MachineSex/results/bridge

bridge — the histogram model reproduces Layer-1 E2 exactly (the HARD GATE)

Claim tested (a plumbing check, not science): does the neural experiment harness, when run with a trivial model, reproduce the Layer-1 math exactly? If not, no later neural result could be trusted.

Setup (Layer 1.5). Same generational loop as every neural experiment — each generation, draw the parent's samples, optionally mix in real data, retrain, measure — but the "model" is a histogram: it just counts which modes appeared and resamples them (no neural net, no smoothing). This reduces the neural setup exactly back to Wright–Fisher drift. K = 200 modes, n = 200, 200 generations, 60 repeats, grounding swept g ∈ {0, …, 0.4}.

Symbols

• mode = one of the K items (Layer-1.5 word for "item"); read off each generated sequence by a zero-error oracle.
• g grounding fraction, g* its critical value, H diversity, H_eq the exact closed-form equilibrium diversity.

The four panels

1. Trajectories. Diversity H per g. g = 0 collapses; g > 0 plateaus — the E2 picture, now produced by the neural runner.
2. Bridge = Layer 1 (the gate). Dots = the neural histogram runner's stationary H vs g; the black dashed curve = the exact H_eq closed form from Layer 1. The dots sit on the curve, and the recovered critical grounding is g* = 0.047 (CI [0.045, 0.052]) — matching Layer-1's 0.048. This equality is what licenses every trained-model result to be read against the analytic core.
3. Tail survival. Fraction of the rare tail retained vs g (item-count red, truth-mass purple) — rises with grounding, deep tail lags, exactly as in E2.
4. Per-rarity band. Survival by rarity band vs g; deep bands need more grounding (m·p*ᵢ ≳ 1).

Takeaway

The harness is faithful: with a memoryless model it reproduces Layer 1 to the decimal. This is a gate, not a finding — passing it is the precondition for the RNN/MLP experiments (collapse, grounding, architectures, recombination), where the model is what changes.