---
slug: nested_epistemic_depth
name: Nested Epistemic Depth
one_liner: "A fact's epistemic standing in a group is graded by how many levels of nested 'knows-that' iterate over it — from private knowledge, through finite mutual knowledge, to unbounded common knowledge."
type: prime
schema_version: 2
status: density_v2_draft
categories: [epistemic, social, cognitive]
origin_domain: epistemic_logic
alternate_origin_domains: [game_theory, distributed_systems, social_coordination]
aliases: [knowledge_iteration_hierarchy, orders_of_knowledge, higher_order_knowledge, depth_of_mutual_knowledge, epistemic_tower]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-05 generalization) — the proposed GENUS of `common_knowledge` (lift: strip the infinite-tower constraint and grade epistemic standing by the NUMBER of nested knows-that levels, so private / mutual-depth-k / common knowledge become points on one axis). Discovered in the op-05 remaining-strata sweep 2026-07-03. Tier A candidate (established genus: mutual knowledge, k-level reasoning, and distributed knowledge are recognized terms in epistemic logic / game theory). STATUS: HOLD pending owner keep/cut."
- "Created 2026-07-03; STUB ONLY (frontmatter + Core Idea + Notes). SI/SF/similarity are PROPOSER ESTIMATES pending triangulation. Body sections (Structural Signature onward) NOT written."
- "Phase-C action: re-similarity-check vs `common_knowledge` (the input) and any existing knowledge / belief / theory-of-mind / recursion prime; confirm it is a genuine genus (covers the input — common knowledge as the depth=∞ limit — AND the mutual-knowledge / k-level-reasoning / distributed-knowledge siblings) and is not a restatement of common_knowledge; draw a subsumption edge common_knowledge→candidate + sibling edges at incorporation; sync one_liner to a v1 draft per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 4
  domain_breadth: 4
  structural_abstraction: 5
  transfer_evidence: 4
  reasoning: 'The bare structure — take a proposition and iterate the operator "___ knows that" over a set of agents to a specified depth, and grade the propositions epistemic standing by that depth (0 = private, 1 = mutual, k = depth-k, ∞ = common) — is a clean, purely relational recursion over a knowledge operator, with maximal structural abstraction. It presupposes only agents, a proposition, and a well-defined knows-that operator, so it applies wherever "agents represent each others representations": human coordination, game-theoretic k-level reasoning, distributed-systems agreement, and theory-of-mind. Transfer is strong and the abstraction is essentially content-free (it is the recursion, not any particular fact, that matters). Held just under a perfect composite only because "knowledge" itself is an epistemic primitive rather than a physical/graph relation, so the axis inherits mild epistemic content; structural_abstraction is a clean 5.'
similarity_to_nearest_existing_prime:
  slug: common_knowledge
  similarity: 0.62
structural_framed:
  label: structural
  aggregate: 0.24
  criteria: {vocab_travels: 0.9, evaluative_weight: 0.05, institutional_origin: 0.2, human_practice_bound: 0.3, import_vs_recognize: 0.2}
  rationale: 'The candidate is a bare recursive-relation structure: iterate a knows-that operator over agents and grade standing by depth. Its vocabulary (depth, order, iteration, mutual, common) travels across epistemic logic, game theory, and distributed systems; it carries essentially no evaluative content and is recognized as a formal structure rather than imported as a doctrine. It reads clearly structural.'
phase_b_passes_completed: []
---
# Nested Epistemic Depth

## Core Idea

Nested epistemic depth is the structure in which *the epistemic standing of a fact within a group is graded by how many levels of nested "knows-that" iterate over it* — private knowledge at depth zero (I know p), mutual knowledge at some finite depth (we each know p, and know that we each know it, up to k rungs), and common knowledge in the unbounded limit (the tower of "everyone knows that everyone knows…" runs without end). The genus is reached from its input, `common_knowledge`, by stripping the single constraint that the tower be *infinite*: common knowledge is one distinguished point — the depth=∞ limit — on a whole axis of iterated-knowledge depths, and the general prime is that axis itself.

The load-bearing anatomy is threefold. First, a *knows-that operator* over a set of agents, which can be applied to a proposition and, crucially, to *another agent's knowing* of that proposition — the operator is iterable. Second, a *depth index* counting how many times the operator has been nested: depth 0 is a bare fact held privately, depth 1 adds "and I know that you know," and each further rung adds another layer of agents representing agents' representations. Third, a *monotone strengthening*: higher depth is strictly stronger — depth-k knowledge implies all shallower depths but not conversely — so the axis is a genuine order from weak (private) to strong (common). The characteristic payoff is *coordination capacity*: which joint actions a group can reliably undertake depends on how high the tower goes, because acting on the expectation that others will act requires knowing that they know that you know, and so on; the depth is the resource that coordination consumes. The characteristic cost/failure is the *coordination cliff*: because the tower can fail silently at any rung, agents routinely believe they share deep common knowledge when they in fact share only shallow mutual knowledge, and the gap is invisible until a coordinated act misfires — the depth you *have* and the depth you *assume* can diverge without any local signal.

Read structurally, the same graded iteration recurs across unlike domains, and the input is only its limit case. *Mutual knowledge* is the finite depth-2 rung — "I know you know" — which suffices for some coordination but provably not for others (the electronic-messaging / coordinated-attack results turn precisely on the gap between finite mutual and infinite common knowledge). *k-level reasoning* in game theory grades players by exactly this depth: a level-k player best-responds to level-(k−1) players, and strategic outcomes hinge on the population's depth distribution. *Distributed knowledge* sits on a companion axis — what a group would know if its members pooled their information — showing the "who holds it" dimension the depth axis abstracts over. These are structurally unlike in setting (conversation, games, information systems) yet all are readings of the same underlying quantity: how many levels of nested knowing stand behind a fact.

## Notes

**Provenance.** Produced by operator 05 (generalization / lift) as the proposed genus of the input prime `common_knowledge`, by stripping the infinite-tower constraint to reach the general depth-graded axis of iterated knows-that. Surfaced in the op-05 remaining-strata sweep on 2026-07-03. It claimed the siblings *mutual knowledge (finite depth-2)*, *k-level reasoning in game theory*, and *distributed knowledge across a group's combined information*. Logged as a Tier-A (A-rest) candidate — an established, verified-absent genus — in the op-05 remaining-strata manifest.

**For owner evaluation (keep/cut).** The case FOR is strong: unlike most op-05-rest candidates, this is **not a coinage** — mutual knowledge, k-level reasoning, and distributed knowledge are all recognized terms in epistemic logic and game theory, so the genus is established, not invented, and its absence (with `common_knowledge` already present as the limit case) is a real anatomy hole. The axis is maximally structural (structural_abstraction ~5): it is pure recursion over a knowledge operator, graded by depth, with content-free transfer across coordination, game theory, and distributed systems. The RISKS Kurt should weigh: (1) **Restatement vs. the input** — estimated similarity to `common_knowledge` is moderate (~0.62); the candidate must earn its keep by covering the *finite-depth* rungs (mutual, k-level) that `common_knowledge` alone does not — if in practice everyone will only ever want the ∞ case, the axis adds little and `common_knowledge` should absorb a "depth axis" note instead. (2) **Sibling heterogeneity** — mutual knowledge and k-level reasoning genuinely live on the *depth* axis, but *distributed knowledge* is arguably a different axis (who holds the information, not how deep the iteration runs); if so, it is a cousin, not a species, and should not be pulled under this genus. **The sharp keep/cut question: is the depth-graded axis worth a distinct prime above `common_knowledge`, and if so, do all three named siblings actually sit on the depth axis, or is distributed_knowledge a separate structure?** (3) **Overlap with a recursion / theory-of-mind / higher-order-belief prime** — if the corpus holds one, draw the boundary on the *epistemic depth grading* specifically.

**At incorporation.** Re-run similarity against `common_knowledge` (the input) and against any knowledge, belief, theory-of-mind, higher-order-belief, or recursion prime. If kept, draw a subsumption edge `common_knowledge` → `nested_epistemic_depth` (common knowledge as the depth=∞ species) and sibling edges to `mutual_knowledge` and k-level reasoning; treat `distributed_knowledge` as a possible separate-axis cousin rather than a species until confirmed. Flag a missing-edge to any recursion / higher-order-representation genus if present. Write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
