:orphan: e7Day MegaTable --- All Axioms, Theorems, and GrowthGarden Candidates ======================================================================== **Model:** e7Day --- Generalized System Construction **Version:** dv_ClaOp46Max_OOv1_2026m03d27 **Purpose:** Comprehensive registry of every axiom, theorem, and candidate ever considered for e7Day. Includes selected (OOv1) and unselected (GrowthGarden) items. Anyone may chip away at GrowthGarden candidates and update their status. **Legend:** - **Sel** = Selected for OOv1 (Y/N) - **Type** = axiom / theorem / theorem(conj.) - **Cred** = Credence (subjective probability of being true and useful) - **VVN** = Who proposed it - **Con-Pro** = Known objections and defenses ---- Cross-Day Meta-Axioms ----------------------- .. list-table:: :header-rows: 1 :widths: 10 28 5 5 5 20 * - ID - Statement - Sel - Type - Cred - Con-Pro / Notes * - mc.ax1 - Every submodel produces a constructive fixpoint (distributed aggregation yield). fix(result(mk)) = result(mk). - Y - ax - 95% - **Pro:** Grounded in type theory. **Con:** "Constructive" may be too strong for m0 (does VOID have a constructive witness?). * - mc.ax2 - OK ↔ fixpoint convergence + scope-creep-excluded. - Y - ax - 90% - **Pro:** Clean formalization of "it was good." **Con:** "scope-creep-excluded" needs formal definition. * - mc.ax3 - Evening-first: process(mk) = evening ▷ morning (via negativa before commitment). - Y - ax - 80% - **Pro:** Connects to HELL methodology. **Con:** May be a design principle (theorem?) rather than axiom. Could be derivable from optimization theory (R6). * - mc.ax4 - Construction cascade: input(mk) ⊇ ∪{result(mj) | j < k}. - Y - ax - 95% - **Pro:** Straightforward filtered sequence. **Con:** m5 depends on m3 AND m4 (multi-dependency), so "j < k" may need refinement to a DAG rather than linear order. Submodel Axioms ----------------- .. list-table:: :header-rows: 1 :widths: 10 28 5 5 5 20 * - ID - Statement - Sel - Type - Cred - Con-Pro / Notes * - m0.ax1 - VOID: ∃Ω with H(Ω) = Hmax ∧ no types in Ω. - Y - ax - 85% - **Con:** Does Ω "exist" constructively? Tension with mc.ax1. * - m1.ax1 - TYPE: Ω = L ⊎ D, irrevocable, all construction within L. - Y - ax - 95% - **Pro:** Clean, foundational. * - m2.ax1 - EQUAL: Types(L) = Int(L) ⊎ Real(L). - Y - ax - 90% - **Pro:** Standard CS distinction. * - m2.ax2 - Every Real→Int mapping is lossy (∀φ: info-loss(φ) ≥ ε > 0). - Y - ax - 95% - **Pro:** Known result in numerical analysis. * - m2.th1 - PERFECT ∧ PERFIDE cannot hold universally. (Reclassified from m2.ax3 at TEMPER: derivable from m2.ax1 + m2.ax2. Proof sketch: applying PERFECT to Real types requires Real→Int mapping, which is lossy by m2.ax2, destroying what PERFECT tries to preserve. Applying PERFIDE to Int types requires Int→Real mapping, adding spurious precision. Neither direction preserves both.) - Y - th - 90% - **Pro:** Maps to known dichotomies (nominal/structural typing, Kant/utilitarian, Arrow's Impossibility). Derivability from m2.ax1+m2.ax2 *strengthens* the claim. **AA:** Formalize proof (AA-TEMPER-a8). Iron Maiden: HELD (9 OK, 1 OKO). * - m3.ax1 - VALUE: Values(L) = Ground(L) ⊎ Ocean(L). - Y - ax - 90% - **Pro:** Follows from m2 residual uncertainty. * - m3.ax2 - Programs: finite decision trees rooted in Ground, taking Water input, producing Ground output. - Y - ax - 90% - **Pro:** Curry-Howard pair. **Note:** "Water" (not "Ocean") leaves open the circulation mechanism (RoL?). * - m3.ax3 - Water Circulation: Water must cycle Ocean → Trees → Ocean. Without circulation, Ground dries and Ocean stagnates. (Promoted from R1.) - Y - ax - 85% - **Pro:** Biologically fundamental. **Con:** Circulation mechanism (RoL?) unspecified. **Note:** Promoted from GrowthGarden R1 by user request. * - m4.ax1 - LOGIC: Processes(L) = DAY(L) ⊎ NIGHT(L). - Y - ax - 90% - **Pro:** Standard foreground/background distinction. * - m4.ax2 - Time is a first-class entity with measurable progress. - Y - ax - 85% - **Con:** "First-class" is vague. Needs sharper type-theoretic definition. * - m5.ax1 - CARE: Conditional-data machines are self-managing and self-replicating. - Y - ax - 85% - **Con:** Self-replication is a strong biological claim. Needs formal specification of replication mechanism. * - m5.ax2 - UMP: noise > θ → channel-capacity-for-signal → 0. - Y - ax - 90% - **Pro:** Grounded in Shannon. * - m6.ax1 - HOPE-p1: Special-purpose world complete but no general intelligence. - Y - ax - 90% - **Pro:** Descriptive, well-supported by construction cascade. * - m6.ax2 - HOPE-p2: Balospe exists with general intelligence, responsible for Balance(L) OLT, recursively endowed. - Y - ax - 85% - **Con:** Recursive endowment ("shares Creator's abilities") is the boldest claim. **Pro:** Self-hosting compiler analogy is precise. * - m6.ax3 - HOPE-p3: OKO(m2) ∧ OKO(m6.2) ∧ designed-to-resolve(m6.2,m2) → OK+(system). Matched OKO pair = self-correcting architecture. - Y - ax - 75% - **Con:** The multiplication metaphor (−1 × −1 = +1) is suggestive but not a proof. Needs rigorous justification. Most vulnerable axiom in the set. * - m6.ax4 - Self-assessment bifurcation (asymmetric): self-assesses(B,OK) → BABL (sufficient; self-reinforcing trap). ZION → self-assesses(B,OKO) (necessary but not sufficient; a free agent can stop at any time, cf. Gen.18 Abraham). OKO enables but does not guarantee ZION. - Y - ax - 82% - **Pro:** Candidate deepest finding. OK→BABL is modally necessary (analytic: "blindly assuming" entails OK). The asymmetry (BABL is stable attractor, ZION requires perpetual maintenance) is realistic. **Con:** OKO→ZION direction weakened to necessity at TEMPER (free agency means any agent can stop). Iron Maiden: HELD (8 OK, 2 OKO). * - m6.th1 - OSCR Collapse: OKO(m2) ∧ self-assesses(B,OK) → ¬self-corrects(B) → ¬designed-to-resolve(B,m2) → KO(system). (Reclassified from m6.ax5 at TEMPER: derivable from m6.ax3 + m6.ax4 in 6 steps. See TEMPER Entry a10, Test II.) - Y - th - 90% - **Pro:** The negative case is MORE defensible than the positive (m6.ax3), because the causal chain (complacency → no correction → system failure) is empirically well-documented. Derivability from m6.ax3+m6.ax4 *strengthens* the claim. **AA:** Formalize derivation (AA-TEMPER-a10). Iron Maiden: HELD (8 OK, 2 OKO). * - m7.ax1 - TRUST: result(m7) = ∪{result(mk) | k=0..6}. Null operation. - Y - ax - 95% - **Pro:** Definitional. * - m7.ax2 - Time = WorkTime ⊎ RestTime (type distinction). - Y - ax - 90% - **Pro:** Heschel's "cathedral in time." Clean. * - m7.ax3 - Fractal periodicity: 6+1 integer-day ratio is the constrained optimum for Earth-like systems, given: circadian biology (integer-day batching), lunar alignment (28÷7=4), innovation-cycle isomorphism (e7Ch, pending), and Schelling-point resistance to BABL erosion. - Y - ax - 75% - **Con:** Universality across non-Earth-like systems unproven. Depends on e7Ch being well-founded. **Pro:** Numerology objection answered (constrained optimization, not cargo cult). Schelling-point argument provides game-theoretic stability. Iron Maiden: initially BREACH, rescued to HELD (7 OK, 3 OKO) after user defense. Theorems ---------- .. list-table:: :header-rows: 1 :widths: 10 28 5 5 5 20 * - ID - Statement - Sel - Type - Cred - Con-Pro / Notes * - th1 - W = L (scope produces the world). PET connection. - Y - th - 90% - **Pro:** Straightforward if e7Day models creation. **Con:** Only applies when constructor = God. * - th2 - Lossiness: Complex(L) → all Real→Int mappings lose info. - Y - th - 95% - **Pro:** Follows directly from m2.ax2. * - th3 - BABL originates in self-assessment (OK→BABL; OKO necessary for ZION). Mechanism behind th8. - Y - th - 85% - **Pro:** Definitional argument (BABL = "blindly assuming" analytically entails OK). Substantially addresses Con-11. **Con:** OKO→ZION weakened to necessity (consistent with m6.ax4 reformulation). Iron Maiden: HELD (9 OK, 1 OKO). * - th4 - System requires Balospe for OLT survival. General intelligence necessary because special-purpose machines cannot handle novel PERFECT/PERFIDE trade-offs. - Y - th - 87% - **Pro:** Derivable from m2.th1 + m5.ax1 + m6.ax1. Substantially addresses Con-22 (volunteer requirement). **Con:** Assumes continuous novelty generation (needs explicit axiom or derivation). Iron Maiden: HELD (8 OK, 2 OKO). * - th5 - Rest (periodic consolidation) is structurally necessary. Omitting consolidation → error accumulation → BABL. - Y - th - 80% - **Pro:** Three defense lines (information-theoretic, thermodynamic, computational). Motivates ax25 (Jubilee). Addresses Con-31 with refinement (concurrent GC still redirects resources). **Con:** Error-accumulation model needs formalization. Iron Maiden: HELD (8 OK, 2 OKO). * - th6 - Dual-Nothing: VOID (max entropy) and TRUST (min new-entropy) are formally dual. e7Day is entropy-reduction arc. (Promoted from R2.) - Y - th - 80% - **Pro:** Follows from m0.ax1 + m7.ax1. Elegant. **Con:** "Formal duality" needs categorical specification (initial/terminal object?). Iron Maiden: HELD (9 OK, 1 OKO). * - th7 - Compassion Capacity Theorem (five-gate form): Gate 1: repair-history prerequisite (OKO provides, OK does not). Gate 2: scope limitation (finite agents ⊂ F_all). Gate 3: other-awareness (state + context + trajectory). Gate 4: channel quality / Compassion UMP (noise > θ → help → 0). Gate 5: perpetual scope-expansion commitment (hero who stops → irrelevant/supervillain; ASON trap; "arrived" = deadlock). God: all gates non-binding. (Major rescue from original single-arrow form at TEMPER.) - Y - th - 70--80% - **Pro:** Connects to 12+ axioms/theorems across PET, JUB, e7Day. Partially derives th8 mechanism. Provides mechanism for m2.th1. Supervillain theorem, eternal-life corollary. **Con:** Gate 5 depends on e7He (unspecified). Game-theoretic stability defensible but not proven. h* transition vulnerability (potential Con-45). **VVN:** iv_LLoL_OOv1 / dv_ClaOp46Max_OOv1. Iron Maiden: HELD (7 OK, 3 OKO). ---- GrowthGarden (GG) --- Candidates Not Selected for OOv1 --------------------------------------------------------- POST code: GG (GrowthGarden). Content that may be valuable but didn't make the cut. Worth revisiting in future rounds or models. .. list-table:: :header-rows: 1 :widths: 5 28 5 5 5 25 * - GG# - Statement - Type - Cred - VVN - Notes / Why not selected * - GG-a1 - **Monotonic Complexity.** Each day strictly increases structural complexity in some well-defined measure. - th - 75% - dv_ClaOp46Max - Unsure which complexity measure. Intuitively obvious but needs the right metric. * - GG-a2 - **PERFECT/PERFIDE Conservation.** Gain in one requires expenditure from the other; total budget conserved. - ax - 70% - dv_ClaOp46Max - Needs economic formalization not yet in model. * - GG-a3 - **Sabbatical Convergence.** 6:1 is the unique optimal ratio for error accumulation vs. consolidation. - th(conj.) - 65% - dv_ClaOp46Max - No proof sketch. May be empirical. * - GG-a4 - **Evening-First Efficiency.** Via negativa converges faster than morning-first processes. - th - 60% - dv_ClaOp46Max - Plausible from optimization theory but no formal model. * - GG-a5 - **Sky-to-Ocean Depletion.** Sky machines converting potential→existing can exhaust future potential. - th - 55% - dv_ClaOp46Max - Needs dynamics model. Connects to ax24 life-friendly cord. * - GG-a6 - **Constructor-Agnosticism.** Same structural result regardless of constructor identity. - th - 55% - dv_ClaOp46Max - Hard to define equivalence class of constructors. * - GG-a7 - **UMP-BABL Equivalence.** UMP = BABL at info scale; BABL = UMP at system scale. - th - 50% - dv_ClaOp46Max - May be analogy not formal equivalence. * - GG-a8 - **Blind Evolution Insufficiency.** Above threshold C*, evolution too slow to prevent BABL. - th - 50% - dv_ClaOp46Max - Threshold C* undefined. Connects to th8 CTMC. * - GG-a9 - **Verdict Counting.** OK count per day is a formal invariant encoding structural information. - ax - 45% - dv_ClaOp46Max - May be coincidence. Needs more models to confirm. * - GG-b10 - **Heschel Theorem.** RestTime has its own internal modality (contemplation, integration, celebration). - ax - 40% - dv_ClaOp46Max - Needs richer time theory. Beautiful if formalizable. * - GG-b11 - **4Be Completeness.** Four capacities are necessary AND sufficient for OLT innovation economy. - ax - 35% - dv_ClaOp46Max - Belongs to e4Be model, not e7Day. Scope boundary. * - GG-b12 - **EDEN Stability.** All 4 EDEN properties must hold to resist BABL. - ax - 35% - dv_ClaOp46Max - EDEN not yet elaborated. Premature to formalize. * - GG-b13 - **WoLC Duality.** e7Day and WoLC are categorically dual. - th(conj.) - 25% - dv_ClaOp46Max - May be poetic not categorical. Needs functor definition. * - GG-b14 - **Balospe Incompleteness.** By Goedel, Balospe always has blind spots (necessary OKO). - th(conj.) - 20% - dv_ClaOp46Max - Goedel-for-agents is contested. High risk of overreach. * - GG-b15 - **Information-Entropy Mirror.** Total info content = H(VOID) - H(m6.3), conserved under trust test. - th(conj.) - 15% - dv_ClaOp46Max - Probably oversimplifies real entropy dynamics. * - GG-b16 - **Scope Expansion Prohibition.** D→L boundary is formally impermeable within a cycle. - th - 15% - dv_ClaOp46Max - May already follow from m1.ax1. * - GG-b17 - **Strange Attractor Periodicity.** ZION frequencies are discrete and 7-based. - ax - 10% - dv_ClaOp46Max - Could be numerology rather than dynamics. * - GG-b18 - **RoL Circulation Axiom.** Distinguished river cycle transporting Water from Ocean through Ground. - ax - 8% - dv_ClaOp46Max - Almost no formal content yet. May grow with RoL model. * - GG-b19 - **Day 2 / Day 6.2 Resonance.** Bidirectional feedback between EQUAL and Balospe. - th(conj.) - 5% - dv_ClaOp46Max - Needs coupling constant/operator. At the junk boundary.