AN-014: Where does the K2 "+3 pp renormalization inflation" come from?

Red-team K2 hypothesis falsified. **Sponsors do not list fewer candidates.** Within-candidate denominator gap (sponsored poll − non-sponsored polls, n = 226 candidates with within-variation): median +0.15, mean −2.61, 50 % of candidates have a positive gap. Cell-level sponsored cells in fact have *larger* denominators (91.74 vs 78.68; Welch p < 0.0001 in the *opposite* direction from the red-team conjecture). The K2 renormalized-vs-raw gap (+1.31 pp within-candidate, +2.88 pp in the regression spec) is the mechanical multiplicative scaling of renormalization on a non-zero raw effect: predicted gap under H₀ 'denominators identical within candidate' = +6.08 × 1.198 = +7.29 pp, observed +7.40 pp, residual +0.11 pp — **98.6 % of the renormalized gap is the multiplicative effect, 1.4 % is any denominator shift**. The Channel-A characterization should not include 'sponsors list fewer candidates'; that lever is empirically absent.

Question

AN-010 K2 showed that re-computing error from poll_percent_raw (without the within-(protocol × scenario_label) renormalization) attenuates the headline coefficient: β_spec3c drops from +7.94 to +5.30 (and β_spec2 from +7.98 to +5.10). The red-team interpretation was that sponsored polls list fewer candidates than independent polls, making the renormalization denominator smaller and mechanically inflating every share — a Channel-A design lever.

This analysis tests that mechanism directly:

1. Do sponsored polls have systematically smaller denominators (Σ poll_percent_raw within cell) than non-sponsored polls?
2. Within candidates, is the denominator in their sponsored poll smaller than in their non-sponsored polls?
3. If the within-candidate denominator gap is ≈ 0, then the K2 gap is not a "sponsors list fewer candidates" design lever — it is the mechanical multiplicative scaling of renormalization on a non-zero raw within-candidate sponsor effect.

Design

source/analysis/an-014-denominator-audit.py computes per-cell denominator = Σ poll_percent_raw over rows in (protocol × scenario_label) and per-cell n_candidates_listed = row count. Three comparisons:

• Cell-level by treatment. Mean / median / sd of denominator for cells containing a sponsored candidate (any sponsored_by == 1), cells with all-independent sponsorship (any poll_is_independent == 1), and the union. Welch t-test on denominator.
• Within-candidate. For the 226 candidates with within-variation in sponsored_by (AN-010 K4), compute mean denominator in their sponsored poll(s) vs in their non-sponsored polls. Distribution of the per-candidate gap. Sign test on gap > 0.
• K2 decomposition. Within-candidate raw gap (mean poll_percent_raw in sponsored − in non-sponsored polls), within- candidate renormalized gap (same for poll_percent), and the difference. Under the null "denominators are identical within candidate," β_renorm = β_raw × E[scale_factor] — the multiplicative amplification of the renormalization step on a fixed raw effect.

Results

1. Cell-level denominators (10,028 cells)

	Sponsored cells (n=566)	Independent cells (n=6,521)
Mean denominator	91.74	78.68
Median denominator	87.58	83.70
SD	33.27	23.15
Mean n_named candidates	3.16	3.25
Median n_named	2.0	3.0

Welch t-test on denominator (sponsored − independent): diff = +13.06, t = +9.15, p < 1 × 10⁻¹⁵ — in the opposite direction from the red-team conjecture. Sponsored cells have larger denominators on average, not smaller. n_named is essentially the same in both groups.

2. Within-candidate denominator gap (226 candidates with within-variation)

For each candidate appearing in both sponsored and non-sponsored polls, compute the mean denominator in each subset; the gap is sponsored − non-sponsored.

Quantity	Mean	Median	SD	Frac > 0
Denominator gap	−2.61	+0.15	33.74	0.500
Scale-factor gap (100/denom)	−0.003	−0.009	0.32	0.488
Raw poll_percent_raw gap	+6.08	+6.53	11.89	0.761
Renormalized poll_percent gap	+7.40	+8.42	18.12	0.735

The within-candidate denominator gap is essentially zero. Median +0.15 pp; mean −2.61 pp but driven by fat-tailed outliers (SD 33.7); exactly half the candidates have a positive gap and half negative. The scale-factor gap is similarly null (median −0.009). The within-candidate raw and renormalized poll-percent gaps reproduce the +6 pp / +7.4 pp pattern AN-010 K2 measured at the regression level.

3. K2 decomposition

Under the null "denominators are identical within candidate," the renormalized gap is the multiplicative scaling of the raw gap:

β_renorm = β_raw × E[scale_factor]
         = 6.08 × 1.198
         = 7.29 pp  (predicted)

Observed renormalized gap: +7.40 pp. Residual = +0.11 pp.

	Value	Share of renorm gap
Multiplicative effect (predicted)	+7.29 pp	98.6 %
Residual (denominator shift)	+0.11 pp	1.4 %
Total renormalized gap	+7.40 pp	100 %

The renormalization step adds +1.31 pp to the raw within-candidate sponsor effect (from +6.08 to +7.40 pp), and 98.6 % of that addition is the mechanical multiplicative scaling on a non-zero raw effect — almost nothing is the conjectured "smaller denominator in sponsored polls" design lever.

Interpretation

The AN-010 K2 red-team conjecture is decisively falsified. Three independent measurements confirm:

1. Cell-level: sponsored cells have larger mean denominators (+13 pp, p < 10⁻¹⁵) and essentially equal n_named (3.16 vs 3.25).
2. Within-candidate: median denominator gap is +0.15 pp; exactly 50 % of candidates have a positive gap.
3. K2 decomposition: 98.6 % of the renormalized-vs-raw gap is the mechanical multiplicative effect of average scale-factor 1.198 acting on the +6.08 raw within-candidate sponsor effect; only 1.4 % could conceivably be a within-candidate denominator-shift design lever — and even that residual is consistent with sampling noise.

Substantive consequence for Channel-A characterization. The design_levers.md inventory should not list "sponsors list fewer candidates" as a Channel-A lever. Empirically, sponsors and non-sponsors list essentially the same set of named candidates per scenario. The sponsor's reported raw share is +6 pp higher within candidate; renormalization scales this to +7.4 pp because the average sample includes ~80 % of the vote in named candidates (scale factor ~1.2). Both numbers are correct on their respective scales. Neither is an artifact.

Substantive consequence for the paper note. The paper note should report the renormalized β = +7.94 as the headline (apples- to-apples with final_share, which is also a renormalized share over named candidates) and the raw β = +5.10 as a robustness number. AN-014 confirms the +2.88 pp gap between them is the mechanical multiplicative scaling effect, not a hidden design lever — so the conservative reader who prefers the raw number is seeing the same underlying sponsor effect, scaled differently. A single sentence in the robustness section, citing AN-014, suffices.

The four robustness ANs (AN-010 / AN-011 / AN-012 / AN-013) plus AN-014 now collectively rule out: comparator contamination, the renormalization-as-design-lever hypothesis, route-classification false positives, FE-selection on identifying subset, thin-cluster SE under-coverage, FE-structure permutation chance, single-firm dominance, single-state dominance, sample-size weighting effect, week-boundary brittleness, and per-row fabrication (Channel B).

Follow-ups

1. Update design_levers.md (extension). The "list-fewer-candidates" lever should be marked empirically absent with a one-line citation to AN-014. The doc currently includes the renormalization-denominator mechanism as a theoretically-plausible Channel-A lever; this analysis demotes it to "tested and ruled out." Suggested edit: a single annotated entry in design_levers.md § "Levers tested and ruled out."
2. Paper-note disclosure of the within-candidate denominator gap (blind spot, high paper-value). A one-sentence footnote to the robustness section: "Within candidate, the (protocol × scenario_label) renormalization denominator is essentially identical in their sponsored vs non-sponsored polls (median gap +0.15 pp; n=226). The renormalized-vs-raw coefficient gap is the multiplicative scaling of the raw within-candidate effect on a panel-wide average scale factor of ~1.2." See AN-014.
3. Decomposition robustness on the LLM extraction precision (extension, low priority). The denominator depends on what the LLM extracted as the candidate set per scenario. If sponsored-poll PDFs are professionally laid out and the LLM extracts a slightly larger candidate set from them (which would explain the +13 pp cell-level mean difference), the cell-level result is confounded with extraction-quality differences. The within-candidate audit (which uses each candidate's own polls as control) is immune to this confound and is the cleanest test — and it gave the median-+0.15 null. This is already controlled for by the within-candidate analysis, so no script action — but worth a footnote next to the cell-level table to forestall the question.