The logical chain from the Meta-Principle to the complete architecture of reality
From the Meta‑Principle (Nothing cannot recognize itself), plus Composability and Finiteness, eight core theorems follow as logical necessities. These form the backbone of the framework.
Statement: Any recognition structure satisfying MP + Composability + Finiteness admits a unique (up to order‑isomorphism) positive, double‑entry ledger with an immutable generator.
Sketch: Construct the ordered abelian ledger via Grothendieck completion; conservation by telescoping; uniqueness by the universal property.
Implication: A single ordered “book” underlies all dynamics and measurements.
Statement: Each tick posts exactly one elementary ledger entry of size δ; concurrent postings at one tick are impossible.
Sketch: Two entries at one tick violate double‑entry granularity or conservation, forcing quantized, countable events.
Implication: Time and action are inherently discrete at the ledger level.
Statement: Under sequential posting of unit entries, additivity and dual‑balance imply a discrete continuity equation; in the coarse‑grained limit, ∂tρ + ∇·J = 0.
Sketch: Double‑entry telescopes on closed chains, yielding a combinatorial divergence theorem and conservation law.
Implication: All conservation laws descend from ledger double‑entry.
Statement: The only analytic, dual‑balanced, ledger‑finite cost functional on ℝ>0 is J(x) = ½(x + 1/x).
Sketch: Symmetry forces a symmetric Laurent form; finiteness kills higher modes; normalization fixes ½.
Implication: A single convex cost governs all recognitions.
Statement: In the recurrence xn+1 = 1 + k/xn, countability and global cost minimization force k = 1; the fixed point is φ.
Sketch: Integer split‑count from atomicity; monotonicity of total cost in k ≥ 1 gives the minimum at 1; solve x = 1 + 1/x.
Implication: φ emerges as the universal scaling constant.
Statement: Stable dual cycles with nontrivial linking require exactly three spatial dimensions.
Sketch: Jordan/planarity forbids links in 2D; ambient isotopy unlinks in d ≥ 4; only d = 3 supports a non‑erasable link.
Implication: 3D space is selected by stability, not assumption.
Statement: On the 3‑cube, any ledger‑compatible recognition has minimal period T = 8; more generally, T = 2^D on QD.
Sketch: Spatial completeness (visit all vertices) + atomic tick ⇒ at least |V| ticks; Gray code gives existence and minimality.
Implication: An 8‑tick rhythm underlies recognition cycles in 3D.
Statement: Atomic ticks on a nearest‑neighbor lattice define causal cones; faster‑than‑neighbor propagation breaks the ledger order, fixing a maximal speed c = Lmin/τ0.
Sketch: The null class is one‑voxel‑per‑tick; order preservation excludes super‑neighbor hops.
Implication: Relativistic kinematics emerge from ledger causality.
Together, these eight results give the spine of the framework: a unique ordered ledger (T1), quantized ticks (T2), conservation (T3), a unique convex cost (T4), φ from self‑similarity (T5), three spatial dimensions (T6), an 8‑tick period (T7), and causal cones with a universal speed (T8).
They are the deductive consequences of MP + Composability + Finiteness and align with the formal statements in the Foundations document.