docs(sor): PROPOSED stage-05 RQ2 per-circuit posterior clarification (blind)
Overseer ruled DIRECTION=A: fill the specification gap the frozen prereg left on RQ2's per-circuit adversary sender-posterior, rather than silently redefine the DV (B) or relabel a confirmatory test (C). Proposes: adversary observes a circuit's exit-bridge/house and forms a uniform posterior over the observation-consistent anonymity set A_i; per-circuit H_i = Miller-Madow(log2 m_i), with S=2^H [Serjantov2002] and d=H/log2 N [Diaz2002] as the cited normalizations. Makes RQ2-P1 (two- sided ΔH) and RQ2-P3 (Spearman) computable OFFLINE from the immutable raw records (per_circuit_seeds + deterministic assemble()). Status PROPOSED — operator ratification pending. Pre-specified BLIND (2026-07-20, battery running, zero RQ2 statistics computed) so this is honest pre-registration completion, not HARKing. Frozen prereg untouched; sign not presumed (honest-shrink reported with equal prominence). Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>
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# Stage-05 analysis clarification — RQ2 per-circuit adversary sender-posterior
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**Status: PROPOSED — operator (Andre) ratification PENDING. NOT ratified; does
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NOT edit the frozen prereg** (`sor-consent-prereg.md`, SHA
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`f22331a72e0d0ccf38b787e63acabbe9d666456ec76076787a6d545c3193425b`).
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**Pre-specified BLIND to RQ2 outcomes — drafted 2026-07-20**, while the confirmatory
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battery is still running and **no RQ2 confirmatory statistic (RQ2-P1/P2/P3) has been
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computed, inspected, or reported.** This is honest *pre-registration completion* of a
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specification gap the frozen prereg left open — not HARKing (no RQ2 result has been
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seen; the rule is fixed before any RQ2 number exists).
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> Distinction from the Holm clarification: that note **restated** a decision the
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> prereg already fixes (family size 7). This note **fills a genuine gap** — the
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> prereg pins the RQ2 DV as a *per-circuit adversary posterior* but gives no
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> explicit construction formula. It therefore requires **operator ratification**
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> before any RQ2 confirmatory number may be computed.
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## The gap
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The frozen prereg fixes the RQ2 dependent variable as a **per-circuit** quantity:
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- **§3 (Variables), RQ2 DV:** *"Shannon entropy of the **adversary's posterior over
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candidate senders per circuit**; effective set size S = 2^H per [Serjantov2002],
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normalized d = H/log2(N) per [Diaz2002]. Estimator: plug-in (MLE) entropy with
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**Miller–Madow** bias correction; 95% CI by **bootstrap over circuits**."*
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- **§4 (Sampling), unit of analysis:** *"RQ2 — the circuit (entropy of the
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per-circuit sender posterior)."*
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- **§6 (Analysis), RQ2-P3:** Spearman ρ between top-k=3 bridge concentration and
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**per-circuit H** — which *requires a per-circuit H value to exist.*
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What the prereg does **not** give is the **construction rule**: how, from one
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observed confirmatory circuit, the adversary's posterior over candidate senders is
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formed. The cited papers supply the **metric** (entropy of the posterior) and the
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**normalizations** (S = 2^H, d = H/log2 N) — not the posterior itself. The built
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instrument (executor `ce68d41`) currently records only a per-*run* Shannon entropy
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of the realized entry-node **frequency**, which is a different quantity and yields
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no per-circuit H (so RQ2-P3 is uncomputable as-collected). This note proposes the
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missing construction, grounded in the cited literature only.
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## Proposed construction (adversary sender-posterior, per circuit)
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**Adversary model (as in [Serjantov2002], a global passive adversary over the lab
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grid).** For each confirmatory circuit *i*, the adversary observes the circuit's
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**exit segment and the federation-path structure it implies** — i.e. the
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exit-bridge / exit-house through which the circuit leaves the federation. It does
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**not** observe the true entry node (that is the anonymity the metric quantifies).
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**Candidate senders = the observation-consistent anonymity set** \(A_i\): the set
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of consenting entry-node candidates whose paths are **consistent with the observed
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exit-bridge/house** under the cell's topology. Concretely:
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- **1-house-N arm:** no federation observation narrows the pool — every one of the
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\(N\) consenting nodes in the single house is consistent ⇒ \(A_i\) = all \(N\).
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- **bridge-federated / directory-federated arms:** the observed exit bridge/house
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partitions candidates — \(A_i\) is the subset of consenting senders whose
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federation path can reach the observed exit under the split-knowledge topology
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(R6 `select_federated_path` / directory constraints). Bridge concentration
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(few willing bridges carrying many circuits) narrows \(A_i\); a well-spread
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federation widens it. **The sign of this effect is not presumed** (§6 two-sided).
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**Mass rule (the §-consistent default): uniform over \(A_i\).** With no side
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information distinguishing candidates in \(A_i\), the maximum-entropy posterior is
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uniform — the standard [Serjantov2002]/[Diaz2002] anonymity-set assumption. The
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per-circuit posterior count vector is therefore \([1,1,\dots,1]\) of length
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\(m_i = |A_i|\).
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**Per-circuit entropy (Miller–Madow, bits):**
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\[
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H_i = \widehat{H}_{\mathrm{MM}}\big(\underbrace{[1,\dots,1]}_{m_i}\big)
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= \log_2 m_i + \frac{m_i-1}{2\,m_i\ln 2},
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\]
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using the existing `stats.miller_madow_entropy_bits`. Effective set size
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\(S_i = 2^{H_i}\) [Serjantov2002]; normalized degree \(d_i = H_i / \log_2 N\)
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[Diaz2002], \(N\) = matched total consenting nodes. Variation in \(H_i\) **across
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circuits** (different circuits route through different bridges/houses ⇒ different
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\(m_i\)) is the distribution that the frozen "**bootstrap over circuits**" resamples.
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This makes the two frozen RQ2 confirmatory tests computable **offline** from the
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immutable raw records (no re-run, no new traffic):
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- **RQ2-P1 (ΔH, two-sided):** \(\Delta H = \mathrm{mean}_i H_i(\text{federated}) -
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\mathrm{mean}_i H_i(\text{single-house, matched } N)\), BCa 95% CI by bootstrap
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over circuits (`confirm.rq2_p1_delta_h`). **grow** if CI > 0, **honest-shrink**
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(reported with equal prominence) if CI < 0, **inconclusive** if it spans 0.
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- **RQ2-P3 (mechanism):** Spearman ρ between per-circuit top-3 bridge concentration
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\(c_i\) and per-circuit \(H_i\), BCa 95% CI (`confirm.rq2_p3_funnel`); negative ρ
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quantifies funnelling.
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## Offline recomputability (why the running battery is not wasted)
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Every input above is recoverable from the **immutable** raw records the executor
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already writes — the rule is applied *after* the fact, not during collection:
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- `battery-results.json` → `runs[].per_circuit_seeds` gives each circuit's seed;
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- `assemble(cell, cseed)` is **deterministic** ⇒ reproduces each circuit's
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`CircuitSpec` (path hops, house labels, bridge used, topology, consenting pool);
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- from that spec the **observation-consistent set \(A_i\)** and top-3 concentration
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\(c_i\) are derived by the (proposed) offline loader — no live circuit needed.
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So under this rule RQ2-P1/P3 are computed once, post-battery, from the sealed data.
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## Honesty / anti-HARKing safeguards
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1. **Blind:** fixed 2026-07-20 with the battery still running and **zero** RQ2
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confirmatory numbers computed or seen. The rule cannot be reverse-engineered
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from a result that does not yet exist.
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2. **Frozen prereg untouched:** the rule lives only here, PROPOSED, until sign-off.
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3. **Sign not presumed:** ΔH is two-sided; a shrink (federation *reduces* the
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anonymity set) is a publishable honest null, reported as prominently as growth.
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4. **Detectors unchanged:** reuses the §5-calibrated `miller_madow_entropy_bits`
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(gate item 4: H = log2(N) for N equiprobable senders); no per-cell tuning.
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5. **Cited-literature grounding only:** entropy-as-anonymity [Serjantov2002],
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normalized degree [Diaz2002]; uniform-over-anonymity-set is their standard model.
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## Operator decision required
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Ratify (or amend) the **observation-consistent-set + uniform-mass** construction
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above. On ratification this note flips to RATIFIED and the offline RQ2 loader
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(mirroring `confirm_load.py`'s RQ1 path) computes RQ2-P1/P3 once from the sealed
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records. **Until then RQ2 stays BLIND — no RQ2 confirmatory statistic is computed,
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inspected, or reported.**
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### Citations
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- **[Serjantov2002]** Serjantov, A., & Danezis, G. (2002). Towards an Information
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Theoretic Metric for Anonymity. *PET 2002*, LNCS 2482, 41–53.
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https://doi.org/10.1007/3-540-36467-6_4
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- **[Diaz2002]** Díaz, C., Seys, S., Claessens, J., & Preneel, B. (2002). Towards
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Measuring Anonymity. *PET 2002*, LNCS 2482, 54–68.
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https://doi.org/10.1007/3-540-36467-6_5
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