X8 — Multi-issuer confidence aggregation: making §12.5 decidable

§12.5 records, non-normatively, that when a show combines attestations from multiple issuers, "a
verifier's confidence against mis-issuance increases multiplicatively with the number of honest,
independent issuers contributing to a show; a corrupted or negligent issuer degrades only its own
contribution" — and leaves whether and how a profile requires this as an open TF decision. This doc
develops the mechanism into a decidable profile choice: the bound stated as a theorem with its
assumptions explicit, an independence register that makes the load-bearing assumption checkable, a ZK
realization that composes with issuer concealment, and an honest account of what k costs to disclose.

Register: X-series (second wave) · leverage 🟠 · Ladder: reference built 2026-07-18 —
Lab — multi-issuer 10/10: independence collapse as connected components
(always weakens the bound), statement carries effectiveK never raw k, tier enforcement at build AND
verify, §12.5 degradation clause executable. k=2 NOW IN-CIRCUIT (Lab — circom-gadget
dual-issuer 7/7: 10,717 constraints = exactly 2× + 1 distinctness constraint; a duplicate issuer
cannot produce a witness; showContext = §15.2 transcript digest; ISSUER_TAG pinned). Threshold
hiding + registry-side semantics stay out-of-circuit. See both NOTES.md
Anchor (cred-spec): trust registry (issuer/role/revocation authority) · multi-credential shows.
Anchor (decision doc): §12.5 aggregation mechanism (the seed) · §12.1 PR-ISS stays binary ·
§12.3 concealment modes · §12.4 snapshots · §2.4 three-parameter claims · §29 "multi-issuer assurance"
open item · §8 assurance allocation · §10.4/§18.4 rarity and combination · §18.2 accreditation reference.

The model made precise (theorem-shaped, assumptions explicit)

Claim. Model each issuer i's attribute-validation process as bounded by a one-sided error parameter
ε_i: the probability that issuer i attests the predicate when it is in fact false; a false attestation is
the only error mode counted (§12.5's "one-sided"). Then for a show carrying attestations from issuers
{1..k}, under assumptions A1–A4 below, the residual probability that the predicate is false yet every
issuer attested it is bounded by ∏ᵢ ε_i. If issuer j is corrupted or negligent, its factor is replaced
by 1 and the bound degrades to ∏_{i≠j} ε_i — "a corrupted or negligent issuer degrades only its own
contribution and does not weaken the assurance carried by the other issuers involved" (§12.5, verbatim).

A1 — one-sided error. ε_i bounds mis-issuance only; availability/rejection errors are out of scope.
A2 — independence. The issuers' validation errors are statistically independent events. This is a
claim about processes (pipelines, vendors, operators), never about set membership — "issuer
independence is asserted, not derived from set membership alone" (§12.5); PR-ISS itself "does not
establish issuer independence" (§12.2).
A3 — distinctness. The k attestations come from k pairwise-distinct issuers in the accepted set.
A4 — same proposition. All k attest the same predicate over the same subject-binding commitment;
aggregating different predicates is composition (X3), not aggregation.

PR-ISS as stated in §12.1 remains a binary set-membership statement (§12.5's first sentence). The bound
is an additional profile-level statement layered on k instances of it. Negative meaning: the bound does
not establish determination correctness by any issuer (§2.2), equal assurance across the set (§12.2), or
anything about issuers outside the show.

The independence register — making A2 checkable

A2 is load-bearing and undeclarable-by-cryptography, so it must be a governed record, not a vibe. The
registry (§27.3) maintains, per issuer, declared dependency classes: shared biometric vendor or model
version; shared identity-proofing pipeline or operator; common corporate parent; common jurisdiction or
legal-compulsion domain; shared enrolment infrastructure. Two issuers sharing a class are correlated
for that failure mode and collapse to one effective factor: the effective k of a show is the number of
distinct independence classes represented, and a class's ε is the class's shared-dependency bound, not
the product of its members. The verifier's independence assumption then becomes checkable against the
register — and is itself stated in §2.4 form: against whom (which dependency classes and collusion
sets the independence claim survives — it does not survive a compromised shared vendor), for how long
(the register snapshot's effective time and the shortest member assurance horizon), alongside what
(the named registry snapshot, the declared class list, and the show's other disclosed fields).

The ZK realization — k distinct concealed issuers

Statement, informally: k pairwise-distinct members of the accepted issuer set at named epoch root R each
signed an attestation over the same subject commitment attesting the predicate — without revealing which
members. Build: k instances of the shared Poseidon-Merkle membership gadget (O2/O4) against the same
named root (§12.4; inconsistent snapshots are a §26.1 rejection), plus distinctness via per-issuer,
per-show nullifiers: N_i = H_domain(issuer_id_i, show-transcript digest). k distinct N_i prove k
distinct issuers while revealing none — the scoped-reuse discipline (§5.12) reused at the issuer level:
the nullifier establishes distinctness-within-this-show, not issuer identity, and the show-transcript
domain separation (§15.2) prevents cross-show issuer correlation. This slots into §12.3 as its fourth
listed mode ("multi-issuer or threshold assurance") and composes with the others: concealed-within-set
aggregation, or aggregation with coarse issuer-class disclosure per class-specific ε floors.

The disclosure cost — k is the disclosure

k is public; the statement is k-of-n. As in X3's degree-leakage seed, k IS the disclosure: a rare k,
or a rare accepted-set/root choice, fingerprints the verifier's risk posture and narrows the holder
population (§10.4, §18.4) — k=7 against a boutique registry is nearly a name. Remedy, as in X3's bundle
profiles: a small governed tier vocabulary, k ∈ {1, 2, 3}, registered per profile; requests outside
the tiers are a §26.1 rejection. Each additional tier spends anonymity set to buy assurance — the same
trade-curve honesty as §2.3, and the joint record for an aggregated show is X3's joint disclosure record
with k and the root as enumerated members.

The accreditation economics — ε as a certification clock

§12.5: this "gives an accreditation framework a purpose beyond maintaining a list." Concretely: extend
§18.2's issuer-accreditation-reference field class with a per-issuer documented confidence parameter
ε_i — set by the accreditation authority from audit evidence, never self-asserted; re-based on a governed
cadence. In X6's terms ε_i is a certification-clock value with an assurance horizon: an audit
establishes it at a time, and it erodes rather than holds — a profile MUST state the horizon past which a
stale ε_i no longer supports the multiplied bound, and registry snapshots (§12.4) carry ε values and
dependency classes with effective times so the whole aggregation claim is evaluable at a named time.

Where aggregation is worth it — honestly

• PR-PER: the natural home. §29 pairs "multi-issuer assurance" with personhood; independent issuers
  running independent enrolment pipelines are genuinely independent evidence about the same durable
  proposition, so A2 is achievable and the multiplication is real.
• PR-LIV: mostly not. Liveness is one determination by one sensor at one moment. A second issuer
  re-attesting the same capture shares the determination layer (§8) — A2 fails there, effective k = 1,
  and the extra factor is decoration. Aggregation helps only for genuinely separate capture events, which
  is a freshness design, not an assurance multiplier.
• Rule of thumb: aggregate where issuers make independent determinations; never count issuers that
  countersign one.

Build plan

• M1 — boundary record: assurance + disclosure pair for the aggregated statement (Appendix B form),
  including the A1–A4 register and the §2.4-form independence claim.
• M2 — independence-register data model: dependency classes, ε fields, effective times — drafted as a
  registry-profile note for the Credentials TF (§27.3), riding O4's M1 epoch-root model.
• M3 — reference model: k-membership + distinctness nullifiers in the lab; negatives: duplicate
  issuer rejected, mixed roots rejected, k outside tier rejected, stale ε rejected (§26.1 style).
• M4 — decision memo: the §12.5 open issue answered as options (not-required / MAY per profile /
  SHOULD for EPP PR-PER) with the trade curve priced; TF picks.

Upstream surface

Decision input for §12.5's open issue ("whether and how... is a task-force decision") plus the
registry-field proposal (ε_i + dependency classes) to the Credentials TF along the §27.3 boundary — they
own institutional legitimacy of the register; we own the statement, the bound, and its fixtures.

Open questions

• Who audits ε_i, and can two accreditation frameworks' ε values be compared at all, or only within one?
• Threshold variants (k-of-m attestations held, k shown) — does hiding m leak less or just differently?
• Does a corrupted issuer discovered later retroactively re-price past aggregated shows (O4's
  retroactivity question, §10.6/§23 row 2), and who propagates the re-priced bound?
• Jurisdiction as a dependency class: is legal compulsion a correlated failure or a separate adversary
  in the against-whom parameter? (Likely both; needs the §19 observer treatment.)