.. _e7he-logics:

****************************************************
e7He --- Formal Disciplines
****************************************************

e7He draws on multiple formal disciplines. The primary logic system
is inherited from PET/JUB (S5 modal logic + Classical Extensional
Mereology + first-order predicate calculus). e7He adds coalgebraic
structure and dynamical systems theory for the perpetual-cycle
formalization.

All 12 disciplines in the FORGE
:doc:`WisdomBase </matheology/compiler/forge/wb/index>` were loaded
during the TEMPER rounds that tested e7He's 24 statements under the
:doc:`Iron Maiden </matheology/compiler/forge/iron-maiden-tests>`
protocol. Each of the 10 Iron Maiden tests draws on specific
disciplines.


Inherited Logics (from PET/JUB)
=================================

- **S5 Modal Logic** --- necessity and possibility operators with
  equivalence-class accessibility. Used in Iron Maiden Test IV
  (Modal Soundness). e7He axioms that use modal operators: m0.ax1
  (uniqueness holds necessarily), m0.ax3 (GOAL existence).

- **Classical Extensional Mereology (CEM)** --- part-whole
  relationships. Used in Iron Maiden Test V (Mereological Coherence).
  e7He inherits the God-World containment from PET (ax1, ax2). Agent
  scope is a mereological part of the system's total scope.

- **First-Order Predicate Calculus** --- the base logic for all
  formal statements. Universal and existential quantification over
  agents, cycles, stages, and time.


Disciplines Added by e7He
============================

- **Coalgebra and Coinduction** --- the hero journey is formalized
  as a coalgebra with the step function ``step(μ_k) = (J_{k+1},
  μ_{k+1})``. Productivity (th4) guarantees non-termination.
  Bisimulation distinguishes genuinely different journeys from
  superficially different ones. See WB Sheet 9
  (:doc:`Coalgebra </matheology/compiler/forge/wb/coalgebra>`).

- **Dynamical Systems Theory** --- the Ie evolution equation
  (th3), ridge stability (th2), and attractor analysis (th5)
  are dynamical systems concepts. Lyapunov-like arguments underpin
  the ridge conditional stability. See WB Sheet 5
  (:doc:`Dynamical Systems </matheology/compiler/forge/wb/dynamical-systems>`).

- **Game Theory and Mechanism Design** --- the Commitment Trichotomy
  (th6) is a game-theoretic result. Schelling commitment theory,
  Spence signaling, Nash equilibrium, and the Prisoner's Dilemma →
  Assurance Game transformation are central. See WB Sheet 3
  (:doc:`Mechanism Design </matheology/compiler/forge/wb/mechanism-design>`)
  and WB Sheet 8
  (:doc:`Social Choice Theory </matheology/compiler/forge/wb/social-choice-theory>`).

- **Information Theory** --- the entropy condition in m0.ax3
  (``H(GOAL(h, t)) > 0``) and the lognormal influence distribution
  (sp3) are information-theoretic. See WB Sheet 11
  (:doc:`Information Theory </matheology/compiler/forge/wb/information-theory>`).

- **Probability and Stochastic Processes** --- the CTMC (continuous-
  time Markov chain) argument in th5 (BABL self-destructs with
  probability 1 as t → ∞) requires stochastic process theory.
  The lognormal distribution (sp3) requires probability theory.


Iron Maiden Test Coverage
===========================

Each of the 10 Iron Maiden tests draws on specific disciplines.
All 24 e7He statements were subjected to all 10 tests during TEMPER.

.. list-table::
   :header-rows: 1
   :widths: 8 35 57

   * - Test
     - Discipline
     - What it checks in e7He
   * - I
     - Consistency (predicate calculus)
     - No axiom contradicts another; joint satisfiability
   * - II
     - Independence (proof theory)
     - Each axiom adds deductive power not derivable from others
   * - III
     - Necessity (proof theory)
     - Each axiom unlocks genuinely new reasoning
   * - IV
     - Modal Soundness (S5)
     - Necessary vs. contingent claims properly distinguished
   * - V
     - Mereological Coherence (CEM)
     - Part-whole relationships respected; no orphan entities
   * - VI
     - Game-Theoretic Stability
     - Equilibria are stable under iteration; no exploitable defection
   * - VII
     - Computability and Decidability
     - Claims can be checked by finite procedure; no hidden halting
   * - VIII
     - Real-World Grounding
     - Claims connect to observable phenomena; not "not even wrong"
   * - IX
     - Cross-Model Coherence
     - Alignment with PET/JUB; no unintentional contradiction
   * - X
     - Known-Attack Resilience
     - Survives existing 33 con objections; no new undefended surface

See :doc:`/matheology/compiler/forge/iron-maiden-tests` for the full
test specifications.