paper: expand results summary to the full study

Rename layer1-summary.md -> results-summary.md and rewrite to cover the
whole body of work with the sexual-reproduction arc as the through-line:
Layer 1 (E1-E6, Wright-Fisher core + closed forms), Layer 1.5 (neural
existence proof: histogram bridge, RNN metric reframing, architecture-
generality, recombination in weights, real-MNIST external validity), the
learning kernel (neutral drift fails both neural models oppositely), the
sexual society (E7-E11: advantage of sex, Fisher-Muller super-parents,
outbreeding depression, directed sex, the dynamic-society vertical claim),
and the Riis positioning (concede collapse=drift; lead with the cure).
Technical + accessible halves; six design rules.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

paper/results-summary.md

@@ -0,0 +1,252 @@
+# The Lamarckian Society — Summary of results
+
+*The complete laptop-reproducible body of work: the analytical core (Layer 1), the
+architecture-general neural existence proof (Layer 1.5, including real MNIST), the learning
+kernel, and the sexual-reproduction society (E7–E11). Two summaries of the same work — one
+technical, one accessible.*
+
+**The arc in one breath.** Model collapse is **asexual, self-consuming degradation**: a lineage
+trained on its own outputs drifts to its own mode and loses the rare tail. We show this is
+*literally* a Wright–Fisher drift process (validated against closed forms), reproduce it in real
+trained neural weights and on real MNIST images, and then establish the **cure** — a **grounded
+sexual society**: reality-checking (grounding) plus **recombination across many decorrelated
+parents** (sexual reproduction, not teacher→pupil copying) plus **quality-diversity** selection.
+The payoff is not merely arrested collapse but a population whose **offspring exceed their
+parents** and whose general capability **climbs** while specialties are re-earned — and removing
+any one operator breaks it. **122 tests pass**; three of the core predictions are exact closed
+forms, so the headline curves sit on analytic targets rather than eyeballing.
+
+---
+
+## A. Technical summary
+
+### 1. The analytical core — collapse as Wright–Fisher drift (Layer 1)
+
+Knowledge is a distribution `p_t` over `K` items on the simplex; a fixed Zipf-tailed truth `p*`;
+the generational step — *sample `n` from the parent, mix in `m` fresh real samples, refit* — is
+**literally a Wright–Fisher process with immigration**, not an analogy. Each safeguard from the
+perspective paper is one operator on that step (grounding `g=m/(n+m)`; region-matched grounding;
+multi-teacher recombination; directional vs quality-diversity selection; re-minting). Because the
+process is Wright–Fisher it inherits **closed-form validation targets**, enforced as
+`<0.5%`-tolerance assertions (the "spine of trust"): neutral heterozygosity decay
+`E[H_t]=H₀(1−1/n)^t`; fixation probability = initial frequency; the *exact* mutation–drift
+equilibrium `H_eq = H*·m(2n+m−1)/(n+2nm+m²)`; the tail-persistence threshold `m·p*ᵢ ≳ 1`; and the
+recombination union coverage `U(K_T,ρ,q)=T[ρq+(1−ρ)(1−(1−q)^{K_T})]`.
+
+Findings **E1–E6**:
+- **E1 — collapse (null).** Neutral drift reproduces the geometric `H` decay to Monte-Carlo error;
+  support collapses `K→1`; forward-KL diverges; tail *items* die ≈10× faster than head items.
+  (Aggregate tail *mass* is a drift martingale — a misleading metric; tail-*item* survival is the
+  honest one.)
+- **E2 — grounding phase boundary (headline).** Stationary `H` tracks the exact `H_eq`; a critical
+  **`g* = 0.048` (CI [0.047, 0.050]) ≪ 1** — as little as one real sample against 200 inherited
+  restores ~68% of the truth's diversity; `g=0.05` reaches 96%. The sharp threshold lives in
+  discrete tail-item survival, not the smooth `H`. The **deep tail is unrescuable** by grounding at
+  feasible budgets (`m* ∼ 1/p_min`) — which is what recombination is for.
+- **E3 — region-matched grounding.** At fixed budget, matched grounding holds the exercised region's
+  tail (0.49) where uniform lets it collapse (0.07). Grounding protects only what it overlaps.
+- **E4 — multi-teacher recombination ("merge, don't average").** Union coverage matches the closed
+  form exactly (recombination *supplies* the tail). **Principal finding:** under mean-mixture
+  distillation surviving tail coverage is **flat in `K_T`** — a conservation law (averaging's `1/K_T`
+  dilution exactly cancels the union gain). The benefit is realised only under a **union-preserving
+  merge** (`max` over teachers, à la M2N2 model-merging). *Merge weights; don't average outputs.*
+- **E5 — QD vs greedy.** Greedy selection fixes (`H≈0.01`); quality-diversity (`w_i ∝ f_i·p_i^{−α}`)
+  holds `H` at a positive plateau (0.48–0.88, rising with novelty `α`).
+- **E6 — re-minting gate.** Re-minting a *collapsed* lineage makes forward-KL to the original
+  diverge (irreversible lock-in); a diversity gate (`H≥H_gate`) prevents it; healthy re-mint is
+  harmless.
+
+### 2. Collapse in real trained weights, and on real images (Layer 1.5)
+
+A re-scoped, cheaper Layer 2: realise the *same* Wright–Fisher abstractions in **real trained
+generative models** on a fully-synthetic sandbox with an exact oracle, then confirm on real MNIST.
+A model's "knowledge" is its oracle-measured distribution over `K` modes; the generational step is
+*train-a-model-on-the-previous-model's-samples + grounding*.
+
+- **The histogram bridge (HARD GATE).** A memoryless histogram model reduces Layer 1.5 *exactly* to
+  Layer 1: run through the neural runner it recovers `g* = 0.047` and sits on the exact `H_eq` curve.
+  This licenses every trained-model result to be read against the analytic core.
+- **Collapse in an RNN, and the metric reframing.** A GRU retrained each generation on its own
+  output drifts from truth (forward-KL climbs) and grounding arrests it — the sign confirmed. But
+  the **operative neural collapse metric is forward-KL, not `H` or tail-survival**: the RNN's
+  *smoothing* inductive bias keeps spurious tail modes alive, so `H` stays ~80% of `H*` and
+  tail-survival is non-monotone in `g`. On forward-KL, half the divergence gap closes by a
+  median-recovery grounding `g≈0.04` (echoing Layer-1's 0.048), but *full* recovery needs `g≈0.19` —
+  the sharp `g*≪1` is an exact-operator feature the trained net *softens*.
+- **Architecture-generality.** Collapse + grounding-rescue appear in the **histogram, GRU, and MLP**
+  alike — the operator is not an artefact of one model class.
+- **Recombination in real weights.** The E4 "merge, don't average" law reproduces: construction-level
+  union rises 0.49→0.96, oracle-guided max-merge surviving coverage rises while mean-distill stays
+  flat — the conservation law holds in trained weights (compressed/noisier, the expected smoothing
+  caveat).
+- **Real-MNIST external validity.** A convolutional VAE (the canonical collapse vehicle) retrained on
+  its own generated digits, modes = digit-class × stroke-thickness (K=30, Zipf), read by a frozen
+  CNN oracle (98.5% mode accuracy, confusion matrix recorded as the noise floor): the **dry lineage
+  collapses to a single mode** (forward-KL 0.5→18, support 30→1, tail wiped out, H→0), while **10%
+  grounding holds all 30 modes**. The eyeball montage shows varied gen-0 digits degenerating into one
+  blurry blob. Collapse and its cure are real on real images — not a synthetic artefact.
+
+### 3. The learning kernel — neutral drift is a null both neural models fail, oppositely
+
+Prompted by revisiting the neural deviations, the refit step is generalised from a pure resample to
+`p_{t+1} = T_θ(counts/n)`: a **learning kernel** with a *smoothing* knob (mutation toward a prior)
+and a *sharpening* knob (mode-competition), both identity by default (so Layer 1 is untouched).
+Result: **neutral Wright–Fisher fails both neural architectures in opposite directions.** The
+**VAE** (large `n`, small `K`): neutral drift is *inert* (no collapse), yet the real VAE collapses
+to one mode — a **sharpening** kernel reproduces it (the estimator *adds* collapse pressure). The
+**RNN**: neutral drift drives `H→0`, but the real RNN only partially collapses — a **smoothing**
+kernel reproduces the `H`-floor (the estimator *removes* collapse pressure). Model collapse in real
+learners = **neutral drift ⊕ an architecture-specific, signed estimator-bias operator**; this
+mechanistically explains the architecture-generality result and the softened neural `g*`.
+
+### 4. The sexual-reproduction society (E7–E11) — from teacher→pupil to sex with unbounded parents
+
+The single-locus, fixed-`p*` world can only express *recovery toward a ceiling*. The society's
+load-bearing claim is **vertical** — capability that *exceeds* any component — which needs
+combinatorial structure. Knowledge becomes a distribution over **genotypes** (`L` biallelic loci,
+fitness = number of correct loci), and the one new operator is **recombination**. This is where the
+frame shifts: **teacher→pupil distillation is asexual copying (caps at the ceiling); recombination
+is sexual reproduction (combinatorial, generative — offspring can exceed both parents), and unlike
+biology there is no two-parent limit.** The celebrated evolution-of-sex theory maps onto the thesis
+exactly (collapse = Muller's ratchet; merging = meiotic reassortment; "exceeding" = the
+Fisher–Muller effect):
+
+- **E7 — the advantage of sex.** A population adapting toward an optimum: the **sexual lineage adapts
+  faster** than the asexual one (clonal interference), keeping loci in linkage equilibrium (LD→0 vs
+  an LD spike). Honest scope: a *speed* advantage, not a permanent gap (the single-population ratchet
+  is subtle).
+- **E8 — the vertical claim (Fisher–Muller), the headline.** Decorrelated *parents* are specialists
+  (expert on their loci, agnostic elsewhere). **Sexual recombination assembles a genotype fitter than
+  any parent, climbing to the optimum (12/12 — a genotype no parent had)** as parent count grows and
+  correlation `ρ→0`, while the best single parent (~8.7) and the mean-mixture "model soup" (~11.6)
+  plateau below.
+- **E9 — landscape robustness ("why sex?").** On rugged (epistatic, NK) landscapes, blindly
+  recombining trained models causes **outbreeding depression** — offspring fall *below* the parents,
+  worse the more entangled the skills and the higher the recombination rate — and the **optimal
+  recombination rate shrinks as ruggedness grows**. Design rule: *merge freely when skills are
+  complementary; sparingly, and with selection, when entangled.*
+- **E10 — directed sex beats biological sex (the AI superpower).** Biology is stuck with two
+  random-mating parents and no offspring preview; an AI can **choose complementary mates, evaluate
+  many recombinant offspring, keep the fittest, over rounds, with unbounded parents**. Random
+  ("biological") sex craters with ruggedness (0.66→0.51, deep outbreeding depression); **directed sex
+  tracks or exceeds the best parent at every ruggedness** — a catastrophe turned into a win, with no
+  biological analog.
+- **E11 — the dynamic Lamarckian society (the C3 vertical claim, realised).** A finite population of
+  agents (genotypes) evolves on a rugged NK landscape that *is* reality, composing all four operators
+  — grounding, directed sex, quality-diversity, mutation. Grounding is made load-bearing by the
+  **consensus-conformity (self-consumption)** mechanism: selection acts on
+  `g·true_fitness + (1−g)·conformity`, so at `g=0` the society optimises fitting-the-crowd rather than
+  reality. A **4-arm ablation (global optimum ≈ 0.79), each breaking distinctly, only the full society
+  climbing:** `full` **0.78** (climbs to the optimum, diversity maintained longest) · `no_sex` 0.77
+  (can't recombine to escape local optima) · `no_diversity`/greedy 0.74 (collapses diversity fastest,
+  stuck at a worse local optimum) · **`no_grounding` 0.48** (self-consumption collapse to an unfit
+  consensus — trains on the crowd, regresses to a confident-but-wrong mean; the population strongly
+  *agrees* while being *wrong*). The society needs **all** of grounding + directed sex + diversity: on
+  a rugged landscape you need diversity to explore basins, sex to recombine them, and grounding to
+  select on reality.
+
+### 5. Positioning — what is prior art, what is ours
+
+An independent 2026 paper (Riis, *Drift and selection in LLM text ecosystems*) rigorously formalises
+**collapse = Wright–Fisher drift** (martingale of minority mass, rare-first extinction, de Bruijn
+fixed points, drift+selection) with n-gram agents. **We concede that framing as prior art and cite
+it — "collapse is drift" is no longer our contribution.** Crucially, Riis's "mixed environment"
+retains the lineage's own *old synthetic* tokens (no injection of external truth), so his headline is
+*pessimistic* (extinction is independent of retention). **Our defensible contributions, ranked:**
+(1) **recombination as sexual reproduction** — the "merge, don't average" law, the Fisher–Muller
+vertical claim, directed sex, and their limits (E4, E8–E10) — an operator Riis lacks; (2) the
+**learning-kernel / estimator-bias axis**, which he *names as future work*; (3) **grounding as
+immigration from a fixed reality**, giving a critical `g*≪1` his closed loop cannot have; (4)
+**architecture-generality in real weights + real images (MNIST)**; and (5) **the integrated dynamic
+society and its vertical claim** (E11), wholly ours. The repositioning: from *"collapse is drift"*
+(diagnosis) to **a population-genetic control theory for sustaining open-ended knowledge** — the
+engineered cure and its integration.
+
+### Design rules that fall out
+
+1. **Never inherit dry, and ground where it matters** — a little reality (`g*≈5%`) protects most
+   diversity, but it protects only what it overlaps, and it can't hold the deep tail.
+2. **Merge, don't average** — union-preserving model-merging realises the multi-teacher benefit;
+   averaging cancels it.
+3. **Sex, with no parent limit** — recombining decorrelated specialists yields offspring that exceed
+   any parent; more, complementary parents climb higher.
+4. **Match recombination to entanglement** — merge freely for complementary skills; sparingly for
+   entangled ones; and *always select offspring* (directed sex), which AI can afford and biology
+   cannot.
+5. **Keep diversity, and stay grounded** — greedy selection or a broken reality-signal both collapse
+   the society; only grounding + sex + diversity together climb.
+6. **Gate irreversible consolidation on diversity** — re-mint a base only while the lineage is
+   healthy.
+
+---
+
+## B. Accessible summary (for ML engineers and neuroscientists)
+
+### The question
+
+Modern AI is trained once and frozen because it cannot keep learning without *catastrophically
+forgetting*. The Lamarckian Society proposes an alternative: **generations** of bounded agents that
+learn, then reproduce — passing on what they learned. But there is a notorious trap: train a model on
+the previous model's outputs, generation after generation, and it suffers **model collapse** — the
+rare cases (the *tail*) vanish first and the model drifts to a bland mode. So the whole scheme lives
+or dies on one question: **when does generational transmission accumulate knowledge instead of
+degrading it?**
+
+### The through-line: collapse is *asexual* degradation; the cure is *sex*
+
+The key reframe of this work is that **teacher→pupil copying is asexual reproduction** — a pupil can,
+at best, recover what its teachers had. That caps out, and left alone it degrades (collapse). The cure
+is **sexual reproduction between agents**: combine *many decorrelated parents* so the offspring
+inherits a *combination* none of them had — and can be **better than any parent**. Unlike biology,
+AI sex has **no two-parent limit** and can *choose* mates and *select* offspring. That is the engine
+that lets a society climb instead of collapse.
+
+### What we found, in plain terms
+
+1. **Collapse is real, and it's math.** Casting generational training as the century-old
+   **Wright–Fisher** drift process (not a metaphor — the same equations) gives exact formulas to
+   check against. The rare stuff dies ~10× faster than the common stuff.
+2. **A little reality goes a long way — but not for everything.** Mixing in even ~5% verified real
+   data restores most of the diversity and holds it (**grounding**). But the *very rarest*
+   capabilities can't be saved by grounding alone — that's what recombination is for.
+3. **Collapse is real in actual neural nets, and on real images.** We reproduced it in trained RNNs,
+   MLPs, and a VAE, and on **real MNIST** — where a VAE trained on its own digits collapses to a
+   single blurry blob, while a little grounding keeps all the digit styles alive. (Honest nuance: real
+   nets *smooth*, so "how many modes are alive" lies to you; "how far from the truth" is the honest
+   ruler.)
+4. **To fight tail collapse with many teachers: merge, don't average.** Averaging their outputs
+   mathematically cancels the benefit; a *merge* that keeps each item's strongest source realises it.
+5. **Sex makes offspring that beat their parents.** Recombining decorrelated specialist models
+   assembles capabilities none of them had, climbing to the optimum as you add more, complementary
+   parents — while averaging ("model soup") and the best single parent plateau below. This is a
+   celebrated evolutionary result (the Fisher–Muller effect), now shown for AI model merging.
+6. **But sex can backfire — and AI has a fix biology lacks.** When skills are *entangled*, blindly
+   merging good models produces *worse* offspring ("outbreeding depression"). The fix is **directed
+   sex**: choose complementary partners, generate many merges, and keep the best — which AI can do
+   and biology can't. Directed sex turns the catastrophe into a win.
+7. **The whole society climbs only with all the pieces.** In an evolving population on a rugged
+   "reality" landscape, the *full* society (grounding + directed sex + diversity) climbs to the top
+   while keeping its specialists; remove **grounding** and it collapses into a confident, wrong
+   consensus (the exact analogue of training on the internet's AI-generated crowd); remove **sex** and
+   it gets stuck; remove **diversity** and it converges too fast to a worse answer. Each failure is
+   distinct; only the full society wins.
+
+### Why it is novel and why it matters
+
+- **It turns model collapse from a warning into a control theory.** Collapse-as-drift is now known
+  (and independently formalised elsewhere). Our contribution is the *cure* and its integration: a
+  grounded, sexually-reproducing, diversity-preserving society that not only avoids collapse but
+  **climbs, with offspring exceeding parents**.
+- **The sexual-reproduction frame is, we believe, genuinely new for AI** — model merging reframed as
+  meiotic recombination, with a rigorous account of when it helps (complementary skills), when it
+  hurts (entangled skills), and how to make it reliably win (directed sex, unbounded parents).
+- **It is validated, not vibes.** Three core predictions are exact closed forms; the neural and
+  image results confirm the *signs* in real trained weights; 122 automated tests pass; the whole
+  study is laptop-reproducible from a seed.
+- **It gives concrete design rules** for anyone building self-improving or model-merging systems:
+  ground where it matters, merge-don't-average, match recombination to skill-entanglement, select
+  offspring, keep diversity, and gate irreversible consolidation on health.
+
+*The remaining frontier is the **LLM instantiation** — realising the grounded sexual society with
+actual language models (LoRA specialists, real model merging, execution-verified grounding), which
+the blueprint frames as the eventual empirical rung.*