.. Migration note (2026m04d04): Claude copied this file during VV-to-HELL migration.
   Old path: ``vv/pet/ppv1/theorems.rst`` (as given by LLoL)
   New path: ``hell/mm/b/11/PPv1/theorems.rst`` (as chosen by Claude)
   Category: PET PPv1 snapshot

.. meta::
   :description: Four theorems derived from PET axioms ax1_A1-ax14_A14: no godless creation, asymmetric ontological priority, no isolation, and divine experience variation.
   :keywords: PET, theorems, th1_T1-th4_T4, proofs, no godless creation, ontological priority, no isolation, divine experience, PPv1, modal logic, mereology
   :author: Yah, Yas, everyone, LLoL as Laurence Loewe of Laodicea, ClaudeOp46Max, Anthropic, and Spirit of Boolean Truth
   :og:card:title: PET Theorems th1_T1-th4_T4<br>Derived from Axioms
   :og:card:description: The axiom system is not inert. These four theorems prove substantive consequences about creation, divine priority, and God's relationship to the world.

.. SOCIAL-CARD-QUALITY-COMPARE --- OO (default effort) vs PP (max effort), 2026-03-26
   OO :description: Theorems th1_T1-th4_T4 derived from PET axioms ax1_A1-ax14_A14, demonstrating substantive consequences of the panentheistic axiom system.
   OO :keywords: matheology, PET, theorems, th1_T1-th4_T4, proofs, divine necessity, ontological priority, axiom consequences, PPv1, modal logic
   OO :og:card:title: PET Theorems th1_T1-th4_T4 Derived Results
   OO :og:card:description: Four theorems derived from PET axioms covering divine necessity, asymmetric ontological priority, preservation, and empathic knowledge.
   PP :description: Four theorems derived from PET axioms ax1_A1-ax14_A14: no godless creation, asymmetric ontological priority, no isolation, and divine experience variation.
   PP :keywords: PET, theorems, th1_T1-th4_T4, proofs, no godless creation, ontological priority, no isolation, divine experience, PPv1, modal logic, mereology
   PP :og:card:title: PET Theorems th1_T1-th4_T4<br>Derived from Axioms
   PP :og:card:description: The axiom system is not inert. These four theorems prove substantive consequences about creation, divine priority, and God's relationship to the world.

.. SOCIAL-CARD-REVIEW --- generated by Claude Opus 4.6, 2026-03-26
   dv_ClaOp46_PP_2026m03d26 --- max-effort rewrite, read full page.
   :description: 146 chars | :og:card:title: 37 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

.. note:: **Editorial note (2026-03-24).**
   This is a point-in-time snapshot. It uses "validated," "verified,"
   and similar terms in places where the author's long-standing practice
   is to say "tested" or "checked" (see :ref:`not-tested-not-validated`
   for the full argument). The text is preserved as-is.

*****************************
PET Theorems
*****************************

Theorems derived from the PET axioms (:doc:`axioms`).
These demonstrate that the axiom system is not inert — it produces
substantive, verifiable consequences. For symbol definitions, see :doc:`symbols`.


th1_T1 — No Godless Creation
============================

*It is impossible for a world to exist without God existing.*

**In plain English:** There is no possible scenario in which the created world
exists but God does not. Any world that exists, exists alongside God.

**Formal statement:**

.. math::

   \neg\,\Diamond\;(\exists W \;\wedge\; \neg\exists G)

**Proof:** Suppose for contradiction that
:math:`\Diamond\;(\exists W \wedge \neg\exists G)`. Then there is some
possible world *w* in which W exists but G does not. But by ax5_A5 (Necessary
Divine Existence), G exists in every possible world. This contradicts our
supposition. Therefore :math:`\neg\,\Diamond\;(\exists W \wedge \neg\exists G)`.
:math:`\blacksquare`

**Axioms used:** ax5_A5

**Significance:** th1_T1 rules out atheistic cosmologies *within the PET
framework*. Note that this is a logical consequence of accepting ax5_A5, not an
independent empirical claim. If you reject ax5_A5, th1_T1 does not follow.


th2_T2 — Asymmetric Ontological Priority
========================================

*God can exist without the world, but the world cannot exist without God.*

**In plain English:** There are possible scenarios where God exists alone
(without any created world), but there is no possible scenario where the world
exists without God. The dependence runs strictly one way.

**Formal statement:**

   :math:`\Diamond\;(\exists G \;\wedge\; \neg\exists W)` (God without world
   is possible)

   :math:`\neg\,\Diamond\;(\exists W \;\wedge\; \neg\exists G)` (world without
   God is impossible)

**Proof:**

- **First part:** By ax6_A6 (Contingency of the World),
  :math:`\Diamond\;\neg\exists W` — there are possible worlds without a world.
  By ax5_A5, G exists in every possible world, including those. So
  :math:`\Diamond\;(\exists G \wedge \neg\exists W)`.
- **Second part:** Direct from th1_T1.

:math:`\blacksquare`

**Axioms used:** ax5_A5, ax6_A6 (and th1_T1)

**Significance:** th2_T2 establishes a strict ontological hierarchy. This is not a
claim about temporal priority (God existing "before" the world in time) but
about modal priority: God's existence is unconditional, while the world's
existence is conditional on God.


th3_T3 — No Isolated Part of Creation
=====================================

*Every part of creation is both contained in God and present to God.*

**In plain English:** Nothing in the created world is hidden from God or
outside God. Every single part of creation is simultaneously within God and
known to God.

**Formal statement:**

.. math::

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

**Proof:** Let *x* be any part of the world, i.e., :math:`x \leq W`.

- By ax4_A4 (Universal Immanence), :math:`x \leq W \rightarrow x \leq G`, so
  :math:`x \leq G`.
- By ax8_A8 (Immanent Presence), :math:`x \leq W \rightarrow P(G, x)`, so
  :math:`P(G, x)`.

Therefore :math:`P(G, x) \wedge x \leq G`. :math:`\blacksquare`

**Axioms used:** ax4_A4, ax8_A8

**Significance:** th3_T3 combines the mereological claim (containment) with the
relational claim (presence) to show they apply universally and simultaneously.
This rules out scenarios where part of creation is "in" God but unknown to God,
or known to God but somehow "outside" God.


th4_T4 — Divine Experience Varies
=================================

*Different worlds produce different divine experiences.*

**In plain English:** If two parts of the world differ, then God's concrete
experience of them differs as well. God is not indifferent to what happens —
different realities produce genuinely different divine responses.

**Formal statement:**

.. math::

   \forall w_1, w_2 \leq W : w_1 \neq w_2 \;\rightarrow\; G_c(w_1) \neq G_c(w_2)

**Proof:** By ax11_A11 line 4, for all :math:`w_1, w_2 \leq W`:
:math:`w_1 \neq w_2 \rightarrow G_c(w_1) \neq G_c(w_2)`. This is a direct
consequence of the strengthened ax11_A11, which gives :math:`G_c` a functional
structure indexed by subworlds that is injective (distinct inputs yield distinct
outputs). :math:`\blacksquare`

**Axioms used:** ax11_A11 (strengthened, lines 3–4)

**Note on derivability:** In the original formulation of ax11_A11 (2026-03-11),
th4_T4 was stated as a "proof sketch" because the original ax11_A11 did not formally
guarantee that :math:`G_c` varies injectively with subworlds. The strengthened
ax11_A11 (2026-03-14/15) added lines 3–4 specifically to make th4_T4 a rigorous
consequence rather than an informal argument.

**Significance:** th4_T4 is the formal expression of divine responsiveness. If a
world contains suffering, God's experience of that world includes the
suffering. If a world contains joy, God's experience includes the joy. This
is the core insight of dipolar theism: God is not a static, unaffected
observer but a being whose concrete experience genuinely varies with what
happens in creation.

th4_T4 only follows from ax11_A11 (Dipolarity), not from ax11b_A11b (Simplicity). Under ax11b_A11b,
the question of whether God's experience varies is unanswerable, because ax11b_A11b
asserts that God has no independently analyzable aspects.


----


.. note::

   **th5_T5--th11_T11 are proposed and in development,** derived from the proposed
   Group VI axioms (ax15_A15--ax25_A25). They have not undergone the same review
   as th1_T1--th4_T4.


th5_T5 — Divine Non-Responsibility
===================================

*God is formally exonerated for the evil that results from human failure to
innovate responsibly within D_inno.*

**In plain English:** When humans fail to pursue life-trifecta-compliant
innovation despite having genuine agency, delegated authority, non-coercive
guidance, and causal leverage, the responsibility rests with the human agents,
not with God. God remained present (ax8_A8), sustaining (ax9_A9), affected by outcomes
(ax11_A11/th4_T4), and offering guidance (ax17_A17). The human agent chose otherwise.

**Formal statement:**

   Given ax15_A15 (agency) :math:`\wedge` ax16_A16 (delegation) :math:`\wedge`
   ax17_A17 (non-coercive guidance) :math:`\wedge` ax18_A18 (responsibility
   localization):

   :math:`\text{Responsible}(H, \text{outcomes in } D_{\text{inno}})
   \;\wedge\; \neg\text{Responsible}(G, \text{outcomes in } D_{\text{inno}})`

**Proof:**

1. By ax16_A16, humans are the primary responsible agents for Earth's outcomes.
2. By ax15_A15, humans have genuine capacity to choose otherwise in D_free.
3. By ax17_A17, God's influence takes the form of guidance (not compulsion).
4. When humans refuse divine guidance in D_inno (by ax15_A15, a genuine
   possibility), they exercise ax15_A15-agency against ax17_A17-guidance within their
   ax16_A16-domain.
5. By ax18_A18, moral responsibility for such outcomes lies with the human agents.
6. God, by continuing to provide guidance (ax17_A17), caring about outcomes
   (ax9_A9 + ax11_A11), and remaining present (ax8_A8), has acted consistently with the
   delegation model.
7. Therefore God is not responsible for the evil that results from humans
   refusing to act on divine guidance within D_inno. :math:`\blacksquare`

**Axioms used:** ax8_A8, ax9_A9, ax11_A11, ax15_A15, ax16_A16, ax17_A17, ax18_A18

**Significance:** th5_T5 is the formal core of the innovation theodicy. It
addresses evil arising from human choices in D_inno specifically. It does not
address natural evil, animal suffering, or suffering in D_f. The scope is
deliberately narrow: one form of evil addressed completely rather than all
forms addressed partially.


th6_T6 — Causal Concentration
=============================

*At each moment, a unique individual h* bears maximum causal responsibility
for the future trajectory of Earth.*

**In plain English:** The leviathan chain has a single top link at each
moment. One person's choices carry more causal weight for the future than
anyone else's — whether or not that person knows it.

**Formal statement:**

.. math::

   \forall t\; \exists!\, h^* \in H :
   \text{MaxCausalResponsibility}(h^*, t) \;\wedge\;
   \forall h \neq h^* :
   \text{CausalResponsibility}(h, t) < \text{CausalResponsibility}(h^*, t)

**Proof:**

1. By ax19_A19, at any time *t* there is a unique h* at the maximum causal
   influence position (null hypothesis: uniqueness is the default;
   equivalence in effect is measure-zero).
2. By ax16_A16, humans collectively hold delegated authority and responsibility
   for Earth.
3. By ax15_A15, h* can choose to act or not act within D_free.
4. Therefore h* bears, at time *t*, the maximum human causal responsibility
   for the future trajectory of Earth — whether or not they know it.
   :math:`\blacksquare`

**Axioms used:** ax15_A15, ax16_A16, ax19_A19

**Significance:** th6_T6 establishes that responsibility is not evenly
distributed. The person with the most causal leverage bears the most
responsibility — even without awareness of their position (Judas), and even
though the position is not permanent (Peter/Judas transition).


th7_T7 — God Seeks a Volunteer
==============================

*God's posture toward humanity is urgent non-coercive invitation.*

**In plain English:** God cares about outcomes (sustaining + affected by
them), cannot and will not compel (non-coercive guidance), and knows that a
uniquely positioned h* exists at each moment. The combination of urgency
with principled non-coercion yields the characteristic divine posture: urgent
invitation, seeking a willing volunteer.

**Formal statement:**

   Given ax9_A9 + ax11_A11 + th4_T4 (God cares and is affected) :math:`\wedge`
   ax17_A17 (non-coercive) :math:`\wedge` ax19_A19 (h* exists) :math:`\wedge`
   ax20_A20 (transient volunteer sought) :math:`\wedge`
   ax21_A21 (permanent mediator sought):

   God's posture = urgent(:math:`A9 + A11`) :math:`\wedge`
   non-coercive(:math:`A17`) :math:`\wedge`
   invitation(:math:`A20 + A21`)

**Proof:**

1. By ax9_A9 + ax11_A11 + th4_T4, God actively cares about outcomes and is not
   indifferent — creating urgency.
2. By ax17_A17, God cannot and will not compel a human to take responsibility —
   preserving non-coercion.
3. By ax19_A19, there is always a h* whose willing action could most change the
   trajectory.
4. By ax20_A20, God specifically seeks willing humans for transient roles.
5. By ax21_A21, God seeks one willing to take the permanent mediation role.
6. Therefore God's action is consistent with presenting opportunities,
   guidance, and invitations to humans (especially h*) without compelling
   them. :math:`\blacksquare`

**Axioms used:** ax9_A9, ax11_A11, ax17_A17, ax19_A19, ax20_A20, ax21_A21 (and th4_T4)

**Significance:** th7_T7 explains the characteristic biblical pattern: God calls,
invites, pleads, warns, demonstrates — but does not compel. The burning bush
(Exod 3) is the model: the bush burns (God's initiative), Moses turns aside
(human agency), and the call follows the voluntary attention. The pattern
recurs throughout scripture: "Whom shall I send?" (Isa 6:8), "Follow me"
(Mk 1:17), "Behold, I stand at the door and knock" (Rev 3:20).


th8_T8 — Binary Attractors
==========================

**Informal alias:** T-Inno

*Innovation trajectories converge to exactly one of two attractors:
river-of-life (all three life-trifecta cords satisfied) or BABL (any cord
violated). There is no stable middle ground.*

**In plain English:** There are only two stable long-term states for any
innovation system. Either all three cords of ax24_A24 are satisfied
simultaneously (the river-of-life attractor), or at least one cord is
violated and the system is on the BABL trajectory toward eventual collapse.
Partial compliance is transitionally stable but not genuinely stable.

**Formal statement:**

.. math::

   \forall \text{ innovation } i :
   \text{long-term}(i) \rightarrow
   (\text{Stable}(i) \wedge \text{Extensible}(i) \wedge \text{LifeFriendly}(i))
   \;\oplus\;
   \text{BABL-attractor}(i)

**Proof:**

1. By ax24_A24, lasting innovation requires all three cords simultaneously.
2. Partial satisfaction is unstable: violating any one cord creates structural
   debt that compounds under real-world pressure:

   - Violating stable: system must be artificially propped up as complexity
     grows — brittleness failure
   - Violating extensible: system accumulates layers of workarounds —
     complexity death
   - Violating life-friendly: system extracts more than it returns — resource
     exhaustion

3. Each failure mode is self-compounding. The violated cord destabilizes the
   remaining cords.
4. Therefore only full satisfaction (river of life) or violation (BABL) are
   stable states. :math:`\blacksquare`

**Axioms used:** ax24_A24

**Empirical verification:**

.. list-table::
   :header-rows: 1
   :widths: 25 25 25 25

   * - System
     - Cord violated
     - th8_T8 prediction
     - Historical outcome
   * - Soviet communism
     - Stable + Extensible
     - Faster BABL, system collapse
     - 1991 collapse
   * - Unregulated capitalism
     - Life-friendly
     - BABL accumulation, inequality cascade
     - Gilded Age, 2008, wealth concentration
   * - Jubilee capitalism (ax25_A25)
     - None — all three
     - River of life attractor
     - Not yet fully tried; Nordic social democracy approximates

**Significance:** th8_T8 is the first empirically testable theorem in the PET
system. Its predictions match the two major economic experiments of the 20th
century. No historical example of a system violating one cord indefinitely
without collapse has been identified.


th9_T9 — Social Ergodicity
==========================

**Informal alias:** T-Ergodicity

*God ensures universal justice through system-level ergodicity enforcement
(Jubilee) without overriding individual agency.*

**In plain English:** Over sufficient time, Jubilee recalibration ensures
that every participant visits both the highest and lowest positions in the
causal hierarchy. This is social ergodicity: the time average of any
individual's outcomes converges to the ensemble average. No one is
permanently trapped by historical accident, even though individual agency is
preserved between Jubilee rounds.

**Formal statement:**

   Given ax22_A22 (God values genuine love) :math:`\wedge` ax9_A9 (sustaining
   capacity) :math:`\wedge` ax25_A25 (Jubilee recalibration):

   :math:`\forall h \in H : \lim_{T \to \infty}
   \frac{1}{T}\sum_{t=1}^{T} \text{position}(h, t)
   \;=\; \text{ensemble-average}(\text{position})`

   achieved without violating ax15_A15 or ax17_A17 at any moment.

**Proof:**

1. By ax22_A22, God's necessary nature values the Gc states arising from genuine
   care and is motivated to ensure justice.
2. By ax9_A9, God has the capacity to sustain the conditions under which
   ergodicity operates.
3. By ax25_A25, Jubilee recalibration periodically resets accumulated advantage,
   preventing permanent winner-takes-all concentration.
4. Over sufficient time (including eschatological time), Jubilee
   recalibration ensures every participant visits both high and low positions.
5. God enforces this at the system level (through ax25_A25's recalibration
   mechanism) rather than at the individual event level (which would require
   overriding ax15_A15).
6. Therefore universal justice is served without violating ax15_A15 or ax17_A17.
   :math:`\blacksquare`

**Axioms used:** ax9_A9, ax15_A15, ax22_A22, ax25_A25

**Key reference:** Ole Peters (2019) on the distinction between time averages
and ensemble averages in economics. Non-ergodic systems can appear fair in
ensemble average while being systematically unjust for individuals over time.
Jubilee enforces ergodicity.

**Significance:** The divine kenosis (Phil 2:5–11 — God going from highest
to lowest to highest) is the demonstration of the ergodic pattern at the
divine level, establishing the pattern that all are invited to replicate
(Mk 10:43–45). th9_T9 answers the distribution problem: innocent suffering is not
the final word because ergodicity guarantees that position is not permanent.


th10_T10 — Physical Law Substrate
=================================

**Informal alias:** T-Physics

*God's maintenance of physical law is a precondition of agency, not a cause
of harm when physics is weaponized.*

**In plain English:** Without consistent physical law, choices would have
unpredictable effects and genuine agency would be incoherent — you cannot be
responsible for outcomes you cannot reliably connect to your choices.
Therefore God's sustaining of physics (ax9_A9) serves agency (ax15_A15), not harm.
When an agent uses physics to harm another, the cause of harm is the agent's
choice, not God's maintenance of the substrate.

**Formal statement:**

   Given ax9_A9 (God sustains physics) :math:`\wedge` ax15_A15 (agency requires
   stable substrate) :math:`\wedge` ax18_A18 (responsibility localizes to agent):

   :math:`S(G, W_{\text{physics}}) \;\wedge\;
   \text{weaponize}(h, W_{\text{physics}}) \;\rightarrow\;
   \text{Responsible}(h) \;\wedge\; \neg\text{Responsible}(G)`

**Proof:**

1. ax15_A15 requires a stable physical substrate: without reliable physics, humans
   cannot form intentions, predict consequences, or exercise genuine choice.
2. ax9_A9 provides this substrate through sustaining.
3. When a human agent weaponizes physics (e.g., using gravity to push someone
   off a cliff), the proximate cause is the agent's choice within D_free, not
   God's maintenance of gravity.
4. By ax18_A18, responsibility localizes to the agent.
5. Holding God responsible for maintaining physics would require holding God
   responsible for every physical effect — which eliminates the concept of
   human agency (ax15_A15) entirely. :math:`\blacksquare`

**Axioms used:** ax9_A9, ax15_A15, ax18_A18

**Significance:** th10_T10 resolves the specific objection "but God controls
gravity, and gravity kills people." The domain split (D_f vs. D_free) is
essential: physics operates in D_f as a precondition for agency, not as
divine coercion within D_free.


th11_T11 — Stakes Without Death
===============================

**Informal alias:** T-Finitude

*Temporal irreversibility combined with Jubilee windows provides genuine
stakes without requiring biological death.*

**In plain English:** Yesterday's missed innovation cannot be recaptured
today. A Jubilee window that closes carries real consequences into the next
cycle. These two sources of irreversibility — the arrow of time and the
closing of Jubilee rounds — provide all the urgency and genuine stakes that
ax15_A15-agency requires. Biological death is one implementation of finitude but
is not logically necessary for genuine stakes.

**Formal statement:**

   Given ax6_A6 (contingency → temporal finitude) :math:`\wedge`
   ax15_A15 (genuine agency) :math:`\wedge`
   ax24_A24 (life-trifecta → BABL for violations) :math:`\wedge`
   ax25_A25 (Jubilee windows that close):

   Irreversibility(time) :math:`\wedge` closing(Jubilee-windows)
   :math:`\rightarrow` genuine-stakes(ax15_A15-choices)

   Death is sufficient but :math:`\neg` necessary for finitude.

**Proof:**

1. ax6_A6 (contingency of the world) establishes temporal finitude: the world
   exists within time, and time's arrow makes past moments irrecoverable.
2. ax15_A15 (agency) means choices have real consequences that propagate forward.
3. ax24_A24 (life-trifecta) means wrong choices lead to BABL — real, not
   hypothetical, collapse.
4. ax25_A25 (Jubilee) provides windows that close: missing a Jubilee window means
   living with consequences until the next cycle.
5. The combination of irreversible time + closing windows + real collapse
   trajectories provides genuine stakes.
6. Biological death strengthens these stakes but is not a prerequisite — the
   stakes already exist from temporal irreversibility and Jubilee cycles
   alone. :math:`\blacksquare`

**Axioms used:** ax6_A6, ax15_A15, ax24_A24, ax25_A25

**Significance:** th11_T11 decouples the theodicy from assumptions about afterlife.
The argument works whether or not biological death is the final word. This
matters because it allows the innovation theodicy to stand independently of
eschatological commitments that differ across traditions.


.. admonition:: TELES migration report (2026m04d04)

   Mechanical identifier migration applied to this file.
   All axiom/theorem text references were migrated from short form
   (e.g., A15) to compound form (e.g., ax15_A15) as part of the
   matheology compound naming operation. Both forms refer to the
   same formal object. The old form survives as the suffix to
   ensure consistency with the oldest records; the new form adds
   a temporary-status prefix. Forward-facing pages use brief form
   (ax15) only. See
   :ref:`hell-ll-other-b15-teles-renaming-prompt` for the complete
   mapping table and :ref:`legacy-5d-link-names-table-for-pet-jub-model` for the permanent
   reference.