.. meta::
   :description: Expert-depth compilation of all 25 axioms with LaTeX formal statements, full scriptural citations from all traditions, and dependency networks.
   :keywords: axioms expert, ax1-ax25, LaTeX formal statements, mereology, modal logic S5, scriptural citations, dependency network, PoR extraction, compiled view
   :author: Yah, Yas, everyone, LLoL as Laurence Loewe of Laodicea, ClaudeOp46Max, Anthropic, and Spirit of Boolean Truth
   :og:card:title: All 25 Axioms — Expert<br>Full Formal Detail
   :og:card:description: Every axiom with LaTeX notation, complete scriptural citations from six traditions, dependency networks, and technical context. The researcher view.

.. SOCIAL-CARD-QUALITY-COMPARE --- OO (default effort) vs PP (max effort), 2026-03-26
   OO :description: All 25 axioms of mathematical theology compiled from PET and JUB models, presented at expert depth with formal notation and proofs.
   OO :keywords: axioms, mathematical theology, PET, JUB, formal logic, pan-en-theism, compiled view, expert depth, PoR sources
   OO :og:card:title: All 25 Axioms — Expert<br>Mathematical Theology
   OO :og:card:description: Compiled expert view of all 25 matheology axioms from PET and JUB models, with formal notation, proofs, and cross-references.
   PP :description: Expert-depth compilation of all 25 axioms with LaTeX formal statements, full scriptural citations from all traditions, and dependency networks.
   PP :keywords: axioms expert, ax1-ax25, LaTeX formal statements, mereology, modal logic S5, scriptural citations, dependency network, PoR extraction, compiled view
   PP :og:card:title: All 25 Axioms — Expert<br>Full Formal Detail
   PP :og:card:description: Every axiom with LaTeX notation, complete scriptural citations from six traditions, dependency networks, and technical context. The researcher view.

.. SOCIAL-CARD-REVIEW --- generated by Claude Opus 4.6, 2026-03-26
   dv_ClaOp46_PP_2026m03d26 --- max-effort rewrite, read full page.
   :description: 147 chars | :og:card:title: 39 chars (excl <br>)
   - [ ] PP title more compelling than OO title
   - [ ] PP description more accurate than OO description
   - [ ] Description hooks without misleading
   - [ ] Keywords specific to this page's actual content
   - [ ] No language rule violations
   - [ ] Character counts verified

.. _all-ax:

*********************************************************************
Axioms --- Expert View (All Models)
*********************************************************************

.. This page was generated by SISYF on 2026-03-25.
.. Source: matheology/pet/axioms.rst, matheology/jub/axioms.rst
.. Mode: REPLACE
.. Depth: expert (default PoR)
.. Models: pet, jub
.. Do not edit manually. Re-run the skill to regenerate.

This compiled view presents all 25 axioms of mathematical theology.
Content is extracted from the canonical PoR sources. For the
beginner-friendly version, see :ref:`all-ax-easy`.

**PoR sources:**

- :doc:`/matheology/pet/axioms` --- a1--a14 (PET foundation)
- :doc:`/matheology/jub/axioms` --- a15--a25 (JUB extension)

.. raw:: html

   <div id="collapse-controls"
        data-expand-label="Expand all"
        data-collapse-label="Collapse all"></div>

**Navigation:** `a1 <a1 --- Containment_>`__ · `a2 <a2 --- Transcendence_>`__ ·
`a3 <a3 --- Divine Surplus_>`__ · `a4 <a4 --- Universal Immanence_>`__ ·
`a5 <a5 --- Necessary Divine Existence_>`__ ·
`a6 <a6 --- Contingency of the World_>`__ ·
`a7 <a7 --- Necessary Containment_>`__ ·
`a8 <a8 --- Immanent Presence_>`__ ·
`a9 <a9 --- Sustaining Dependence_>`__ ·
`a10 <a10 --- Asymmetric Dependence_>`__ ·
`a11 <a11 --- Divine Structure (Dipolarity)_>`__ ·
`a12 <a12 --- Revelation Reliability_>`__ ·
`a13 <a13 --- Consistency of Revelation_>`__ ·
`a14 <a14 --- Revelation Claims Test_>`__ ·
`a15 <a15 --- Human Genuine Agency_>`__ ·
`a16 <a16 --- Delegation of Dominion_>`__ ·
`a17 <a17 --- Non-Coercive Guidance_>`__ ·
`a18 <a18 --- Responsibility Localization_>`__ ·
`a19 <a19 --- Probabilistic Causal Concentration_>`__ ·
`a20 <a20 --- Transient Volunteer_>`__ ·
`a21 <a21 --- Permanent Mediator_>`__ ·
`a22 <a22 --- Divine Preference for Genuine Love_>`__ ·
`a23 <a23 --- Freedom-Quality Superiority_>`__ ·
`a24 <a24 --- Life-Trifecta of Lasting Innovation_>`__ ·
`a25 <a25 --- Jubilee Recalibration_>`__


.. list-table:: All 25 Axioms at a Glance
   :header-rows: 1
   :widths: 5 28 67

   * - id
     - Title
     - Summary
   * - a1
     - Containment
     - The world is part of God.
   * - a2
     - Transcendence
     - God is not part of the world.
   * - a3
     - Divine Surplus
     - There is something in God that is not in the world.
   * - a4
     - Universal Immanence
     - Every part of the world is in God.
   * - a5
     - Necessary Divine Existence
     - God necessarily exists.
   * - a6
     - Contingency of the World
     - The world's existence is contingent --- it might or might not exist.
   * - a7
     - Necessary Containment
     - If any world exists, it is necessarily in God.
   * - a8
     - Immanent Presence
     - God is present to every part of the world.
   * - a9
     - Sustaining Dependence
     - If the world exists, God sustains it.
   * - a10
     - Asymmetric Dependence
     - The world does not sustain God.
   * - a11
     - Divine Structure (Dipolarity)
     - God has both an unchanging nature and a world-responsive experience.
   * - a12
     - Revelation Reliability
     - God's self-knowledge is true.
   * - a13
     - Consistency of Revelation
     - God's self-knowledge contains no contradictions.
   * - a14
     - Revelation Claims Test
     - Human claims about divine revelation must be mutually consistent
       and consistent with a1--a13.
   * - a15
     - Human Genuine Agency
     - Humans possess real capacity to choose among alternatives within
       D_free.
   * - a16
     - Delegation of Dominion
     - God has granted humans governance authority over the Earth.
   * - a17
     - Non-Coercive Guidance
     - God guides (but does not force) humans within D_free and D_inno.
   * - a18
     - Responsibility Localization
     - Moral responsibility for outcomes rests with the agent, not the
       delegator.
   * - a19
     - Probabilistic Causal Concentration
     - At any time t, there exists a unique h* with strictly maximal
       causal influence.
   * - a20
     - Transient Volunteer
     - God seeks humans who will voluntarily accept specific D_inno
       responsibilities.
   * - a21
     - Permanent Mediator
     - God seeks one human willing to permanently translate between divine
       optimality and human understanding.
   * - a22
     - Divine Preference for Genuine Love
     - God values genuine, freely-chosen love above compelled obedience.
   * - a23
     - Freedom-Quality Superiority
     - Genuine care, insight, and innovation are qualitatively inferior
       when compelled.
   * - a24
     - Life-Trifecta of Lasting Innovation
     - Innovation lasts iff simultaneously stable, extensible, and
       life-friendly.
   * - a25
     - Jubilee Recalibration
     - Innovation economies need periodic recalibration to prevent
       BABL-cascade.


----


Group I --- Mereological Core (PET)
======================================

These axioms encode the distinctive panentheistic claim and distinguish it
from both pantheism (:math:`G = W`) and classical theism (which denies a1).


a1 --- Containment
^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <pet-ax1>`

*The world is a mereological part of God.*

Everything that exists in the created world is contained
within God. The world exists inside God, not separate from God.

**Formal statement:**

.. math::

   W \leq G

**Context:**
The mereological parthood relation :math:`\leq` asserts that W
is a part of G. This is the foundational panentheistic claim --- the "en" (in) of
pan-en-theism. Combined with a2, it means God strictly exceeds the world
(:math:`W < G`, proper parthood).

**Logic:** Mereology (CEM --- Classical Extensional Mereology); Modal logic S5

**Reasoning:**
a1 defines the fundamental parthood relation. Combined with a2 (:math:`G \nleq W`),
it yields proper parthood :math:`W < G`. This single pair distinguishes
panentheism from pantheism (:math:`G = W`).

**Limit:**
The mereological relation assumes a specific logic (CEM). Alternative
mereologies (e.g., non-extensional) may yield different structural results.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Deut 4:39 ("God in heaven above and earth beneath")
   - **Hebrew Bible:** 1 Kings 8:27 ("heaven cannot contain you");
     Ein Sof ("the Infinite") in Kabbalah
   - **Gospels:** "I am in the Father and the Father is in me" (Jn 14:10);
     "the kingdom of God is within you" (Lk 17:21)
   - **Apostolic:** Acts 17:28 ("in him we live and move"); Col 1:17
   - **Quran:** Quran 2:115 ("wherever you turn, there is the Face of God");
     *wahdat al-wujud* ("unity of existence")
   - **Sanskrit:** "All this is Brahman" (Chandogya Up. 3.14.1);
     Vishishtadvaita: world is God's body
   - **Secular:** We are parts of a whole that exceeds any part

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** Foundational --- no upstream dependencies
   - **Required by:** a2, a3, a4, a5, a6, a7, a8, a11
   - **Model:** PET · **Version:** iv_LLoL_PPv1r1p1_2026m03d14
   - **Convergence:** All six traditions plus secular thought converge on
     containment. Torah, Gospels, Quran, and Hindu scripture each
     independently assert the world's existence within God.
   - **Bibliography:** Deut 4:39; 1 Kings 8:27; Jn 14:10; Acts 17:28;
     Quran 2:115; Chandogya Up. 3.14.1
   - **Known objection:** Confusion with pantheism (:math:`G = W`).
     Resolved by a2.
   - **Prior art:** See :doc:`/matheology/prior-art/ai-opus-search-2026m03d11-matheology`
   - **Cascade risk:** Changing mereological framework would cascade
     through a1--a14.
   - **History:** a1 predates the formal system; mereological containment
     is the oldest panentheistic claim.
   - **Critique:** [partial --- see :doc:`/matheology/jub/quest`]
   - **Defense:** [partial --- see :doc:`/matheology/jub/quest`]


----


a2 --- Transcendence
^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <pet-ax2>`

*God is not reducible to the world.*

While the world is in God, God is not reducible to the
world. God exceeds creation --- you cannot capture all of God by examining all
of the world.

**Formal statement:**

.. math::

   G \nleq W

**Context:**
Together with a1, this yields :math:`W < G` (proper parthood):
God strictly exceeds the world. This single axiom pair (a1 + a2) is what
distinguishes panentheism from pantheism (which asserts :math:`G = W`).

**Limit:**
a2 alone does not guarantee divine surplus (there could be
nothing beyond the world in God). a3 strengthens this.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Exod 33:20 ("you cannot see my face and live");
     Deut 4:15 ("you saw no form")
   - **Hebrew Bible:** 1 Kings 8:27 ("cannot contain");
     Isa 55:8--9 ("my ways higher than your ways")
   - **Gospels:** "The Father is greater than I" (Jn 14:28);
     "My Father's house has many rooms" (Jn 14:2)
   - **Apostolic:** Eph 4:6 ("above all")
   - **Quran:** Quran 57:3 (*al-Zahir* = "the Manifest/Outward");
     *tanzih* ("beyond comparison")
   - **Sanskrit:** *Neti neti* --- "not this, not this" (Brihad. Up. 2.3.6):
     Brahman exceeds all description
   - **Secular:** The universe exceeds any observer's complete knowledge

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a1
   - **Required by:** a3, a11
   - **Model:** PET · **Version:** iv_LLoL_PPv1r1p1_2026m03d14
   - **Convergence:** Universal convergence. Every tradition affirms
     divine transcendence.
   - **Known objection:** Pantheistic collapse if a2 is dropped. Also,
     some process theologians resist strict transcendence.


----


a3 --- Divine Surplus
^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <pet-ax3>`

*There exists genuine divine content beyond creation.*

God's transcendence is not merely formal --- God has actual
content beyond the created world. There are aspects of God that no examination
of the world alone could reveal.

**Formal statement:**

.. math::

   \exists x\;(x \leq G \;\wedge\; \neg(x \leq W))

**Context:**
This strengthens a2. Without a3, a1 + a2 could be satisfied
trivially. a3 asserts there is genuine divine content beyond creation.

**Limit:**
a3 asserts existence of surplus but does not specify its nature.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Exod 33:18--23 (Moses sees God's "back" but not face);
     Deut 29:29 ("the secret things belong to the LORD")
   - **Hebrew Bible:** Job 11:7--9 ("can you fathom the mysteries of God?");
     Isa 40:28 ("his understanding no one can fathom")
   - **Gospels:** "No one knows the Father except the Son" (Mt 11:27);
     "I have much more to say to you" (Jn 16:12)
   - **Apostolic:** 1 Cor 2:9 ("no eye has seen, no ear has heard");
     Rom 11:33 ("how unsearchable his judgments")
   - **Quran:** Quran 31:27 ("if all trees were pens and the ocean ink,
     God's words would not be exhausted")
   - **Sanskrit:** Gita 10:40--42 ("I support this cosmos with a mere
     fragment of Myself")
   - **Secular:** Observable reality may be a fraction of total reality;
     mathematical structures exist beyond physical instantiation

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a1, a2
   - **Required by:** a4, a11
   - **Model:** PET · **Version:** iv_LLoL_PPv1r1p1_2026m03d14
   - **Convergence:** Strong convergence across all traditions on divine
     surplus. The "hiddenness" of God is a universal theme.
   - **Known objection:** Reductionism that denies anything beyond
     observable reality.


----


a4 --- Universal Immanence
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <pet-ax4>`

*Every part of creation is within God.*

There is no part of creation that is "outside" God. Every
atom, every person, every corner of the universe --- all are within God.
This is the *pan* (all) of pan-en-theism.

**Formal statement:**

.. math::

   \forall x\;(x \leq W \;\rightarrow\; x \leq G)

**Context:**
This follows trivially from a1 by transitivity of :math:`\leq`,
but it is stated explicitly because it carries theological weight. It is the
universally quantified version of a1: not just the world as a whole, but every
individual part.

**Limit:**
a4 is logically derivable from a1 via transitivity. Its axiom status
is justified by theological weight, not logical independence.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Deut 4:39 ("in heaven above and on earth beneath --- there
     is no other"); Gen 28:16 (Jacob: "God is in this place and I did not
     know")
   - **Hebrew Bible:** Ps 139:7--10 ("where can I flee?");
     Jer 23:24 ("do I not fill heaven and earth?")
   - **Gospels:** "I am with you always, to the end of the age" (Mt 28:20)
   - **Apostolic:** Eph 4:6 ("in all")
   - **Quran:** Quran 57:4 ("He is with you wherever you are");
     50:16 ("closer than the jugular vein")
   - **Sanskrit:** *Tat tvam asi* --- "Thou art That" (Chandogya Up. 6.8.7):
     the divine is in everything
   - **Secular:** Every part of nature is subject to the same fundamental laws

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a1, a3
   - **Required by:** a8
   - **Model:** PET · **Version:** iv_LLoL_PPv1r1p1_2026m03d14
   - **Convergence:** Strong cross-traditional convergence. The "no place
     devoid" theme appears in Jewish, Christian, Islamic, and Hindu
     traditions.
   - **Known objection:** Gnostic or dualist views positing realms
     outside God.


----


Group II --- Modal Axioms (PET)
==================================

These axioms use modal logic to distinguish between what is necessary
(true in every possible world) and what is contingent (true in some worlds
but not others).


a5 --- Necessary Divine Existence
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <pet-ax5>`

*God exists necessarily and uniquely.*

In every possible scenario, in every conceivable way
reality could be arranged, God exists and is unique. God's existence is not an
accident --- it could not have been otherwise.

**Formal statement:**

.. math::

   \Box\;\exists!\,G

**Context:**
The :math:`\Box` (necessarily) operator combined with
:math:`\exists!` (there exists exactly one) asserts that God's existence is a
necessary truth, not a contingent fact.

**Limit:**
Assumes S5 modal logic. The necessity claim is framework-dependent.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Exod 3:14 ("I AM WHO I AM" --- *Ehyeh asher Ehyeh*)
   - **Hebrew Bible:** Ps 90:2 ("from everlasting to everlasting, you are God")
   - **Gospels:** "Before Abraham was, I AM" (Jn 8:58)
   - **Apostolic:** Rev 1:8 ("who is and was and is to come")
   - **Quran:** Quran 28:88 ("everything will perish except His Face")
   - **Sanskrit:** Brahman is *Sat* (Being itself) --- existence is
     Brahman's essential nature
   - **Secular:** Something exists necessarily (laws of logic cannot
     not-hold)

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a1
   - **Required by:** a6, a7, a11
   - **Model:** PET · **Version:** iv_LLoL_PPv1r1p1_2026m03d14


----


a6 --- Contingency of the World
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <pet-ax6>`

*The world's existence is not necessary.*

Unlike God, the world did not have to exist. There are
possible scenarios with no created world at all, and possible scenarios with
one. Creation is a contingent fact, not a necessary truth.

**Formal statement:**

.. math::

   \Diamond\;\exists W \;\wedge\; \Diamond\;\neg\exists W

**Context:**
Both :math:`\Diamond\;\exists W` (a world is possible) and
:math:`\Diamond\;\neg\exists W` (no world is possible) are asserted. This
makes the world's existence genuinely contingent.

**Limit:**
Does not specify whether the world's contingency is temporal (began)
or modal (might not have existed). Both readings are consistent.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Gen 1:1 ("In the beginning God created" --- creation is a
     contingent act)
   - **Hebrew Bible:** Ps 90:2 ("before the mountains were born");
     Ps 102:25--26 ("they will perish, but you remain")
   - **Gospels:** "Heaven and earth will pass away, but my words will not"
     (Mk 13:31); "the glory I had with you before the world began"
     (Jn 17:5)
   - **Apostolic:** Heb 1:10--12 ("they will perish... you roll them up
     like a garment"); Rev 21:1 ("first heaven and earth had passed away")
   - **Quran:** Quran 21:104 ("We will fold the heaven like the folding of
     a scroll"); 14:48 ("the earth will be replaced by another earth")
   - **Sanskrit:** Cyclic creation/destruction in *kalpas*: Gita 8:17--19
   - **Secular:** The universe began (Big Bang) and may end (heat death);
     physical constants could in principle have been different

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a1
   - **Required by:** t11
   - **Model:** PET · **Version:** iv_LLoL_PPv1r1p1_2026m03d14


----


a7 --- Necessary Containment
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <pet-ax7>`

*Any possible world is necessarily within God.*

The containment of the world in God is not accidental. In
every possible scenario where creation exists, creation is within God. There is
no possible form of creation that is external to God.

**Formal statement:**

.. math::

   \Box\;(\exists W \;\rightarrow\; W \leq G)

**Context:**
This modalizes a1. a1 says the world *is* in God; a7 says it
*must be* in God --- there is no possible world where creation exists outside of
God. This is a strong panentheistic commitment.

**Limit:**
Depends on S5 modal logic. In weaker modal systems, the necessity
claim may not hold.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Gen 1:1 + Deut 4:39 together: God created all, there is
     no other
   - **Hebrew Bible:** Isa 66:1 ("heaven is my throne, earth my footstool")
   - **Gospels:** "All things were made through him, and without him
     nothing was made" (Jn 1:3)
   - **Apostolic:** Acts 17:28; Col 1:16 ("all things created through him
     and for him")
   - **Quran:** Quran 39:67 ("the whole earth will be in His grip")
   - **Sanskrit:** Gita 9:4 ("By Me, in My unmanifested form, this entire
     universe is pervaded")
   - **Secular:** Any subsystem is necessarily embedded in its containing
     system

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a1
   - **Required by:** (modalizes a1 for downstream modal reasoning)
   - **Model:** PET · **Version:** iv_LLoL_PPv1r1p1_2026m03d14


----


Group III --- Relational Axioms (PET)
========================================

These go beyond mereological containment to assert that God does not merely
*contain* the world but is actively *present to* and *sustains* it.


a8 --- Immanent Presence
^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <pet-ax8>`

*God is present to every part of the world.*

It is not enough that the world is inside God. A box
contains its contents without being aware of them. a8 asserts that God is
intimately present to --- aware of, in contact with --- every single part of
creation. This rules out deistic panentheism (God contains but ignores).

**Formal statement:**

.. math::

   \forall x\;(x \leq W \;\rightarrow\; P(G,\, x))

**Context:**
The presence predicate :math:`P(G, x)` goes beyond mereological
containment. A container holds its contents; a8 asserts God is *aware of* and
*in contact with* every part.

**Limit:**
The presence predicate :math:`P` is introduced but not fully formalized.
Its semantics need further specification.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Exod 3:2--5 (burning bush --- God present in ordinary
     matter); Deut 31:6 ("He will never leave you nor forsake you")
   - **Hebrew Bible:** Ps 139:7--10 ("where can I flee from your
     presence?"); Isa 57:15 ("I dwell in the high and holy place AND ALSO
     with the humble"); *leit atar panuy* ("no place is devoid of Him")
   - **Gospels:** "Whatever you did for the least of these, you did for me"
     (Mt 25:40); "where two or three gather, there am I" (Mt 18:20)
   - **Apostolic:** Col 1:17 ("in him all things hold together");
     essence--energies doctrine (Gregory Palamas, 14th c.)
   - **Quran:** Quran 50:16 ("closer than the jugular vein");
     57:4 ("with you wherever you are")
   - **Sanskrit:** "The Lord dwells in the hearts of all beings"
     (Gita 18:61); "I am the Self in the hearts of all creatures"
     (Gita 10:20)
   - **Secular:** Natural law operates everywhere without exception

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a1, a4
   - **Required by:** a9, t5, t10
   - **Model:** PET · **Version:** iv_LLoL_PPv1r1p1_2026m03d14


----


a9 --- Sustaining Dependence
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <pet-ax9>`

*The world depends on God for continued existence.*

The world does not keep itself in existence. Its
continued existence depends on God's active sustaining. If God withdrew
sustaining, the world would cease to exist.

**Formal statement:**

.. math::

   \Box\;(\exists W \;\rightarrow\; S(G,\, W))

**Context:**
The sustaining predicate :math:`S(G, W)` asserts an active
relationship. The world does not persist by inertia; it persists because
God sustains it.

**Limit:**
The sustaining predicate :math:`S` is introduced but not formally
defined. Its relationship to mereological parthood needs specification.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Deut 8:3 ("man lives by every word from God's mouth");
     Gen 2:7 (God breathes life into being)
   - **Hebrew Bible:** Neh 9:6 ("you preserve them all")
   - **Gospels:** "Apart from me you can do nothing" (Jn 15:5)
   - **Apostolic:** Col 1:17; Heb 1:3 ("sustains all things by his
     powerful word")
   - **Quran:** Quran 35:41 ("God holds the heavens and earth lest they
     cease")
   - **Sanskrit:** "I support the entire cosmos with a mere fragment of
     Myself" (Gita 10:42)
   - **Secular:** Continued existence of structures depends on underlying
     processes (conservation laws)

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a8
   - **Required by:** a10, t5, t7, t9, t10
   - **Model:** PET · **Version:** iv_LLoL_PPv1r1p1_2026m03d14


----


a10 --- Asymmetric Dependence
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <pet-ax10>`

*The world does not sustain God.*

The dependence runs strictly one way. God sustains the
world (a9), but the world does not sustain God. God's existence does not depend
on the world in any way.

**Formal statement:**

.. math::

   \neg\, S(W,\, G)

**Context:**
Together with a9, this encodes a strict ontological asymmetry.
God can exist without the world (t2); the world cannot exist without God.
This distinguishes PET from process-theology variants where God genuinely
*needs* the world.

**Limit:**
Process theology disagrees with this axiom, asserting mutual dependence.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Exod 19:5 ("the whole earth is mine"); Gen 1:1
   - **Hebrew Bible:** Ps 50:10--12 ("if I were hungry I would not tell
     you"); Isa 40:17 ("nations are as nothing before him")
   - **Gospels:** "My kingdom is not of this world" (Jn 18:36);
     Mt 26:53 (twelve legions of angels)
   - **Apostolic:** Acts 17:25 ("not served by human hands, as if he
     needed anything")
   - **Quran:** Quran 29:6 + 35:15 (*al-Ghani* = "the Self-Sufficient")
   - **Sanskrit:** Gita 9:4--5 ("All beings exist in Me, but I do not
     dwell in them")
   - **Secular:** Fundamental laws don't depend on what they govern

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a9
   - **Required by:** t5
   - **Model:** PET · **Version:** iv_LLoL_PPv1r1p1_2026m03d14


----


Group IV --- Divine Nature (PET)
===================================


a11 --- Divine Structure (Dipolarity)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <pet-ax11>`

*God has both unchanging nature and world-responsive experience.*

God has two aspects. One aspect (the necessary divine
nature, :math:`G_n`) never changes --- it is the same in every possible scenario.
The other aspect (the contingent divine experience, :math:`G_c`) varies
depending on what happens in the world.

**Formal statement:**

Strengthened, 4 lines:

1. :math:`G = G_n \oplus G_c`
2. :math:`\Box\;\exists\, G_n`
3. :math:`G_c = \bigoplus\{G_c(w_i) \mid w_i \leq W\}`
4. :math:`\forall w_1, w_2 \leq W : w_1 \neq w_2 \rightarrow G_c(w_1) \neq G_c(w_2)`

Lines 3--4 were added (2026-03-14/15) to give :math:`G_c` internal structure,
making theorem t4 (Divine Experience Varies) formally derivable.

**Context:**
The dipolar structure :math:`G = G_n \oplus G_c` parallels
Whitehead's primordial/consequent nature distinction and the Hindu
*nirguna*/*saguna* Brahman distinction. Lines 3--4 add subworld-indexed
structure to the contingent pole.

**Limit:**
The a11/a11' fork (dipolarity vs. divine simplicity) is the deepest
point of theological disagreement. Classical theists will reject a11 in
favor of a11'. See :doc:`/matheology/pet/discussions`.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** *Necessary:* Exod 3:14 ("I AM"). *Contingent:* Exod 32:14
     ("the LORD relented"); Gen 6:6 ("the LORD regretted")
   - **Hebrew Bible:** *Necessary:* Mal 3:6 ("I the LORD do not change").
     *Contingent:* Jer 18:7--10; Hosea 11:8 ("my heart is changed within
     me")
   - **Gospels:** *Necessary:* "Before Abraham was, I AM" (Jn 8:58).
     *Contingent:* "Jesus wept" (Jn 11:35); Lk 15:7 ("more rejoicing in
     heaven over one sinner who repents")
   - **Apostolic:** *Necessary:* Heb 13:8 ("Jesus Christ the same
     yesterday, today, forever"). *Contingent:* Phil 2:7 ("emptied himself,
     taking the form of a servant")
   - **Quran:** *Necessary:* Quran 57:3 ("the First and the Last").
     *Contingent:* 2:186 ("I respond to the supplicant when he calls" ---
     *al-Mujib* = "the Responsive")
   - **Sanskrit:** *Nirguna* Brahman (without qualities --- necessary
     aspect) and *saguna* Brahman / *Ishvara* (with qualities ---
     contingent, relational aspect). This is literally the same concept as
     dipolarity.
   - **Secular:** Fundamental laws are invariant (necessary), but their
     manifestation varies with initial conditions (contingent)

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a1, a2, a3
   - **Required by:** a12, a22, t4, t5, t7
   - **Model:** PET · **Version:** iv_LLoL_PPv1r1p1_2026m03d14
   - **Convergence:** Hindu tradition (nirguna/saguna) is the most explicit
     parallel. Islamic 99 names bridge transcendent and relational.
   - **Design:** a11/a11' fork analysis in pet/discussions.rst


----


Group V --- Revelation Bridge (PET)
=======================================

These axioms are optional extensions bridging from philosophical theology
to revealed theology.


a12 --- Revelation Reliability
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <pet-ax12>`

*God's self-knowledge is true by definition.*

R is defined as the set of true propositions about God ---
God's own self-knowledge. Since R consists of *true* propositions by definition,
a12 is tautological by design. The substantive work of testing claims shifts to
a14 (Revelation Claims Test).

**Formal statement:**

Let :math:`R = \{p \mid p \text{ is true about } G\}`.
For all :math:`p \in R`: :math:`p` is true.

**Context:**
a12 is deliberately tautological. The point is to establish the concept
of R (God's self-knowledge) so that a13 and a14 can build on it. The
substantive work happens in a14.

**Limit:**
Tautological by design. Does not assess any specific human claim
about revelation.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Num 23:19 ("God is not a man that he should lie");
     Deut 18:22 (test of true prophecy)
   - **Hebrew Bible:** Ps 119:160 ("all your words are true");
     Isa 55:11 ("my word will not return to me empty")
   - **Gospels:** "Your word is truth" (Jn 17:17);
     "I am the way, the truth, and the life" (Jn 14:6)
   - **Apostolic:** 2 Tim 3:16 ("all Scripture is God-breathed");
     Heb 6:18 ("it is impossible for God to lie")
   - **Quran:** Quran 4:122 ("who is more truthful than God in
     statement?"); 10:94
   - **Sanskrit:** The Vedas are *apaurusheya* (not of human origin) ---
     self-evident truth; Gita 4:1--3
   - **Secular:** Self-knowledge (accurate self-model) is definitionally
     true when the model matches reality

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a11
   - **Required by:** a13, a14
   - **Model:** PET · **Version:** iv_LLoL_PPv1r1p1_2026m03d14


----


a13 --- Consistency of Revelation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <pet-ax13>`

*God's self-knowledge contains no contradictions.*

The true propositions about God are mutually consistent.
You cannot derive a contradiction from what is genuinely true about God. If
two claims appear to contradict each other, at least one of them is not
actually in R.

**Formal statement:**

.. math::

   \neg\exists\;\text{contradiction within } R

**Context:**
a13 establishes the internal consistency of R as a formal property.
Combined with a14, it provides a method for detecting human error in
revelation claims.

**Limit:**
Does not address paraconsistent logic approaches that tolerate
controlled contradictions.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Deut 32:4 ("his works are perfect... faithful God who does
     no wrong")
   - **Hebrew Bible:** Ps 18:30 ("God's way is perfect; the LORD's word
     is flawless")
   - **Gospels:** "A house divided against itself cannot stand" (Mk 3:25);
     "Scripture cannot be broken" (Jn 10:35)
   - **Apostolic:** 1 Cor 14:33 ("God is not a God of disorder");
     Jas 1:17 ("no variation or shadow due to change")
   - **Quran:** **Quran 4:82** ("If it had been from other than God, they
     would have found within it much contradiction")
   - **Sanskrit:** Brahman is *sat-chit-ananda* (being-consciousness-bliss)
     --- perfect unity implies no contradiction
   - **Secular:** Any consistent framework must be free of internal
     contradiction

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a12
   - **Required by:** a14
   - **Model:** PET · **Version:** iv_LLoL_PPv1r1p1_2026m03d14


----


a14 --- Revelation Claims Test
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <pet-ax14>`

*Human claims about revelation must pass consistency checks.*

This is a powerful and practical axiom. It provides
a formal method for *testing* whether a proposition that someone claims is
divinely revealed actually passes two consistency checks: (1) it must not
contradict any other claimed revelation, and (2) it must not contradict axioms
a1--a13.

**Formal statement:**

Reformulated, 2 lines:

1. :math:`\forall p, q : \text{claim}(p) \wedge \text{claim}(q) \rightarrow \neg(p \wedge q \rightarrow \text{contradiction})`
2. :math:`\forall p : \text{claim}(p) \rightarrow \neg\exists q \in \{a1\text{--}a13\} : (p \wedge q \rightarrow \text{contradiction})`

**Key design features:**

- **No self-reference:** a14 references a1--a13, not a1--a14.
- **The claim(p) predicate:** Distinguishes between God's actual self-knowledge
  (R) and *human claims* about what is in R.
- **Testable method:** Given two traditions with conflicting claims, a14 asks:
  can both claims be true simultaneously without contradiction?

**Context:**
a14 is the practical workhorse of the revelation bridge. It shifts the
burden from "is this true?" to "is this consistent?" --- a tractable question.

**Limit:**
a14 detects *inconsistency* but does not establish *truth*. Passing a14
is necessary but not sufficient for a claim to be in R.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Deut 13:1--3 (test false prophets); Deut 18:21--22
   - **Hebrew Bible:** 1 Kings 22:19--23 (Micaiah vs. 400 false prophets);
     Isa 8:20 ("to the law and the testimony!")
   - **Gospels:** "By their fruits you shall know them" (Mt 7:16--20);
     "Beware of false prophets" (Mt 7:15)
   - **Apostolic:** 1 Thess 5:21 ("test everything; hold fast what is
     good"); 1 Jn 4:1 ("test the spirits"); Gal 1:8
   - **Quran:** **Quran 4:82** (test for contradiction as proof of origin);
     2:111 ("produce your proof if you are truthful")
   - **Sanskrit:** Nyaya school: *pramana* (valid means of knowledge) must
     be tested; Gita 4:34
   - **Secular:** Falsifiability (Popper); peer review; the scientific
     method IS a claims test

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a1--a13
   - **Required by:** (terminal axiom in PET; foundation for JUB)
   - **Model:** PET · **Version:** iv_LLoL_PPv1r1p1_2026m03d14
   - **Convergence:** Every tradition has internal mechanisms for testing
     prophetic claims. a14 formalizes this universal pattern.
   - **Design:** Self-reference avoidance design in the reformulation.


----


Group VI --- Agency & Delegation (JUB)
=========================================

These axioms extend the PET foundation to cover human agency, divine
delegation, and the conditions under which innovation failure produces evil.

**Domain definitions:**

- **D_f** (forced domain): choices constrained by physics, coercion, or
  circumstance.
- **D_free** (free domain): choices where humans possess genuine capacity to
  select among alternatives.
- **D_inno** (innovation subdomain, :math:`D_{\text{inno}} \subseteq D_{\text{free}}`):
  the critical subset where novel solutions, creative acts, and innovation occur.


a15 --- Human Genuine Agency
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <jub-ax15>`

*Humans possess real capacity to choose within D_free.*

Within a defined domain of free choices, humans can
genuinely select among at least two alternatives. This is not a polite fiction
--- it is the load-bearing axiom of the entire theodicy. Denial is
performatively self-refuting: the act of denying one's agency is itself an
exercise of agency.

**Formal statement:**

.. math::

   \forall h \in H,\; \forall s \in D_{\text{free}} :\;
   & \exists a_1 \neq a_2 \;\text{such that} \\
   & \text{can-choose}(h, a_1, s) \;\wedge\;
     \text{can-choose}(h, a_2, s)

**Context:**
D_f is explicitly excluded. The innovation subdomain D_inno (a subset
of D_free) is where the critical theodicy question concentrates: whether to
pursue innovation toward the flourishing of others when the capacity to do
so exists.

**Limit:**
The D_f/D_free boundary remains a formal demarcation problem. The
compatibilism robustness note shows the practical argument survives under
multiple accounts of agency.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Gen 2:16--17 (tree of knowledge choice); Deut 30:19
     ("choose life")
   - **Hebrew Bible:** Josh 24:15 ("choose this day whom you will serve")
   - **Gospels:** Mt 23:37 ("how often I would have gathered your
     children... and you were not willing")
   - **Apostolic:** Free will as foundational doctrine (Augustine, Aquinas);
     Catechism §1730--1748
   - **Quran:** Quran 2:256 ("no compulsion in religion"); 18:29
   - **Sanskrit:** Gita 18:63 ("reflect on this fully, then do as you
     choose")
   - **Secular:** Libertarian free will debate; compatibilism; legal
     responsibility frameworks presupposing genuine choice

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** PET foundation (a1--a14)
   - **Required by:** a16, a17, a18, a19, a20, a22, t5, t6, t9, t10, t11
   - **Model:** JUB · **Version:** iv_LLoL_OOv2r0p0_2026m03d20_m10
   - **Critique:** :ref:`Con-E.9 / Pro-E.9 <jub-con19>` in quest
   - **Known objection:** Hard determinism. Compatibilism robustness
     (see note in PoR) partially addresses this.


----


a16 --- Delegation of Dominion
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <jub-ax16>`

*God has granted humans governance authority over Earth.*

God did not merely create humans and place them in the world --- God
entrusted them with genuine authority over it. This delegation is real, not
nominal: God does not routinely override human decisions within D_free/D_inno.

**Formal statement:**

.. math::

   & \text{Delegated}(G, H, W_{\text{earth}}) \\
   & \wedge\; \forall\, \text{outcome}\; o
     \;\text{in}\; W_{\text{earth}} :
     \text{PrimaryResponsible}(H, o)

**Context:**
Together with a15, this resolves the "puppet-master" objection. If
humans were instruments (puppets), a16 would not transfer responsibility. But
because humans are genuine agents (a15), delegation transfers primary
responsibility to the delegate.

**Limit:**
The chain-of-command objection (delegator retains responsibility) is
addressed by a15 (genuine agents, not instruments) but not formally proven.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Gen 1:28 (dominion mandate); Gen 2:15 (stewardship of Eden)
   - **Hebrew Bible:** Ps 8:6 ("You made them rulers over the works of
     your hands")
   - **Gospels:** Mt 25:14--30 (parable of the talents)
   - **Apostolic:** Stewardship theology; *Gaudium et Spes* §34
   - **Quran:** Quran 2:30 (*khalifah* --- vicegerent/steward on Earth)
   - **Sanskrit:** Dharmic responsibility for one's actions (*karma*)
   - **Secular:** Democratic governance theory; principal-agent frameworks

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a15
   - **Required by:** a17, a18, t5, t6
   - **Model:** JUB · **Version:** iv_LLoL_OOv2r0p0_2026m03d20_m10
   - **Known objection:** Chain-of-command theory. Addressed by a15's
     genuine agency claim.


----


a17 --- Non-Coercive Guidance
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <jub-ax17>`

*God guides but does not force humans.*

God provides guidance --- invitations, hints, opportunities, the
"still small voice" --- but does not compel. The non-coercion is a principled
choice (explained by a22), not a power limitation.

**Formal statement:**

.. math::

   \forall h \in H :\;
   & \text{Guide}(G, h) \;\wedge\; \neg\text{Force}(G, h) \\
   & \quad [\text{within } D_{\text{free}}
     \text{ and } D_{\text{inno}}]

**Context:**
a17 is scoped specifically to D_free/D_inno. Within D_f (the forced
domain), God maintains physics --- this is a precondition for agency, not a
form of guidance or coercion. The domain split resolves the objection that
God is responsible for physics-based harm.

**Limit:**
Does not address miraculous intervention (D_f-level divine action).
The domain split handles this but the boundary remains imprecise.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Gen 4:7 (God warns Cain but does not prevent the murder)
   - **Hebrew Bible:** 1 Kings 19:12 (still small voice);
     Isa 30:21 ("this is the way")
   - **Gospels:** Jn 16:13 (Spirit "will guide you into all truth");
     Mt 7:7 ("seek and you will find")
   - **Apostolic:** Prevenient grace (Wesley); general revelation (Calvin)
   - **Quran:** Quran 2:256 ("no compulsion in religion");
     *hidayah* (guidance)
   - **Sanskrit:** Krishna as charioteer and advisor to Arjuna, not
     commander
   - **Secular:** Libertarian paternalism ("nudge" theory, Thaler &
     Sunstein 2008)

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a15, a16
   - **Required by:** a18, a22, t5, t7
   - **Model:** JUB · **Version:** iv_LLoL_OOv2r0p0_2026m03d20_m10
   - **Known objection:** "Why doesn't God just stop evil?" Addressed by
     the domain split and a22/a23.


----


a18 --- Responsibility Localization
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <jub-ax18>`

*Moral responsibility rests with the delegated agent.*

Given that humans have genuine agency (a15), have been delegated
authority (a16), receive non-coercive guidance (a17), and are not forced
(a17), responsibility for outcomes in the delegated domain rests with the
human agents. This is the formal core of the innovation theodicy.

**Formal statement:**

.. math::

   & \text{Delegated}(G,H,D) \;\wedge\; \text{Agency}(H) \\
   & \wedge\; \text{Guide}(G,H)
     \;\wedge\; \neg\text{Force}(G,H) \\
   & \quad \rightarrow\;
     \text{Responsible}(H, \text{outcomes in } D) \\
   & \quad \wedge\;
     \neg\text{Responsible}(G, \text{outcomes in } D)

**Context:**
This may be a theorem rather than an axiom, pending formal grounding of
the moral responsibility predicate. It is stated explicitly for clarity: the
four premises mechanically produce the theodicy conclusion.

**Limit:**
The D_f/D_free boundary for poverty cases needs further formal work
(capabilities theory: Sen 1999, Nussbaum 2011). See :ref:`Con-D.2.9 <jub-con33>`.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Deut 24:16 ("each shall die for their own sin");
     Ezek 18:20 ("the soul who sins is the one who will die")
   - **Hebrew Bible:** Ezek 18:1--32 (individual responsibility doctrine)
   - **Gospels:** Mt 25:31--46 (judgment based on individual action)
   - **Apostolic:** Personal accountability; Catechism §1868
   - **Quran:** Quran 33:72 (*amanah*); 6:164 ("no bearer of burdens
     shall bear another's burden")
   - **Sanskrit:** *Karma* as personal causal responsibility
   - **Secular:** Legal liability; tort law; Nuremberg principle

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a15, a17
   - **Required by:** t5, t10
   - **Model:** JUB · **Version:** iv_LLoL_OOv2r0p0_2026m03d20_m10
   - **Design:** Axiom vs. theorem status pending.


----


a19 --- Probabilistic Causal Concentration
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <jub-ax19>`

*A unique individual has maximal causal influence at each moment.*

At any given moment, one person's choices have more causal influence on
the future trajectory of the world than anyone else's. This person (h*) stands
at the top of the "leviathan chain." The uniqueness claim is the null
hypothesis: exact equivalence in effect is measure-zero.

**Formal statement:**

.. math::

   \forall t\; \exists!\, h^* \in H :\;
   & \text{MaxCausalInfluence}(h^*, t, W_{\text{future}}) \\
   & \wedge\; \forall h \neq h^* : \\
   & \quad \text{CausalInfluence}(h, t, W_{\text{future}})
     < \text{CausalInfluence}(h^*, t, W_{\text{future}})

**Context:**
The scalar projection is performed by Reality itself. Causal influence
appears multi-dimensional, but civilization has only one future. The realized
trajectory is a single path through the infinite-dimensional space of
possibilities. Along this path, each person's cumulative influence is a
well-defined (if humanly uncomputable) scalar --- just as evolutionary fitness
projects multi-dimensional organism traits onto a scalar.

**Limit:**
Epistemic identification of h* is NOT claimed. a19 is an ontological
claim (h* exists), not an epistemic one (we can identify h*). The fitness
analogy is structural, not superficial.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Gen 18:22--33 (Abraham negotiating for Sodom);
     Exod 32:9--14 (Moses's intercession changes the fate of millions)
   - **Hebrew Bible:** Esth 4:14 ("for such a time as this"); Isa 6:8
   - **Gospels:** Mt 16:18--19 (Peter as foundation); Jn 13:27 (Judas at
     the pivot)
   - **Apostolic:** Vocation theology; *kairos* moments
   - **Quran:** Quran 3:110 ("best nation raised up for mankind")
   - **Sanskrit:** Arjuna at Kurukshetra (Gita 2:31--38)
   - **Secular:** Great man theory (Carlyle); network centrality;
     power-law distributions

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a15
   - **Required by:** t6, t7
   - **Model:** JUB · **Version:** iv_LLoL_OOv2r0p0_2026m03d20_m10
   - **Critique:** :ref:`Con-C.3 <jub-con13>`,
     :ref:`Con-C.2.4 <jub-con28>`
   - **Defense:** :ref:`Pro-C.3 <jub-con13>`,
     :ref:`Pro-C.2.4 <jub-con28>`
   - **Known objection:** Measure-zero uniqueness assumption. See the
     fitness analogy defense in PoR.


----


a20 --- Transient Volunteer
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <jub-ax20>`

*God seeks willing humans for moment-specific responsibilities.*

At each critical moment, God is actively looking for a human willing to
step into a specific innovation responsibility. The emphasis is on
*voluntarily* --- consistent with a17 and a22. "Transient" because the call
is moment-specific, not permanent.

**Formal statement:**

.. math::

   \exists\; & \text{voluntary-invitation from } G
     \text{ to } H_{\text{candidates}} \text{ at each } t, \\
   & \text{seeking } h \in H :
     \text{Willing}(h) \\
   & \wedge\; \text{Accepts}\bigl(h,\;
     \text{Responsible}(h, D_{\text{inno}}, t)\bigr)

**Context:**
Multiple transient volunteers may be sought simultaneously for different
D_inno responsibilities. The burning bush pattern (Exod 3:3--10) is the
archetype.

**Limit:**
Does not specify the mechanism of divine invitation.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Exod 3:4--10 (Moses at the burning bush)
   - **Hebrew Bible:** Isa 6:8 ("Whom shall I send?"); Jonah (reluctant
     volunteer)
   - **Gospels:** Mk 1:17 ("Follow me" --- invitation, not command)
   - **Apostolic:** Calling/vocation theology (Luther, Calvin)
   - **Quran:** Prophetic calling (*nubuwwah*) as divine invitation
   - **Sanskrit:** Divine assignments in the Mahabharata and Ramayana
   - **Secular:** Whistleblower moments; moral exemplars (Bonhoeffer,
     Mandela)

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a15
   - **Required by:** t7, t8
   - **Model:** JUB · **Version:** iv_LLoL_OOv2r0p0_2026m03d20_m10


----


a21 --- Permanent Mediator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <jub-ax21>`

*God seeks one willing permanent translator between divine and human.*

Beyond the moment-specific calls of a20, God seeks one person willing
to permanently accept the role of translating between what God knows is
optimal and what humanity currently understands. The output quality depends
on it being freely chosen --- a compelled translator cannot produce genuine
translation.

**Formal statement:**

.. math::

   \exists\; & \text{voluntary-invitation from } G
     \text{ seeking } h \in H : \\
   & \text{Willing}(h) \\
   & \wedge\; \text{Permanent}\bigl(h,\;
     \text{TranslatorRole}(G, H, W_{\text{future}})\bigr) \\
   & \wedge\; \text{OutputQuality depends on }
     \neg\text{Force}(G, h)

**Context:**
a21 is the individual permanent analog of Quran 33:72 (*amanah*).
a20 handles transient individual calls; a21 handles the permanent mediation
role.

**Limit:**
The singularity of the permanent role is a strong claim. Multiple
sequential mediators may be the historical pattern.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Exod 20:19 (Moses as mediator); Deut 18:15--18
   - **Hebrew Bible:** Isa 53 (suffering servant); Mal 3:1 (messenger of
     the covenant)
   - **Gospels:** Mk 10:45 ("came not to be served but to serve");
     Jn 14:6; Heb 8:6
   - **Apostolic:** Christology; the permanent high priest (Heb 7:24)
   - **Quran:** Quran 33:72 (*amanah*); 33:40 (seal of prophets)
   - **Sanskrit:** *Avatar* concept --- divine descent to mediate
   - **Secular:** Public intellectual; systems translator;
     interdisciplinary bridge-builder

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a20
   - **Required by:** t7, t8
   - **Model:** JUB · **Version:** iv_LLoL_OOv2r0p0_2026m03d20_m10


----


a22 --- Divine Preference for Genuine Love
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <jub-ax22>`

*God's necessary nature values freely-chosen care above forced compliance.*

This is not merely a preference --- it is part of God's necessary
nature (Gn). It explains *why* God uses non-coercive guidance (a17) when
having the power to compel: compelled care produces a qualitatively different
divine experience (Gc) that the necessary divine nature (Gn) values less.
Forced love is not love, and God knows the difference.

**Formal statement:**

.. math::

   \forall w_1, w_2 \leq W \;\text{where}\;
   & w_1 = \text{freely-chosen-care}, \\
   & w_2 = \text{forced-compliance}, \\
   & \text{physical}(w_1) = \text{physical}(w_2) : \\[6pt]
   & G_n\text{-valuation}(G_c(w_1))
     > G_n\text{-valuation}(G_c(w_2))

**Context:**
This axiom connects to the a11 dipolar structure. Gc is affected by the
*quality* of human response --- freely-chosen love produces a qualitatively
different Gc state than forced compliance, even when the physical actions are
identical.

**Limit:**
Assumes God's Gn includes a valuation ranking. This is a substantive
metaphysical claim about divine nature.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Deut 6:5 ("love the LORD your God with all your heart")
   - **Hebrew Bible:** Ps 22:24; Hos 6:6 ("I desire mercy, not sacrifice")
   - **Gospels:** 1 Jn 4:8 ("God is love"); Jn 15:12--15 ("I have called
     you friends")
   - **Apostolic:** Agape theology; kenosis (Phil 2:5--11)
   - **Quran:** *Al-Wadud* (The Loving); Quran 5:54
   - **Sanskrit:** *Bhakti* tradition; *prema* (divine love) as highest
     value
   - **Secular:** Intrinsic vs. extrinsic motivation research; authenticity
     in relationships

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a15, a17
   - **Required by:** a23, t7, t9
   - **Model:** JUB · **Version:** iv_LLoL_OOv2r0p0_2026m03d20_m10
   - **Known objection:** "How do you know what God values?" Addressed by
     the cross-traditional convergence on love > compulsion.


----


a23 --- Freedom-Quality Superiority
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <jub-ax23>`

*Genuine care and innovation require freedom to achieve full quality.*

This is the empirical backbone of a22's explanation. Genuine care,
creative insight, and lasting innovation are qualitatively degraded or
destroyed by compulsion. A compelled factory worker can produce identical
widgets; a compelled poet cannot produce genuine poetry.

**Formal statement:**

.. math::

   \forall o \in O_{\text{genuine}} :
   \text{quality}(o \mid \text{compelled}) <
   \text{quality}(o \mid \text{freely-chosen})

**Context:**
Self-Determination Theory (Ryan & Deci, 2000) provides the secular
empirical backbone: autonomously motivated behavior produces higher quality
outcomes, greater creativity, more durable results, and better wellbeing.

**Limit:**
Applies specifically to outputs requiring creativity, empathy, or
genuine engagement --- not to all outputs.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Exod 35:21 ("everyone whose heart stirred them" ---
     voluntary contributions for the Tabernacle)
   - **Hebrew Bible:** Ps 23 (shepherd imagery --- "he leads me," not "he
     drives me"); Ps 110:3
   - **Gospels:** 2 Cor 9:7 ("God loves a cheerful giver"); Philemon
   - **Apostolic:** Theology of gift; worship as free response
   - **Quran:** *Ikhlas* (sincerity) as prerequisite for valid worship;
     Quran 2:256
   - **Sanskrit:** *Nishkama karma* (selfless action) vs. *sakama karma*
   - **Secular:** Self-Determination Theory (Ryan & Deci 2000); intrinsic
     motivation; the open-source movement

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a22
   - **Required by:** a24
   - **Model:** JUB · **Version:** iv_LLoL_OOv2r0p0_2026m03d20_m10
   - **Known objection:** "Some compelled systems produce good results"
     --- addressed by scoping to genuine/creative outputs only.


----


a24 --- Life-Trifecta of Lasting Innovation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <jub-ax24>`

*Innovation lasts iff stable, extensible, and life-friendly simultaneously.*

Three properties must be satisfied simultaneously --- violating any one
places the innovation on the BABL attractor (Blindly Assuming Blind
Leveraging), leading to eventual collapse. Three formulations:
ethical (Gentle/Kind/Reasonable), engineering (Stable/Extensible/Life-friendly),
negative (not oversimplifying / not overcomplicating / not overreaching).

**Formal statement:**

.. math::

   \text{Lasting}(i) \;\leftrightarrow\;
   \text{Stable}(i) \;\wedge\; \text{Extensible}(i) \;\wedge\;
   \text{LifeFriendly}(i)

**Context:**
BABL (Blindly Assuming Blind Leveraging) is the self-destructive
attractor. The river of life (Ezek 47, Rev 22:1--2) is the positive
attractor; the millstone/sea of destruction (Mt 18:6) is BABL. Bezalel
(Exod 31:2--5) is the positive Torah archetype.

**Limit:**
The predicates Stable, Extensible, LifeFriendly lack formal semantics.
The 7TrackRole model is identified as the formalization path but not yet
executed.

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Exod 31:2--5 (Bezalel: wisdom + understanding + knowledge);
     Gen 11:1--9 (Babel as BABL archetype)
   - **Hebrew Bible:** Prov 8 (wisdom as master craftsman); Eccl 3:1--8
   - **Gospels:** Mt 7:24--27 (house on rock vs. sand); Mt 18:6
     (millstone)
   - **Apostolic:** Common good theology; *Laudato Si'*
   - **Quran:** *Mizan* (balance); Quran 55:7--9
   - **Sanskrit:** *Rta* (cosmic order); dharma as sustainable pattern; the
     three *gunas*
   - **Secular:** Sustainability science; resilience theory; complex
     adaptive systems.
     **Capitalist stream:** Market competition as discovery procedure
     (Hayek); creative destruction (Schumpeter) --- but unregulated markets
     violate life-friendly.
     **Communist/Socialist stream:** Collective planning for life-friendly
     outcomes --- but central planning violates stable + extensible.

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a20, a21
   - **Required by:** a25, t8, t11
   - **Model:** JUB · **Version:** iv_LLoL_OOv2r0p0_2026m03d20_m10
   - **Critique:** :ref:`Con-D.3.5 <jub-con41>` (paradigm diversity)
   - **Defense:** :ref:`Pro-D.3.5 <jub-con41>`
   - **Known objection:** Vagueness of the three cords. Formal semantics
     are future work.
   - **Design:** Paradigm diversity note (see Kuhn 1962).


----


a25 --- Jubilee Recalibration
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

:ref:`PoR source <jub-ax25>`

*Innovation economies need periodic recalibration to prevent BABL-cascade.*

Even initially life-trifecta-compliant economies accumulate
concentration over time. Without periodic recalibration, the life-friendly
cord is eventually violated. The biblical Jubilee year (Lev 25, Deut 15) is the
Torah instantiation. a25 synthesizes capitalism's insight (incentive structures
between rounds) with communism's insight (redistribution mechanism), avoiding
either's fatal flaw.

**Formal statement:**

.. math::

   \forall\; & \text{innovation economy } E \text{ under a24} : \\
   \exists\; & \text{recalibration } R : \\
   & \text{periodic}(R) \;\wedge\;
     \text{redistributes}(R, E) \\
   & \wedge\; \text{resets}(R, E)
     \;\rightarrow\; \text{sustainable}(E)

**Context:**
The Jubilee System preserves property rights and incentives *between* rounds
(capitalism's contribution) while resetting accumulated advantages *at* each
round (communism's contribution). Neither ideology alone satisfies the
life-trifecta.

**Limit:**
Cross-traditional support requires qualification --- only Torah directly
supports the periodic reset mechanism. Arrow's impossibility constrains but
does not prohibit Jubilee System design (see :ref:`Con-E.2.11 <jub-con35>`).

.. dropdown:: Independent Support
   :class-title: sd-font-weight-normal

   - **Torah:** Lev 25 (Jubilee --- 50-year land return); Deut 15
     (Sabbatical year debt release)
   - **Hebrew Bible:** Isa 61:1--2 (proclaim Jubilee); Neh 5 (Nehemiah's
     debt reform)
   - **Gospels:** Lk 4:18--19 (Jesus reads Isa 61 --- programmatic Jubilee
     announcement)
   - **Apostolic:** Social encyclicals (*Rerum Novarum*, *Centesimus
     Annus*); liberation theology; universal destination of goods
   - **Quran:** *Zakat* (mandatory redistribution --- 2.5% annual wealth
     tax); prohibition of *riba* (usury)
   - **Sanskrit:** *Dana* (generosity as dharmic duty); cyclical renewal
     (*yugas*)
   - **Secular:** Progressive taxation; antitrust law; Piketty (r > g
     concentration dynamic).
     **Capitalist stream:** "Rules of the game" (Friedman) but resists
     systematic redistribution.
     **Communist/Socialist stream:** Redistribution as core principle but
     historical implementations destroyed incentive structures.

.. dropdown:: Network & Dependencies
   :class-title: sd-font-weight-normal

   - **Depends on:** a24
   - **Required by:** t8, t9, t11
   - **Model:** JUB · **Version:** iv_LLoL_OOv2r0p0_2026m03d20_m10
   - **Critique:** :ref:`Con-A.2.1 <jub-con25>`,
     :ref:`Con-A.2.2 <jub-con26>`,
     :ref:`Con-E.2.10 <jub-con34>`,
     :ref:`Con-E.2.11 <jub-con35>`
   - **Defense:** :ref:`Pro-A.2.1 <jub-con25>`,
     :ref:`Pro-A.2.2 <jub-con26>`,
     :ref:`Pro-G.2.10 <jub-con34>`,
     :ref:`Pro-E.2.11 <jub-con35>`
   - **Convergence:** Only Torah (Lev 25) directly supports periodic
     comprehensive recalibration. Islamic Zakat is continuous, Hindu *dana*
     is voluntary, secular perspectives address redistribution in general.
     This equivocation is honestly acknowledged.
   - **Known objection:** Arrow's impossibility theorem; political
     feasibility; historical absence of full Jubilee implementation.
   - **Design:** 2-leg Jubilee cycle design; commons-tragedy convergence.


----


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