neural: grounding refinement + all five Layer-1.5 figures

Grounding refinement (18 reps): forward-KL is the operative neural
collapse metric, not H or tail-survival. The RNN's smoothing keeps
spurious tail modes alive, so tail_truth_mass_alive is flat/non-monotone
in g and H stays ~0.8 of H*; only forward-KL falls monotonically (dry
2.08 -> g=0.2: 0.75, paired t up to 3.3). The sharp g* << 1 is an
exact-operator feature carried by the histogram bridge (0.047); the
trained RNN confirms the SIGN and softens the sharpness (half the KL gap
closes by g~0.04, but full recovery needs g~0.19). Blueprint 3.5's
directional claim holds; the pre-registered 95%-of-H*/tail falsifier is
not met because those are the wrong metrics for a smoothing model.

Robustness: a fully-degenerate RNN can emit only invalid codewords, so
measure_distribution now returns a terminal-collapse sentinel (fixation
on the dominant mode) instead of crashing a long sweep. Edge test added
(94 tests green).

Figures: plot_{bridge,collapse,grounding,architectures,recombination}.py,
each a pure function of its committed bundle, wired into `make figures`
(glob plot_*.py minus plot_E[1-6]/_*). bridge sits on the exact H_eq
curve (g*=0.047); recombination shows max-merge rising while mean-distill
stays flat; architectures shows the collapse/rescue signs across
histogram/GRU/MLP.

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

CLAUDE.md

@@ -14,10 +14,11 @@ This file provides guidance to Claude Code (claude.ai/code) when working with co
   fully-synthetic sandbox with an exact oracle, plus real MNIST as a later secondary tier. See
   `tasks/todo.md` for status and `~/.claude/plans/we-are-going-to-cheerful-fog.md` for the plan.
   **Done:** scaffold, the histogram bridge gate (reproduces Layer 1 exactly), `bridge` (neural
-  g*=0.047 ≈ Layer 1), `collapse` (in RNN weights), `grounding` (neural phase boundary),
-  `architectures` (architecture-generality), `recombination` (the E4 "merge, don't average" finding
-  reproduced in real weights). **Remaining:** `grounding` refinement, `region_matched`, `remint`,
-  figures, the MNIST tier. The LLM/LoRA rung and the C3 vertical claim are deferred. Experiments are
+  g*=0.047 ≈ Layer 1), `collapse` (in RNN weights), `grounding` (refined; sign confirmed, threshold
+  softened by neural smoothing — see finding below), `architectures` (architecture-generality),
+  `recombination` (the E4 "merge, don't average" finding reproduced in real weights), and all five
+  neural figures (`figures/plot_{bridge,collapse,grounding,architectures,recombination}.py`, wired
+  into `make figures`). **Remaining:** `region_matched`, `remint`, the MNIST tier, VAE fidelity. The LLM/LoRA rung and the C3 vertical claim are deferred. Experiments are
   named descriptively (`configs/neural/<name>.yaml`), not by code.
 
 The two design documents are the source of truth for intent:
@@ -69,6 +70,8 @@ E4's whole purpose is to isolate the effect of teacher **decorrelation ρ**, so
 
 **Finding (2026-07-04, E4) — the recombination operator matters, and mean-mixture distillation does not realise the benefit.** Under the blueprint's mean-mixture pupil (`p̄ = mean(teachers)`), surviving tail coverage is **flat in K_T** — a conservation law: averaging preserves expected pupil tail mass at `q·(tail mass of p*)` regardless of K_T, and in the rare-tail (linear-survival) regime the 1/K_T dilution exactly cancels the union gain. The recombination benefit is realised only under a **union-preserving merge** (`max` over teachers, à la M2N2), where surviving rises with K_T and decorrelation. So E4 reports surviving under **both** operators (`surviving_mean`, `surviving_max`): union = supply (validated vs closed form), max-merge = realised benefit, mean-distill = the null that motivates why merging/grounding is needed. GG decision: report both. This sharpens rather than refutes the thesis, but the paper's recombination claim rests on the *merge* operator, not naive mean distillation — worth carrying into Layer 2 (C4) and the write-up.
 
+**Finding (2026-07-05, neural `grounding`) — grounding's SIGN transfers to trained RNN weights, but the sharp `g*` does not; and tail-survival is the *wrong* neural collapse metric.** Re-ran the phase-boundary sweep at 18 replicates. Two results: (1) **forward-KL is the operative neural collapse metric, not H or tail-survival.** The RNN's smoothing inductive bias keeps *spurious* tail modes alive (it generalises to unseen codewords), so `tail_truth_mass_alive` is flat/**non-monotone** in g (dry 0.54 > most grounded points) and H stays ~0.77–0.85 of H\* throughout — neither shows a threshold. Stationary **forward-KL** falls monotonically (dry 2.08 → g=0.2: 0.75), significant at g≥0.05 (paired t up to 3.3; 89% of lineages improve at g=0.2). This *refines* the earlier "forward-KL AND tail survival" note: for a smoothing model, support-counting decouples from closeness-to-truth. (2) **The sharp `g*≪1` is an exact-operator feature, softened by neural inductive bias.** Half the achievable KL reduction closes by a *median-recovery* grounding g≈0.04 (bootstrap CI [0.004, 0.116]) — a striking echo of Layer-1's 0.048 — but full (95%) recovery needs g≈0.19, far more than the histogram bridge, because smoothing both caps dry collapse (KL~2, not ∞) and slows full recovery. So the quantitative `g*≪1` claim rests on the **histogram bridge** (g\*=0.047, exact reduction to Layer 1), which the trained RNN confirms in *sign* and softens in sharpness. Honest note: the pre-registered 95%-of-H\*/tail-survival falsifier is not met, but that is because those are the wrong metrics for a smoothing model, not because grounding fails — the blueprint §3.5 directional claim (grounding arrests collapse) holds robustly. Robustness fix landed alongside: a fully-degenerate RNN can emit only invalid codewords, so `measure_distribution` returns a terminal-collapse sentinel (fixation on the dominant mode) instead of crashing a long sweep.
+
 ## Build order (blueprint §7) — respect the gate
 
 1. Scaffold: repo layout (§5), container, pytest skeleton, config system, seeding utils. `make test` green.