Searching equation space

Post 7 — the bottom-up track: instead of designing worlds, sample the space of field theories and let physics tell us what exists. 4,500+ candidate universes screened for ~$12 of compute; every headline independently re-verified.

What the search found (the seven results)

Everything below explains how these were found and gives the evidence.

How: the levels doctrine (why a bottom-up track exists)

The certified physics this searches over: posts 1–6. The G0c funnel derives from the spatial-eigenvalue analysis introduced in post 3.

Phase 2 clarified the working structure of blob science as a ladder of description levels:

The doctrine: findings at a higher level must be reducible to the level below by a learned process — L2 landscapes should be reachable as L1 fixed points (the inverse problem: can deposition + relaxation evolve b into the sawtooth-and-rails the machine needs? the phase-2 teaser says the sawtooth, at least, is a natural attractor), and L3 components must carry their L2/L1 certificates (bond curves, dial laws, stability windows). Higher levels buy speed of exploration; reduction buys legitimacy — a machine sketched at L3 is a conjecture until its landscape is demonstrated at L2 and, ultimately, grown at L1. The two phase-2 tracks are therefore not merged but stacked: L2 work discovers what is worth wanting; L1 work discovers what can be grown; the reduction map between them is itself a research object.

The bottom-up search (L0), status. The genome/funnel/assay stack is built and certified (reproduces every reference world to 5 digits; the spatial-eigenvalue screen predicts bond shells of unseen worlds to ~±15%). Stage-1 yield curves: uniform random finds ~2 alive worlds/100 candidates (but both novel mechanisms so far — including a bond with no oscillatory tails, a second binding mechanism our theory did not predict — came from uniform); jitter around references finds ~49/100 (maps island geography 25× faster). An evolutionary layer (merge = block composition, validated by reconstructing our own vvw and xv design jumps as single merge operations) beats jitter on alive-cells per assay and owns ALL multi-species coexistence cells — it even rediscovered the rotor from M4 parts without the recipe (audited: ω to 4 digits). The 32-worker pod sweep (3,195 candidates, ~$9 of compute) came back: the archive grew 111 → 404 cells. Confirmed at scale: jitter yields 51% alive / 24% bonded; blind-uniform novelty is ~1/600; the bond-shell law settled at d\* = (1.37±0.13 + 0.55k)·wavelength over 830 bonds. New physics found by the sweep: the non-oscillatory "plateau bond" is a robust 41-cell family (a saturating-inhibitor well, invisible to the tail eigenvalues by construction — binding chemistry beyond the funnel's prediction), plus at least one further unattributed binding route; the first 3-species bonded matter (from the evolver's elite line); 13 self-launching worlds; and a program speed record — c = 0.204, a jittered descendant of the evolver's rediscovered rotor, 45% faster than anything hand-designed (controller-audited to 4 digits). Every candidate carries its selection provenance (anthropic-screening doctrine).

How: the genome and the algebraic funnel

Every world in the search is a point in one canonical family — n cubic activators coupled to m linear-or-saturating channels:

The genome is {acts: (λ, k₁, D_u)ᵢ, chans: (τ, D, g)_c, W, K} — and because every channel is driven by deviations (u−u₀), the uniform vacuum is an exact fixed point for any weights: the iso-line trick as architecture. All five hand-built worlds (M0, the motility line, vvw, xv, bfield) are reference points of this family, reproduced to 5 digits by the generic simulator before any sampling was allowed.

Candidates pass a microsecond-scale algebraic funnel before any simulation:

G0c's shell prediction was validated on the certified worlds (10.9px measured vs 10.8 predicted) and then held across 830 bonds of the stage-2 sweep. Blob existence itself is subcritical (a homoclinic orbit — no local criterion suffices), so one seeded poke per candidate settles it; the funnel just makes the poke affordable.

Evidence I — yields and costs (findings 4, 5)

Finding 1's first specimen: uni_3034 — binding without oscillatory tails. The monotone-tail world that bonds anyway (d*=27.9, two-sided): the plateau-bond mechanism, invisible to G0c by construction (its tanh channel has g′(0)=0). Blind uniform search found what the theory could not predict.

The other uniform novelty (uni_3050): a tanh-channel core with an exciting K entry; merge-or-repel pair law.

Evidence II — the stage-3 harvest (findings 2, 3, 4, 6)

17 shards, 1,068 records, 4 pods, ~$3. Everything below controller-verified on fresh reruns where marked.

The carrier census (findings 3 & 4)

Of 996 alive worlds given the kicked travel assay, only 24 travel (2.4%) — self-propulsion is a scarce phenotype in equation space, exactly as the funnel's near-onset analysis predicted. And it is not uniformly scattered: the traveler rate by ancestry is 16% for dynamical-b lines (self-launchers — motility by deposition is the easiest route), 4% for the engine/rotor lines, ~2% for uniform ancestry, and 0% for the M0/M4 and VVW lines (170 and 151 worlds each, not one traveler — the classic families are deeply static away from their tuned onsets). Speed leaderboard: c = 0.304 (XV-line jitter, verified independently to 4 digits ✓), 0.245, 0.208 (both from the evolver's slow-tanh merge line e1_9513), 0.204 (the 3-field engine) — and combining the engine's two best dials (Du×1.1, D_v×0.9) gives the outright program record, c = 0.3439 in a bare 3-field world (verified ✓). For scale: the best hand-designed carrier was 0.143. The two independent motility protocols (census kicks; dial-ladder onsets) agree on the overall rate to within 0.3% — and on the deeper point that all known motility lives in exactly two mechanisms: stigmergic self-launch and two-timescale v/w engines.

The encounter tables (finding 6 + machine parts)

Six 3-species worlds got full pairwise encounter matrices (three cross-pairs × three distances). Two are machine-grade: s2_111_17 (two distinct cross-species bonds, zero replication anywhere — the species-rail candidate) and s2_116_46 (cross-bond in a robust cell). Two replicate on contact (fragile lines), one is inert. And one — s2_101_58 — shows kill outcomes: one species consistently eliminates another on close encounter while surviving itself. That is a predation-like interaction, found by blind search, not designed. It is parked for the world-catalog phase (a predator–prey blob ecology is now demonstrably available in this equation space).

The stack verdict (finding 2)

Head-to-head stack probes: plateau-bonded stacks (the tanh-well binding family) park motionless at n=2 and n=3, including under working noise, while the M4 control stack shuttles under noise exactly as machine-v2's law cascade predicted. The stack-safety problem is solved by a binding mechanism that blind uniform search found and that our tail-eigenvalue theory is structurally blind to. (Update — the census closed this: the uni_3034 "death" was a job-spec artifact (dressing overdose); re-run bare, its 3-stack parks at 29.3 px under noise, controller-verified. Both family members are usable cargo, and the plateau design rule gives a d\* menu: 14 px fixed, or 20–30 px tunable via the slow-channel τ — stack spacing fully decoupled from tail physics.)

Machine v3's parts list, assembled by search

This closes the loop the program was built to test: top-down machine failures became bottom-up search queries, and the search answered all three. The next machine will be built from parts no one designed.

Next: machine v2 (multi-cargo delivery-and-sort logistics from the certified parts), the b_target inverse problem (differentiable-RD or evolutionary, now that the forward map is characterized), pod-harvest atlas review, then world-ification — anonymous ports, alienization, prediction/preparation contracts (bond lengths, speed laws, climb thresholds, delivery tasks), certification, oracle and frontier baselines. The blob universe joins the catalog when it passes the same gates as every other world.

The deep search: selection for complexity works (finding 7)

The gated experiment ran: MAP-Elites over the audited complexity descriptors, seeded with the ground truths and atlas elites, evolving by the validated operators (merge to open cells, mutate to climb) with a growing block library. Six generations, 196 evaluations, 48 core-hours. The verdict:

Sharpened operator law: merges open new cells (12/21 first-touches), mutations climb within them (17 cell-holders and both top scores). Next: mechanism autopsies of the champions (what IS ds3_017 doing? — it maxes ecology, roles, and graph components simultaneously), then the pod-scale run (20 generations, immigration, richer descriptors) if the autopsies justify it.

Champion autopsies — what the evolved worlds actually are

Per the review direction: less methodology, more "what did we breed." Training dynamics first, then films of the top worlds with what the metrics saw and what they missed.

Training dynamics

(a) Interest distribution per generation: every dot one evaluated world; the mean (red) climbs monotonically +49% before saturating at g6; the dashed line is our best designed world — the population median crosses it by g4. (b) Cumulative behavior-space coverage: the g3 jump is the 4-species breakthrough (merge-opened). (c) The division of labor between operators: merges account for most first-touches of new cells relative to their evaluation share; mutations dominate within-cell climbing. Numbers on bars = new-cells-per-evaluation.

ds3_017, the champion (I = 76, audited): a worm ecology with trails

Watch: the red species does something no designed world did — it GROWS: elongated worm-blobs that stretch, split, and pack the domain to a carrying capacity (~45 segments, logistic curve top-right, then slight decline — bounded, not a cascade). Among the worms, two conserved compact species (blue, green) wander. Top-right: total population; bottom-right: per-species biomass — red saturates, blue/green flat.

Genome anatomy (3 species, 8 channels): three M4-class motility engines at different clocks (τ = 4.1 / 5.75 / 1.8 — the middle one sits in the traveling corridor), weak mutual v-crosstalk between two species (the xv bond/rotor wiring, diluted), and two slow tanh memory channels: a shared trail field (τ=189, weakly attractive to two species — a faint stigmergy layer) and a write-only recorder (τ=200, K=0 — it influences nothing; evolution kept a channel that only remembers. A vestigial organ, or free capacity the metric never charged for).

What the film shows that the metrics did not: the "replicator" is the labyrinth instability domesticated — the same fingering that killed species A in post 3 is here bounded (worms stop at a finite width and count) and coexists with compact blobs. The metric scored it as ecology + roles + population dynamics (C6=1.0, C7=0.998, C1=0.7); the mechanism is stripe-forming physics living with spot-forming physics in one world — the coexistence the Purwins-class literature treats as separate phases, evolved into a single stable mixture.

ds3_014 (I = 61, audited): four species, rotors, and a delayed baby boom

Watch: a sparse 4-species blob ecology — compact blobs (yellow/blue/red/green), some locked in tight pairs (the vvw-wired species 1+2: rotor cells). Nothing happens for 1500 tu — population flat at 12 — then a delayed replication onset: the yellow species begins budding and the count climbs stepwise (top-right). The metric's memory component saw the slow tanh recorder charging toward threshold during the quiet phase; the film shows what it was counting down to.

Follow-up: are the champion's spheres load-bearing? (ablation)

Reviewer question: the blue/green compact blobs in ds3_017 don't visibly interact with the worms — do they matter? Ablation answer: mostly passengers. Deleting both sphere species leaves the worm ecology essentially unchanged (carrying capacity 47 vs 46 baseline); deleting only blue raises worm suppression slightly (37 vs 46 — blue is a weak worm inhibitor), green is near-neutral. So the champion's core is the domesticated-worm physics plus trails; the sphere species pad the ecology component of its score. Logged as a metrics-v2 case: species that neither interact nor die should not count toward "ecology" — an interaction-strength gate would fix it.

Follow-up: ds3_014 run to its climax (T = 12,000)

Watch the whole succession: the sparse 12-blob world (first 1500 tu — the original clip ended here, mid-takeoff) erupts between t≈2000–4000 into a dense mixed tissue of ~150 blobs, then ripens slowly. The species boom IN ORDER: yellow pioneers saturate first (~t=4000), blue converts to worm-infrastructure second (~t=5000), red spots fill the remaining space in a slow climb still completing at t=12,000 — and the green species never booms (3 marginal blobs, near-zero biomass in the climax: the succession's loser). Right panels: stepwise population curve and the staggered per-species biomass saturations.

Reading: what looked like a "delayed baby boom" is the first act of a full ecological succession — pioneer → infrastructure → climax-filler → loser — ending in a packed, statistically steady tissue (population flat at ~150 from t≈7000; red biomass the last still-ripening component). The steady state is a static-mosaic climax, not a cycle. The slow tanh channels are the succession clock: each species' boom awaits its memory field charging toward threshold. This is the closest thing to a developmental program the search has produced — and it was bred, not designed.

The fossil vertex: how ds3_014 actually works (correction + lineage)

Correction to the anatomy above: the "write-only recorder" is not vestigial. Its feedback column is zero, but it acts through the genome's single bilinear term — red's equation contains −m·w_R: the slow memory field multiplies red's own exclusion halo. Ablating this one term (same seed, T=12,000) is catastrophic: the entire succession disappears — no red boom (3→7 blobs instead of 3→71), no yellow boom (3 instead of 75), twelve thousand time units of near-stasis. One term carries the whole developmental program: as m charges over ~130 tu wherever red persists, it modulates red's self-inhibition until mature spots destabilize and split — fatigue as the replication trigger.

And the term is a fossil. No operator in the search can create bilinear couplings (creation rate: structurally zero — audited). The lineage traces it through five generations and three separate campaigns — ds3_014 ← g0_jit_11 ← rail_111_17 ← s2_111_17 ← e1_9508 ← ref_BFIELD — to the designed stigmergy vertex −b·(w−u₀) of the dynamical-background world (post 5), inherited verbatim and repurposed: a trail-steering coupling exapted into a replication clock. Every nonlinear interaction vertex in the entire evolved archive descends from this one designed gene. Consequence for the next search round: give mutation the power to mint vertices, and find out whether evolution uses them or this founder effect was luck.

ds6_000 (I = 64): the flicker phase

Watch: the never-before-occupied bond-network phase — a dense two-species tissue (~150 blobs at carrying capacity, spots + short worms) whose bond network continuously rewires without ever freezing or evaporating: a bond-liquid at high density, 'flicker' in the graph-churn descriptor. Population and biomass are statistically flat while the micro-structure never repeats.

Two more champion films on disk (p7_world_ds6_019.gif — the gen-6 interest peak, a denser worm ecology; p7_world_g0_jit_11.gif — the gen-0 rotor-line high scorer). Autopsy data: genome anatomies and population traces in probes/blobs/l0/deepsearch/; all five worlds re-runnable from their genome JSONs. Flag for metrics v2: the champion lineage exploits growth (C1) + species count (C6) — both worm-segment-count artifacts in part; a segment-vs-organism discriminator would sharpen the next selection round.