# E13 — Real-weight model speciation (the alignment residual, now modulo the full symmetry group)

**Claim tested.** E12 predicts model *speciation* analytically: as two lineages diverge, recombination
(merging) fails, via Bateson–Dobzhansky–Muller incompatibilities. E13 confirms it in **real trained
weights**, separating the part of the merge barrier that is a mere **coordinate artefact** (removable
by re-coordinating hidden units) from the **residual** that no alignment can remove — the true
reproductive-isolation signal.

**E13c hardening (2026 PNAS campaign).** Recent work shows symmetry groups *richer than permutations*
remove more of the barrier between independently trained transformers (arXiv:2606.23607;
neuron-identifiability LMC). We therefore align modulo the **full function-preserving unit symmetry
group of a plain ReLU MLP** — per-unit positive rescaling (scale canonicalisation, exact) *composed
with* Git Re-Basin permutation matching (`neural/rebasin.py`; the sanity gate recovers a permuted
**and rescaled** copy to exact weight identity). Both residuals are reported: `residual` (permutation
only) and `residual_scale` (full group).

**Setup.** No-BatchNorm MLPs (784–512–512–10) on MNIST. Children forked/trained per condition;
weight-average merge; linear-mode-connectivity error barrier before/after alignment; midpoint
(merged-model) accuracy recorded alongside. 3 replicates (decomposition/cliff), 4 (emergent).
Statistically reproducible (seeded); the alignment itself is deterministic NumPy/scipy.

### Results — the decomposition (mean over divergence, reps)
| condition | naive barrier | residual (permutation) | **residual (full symmetry group)** | merged acc |
|---|---|---|---|---|
| `shared` (same task, shared fork) | 0.000 | 0.000 | **0.000** | 0.964 |
| `independent` (same task, different init) | 0.044 | 0.001 | **0.001** | 0.960 (= parents) |
| `conflict` (contradictory label maps) | 0.502 | 0.502 | **0.497** | **0.037 (inviable)** |

- **`independent`**: the barrier is a coordinate artefact — permutations already remove ~98%, and the
  full symmetry group confirms (residual 0.001). The aligned merge performs **at parent level**
  (0.960): same species, different basis.
- **`conflict`**: permutation+rescaling alignment removes essentially nothing (0.502 → 0.497), and the
  hybrid is functionally dead (accuracy 0.037). The tested alignment removes the same-task barrier but
  leaves the conflict-associated barrier largely unchanged — supporting a functional-conflict
  interpretation without proving optimal alignment (control recovery validates a special case;
  removable = lower bound, residual = upper bound).
- Formal floor (SI note S1, `paper/si-notes.md`): for label maps conflicting on mass `μ(S)`, *any*
  single merged model errs at rate ≥ `μ(S)/2` against at least one parent, under *any* alignment
  group and merge operator — hybrid disadvantage is information-theoretic, and endpoints/chord are
  invariant to all function-preserving transformations.

### Results — the isolation cliff (`speciation_real_cliff/`)
Sweeping the fraction of conflicting classes (residual = full-symmetry alignment; `t_div=800`):

| conflict fraction | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 |
|---|---|---|---|---|---|---|
| residual barrier | 0.000 | 0.122 | 0.187 | 0.278 | 0.406 | 0.506 |
| **merged (hybrid) accuracy** | 0.968 | 0.764 | 0.586 | 0.396 | 0.199 | 0.034 |

`residual_scale ≈ residual` at every point (±0.005): the cliff is functional isolation under the full
symmetry group. Read as **hybrid fitness**, the merged model's accuracy falls 0.97 → 0.03 — the
real-weight image of E12's *compatible → outbreeding depression → hybrid inviability* trajectory.

### Results — emergent divergence does NOT speciate (`speciation_real_emergent/`, E13b)
The conflict condition *imposes* contradiction; a true BDM incompatibility is *emergent*. Two
pre-registered conditions with **no conflicting training signal anywhere**: `disjoint` (child A trains
only on classes 0–4, child B on 5–9) and `augment` (same labels, inputs rolled ±3 px), swept to
`t_div = 3200` (children trained 6.4× longer than the shared base):

- **Residual barrier = 0.000 at every divergence, both conditions** (naive barrier is 0 too — the
  children never leave the shared basin).
- The `disjoint` parents decay to 0.535/0.474 on the full task (each forgets the other's classes),
  while the **merged model holds ≈ 0.955 at every divergence** — a sustained ~40-point
  **Fisher–Muller rescue** of two catastrophically-forgetting specialists, at zero barrier.
  `augment` shows the same shape (parents 0.65/0.73, merge ≈ 0.90).

**Honest conclusion (the pre-registered second reading):** in this regime — shared ancestry, same
architecture, compatible tasks, divergence up to 3200 steps — **model speciation does not emerge
spontaneously; reproductive isolation requires functional conflict.** Trained networks are *more*
merge-compatible than the biological analogy predicts, and the design rule sharpens: *merge freely
across divergently-specialised lineages of shared ancestry — the danger is conflicting conventions,
not specialisation per se.* Scope caveat: small MLPs, one fork depth; whether long-horizon
over-specialisation at LLM scale erodes mergeability (as the expert-training-duration literature
suggests, arXiv:2607.11997) is exactly the Phase-3 `llm_speciation` question.

### Positioning
Git Re-Basin / Entezari (barriers as coordinate artefacts), the richer-symmetry LMC results
(2606.23607 and neuron-identifiability, 2026), Frankle (fork instability), Pari 2024 (route don't
fuse), Zhou 2026 / 2601.22285 (predict mergeability from divergence/geometry), 2603.09463
(merge-collapse capacity theory). E13's contribution is the synthesis they lack: a controlled
decomposition where alignment — *modulo the full symmetry group* — cleanly partitions the merge
barrier into a removable coordinate artefact and a **functional reproductive-isolation residual** that
rises with task conflict, is absent under compatible specialisation, and carries an
information-theoretic floor. **Falsifiers (none triggered):** alignment failing on `independent`
(would invalidate the residual); conflict showing no residual; the richer symmetry group dissolving
the conflict residual (it removed 0.005 of 0.502); emergent conditions showing residual attributable
to alignment failure.