.. meta:: :description: e7Day axioms: 21 formal axioms for generalized system construction, grouped by submodel (cross-day meta, VOID, TYPE, EQUAL, VALUE, LOGIC, CARE, HOPE, TRUST). :keywords: e7Day axioms, formal statements, mc, VOID, TYPE, EQUAL, VALUE, LOGIC, CARE, HOPE, TRUST, OOv1 :author: LLoL as Laurence Loewe of Laodicea and Claude Opus 4.6 **************************************************** e7Day Axioms **************************************************** The axioms of the e7Day model (Generalized System Construction), organized by submodel. Each axiom is presented with its formal statement and a brief note. .. vvnow dv_ClaOp46Max_OOv1_2026m03d27 This is version **dv_ClaOp46Max_OOv1_2026m03d27** --- TEMPER-updated formulations. For the symbol dictionary, see :doc:`symbols`. For derived theorems, see :doc:`theorems`. For GrowthGarden candidates, credences, and con-pro notes, see :doc:`gg/megatable`. .. contents:: On this page :depth: 2 :local: .. E7DAY-AXIOMS-CONTENT-START ---- Cross-Day Meta-Axioms ======================= Four axioms governing how submodels compose. These are not tied to any single day but constrain the entire construction arc. .. _e7day-mc-ax1: mc.ax1 --- Constructive Fixpoint ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \text{fix}(\text{result}(m_k)) = \text{result}(m_k) Every submodel produces a constructive fixpoint (distributed aggregation yield). The possibly long, complex path to each day's conclusion does not matter; only the robust end result. .. _e7day-mc-ax2: mc.ax2 --- OK Convergence ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \text{OK} \;\leftrightarrow\; \text{fixpoint-convergence} \;\wedge\; \text{scope-creep-excluded} Formalizes "it was good" as convergence to a fixpoint without scope creep. .. _e7day-mc-ax3: mc.ax3 --- Evening-First ^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \text{process}(m_k) = \text{evening} \triangleright \text{morning} Via negativa before commitment. The elimination phase (evening) precedes the construction phase (morning). .. _e7day-mc-ax4: mc.ax4 --- Construction Cascade ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \text{input}(m_k) \supseteq \bigcup\{\text{result}(m_j) \mid j < k\} Each day's input includes all prior days' results. The cascade is cumulative and filtered. ---- m0 --- VOID ============== .. _e7day-m0-ax1: m0.ax1 --- Maximum-Entropy Pre-Partition ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \exists\,\Omega \;:\; H(\Omega) = H_{\max} \;\wedge\; \neg\exists\;\text{types in}\;\Omega Pre-partition chaos: maximum entropy, no types, no structure. Tohu va-vohu. ---- m1 --- TYPE ============== .. _e7day-m1-ax1: m1.ax1 --- Binary Scope Partition ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \Omega = L \uplus D \qquad\text{(irrevocable; all construction within } L\text{)} Scope partition into L (light, in-scope) and D (dark, out-of-scope). The partition is irrevocable within a construction cycle. ---- m2 --- EQUAL =============== .. _e7day-m2-ax1: m2.ax1 --- Integer/Real Type Split ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \text{Types}(L) = \text{Int}(L) \uplus \text{Real}(L) Types within scope partition into indivisible (Integer) and divisible (Real). .. _e7day-m2-ax2: m2.ax2 --- Lossy Mapping ^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \forall\,\varphi : \text{Real}(L) \to \text{Int}(L) \;:\; \text{info-loss}(\varphi) \geq \varepsilon > 0 Every Real-to-Integer mapping is lossy. No lossless reduction from divisible to indivisible types exists. ---- m3 --- VALUE =============== .. _e7day-m3-ax1: m3.ax1 --- Ground/Ocean Value Partition ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \text{Values}(L) = \text{Ground}(L) \uplus \text{Ocean}(L) Values within scope partition into unconditionally known (Ground) and conditionally known (Ocean). .. _e7day-m3-ax2: m3.ax2 --- Programs as Decision Trees ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Programs are finite decision trees rooted in Ground, taking Water input and producing Ground output. Curry-Howard pair. "Water" (not "Ocean") leaves the circulation mechanism open. .. _e7day-m3-ax3: m3.ax3 --- Water Circulation ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Water must cycle Ocean → Trees → Ocean. Without circulation, Ground dries and Ocean stagnates. Promoted from GrowthGarden (R1). ---- m4 --- LOGIC =============== .. _e7day-m4-ax1: m4.ax1 --- DAY/NIGHT Process Partition ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \text{Processes}(L) = \text{DAY}(L) \uplus \text{NIGHT}(L) Processes within scope partition into foreground (DAY: directed activity) and background (NIGHT: nondeterministic guidance). .. _e7day-m4-ax2: m4.ax2 --- First-Class Time ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Time is a first-class entity with measurable progress. Time has type-theoretic standing within L, not merely a parameter. ---- m5 --- CARE ============== .. _e7day-m5-ax1: m5.ax1 --- Self-Managing Machines ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Conditional-data machines are self-managing and self-replicating. Machines operating on conditional data (Ocean and Sky machines) maintain and reproduce themselves. .. _e7day-m5-ax2: m5.ax2 --- Unimportant Message Problem (UMP) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \text{noise} > \theta \;\rightarrow\; \text{channel-capacity-for-signal} \to 0 When noise exceeds threshold θ, the channel capacity for meaningful signal collapses to zero. Grounded in Shannon information theory. ---- m6 --- HOPE ============== .. _e7day-m6-ax1: m6.ax1 --- Special-Purpose Completion (HOPE-p1) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The special-purpose world is complete but contains no general intelligence. The construction cascade (m0--m5) produces a functionally complete world of self-managing machines without general problem-solving capability. .. _e7day-m6-ax2: m6.ax2 --- Balospe (HOPE-p2) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Balospe exists with general intelligence, responsible for Balance(L) OLT, recursively endowed. Balance-o-stat species: the unique agent type with general intelligence, tasked with maintaining long-term balance within L. "Recursively endowed" means the constructor's pattern is replicated within the construct (self-hosting compiler analogy). .. _e7day-m6-ax3: m6.ax3 --- Matched OKO Self-Correction (HOPE-p3) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \text{OKO}(m2) \;\wedge\; \text{OKO}(m6.2) \;\wedge\; \text{designed-to-resolve}(m6.2, m2) \;\rightarrow\; \text{OK}^+(\text{system}) Two matched OKO verdicts produce a self-correcting architecture. Balospe (m6.2, OKO) is specifically designed to resolve the EQUAL ambiguity (m2, OKO). .. _e7day-m6-ax4: 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) OK→BABL is modally necessary (analytic: "blindly assuming" entails OK). OKO enables but does not guarantee ZION; the asymmetry reflects that BABL is a stable attractor while ZION requires perpetual maintenance. ---- m7 --- TRUST =============== .. _e7day-m7-ax1: m7.ax1 --- Null Aggregation ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \text{result}(m_7) = \bigcup\{\text{result}(m_k) \mid k = 0..6\} TRUST adds no new content; it aggregates all prior results. Null operation. .. _e7day-m7-ax2: m7.ax2 --- WorkTime/RestTime Partition ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \text{Time} = \text{WorkTime} \uplus \text{RestTime} Type distinction within time. Heschel's "cathedral in time." .. _e7day-m7-ax3: m7.ax3 --- Fractal Periodicity ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The 6+1 integer-day ratio is the constrained optimum for Earth-like systems. Four constraints: circadian biology (integer-day batching), lunar alignment (28 ÷ 7 = 4), innovation-cycle isomorphism (e7Ch, pending), and Schelling-point resistance to BABL erosion. .. E7DAY-AXIOMS-CONTENT-END ---- | ------------------------------------------------------------------------------------------ | If you wish to extend such analyses and to explore many more such questions to replace eschatological warfare between nuclear and other nations on Earth with mathematical clarity, then please consider :doc:`supporting this work by buying in ` towards scaling up a ResearchCity that can actually do it (see box below).