Correct #823 Takeaway 4: the persona-ladder decline is not denominator arithmetic (denominator-free paired read refutes the mixture-floor interpretation)

Goal

Correct a refuted bolded Takeaway in the promoted body of task #823, and decide whether the parent round's mixture_floor interpretation needs a re-read.

What is refuted

#823's promoted ## Takeaways bullet 4 currently reads, in part:

the mechanical mixture penalty implied by between-persona target variance accounts for essentially all of the decline (implied 0.171 vs observed 0.156 at layer 14; 0.192 vs 0.202 at layer 26): mixing target origins caps attainable R² mechanically, with no evidence the map itself becomes harder to learn.

The ## Results section "The implied mechanical mixture penalty accounts for essentially all of the persona-count decline" carries the same reading:

So the measured degradation under inconsistent origins is essentially what mixed targets mechanically force, not evidence the context→content association weakens.

A denominator-free paired read run 2026-08-22 contradicts the "no evidence the map itself becomes harder to learn" clause.

Refuting evidence

Producer: scripts/issue823_shared_persona_paired.py Artifact: eval_results/issue_823/inconsistent_origin_ladder/shared_persona_paired.json

Method. Generation in the ladder round was deduplicated per (context, persona) — 14,996 unique pairs == n_pairs == registered_total_pairs_full. Under the registered rule persona(i, k) = i mod k, a context with i mod K == 0 is assigned persona 0 in BOTH the k=1 arm and the k=K arm, so on those contexts the two maps predict the IDENTICAL target vector, scored in the SAME test fold (KFold(5, shuffle=True, random_state=0) depends only on n) from the same training context indices. Their held-out per-context ss_res is therefore directly comparable with no denominator. Both arms' errors are out-of-fold (scripts/issue823_ladder_fits.py:1903 writes p1_sres[..., te] at test indices only).

Result, k=16 vs k=1 on 310 shared contexts (all three read-out layers: evil 14, sycophancy 26, hallucination 17):

	L14	L26	L17
ss_res ratio pooled/reference	1.310	1.303	1.305
mean paired diff (bootstrap 95% CI)	+62.0 [+51.2, +73.1]	+2352.8 [+1891.3, +2806.4]	+84.9 [+69.3, +101.3]
contexts where pooled is worse	85.2%	85.2%	87.1%
Wilcoxon p	8.7e-36	4.1e-32	4.1e-35
common-denominator R², reference	0.514	0.513	0.527
common-denominator R², pooled	0.364	0.366	0.383
ss_tot ratio pooled/reference	1.058	1.052	1.054

The common-denominator R² gap on the shared subset is ~0.15, essentially the whole headline pooled drop (0.156 / 0.202 / 0.150). The denominator moves only 5–6% on this subset. So the decline lives in the NUMERATOR — held-out prediction error on clean single-origin targets — not in the scoring denominator.

Monotone in k, same direction at every layer: ss_res ratio 1.10–1.17 (k=2, n=2,186) → 1.19–1.27 (k=4, n=1,223) → 1.25–1.28 (k=8, n=615) → 1.30–1.31 (k=16, n=310).

Representativeness: the k=1 arm's own mean ss_res on each shared subset is 0.947–0.990 of its mean over the whole 4,629-context mask, so the i mod K == 0 slice is typical, not an easy subset.

Offset-bias control (the discriminator)

Write v_j(x) = m(x) + p_j with persona 0 as reference. If origins share one map and differ only by a constant offset, a pooled fit converges to m(x) + p_bar and its excess squared error on persona-0 contexts is ||p_bar||² ≈ E/k, where E is the between-persona mean-shift energy the parent round already computed (ladder_analysis_summary.json → mixture_floor.implied_mixture_penalty).

Measured excess is 3.4–12.8× that prediction, and the discrepancy WIDENS with k (3.4× at k=2 → 12.3× at k=16) — the opposite of what the offset mechanism implies. Instead the excess tracks the FULL energy E: ratio 0.73–0.80 at k=8 and k=16. All 12 arm×layer cells return excess-tracks-full-between-persona-energy.

Reading: the mixture-floor MAGNITUDE the parent round computed is the right scale, but it is not acting as a denominator cap. The pooled fit appears to absorb persona variation into its coefficients rather than its intercept, so held-out predictions degrade on clean single-origin targets at roughly full E.

Named confound (do not drop from the correction)

The underlying ladder fits sit at n_over_d_ratio 1.033 (n_train_per_fold [3703, 3704] vs d 3584) — the interpolation threshold — with g2_parity_gate.pass: false and sensitivity_dof_capped: null. Near interpolation a ridge can nearly fit its training targets including their persona offsets, which then generalize as noise; a well-conditioned fit (n >> d) would average those offsets into the intercept and pay only ≈ E/k. So the degradation measured here may be specific to the near-interpolation regime rather than a structural property of the context→answer map.

That is a falsifiable prediction worth recording: at larger n/d, measured_excess / E should fall toward 1/k. Testing it needs either dimensionality reduction or more contexts, so it is NOT part of this correction task.

Deliverables

1. Amend #823's Takeaway 4 and the "implied mechanical mixture penalty" Results section so the "no evidence the map itself becomes harder to learn" clause is replaced by the denominator-free finding plus the near-interpolation confound. #823 is at awaiting_promotion; runs.classification must NOT be touched.
2. Decide whether the paper paragraph "Consistent origin is what matters, not on-policy text" (~/overleaf-6a59c927/sections/results/c1_linear.tex:93) should cite this read. Its % TODO-VERIFY comment still says the inconsistent-origin arm is unrun/in-flight; the arm landed 2026-08-20 and this read is its denominator-free follow-up.
3. Refresh docs/paper_context_answer_map/claims.md row K4 and the register line, both of which still say the arm is IN-FLIGHT.

Provenance

Surfaced by a user-chat inline free analysis on #823 (0 GPU-h, existing artifacts), 2026-08-22. Dispatch and completion notes are epm:progress markers on #823.