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 Not Validated but Tested for the full argument). The text is preserved as-is.
PET Theorems#
Theorems derived from the PET axioms (PET Axioms ax1_A1–ax14_A14). These demonstrate that the axiom system is not inert — it produces substantive, verifiable consequences. For symbol definitions, see PET Symbol Dictionary.
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:
Proof: Suppose for contradiction that \(\Diamond\;(\exists W \wedge \neg\exists G)\). 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 \(\neg\,\Diamond\;(\exists W \wedge \neg\exists G)\). \(\blacksquare\)
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:
\(\Diamond\;(\exists G \;\wedge\; \neg\exists W)\) (God without world is possible)
\(\neg\,\Diamond\;(\exists W \;\wedge\; \neg\exists G)\) (world without God is impossible)
Proof:
First part: By ax6_A6 (Contingency of the World), \(\Diamond\;\neg\exists W\) — there are possible worlds without a world. By ax5_A5, G exists in every possible world, including those. So \(\Diamond\;(\exists G \wedge \neg\exists W)\).
Second part: Direct from th1_T1.
\(\blacksquare\)
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:
Proof: Let x be any part of the world, i.e., \(x \leq W\).
By ax4_A4 (Universal Immanence), \(x \leq W \rightarrow x \leq G\), so \(x \leq G\).
By ax8_A8 (Immanent Presence), \(x \leq W \rightarrow P(G, x)\), so \(P(G, x)\).
Therefore \(P(G, x) \wedge x \leq G\). \(\blacksquare\)
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:
Proof: By ax11_A11 line 4, for all \(w_1, w_2 \leq W\): \(w_1 \neq w_2 \rightarrow G_c(w_1) \neq G_c(w_2)\). This is a direct consequence of the strengthened ax11_A11, which gives \(G_c\) a functional structure indexed by subworlds that is injective (distinct inputs yield distinct outputs). \(\blacksquare\)
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 \(G_c\) 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) \(\wedge\) ax16_A16 (delegation) \(\wedge\) ax17_A17 (non-coercive guidance) \(\wedge\) ax18_A18 (responsibility localization):
\(\text{Responsible}(H, \text{outcomes in } D_{\text{inno}}) \;\wedge\; \neg\text{Responsible}(G, \text{outcomes in } D_{\text{inno}})\)
Proof:
By ax16_A16, humans are the primary responsible agents for Earth’s outcomes.
By ax15_A15, humans have genuine capacity to choose otherwise in D_free.
By ax17_A17, God’s influence takes the form of guidance (not compulsion).
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.
By ax18_A18, moral responsibility for such outcomes lies with the human agents.
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.
Therefore God is not responsible for the evil that results from humans refusing to act on divine guidance within D_inno. \(\blacksquare\)
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:
Proof:
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).
By ax16_A16, humans collectively hold delegated authority and responsibility for Earth.
By ax15_A15, h* can choose to act or not act within D_free.
Therefore h* bears, at time t, the maximum human causal responsibility for the future trajectory of Earth — whether or not they know it. \(\blacksquare\)
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) \(\wedge\) ax17_A17 (non-coercive) \(\wedge\) ax19_A19 (h* exists) \(\wedge\) ax20_A20 (transient volunteer sought) \(\wedge\) ax21_A21 (permanent mediator sought):
God’s posture = urgent(\(A9 + A11\)) \(\wedge\) non-coercive(\(A17\)) \(\wedge\) invitation(\(A20 + A21\))
Proof:
By ax9_A9 + ax11_A11 + th4_T4, God actively cares about outcomes and is not indifferent — creating urgency.
By ax17_A17, God cannot and will not compel a human to take responsibility — preserving non-coercion.
By ax19_A19, there is always a h* whose willing action could most change the trajectory.
By ax20_A20, God specifically seeks willing humans for transient roles.
By ax21_A21, God seeks one willing to take the permanent mediation role.
Therefore God’s action is consistent with presenting opportunities, guidance, and invitations to humans (especially h*) without compelling them. \(\blacksquare\)
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:
Proof:
By ax24_A24, lasting innovation requires all three cords simultaneously.
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
Each failure mode is self-compounding. The violated cord destabilizes the remaining cords.
Therefore only full satisfaction (river of life) or violation (BABL) are stable states. \(\blacksquare\)
Axioms used: ax24_A24
Empirical verification:
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.
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) \(\wedge\) ax15_A15 (agency requires stable substrate) \(\wedge\) ax18_A18 (responsibility localizes to agent):
\(S(G, W_{\text{physics}}) \;\wedge\; \text{weaponize}(h, W_{\text{physics}}) \;\rightarrow\; \text{Responsible}(h) \;\wedge\; \neg\text{Responsible}(G)\)
Proof:
ax15_A15 requires a stable physical substrate: without reliable physics, humans cannot form intentions, predict consequences, or exercise genuine choice.
ax9_A9 provides this substrate through sustaining.
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.
By ax18_A18, responsibility localizes to the agent.
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. \(\blacksquare\)
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) \(\wedge\) ax15_A15 (genuine agency) \(\wedge\) ax24_A24 (life-trifecta → BABL for violations) \(\wedge\) ax25_A25 (Jubilee windows that close):
Irreversibility(time) \(\wedge\) closing(Jubilee-windows) \(\rightarrow\) genuine-stakes(ax15_A15-choices)
Death is sufficient but \(\neg\) necessary for finitude.
Proof:
ax6_A6 (contingency of the world) establishes temporal finitude: the world exists within time, and time’s arrow makes past moments irrecoverable.
ax15_A15 (agency) means choices have real consequences that propagate forward.
ax24_A24 (life-trifecta) means wrong choices lead to BABL — real, not hypothetical, collapse.
ax25_A25 (Jubilee) provides windows that close: missing a Jubilee window means living with consequences until the next cycle.
The combination of irreversible time + closing windows + real collapse trajectories provides genuine stakes.
Biological death strengthens these stakes but is not a prerequisite — the stakes already exist from temporal irreversibility and Jubilee cycles alone. \(\blacksquare\)
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.
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 TELES Axiom/Theorem Compound Naming — Execution Prompt for the complete mapping table and DD b12 — Legacy Naming for PET/JUB Axioms and Theorems for the permanent reference.