R95 methods: monotone completion shape and exact sharp population mean sets¶

Algebraic, not empirical; 23 October 2026

Population experiment¶

Condition on a fixed originally missing binary outcome \(Z\in\{0,1\}\). \(p=P(Z=1)\) is latent, randomized audit invitation \(I\) has known \(g=P(I=1)>0\), and actual completed return \(R\) has unknown outcomes-specific \(c_0=P(R=1\mid Z=0,I=1)\) and \(c_1=P(R=1\mid Z=1,I=1)\). Observed returned-outcome masses conditional on invitation are \(q_0=(1-p)c_0\), \(q_1=pc_1\), total invited nonresponse \(\delta=1-q_0-q_1\).

The four mutually exclusive observable events and their probabilities are: \(I=0\) with mass \(1-g\); returned \(Z=0\) with mass \(gq_0\); returned \(Z=1\) with mass \(gq_1\); invited nonreturn with mass \(g\delta\). Each latent completion model in the exact research archive reproduces these four masses exactly.

Exact shape-conditioned identification¶

The unrestricted sharp mean interval is \(p\in[q_1,1-q_0]\). Positive response probabilities imply

Therefore, for an externally justified known direction:

If the stronger positive directional ratio bound \(1\le c_1/c_0\le1+\delta\) is also externally justified, the sharp set becomes

An unknown sign of monotonicity is the union of the two directional sets, hence the unrestricted region; one must not report it as informative. Every feasible \(p\) is witnessed by the exact latent completion probabilities \(c_0=q_0/(1-p)\), \(c_1=q_1/p\), establishing sharpness.

Independent unseen-mass derivation¶

R95-A independently defines unknown unreturned \(Z=1\) mass \(t\in[0,\delta]\) so \(p=q_1+t\). \(c_1\ge c_0\) becomes \(t\le q_1\delta/(q_0+q_1)\); reverse direction requires \(\ge\); equality sets \(t\) to that value. The additional upper ratio bound gives

Intersect those inequalities with \([0,\delta]\), then return \(p=q_1+t\) at both endpoints. This independent derivation does not import the original R95 source module or a production package estimator.

Frozen design and arithmetic verification¶

Three exact invited nonreturn fractions \(1/1000,1/100,1/20\), three invitation probabilities \(1/4,1/2,1\), and six externally stipulated shape regimes yield 54 deterministic exact rational records. Each has \(q_0=q_1=(1-\delta)/2\). The six regimes are unrestricted, \(c_1\ge c_0\), \(c_0\ge c_1\), \(c_1=c_0\), \(1\le c_1/c_0\le1+\delta\), and direction unknown. The last regime is a deliberately negative restriction-control: it must never tighten the unrestricted set.

Three Python-version original identity/integrity checks precede the immutable original SHA256 evidence artifact. Independently programmed R95-A contract checks precede mandatory full SHA256 replay of the original archived 54 cells. No random seeds, simulated participants, confidence coverage evaluations or sample replicates occur. A case is a mathematical condition, not an independent unit of evidence.

In this frozen symmetric population law, the six widths are \(\delta,\delta/2,\delta/2,0,\delta/[2(2+\delta)],\delta\) respectively. The sharp set is invariant to \(g\) because the returned masses are conditional on being invited, though observable joint event probabilities depend on \(g\).

Qualification boundary¶

The identified sets are conditional on externally warranted sign/equality/ratio restrictions. The observed likelihood alone does not establish any direction. These bounds do not generalize automatically to functional dependent gaze trajectories, multitime missingness processes or simultaneous 90% functional slope confidence bands. A mathematically zero-width identified set under the strong equality restriction is not a validated real-data estimator.

Both research PRs remain draft/unmerged. Release blockers #289/#296 remain open; no public package API update, v1.2 release, tag or PyPI action is authorized.