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.

This is version dv_ClaOp46Max_OOv1_2026m03d27 — TEMPER-updated formulations.

For the symbol dictionary, see e7Day Symbol Dictionary. For derived theorems, see e7Day Theorems. For GrowthGarden candidates, credences, and con-pro notes, see e7Day MegaTable — All Axioms, Theorems, and GrowthGarden Candidates.

Cross-Day Meta-Axioms #

Four axioms governing how submodels compose. These are not tied to any single day but constrain the entire construction arc.

mc.ax1 — Constructive Fixpoint #

\[\text{fix}(\text{result}(m_k)) = \text{result}(m_k)\]

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.

mc.ax2 — OK Convergence #

\[\text{OK} \;\leftrightarrow\; \text{fixpoint-convergence} \;\wedge\; \text{scope-creep-excluded}\]

Formalizes “it was good” as convergence to a fixpoint without scope creep.

mc.ax3 — Evening-First #

\[\text{process}(m_k) = \text{evening} \triangleright \text{morning}\]

Via negativa before commitment. The elimination phase (evening) precedes the construction phase (morning).

mc.ax4 — Construction Cascade #

\[\text{input}(m_k) \supseteq \bigcup\{\text{result}(m_j) \mid j < k\}\]

Each day’s input includes all prior days’ results. The cascade is cumulative and filtered.

m0 — VOID #

m0.ax1 — Maximum-Entropy Pre-Partition #

\[\exists\,\Omega \;:\; H(\Omega) = H_{\max} \;\wedge\; \neg\exists\;\text{types in}\;\Omega\]

Pre-partition chaos: maximum entropy, no types, no structure. Tohu va-vohu.

m1 — TYPE #

m1.ax1 — Binary Scope Partition #

\[\Omega = L \uplus D \qquad\text{(irrevocable; all construction within } L\text{)}\]

Scope partition into L (light, in-scope) and D (dark, out-of-scope). The partition is irrevocable within a construction cycle.

m2 — EQUAL #

m2.ax1 — Integer/Real Type Split #

\[\text{Types}(L) = \text{Int}(L) \uplus \text{Real}(L)\]

Types within scope partition into indivisible (Integer) and divisible (Real).

m2.ax2 — Lossy Mapping #

\[\forall\,\varphi : \text{Real}(L) \to \text{Int}(L) \;:\; \text{info-loss}(\varphi) \geq \varepsilon > 0\]

Every Real-to-Integer mapping is lossy. No lossless reduction from divisible to indivisible types exists.

m3 — VALUE #

m3.ax1 — Ground/Ocean Value Partition #

\[\text{Values}(L) = \text{Ground}(L) \uplus \text{Ocean}(L)\]

Values within scope partition into unconditionally known (Ground) and conditionally known (Ocean).

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.

m3.ax3 — Water Circulation #

Water must cycle Ocean → Trees → Ocean. Without circulation, Ground dries and Ocean stagnates.

Promoted from GrowthGarden (R1).

m4 — LOGIC #

m4.ax1 — DAY/NIGHT Process Partition #

\[\text{Processes}(L) = \text{DAY}(L) \uplus \text{NIGHT}(L)\]

Processes within scope partition into foreground (DAY: directed activity) and background (NIGHT: nondeterministic guidance).

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 #

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.

m5.ax2 — Unimportant Message Problem (UMP)#

\[\text{noise} > \theta \;\rightarrow\; \text{channel-capacity-for-signal} \to 0\]

When noise exceeds threshold θ, the channel capacity for meaningful signal collapses to zero. Grounded in Shannon information theory.

m6 — HOPE #

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.

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).

m6.ax3 — Matched OKO Self-Correction (HOPE-p3)#

\[\text{OKO}(m2) \;\wedge\; \text{OKO}(m6.2) \;\wedge\; \text{designed-to-resolve}(m6.2, m2) \;\rightarrow\; \text{OK}^+(\text{system})\]

Two matched OKO verdicts produce a self-correcting architecture. Balospe (m6.2, OKO) is specifically designed to resolve the EQUAL ambiguity (m2, OKO).

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 #

m7.ax1 — Null Aggregation #

\[\text{result}(m_7) = \bigcup\{\text{result}(m_k) \mid k = 0..6\}\]

TRUST adds no new content; it aggregates all prior results. Null operation.

m7.ax2 — WorkTime/RestTime Partition #

\[\text{Time} = \text{WorkTime} \uplus \text{RestTime}\]

Type distinction within time. Heschel’s “cathedral in time.”

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.

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