: bisection of κ_c in interval (0.02, 0.073). First run: κ = 0.04. Simultaneous simulation at T = 0.0273 (analog CMB temperature) to characterize phase coherence dependence on thermal noise (Program QS). Both running on v11.0. S26 kearly = 27 on κ = 0.02: sensitivity correction (L3). Numerical verification L4 cross-κ. Multi-seed ensemble at κ_c (L5). Long-term Directions 2D phase diagram (γ, T) independently — the ratio κ = γ/T is the current primary parameter, but the individual effects of γ and T on the system are unexplored. v11.0 separates them explicitly. Critical exponents (ν, β) at κ_c — to establish universality class and connection to Anderson localization [5,6], quantum graphity [7], and 2D quantum gravity [8]. Search for quantum-like phenomena (FQ1–FQ4): stable phase coherence, non-local correlations, amplitude conservation in transfer, partial norm conservation — as functions of (T, γ, λ). Program QS. §14 Methodology and Reproducibility All analyses follow 18 working rules developed during sessions GC28–GC31 in response to identified methodological errors (CFT_REGULI_DE_LUCRU_v1_0.txt). Essential principles: (R01) all scripts read directly from simulator pkl files, without intermediaries; (R03) no script applies filters without first reporting the complete unfiltered distribution; (R04) the script measures, the researcher interprets; (R05) complete distributions reported (p10–p99); (R18) any system property is confirmed at a minimum of 2 κ values. Simulator reference: cft_genesis_v9_0.py [3]. Hardware: NVIDIA RTX A3000 6GB on Dell Precision 7560 (κ = 0.02 @50M steps) and NVIDIA GTX 1660 6GB on H310 Gaming Infinite S MS-B928 (κ = 0.073/0.10 @15M steps). v10.g and v11.0 simulators extend v9.0 with new observables without modifying the physics core. All identified technical anomalies are documented with numbers [A1–A21] in SESSION_SUMMARY GC28–GC39. All principal v9.0 runs used a free, unfixed seed — a deliberate design choice: seed fixation would introduce a preferred initial condition, inconsistent with the elimination philosophy of the project. Each run is therefore an independent stochastic realization of the same dynamics. Consequently, cross-κ comparison combines dynamical regime variation with independent stochastic realizations. A multi-seed ensemble (minimum 3 realizations at κ_c = 0.04, v11.0) is the immediate next step for quantifying statistical spread (L5). Code, working rules, and data are available on Zenodo: https://doi.org/10.5281/zenodo.20432971 [3]. Acknowledgments The author thanks Prof. S. Solomon (Racah Institute of Physics, Hebrew University of Jerusalem) for pointing out connections to 2D quantum gravity models and the AB model (private communication, March 2026). No funding was received for this work. Conflict of interest: none. References [1] Ungureanu, S.D. (2026). Emergent Persistent Modes in an Adaptive Field-Topology System I. Zenodo. https://doi.org/10.5281/zenodo.19163680 [2] Ungureanu, S.D. (2026). Emergent Persistent Modes in an Adaptive Field-Topology System II: Complex Field and Phase Degrees of Freedom. Zenodo. https://doi.org/10.5281/zenodo.19208318 [3] Ungureanu, S.D. (2026). CFT Genesis v9.0 — Simulator and working rules. Zenodo. https://doi.org/10.5281/zenodo.20432971 [4] Gross, T., & Blasius, B. (2008). Adaptive coevolutionary networks: a review. Journal of the Royal Society Interface, 5(20), 259–271. [5] Anderson, P.W. (1958). Absence of diffusion in certain random lattices. Physical Review, 109(5), 1492–1505. [6] Evers, F., & Mirlin, A.D. (2008). Anderson transitions. Reviews of Modern Physics, 80(4), 1355– 1417. [7] Konopka, T., Markopoulou, F., & Severini, S. (2008). Quantum graphity. Physical Review D, 77, 104029. [8] Ambjørn, J., Durhuus, B., & Jonsson, T. (1997). Quantum Geometry: A Statistical Field Theory Approach. Cambridge University Press. [9] Watts, D.J., & Strogatz, S.H. (1998). Collective dynamics of 'small-world' networks. Nature, 393, 440– 442. [10] Barabási, A.L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286(5439), 509–512. Appendix — Analysis Library (GC39 status) Script Question measured κ covered Key result S02 census What exists, when, how much 0.02/0.073/0.10 density 16.9/9.96/8.60% S03 identity W Jaccard neighbors at reappearance 0.02/0.073/0.10 Jaccard=0 in 99–100% S06 longevity span ~ amplitude 0.02/0.073/0.10 no power law, R²<0.07 S08 memory Mode vs. its own past 0.02/0.073/0.10 signal < baseline, Δ negative S09 reappearance Mode at reappearance vs. itself 0.02/0.073/0.10 median=0.57 cross-κ S10 similarity New mode vs. donors from field 0.02/0.073/0.10 median=0.86, >99% >0.7 S_ANAT anatomy Portrait of a mode 0.02/0.073/0.10 amp_entry median ~4.0–4.7; degree_entry=1 invariant S25 context/gap Local context at appearance and gap 0.02/0.073/0.10 100% reorganization over gap S18 proximity Topological distance mode vs. random 0.02/0.073/0.10 Δ median=+2.0 invariant S24 contrast Persistent vs. non-persistent 0.02/0.073/0.10 ~27× amplitude ratio S26 founders Early founders / never_died 0.02/0.073/0.10 1 structural never_died S11 τ_amp Amplitude autocorrelation 0.02/0.073/0.10 quasi-periodic ACF, τ~213–244 S12 IPR Spectral localization Laplacian W 0.02/0.073/0.10 P1–P5 confirmed cross-κ S22 phase diagram Density vs κ power law 0.02/0.073/0.10 density ≈ 0.00258×κ⁻²·⁰⁵⁵, r=−0.9999 S19 transfer Amplitude transfer BFS κ=0.02 (cross-κ pending) r=0.283, BFS med=5, ratio=1.001 S20 conservation E_total, E_persistent evolution κ=0.02 (cross-κ pending) frac_E_pers p50=0.064, r(E_tot,nnz)=−0.946 S21 phase corr |Δφ| distribution, stable pairs κ=0.02 (cross-κ pending) peak 90.9°, 70.8% stable pairs r>0.5 Script Question measured κ covered Key result S27 perturbation E1: shock κ=0.02→0.10 @50M κ=0.02 33.8% survival @3.2M, ratio_new=1.41 S_ENTROPY local_entropy, local_w_flux κ=0.02 (cross-κ pending) r(entropy,flux)=0.56; no corr with amp S_PHASE global_coherence, local_tension κ=0.02 (cross-κ pending) r(coh,amp)=+0.863; tension +42% persistent Stefan D. Ungureanu | ORCID: 0009-0007-6242-6305 | June 2026