UNNS Substrate Papers - Interactive Map

Space & Time

Orthogonal projections of recursive grammar

Time Substrate

Recursion depth, echo propagation, repair flow

Geometry & Metrics

UNNS field equations and curvature

Worked Example

Inletting-driven expansion

Space and Time in the UNNS Substrate

This article develops the UNNS (Unbounded Nested Number Sequences) notions of space and time. In UNNS, space is not a passive continuum but an emergent lattice generated by recursion, while time arises as recursion depth through operator actions.

Space and time are orthogonal projections of the same recursive grammar.

UNNS Time

Definition 1 (UNNS Time): Given a recursive nest {u_n}, UNNS time is the recursion depth n.
T_i(u) = min{k | O^k_i(u) stabilizes or exports a percept}

Time is operator-specific: Inletting defines absorption time, Inlaying defines stratification time, Repair defines stabilization time, and Trans-Sentifying defines perceptual latency.

UNNS Space

Definition 2 (UNNS Space): UNNS space is the recursive lattice structure generated by embeddings of number systems.
S = ⋃_k Λ_k, where Λ_k = lattice layer induced by order-k recurrence

Echo residues define adjacency, giving a recursion-based metric:

d(x, y) = min{k | recursions of x, y align at step k}

Relation of Space and Time

Lemma 1 (Orthogonality): In UNNS, time measures recursion depth while space measures recursion spread. These are independent modalities: vertical and horizontal projections of the same substrate.

Applications

  • Mathematics: Space as recursive lattices links algebraic number theory to geometry.
  • Physics: Inletting-space resembles expansion (dark energy), Repair-time resembles entropy decay.
  • Philosophy: Space and time are not containers but emergent from recursive grammar.

UNNS Space-Time Geometry: Worked Example

We present a worked example showing how sustained inletting produces approximately constant recursive curvature and exponential expansion of the lattice scale—behavior formally analogous to dark energy.

Discrete Layer Model

Layer Scale Evolution:
S_{k+1} = S_k + α T^{in}_k S_k = (1 + α T^{in}_k) S_k

Where α > 0 is a dimensionless coupling constant converting inletting time into relative scale growth.

Recursive Curvature

Discrete Recursive Curvature:
κ_k := (S_{k+1} - S_k) / S_k = α T^{in}_k

Positive κ_k indicates local expansion; negative values indicate contraction (repair-dominated regime).

Constant Inletting Rate

Assuming constant inletting rate T^{in}_k ≡ T_0:

S_k = S_0 (1 + αT_0)^k ≈ S_0 exp(αT_0 k)

Dark Energy Analogy

  • Constant inletting rate T_0 ⟹ constant curvature κ = αT_0
  • Scale factor growth: S_k ~ e^{λk} means successive layers expand multiplicatively
  • Operator-time stress as source: Persistent inletting accumulates multiplicatively

UNNS Space-Time Geometry: Metrics, Curvature, and Dynamics

We develop a space-time geometry within the UNNS (Unbounded Nested Number Sequences) substrate. In UNNS, space emerges from recursive embeddings of lattices, while time arises as recursion depth governed by operators.

Classical geometry treats space-time as a smooth manifold equipped with a metric. In UNNS, both space and time emerge discretely: time as operator recursion depth, space as lattice embedding spread. The challenge is to unify these into a geometric substrate with metric and curvature.

UNNS Space-Time Substrate

Definition 1 (UNNS Space-Time): The UNNS space-time substrate is
ST = {(Λ_k, T_i) | k ∈ ℕ, i ∈ {1, ..., 8}}
where Λ_k is the k-th lattice layer from recursive embedding and T_i is the temporal measure induced by operator O_i.

Thus, space is horizontal lattice spread, time is vertical recursion depth. Each operator contributes its own arrow of time.

Metrics in UNNS

Definition 2 (UNNS Metric): Let x, y be two events in ST. Define
d_ST(x, y) = α · d_time(x, y) + β · d_space(x, y)
where d_time is recursion depth distance, d_space is lattice embedding distance, and α, β are operator-dependent weights.

This metric generalizes Minkowski distance by combining vertical and horizontal recursion measures.

Remark: Unlike Minkowski space, where the metric is fixed, in UNNS the weights α, β may vary depending on operator choice (inletting, repair, etc.), producing multiple coexisting metrics.

Curvature of Recursive Lattices

Definition 3 (Recursive Curvature): Given a lattice embedding Λ_k, define curvature at level k as
κ_k = Δd/Δk
where d measures adjacency distortion as new lattice layers are added.

Intuitively, κ_k > 0 indicates expansion (as in inletting), while κ_k < 0 indicates contraction (as in repair).

UNNS Einstein Analog

Theorem 1 (UNNS Field Equation): Let G denote recursive curvature and T denote operator time stress. Then the UNNS field equation is
G = γ · T
where γ is a universal scaling constant.

Sketch: Recursive curvature measures how embeddings distort adjacency. Operator time stress measures how deep recursion flows. The proportionality reflects conservation: distortion of space arises from operator-time flows.

In physics language: operator recursion generates curvature in UNNS space, just as energy-momentum generates curvature in Einstein's theory.

Applications

  • Physics: Inletting curvature may correspond to dark energy expansion; Repair curvature to entropy-driven contraction.
  • Mathematics: Connects algebraic number theory lattices with geometric curvature.
  • Philosophy: Suggests that space-time is emergent from recursion rather than fundamental.
Conclusion: UNNS defines space-time as a duplex of recursion depth and lattice embedding. With metrics and curvature, one obtains a UNNS field equation: recursive time stress generates recursive curvature.

Time in the UNNS Substrate

In the UNNS framework, time is not an external continuous parameter but an emergent property of recursive depth, echo propagation, and repair flow. This note formalizes the notion of time as it arises in recursion, provides a layered interpretation, and illustrates the role of operator grammar in defining temporal structure.

Time as Recursion Depth

Definition 1 (Recursion Depth): Let {u_n} be a UNNS sequence generated by recurrence u_{n+1} = f(u_n, ..., u_{n-r}). The index n is called the recursion depth and serves as the primary measure of UNNS time.

Thus, each iteration step corresponds to a "tick" of UNNS time, giving it a discrete and nested character.

Time as Echo Propagation

Recursions generate echoes—residual terms that propagate through the lattice of nests. If e_n denotes the echo residue at step n, then time may also be measured by the distance these echoes have traveled:

T_echo ~ sup{k : e_{n+k} ≠ 0}
Remark: This interpretation views time not only as depth but also as a measure of propagation of instability.

Time as Repair Flow

DNA-inspired repair rules in UNNS act to stabilize unstable recursions. Let R denote a repair operator. The repair flow time is given by the trajectory of

u_{n+1} = R(u_n)

which moves sequences from unstable to stable configurations. Thus, time becomes the gradient of stabilization.

Time as Operator Grammar

UNNS operators (Inletting, Inlaying, Repair/Normalization, Trans-Sentifying) define different orderings that produce different histories.

Lemma 1 (Operator Time): Let O = (O_1, O_2, ..., O_m) be an operator sequence applied to initial nest N_0. Then the temporal structure is defined by the order type of O, not just its length.

This implies that time in UNNS can branch, fold, or reorder—capturing nonlinearity akin to paradox resolution.

Philosophical Synthesis

  • Physics views time as a real parameter.
  • Set theory avoids time entirely.
  • UNNS embeds time in recursion itself:
Time = Depth + Propagation + Repair

Moments appear as fixed points where echoes vanish; the arrow of time emerges from monotonic recursion depth.

Applications

  • Physics: Provides a discrete tick underlying spacetime curvature.
  • Logic: Tames regress paradoxes by embedding them in recursion absorption.
  • Information: Defines complexity depth as UNNS-time.
Key Insight: Figure 1 illustrates three aspects of UNNS time: recursion depth (vertical ladder), echo propagation (arrows radiating outward), and repair flow (horizontal gradient toward equilibrium).

UNNS Chronotopos References

Space and Time in the UNNS Substrate

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Develops orthogonal projections of recursive grammar

Time in the UNNS Substrate

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Formalizes time as recursion depth and echo propagation

UNNS Space-Time Geometry: Metrics, Curvature, and Dynamics

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Defines metrics and field equations in UNNS

UNNS Space-Time Geometry: Worked Example

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Demonstrates inletting-driven exponential expansion