Theorems — Expert View (All Models)#

This compiled view presents all 11 theorems of mathematical theology — derived consequences that follow logically from the axiom system. Theorems are not assumptions; they are results. If the axioms hold, the theorems hold. For the beginner-friendly version, see Theorems — What the Axioms Prove.

PoR sources:

PET Theorems — t1–t4 (PET foundation theorems)
JUB Theorems — th5–th11 — t5–t11 (JUB extension theorems)

Navigation: t1 · t2 · t3 · t4 · t5 · t6 · t7 · t8 · t9 · t10 · t11

All 11 Theorems at a Glance#
id	Title	Summary
t1	No Godless Creation	It is impossible for a world to exist without God existing.
t2	Asymmetric Ontological Priority	God can exist without the world, but the world cannot exist without God.
t3	No Isolated Part of Creation	Every part of creation is both contained in God and present to God.
t4	Divine Experience Varies	Different worlds produce different divine experiences.
t5	Divine Non-Responsibility	God is not responsible for evil resulting from human failure to innovate in D_inno.
t6	Causal Concentration	At each moment, a unique individual bears maximum causal responsibility for Earth’s trajectory.
t7	God Seeks a Volunteer	God’s posture toward humanity is urgent non-coercive invitation.
t8	Binary Attractors	Innovation converges to river-of-life or BABL; no stable middle ground.
t9	Social Ergodicity	Jubilee enforces universal justice through system-level ergodicity without overriding agency.
t10	Physical Law Substrate	Physics is a precondition of agency, not a cause of harm when weaponized.
t11	Stakes Without Death	Temporal irreversibility plus Jubilee windows provides genuine stakes without requiring death.

Group A — Consequences of God Containing the World (PET)#

These four theorems follow from the first 14 axioms (a1–a14). They establish that creation cannot exist without God, that God has ontological priority, that nothing in creation is hidden from God, and that God is genuinely affected by what happens.

t1 — No Godless Creation#

PoR source

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

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

Formal statement:

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

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 (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

Significance: th1 rules out atheistic cosmologies within the PET framework. Note that this is a logical consequence of accepting ax5, not an independent empirical claim. If you reject ax5, th1 does not follow.

t2 — Asymmetric Ontological Priority#

PoR source

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

Context: 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 (Contingency of the World), \(\Diamond\;\neg\exists W\) — there are possible worlds without a world. By ax5, G exists in every possible world, including those. So \(\Diamond\;(\exists G \wedge \neg\exists W)\).
Second part: Direct from th1.

\(\blacksquare\)

Axioms used: ax5, ax6 (and th1)

Significance: th2 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.

t3 — No Isolated Part of Creation#

PoR source

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

Context: 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:

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

Proof: Let x be any part of the world, i.e., \(x \leq W\).

By ax4 (Universal Immanence), \(x \leq W \rightarrow x \leq G\), so \(x \leq G\).
By ax8 (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, ax8

Significance: th3 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.

t4 — Divine Experience Varies#

PoR source

Different worlds produce different divine experiences.

Context: 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:

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

Proof: By ax11 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, which gives \(G_c\) a functional structure indexed by subworlds that is injective (distinct inputs yield distinct outputs). \(\blacksquare\)

Note on derivability: In the original formulation of ax11 (2026-03-11), th4 was stated as a “proof sketch” because the original ax11 did not formally guarantee that \(G_c\) varies injectively with subworlds. The strengthened ax11 (2026-03-14/15) added lines 3–4 specifically to make th4 a rigorous consequence rather than an informal argument.

Axioms used: ax11 (strengthened, lines 3–4)

Significance: th4 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 only follows from ax11 (Dipolarity), not from ax11b (Simplicity). Under ax11b, the question of whether God’s experience varies is unanswerable, because ax11b asserts that God has no independently analyzable aspects.

Group B — The Innovation Theodicy (JUB)#

These seven theorems follow from all 25 axioms (a1–a25). They build the formal case that evil results from human failure to innovate, not from divine indifference — and that a Jubilee-based system is the structural solution.

Note

Proto-formal status. th5–th11 are derived from the proposed Group VI axioms (ax15–ax25). Unlike th1–th4 (which use mereological and modal apparatus with well-defined CEM/S5 semantics), th5–th11’s predicates (Stable, Extensible, LifeFriendly, etc.) lack formal semantics and the proofs are not yet machine-checkable. See the JUB PoR status note for the full formalization roadmap.

t5 — Divine Non-Responsibility#

PoR source

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

Context: 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), sustaining (ax9), affected by outcomes (ax11/th4), and offering guidance (ax17). The human agent chose otherwise.

Formal statement:

Given ax15 (agency) \(\wedge\) ax16 (delegation) \(\wedge\) ax17 (non-coercive guidance) \(\wedge\) ax18 (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, humans are the primary responsible agents for Earth’s outcomes.
By ax15, humans have genuine capacity to choose otherwise in D_free.
By ax17, God’s influence takes the form of guidance (not compulsion).
When humans refuse divine guidance in D_inno (by ax15, a genuine possibility), they exercise ax15-agency against ax17-guidance within their ax16-domain.
By ax18, moral responsibility for such outcomes lies with the human agents.
God, by continuing to provide guidance (ax17), caring about outcomes (ax9 + ax11), and remaining present (ax8), 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, ax9, ax11, ax15, ax16, ax17, ax18

Significance: th5 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.

t6 — Causal Concentration#

PoR source

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

Context: 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:

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

Proof:

By ax19, 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, humans collectively hold delegated authority and responsibility for Earth.
By ax15, 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, ax16, ax19

Significance: th6 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).

t7 — God Seeks a Volunteer#

PoR source

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

Context: 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 + ax11 + th4 (God cares and is affected) \(\wedge\) ax17 (non-coercive) \(\wedge\) ax19 (h* exists) \(\wedge\) ax20 (transient volunteer sought) \(\wedge\) ax21 (permanent mediator sought):

God’s posture = urgent(\(A9 + A11\)) \(\wedge\) non-coercive(\(A17\)) \(\wedge\) invitation(\(A20 + A21\))

Proof:

By ax9 + ax11 + th4, God actively cares about outcomes and is not indifferent — creating urgency.
By ax17, God cannot and will not compel a human to take responsibility — preserving non-coercion.
By ax19, there is always a h* whose willing action could most change the trajectory.
By ax20, God specifically seeks willing humans for transient roles.
By ax21, 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, ax11, ax17, ax19, ax20, ax21 (and th4)

Significance: th7 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).

Note

Secular convergence. Even in a purely secular framing, th7’s functional conclusion stands: structural reform requires someone willing to go first — champions who accept personal political risk to advance a collective good. Whether called “volunteers responding to divine invitation” or “moral leaders with unusual courage,” the need for individual initiative is the same. See Con-E.12 / Pro-F.12 in Quest: Jubilee-Based Innovation Economy.

t8 — Binary Attractors#

PoR source

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.

Context: There are only two stable long-term states for any innovation system. Either all three cords of ax24 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:

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

Proof:

By ax24, 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. The formal basis:

3a — Why oscillation cannot persist. An economy that oscillates between cord-compliance and cord-violation (the Kuznets wave scenario) periodically approaches the BABL boundary. Each oscillation is a repeated game against stochastic extinction. In individual-based stochastic systems (as opposed to continuous deterministic ODE models), zero is an absorbing state — once a critical threshold is crossed, the system cannot recover:

\[P(\text{survive } N \text{ cycles}) = \prod_{k=1}^{N} p_k \;\to\; 0 \quad \text{as } N \to \infty\]

Even if each cycle’s survival probability \(p_k\) is close to 1, eventual absorption is certain. The “stable middle ground” is a metastable state with finite lifetime.

3b — Technological amplification accelerates convergence. Crucially, \(p_k\) is not constant but decreasing over time: nuclear weapons, AI capabilities, planetary-scale environmental modification, and other emerging technologies amplify the damage potential of each oscillation trough. The RiskyMADorMAP CTMC model (formally equivalent to Michaelis-Menten enzyme kinetics; see SD1) estimates median time to BABL absorption at ~19 years (middle estimate) from Cold War data (4 near-miss crises in 40 years). Nuclear risk is only one pathway; AI risk, climate tipping points, engineered pandemics, and the other 7DUIs add additional independent or correlated extinction pathways.

3c — Civilization as a single coupled system. th8’s “for all innovation i” applies to civilization-as-a-whole because civilization is not a portfolio of independent innovations but a single tightly coupled system with shared infrastructure (energy grids, financial systems, supply chains, communication networks, ecosystem services). Failure in one subsystem cascades through the entire network (Helbing 2013; Buldyrev et al. 2010).
Therefore only full satisfaction (river of life) or violation (BABL) are stable states. The oscillatory middle ground is metastable with finite — and alarmingly short — expected lifetime. \(\blacksquare\)

Axioms used: ax24

Empirical testing:

System	Cord violated	th8 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-based capitalism (ax25)	None — all three	River of life attractor	Not yet implemented; theoretical prediction

Note

Illustrative, not confirmatory. The examples above are retrospective categorizations, not ex ante predictions. The theoretical argument (absorbing CTMC model) carries the weight of th8’s defense; these historical parallels are illustrative of the pattern, not confirmatory evidence for it. See Quest: Jubilee-Based Innovation Economy for the full critique-response exchange (Con-A.2 / Pro-D.2).

Significance: th8 is a central theorem — perhaps the central theorem — of the Jubilee-based innovation economy framework. Its defense rests on the absorbing CTMC model (steps 3a–3b), which proves from first principles that cord-violating systems have finite expected lifetime — the theoretical argument does not require empirical confirmation, any more than the theorem that absorbing Markov chains eventually absorb requires empirical confirmation. The historical parallels in the table above are illustrative of the pattern, not ex ante predictions (see Con-A.2 / Pro-D.2 in Quest: Jubilee-Based Innovation Economy). th8 IS empirically testable in principle — by specifying measurable indicators for each cord and making prospective predictions — but this falsification framework remains future work.

Note

Competitive-inhibitor model and commons-tragedy convergence. The causal gap objection (Con-A.2.1: RiskyMADorMAP proves extinction risk but not Jubilee necessity) is addressed by the Michaelis-Menten competitive-inhibitor model: ResearchCity introduces an alternative reaction pathway (Earth + ResearchCity → GlobalCooperation → MAP) that competes with the MAD pathway. A Jubilee system does not change the MAD rate parameters directly but creates an alternative trajectory. Furthermore, the multi-pathway objection (Con-A.2.2: Jubilee cannot address all extinction risks) is resolved by showing all major existential risks (nuclear, AI, climate, pandemic) are variants of the tragedy of the commons, addressable through the global coordination infrastructure that a Jubilee-based ResearchCity provides. See Con-A.2.1 / Pro-A.2.1 and Con-A.2.2 / Pro-A.2.2 in Quest: Jubilee-Based Innovation Economy.

t9 — Social Ergodicity#

PoR source

Informal alias: T-Ergodicity

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

Context: 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 (God values genuine love) \(\wedge\) ax9 (sustaining capacity) \(\wedge\) ax25 (Jubilee recalibration):

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

achieved without violating ax15 or ax17 at any moment.

Proof:

By ax22, God’s necessary nature values the Gc states arising from genuine care and is motivated to ensure justice.
By ax9, God has the capacity to sustain the conditions under which ergodicity operates.
By ax25, Jubilee recalibration periodically resets accumulated advantage, preventing permanent winner-takes-all concentration.
The 7TrackRole system (7 functional roles × 7 change stages = 49 configurations) models society as a finite-state Markov chain. Jubilee recalibration (ax25) acts as a mixing perturbation ensuring this chain is irreducible: every role can transition to every other role over sufficient rounds. By the Markov chain convergence theorem (Levin, Peres & Wilmer 2009, Theorem 4.9), an irreducible, aperiodic finite Markov chain converges to its stationary distribution in finite expected time (bounded by the chain’s mixing time).

4a — Why irreducibility is existential. Without the Jubilee System, accumulated advantage creates absorbing classes: AMO consolidates permanently at the top, GIR is permanently marginalized at the bottom. This role collapse is itself a prediction of societal “eschatological time” — a society reduced to only AMO and GIR cannot function, because the remaining five roles (HIT, CAN, PHE, JEB, HIV) provide essential functions no society can sustain without: enforcement, trade, open expertise, administration, and niche-building. Role collapse therefore entails societal collapse.

4b — Jubilee cycles prevent, not cause, eschatological collapse. A Jubilee-based recalibration is not “eschatological time” in the apocalyptic sense — it is the mechanism that prevents eschatological collapse. Continuous lower-level tuning (progressive taxation, regulation, social programs) addresses minor role imbalances. The Jubilee System provides the higher-order reset when accumulated complexity exceeds what continuous tuning can correct, keeping the full 7-role structure functional.

4c — Historic absence and current urgency. The prolonged historical absence of a Jubilee system has allowed structural problems to accumulate to the point where the world now faces conditions resembling “eschatological time” in the traditional sense: options for averting self-destruction are narrowing rapidly. This is not inevitable but is the predicted consequence of a society operating without the higher-order reset prescribed by ax25.
God enforces this at the system level (through ax25’s recalibration mechanism) rather than at the individual event level (which would require overriding ax15).
Therefore universal justice is served without violating ax15 or ax17. \(\blacksquare\)

Axioms used: ax9, ax15, ax22, ax25

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.

Note

Remaining gap: transition probabilities. Step 4’s Markov chain model provides the structural argument for ergodicity, but the transition probabilities between the 49 role-stage configurations are not yet specified. Estimating these from historical data is a significant empirical project. Until done, the ergodicity claim rests on the structural argument (Jubilee cycles ensure irreducibility) rather than a fully quantitative model. See Quest: Jubilee-Based Innovation Economy (Con-C.5 / Pro-C.5) for the full critique-response exchange.

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 answers the distribution problem: innocent suffering is not the final word because ergodicity guarantees that position is not permanent.

t10 — Physical Law Substrate#

PoR source

Informal alias: T-Physics

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

Context: 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) serves agency (ax15), 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 (God sustains physics) \(\wedge\) ax15 (agency requires stable substrate) \(\wedge\) ax18 (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 requires a stable physical substrate: without reliable physics, humans cannot form intentions, predict consequences, or exercise genuine choice.
ax9 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, 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) entirely. \(\blacksquare\)

Axioms used: ax9, ax15, ax18

Significance: th10 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.

t11 — Stakes Without Death#

PoR source

Informal alias: T-Finitude

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

Context: 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-agency requires. Biological death is one implementation of finitude but is not logically necessary for genuine stakes.

Formal statement:

Given ax6 (contingency → temporal finitude) \(\wedge\) ax15 (genuine agency) \(\wedge\) ax24 (life-trifecta → BABL for violations) \(\wedge\) ax25 (Jubilee windows that close):

Irreversibility(time) \(\wedge\) closing(Jubilee-windows) \(\rightarrow\) genuine-stakes(ax15-choices)

Death is sufficient but \(\neg\) necessary for finitude.

Proof:

ax6 (contingency of the world) establishes temporal finitude: the world exists within time, and time’s arrow makes past moments irrecoverable.
ax15 (agency) means choices have real consequences that propagate forward.
ax24 (life-trifecta) means wrong choices lead to BABL — real, not hypothetical, collapse.
ax25 (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, ax15, ax24, ax25

Significance: th11 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.