Connectivity Margin as a Boundary Distance Coordinate

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UNNS Substrate Research Program · April 2026 · Geometric Foundations

Margin as a Coordinate

The connectivity margin—previously an empirical ordering parameter—is shown to be a genuine distance-to-boundary coordinate in realizability space. Physical interactions, trajectory irreversibility, and the canonical ladder principle emerge as geometric consequences.
93 Datasets 22,817 Evaluations 11 Physical Domains 5 Theorems Zero Monotonicity Violations PRD/JSTAT Submission Ready
Status: Manuscript complete · April 2026 Corpus: Phase Mapping + Voyager + Ising + CMB + Atomic Key result: m(L) ≍ d(L, ∂ΩL) — Bi-Lipschitz equivalence established locally and globally under non-degeneracy
Primary Manuscript
Connectivity Margin as a Coordinate of Realizability Space: Monotonicity, Boundary Geometry, and Canonical Structure in the UNNS Substrate
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Executive Summary

The UNNS Substrate has been built on the observation that physical ordered sequences — energy spectra, vibrational modes, spatial harmonics — cluster deep inside stable structural regimes, never crossing realizability boundaries at physical parameter values. The connectivity margin m(L) has been the numerical signature of this stability: larger margin, deeper interior; smaller margin, closer to a phase boundary.

But until now, that relationship was empirical. No proof existed that m(L) behaves as a distance. It could have been a useful statistic that happened to correlate with stability — not a coordinate of the space itself.

This manuscript closes that gap. It introduces the distance-to-boundary functional d(L, ∂ΩL), proves four theorems of increasing strength establishing that m(L) and d(L, ∂ΩL) are equivalent, and derives the consequences: trajectory monotonicity is irreversible, the maximum-margin selection is geometrically grounded, and the interaction hierarchy is a coordinate ordering — not a classification.

"The connectivity margin is a coordinate of realizability space. The interaction hierarchy is its asymptotic footprint."

🎯 The Gap That Was Filled

The Interaction Unification manuscript established that all four known physical interactions can be classified through the connectivity margin m(L) as asymptotic regimes of a common functional. It identified the monotonicity of m(L) with respect to realizability-class boundaries as "the highest-priority open problem in the framework" — because without it, the maximum-margin selection principle remained an empirical rule rather than a structural theorem.

Before This Manuscript

  • m(L) was a classification parameter
  • Maximum-margin principle was empirically validated
  • Interaction hierarchy was a post-hoc classification
  • Scaling correspondences were analogies
  • Encoding invariance was corpus-consistent, unproved

After This Manuscript

  • m(L) is a geometric coordinate on realizability space
  • Maximum-margin principle has a geometric foundation
  • Interaction hierarchy is a consequence of ordering
  • Scaling correspondences are ordering consequences
  • Encoding invariance is proved under non-degeneracy

The Central Claim

m(L) induces a global ordering of admissible ladders equivalent to their ordering by distance to the nearest realizability-class boundary. This is not a local regularity result — it is a statement that the margin is the correct coordinate system for realizability space.

📐 Four Theorems of Increasing Strength

The manuscript establishes four results, each building on the previous, under explicit regularity and non-degeneracy conditions stated precisely in the text.

Theorem 4.1 · Local Equivalence

m(L) ≍ d(L, ∂ΩL)

In any rigidity neighbourhood, the connectivity margin and the distance-to-boundary functional are bi-Lipschitz equivalent. The Lipschitz constants c₁, c₂ are determined by the local gap geometry.

Theorem 4.2 · Normal Coordinate Form

d(L, ∂ΩL) = m(L)/√2 + O(m²)

In the canonical normal coordinate aligned with the decisive direction, the margin is asymptotically isometric to boundary distance. Anisotropy collapses: c₂/c₁ = 1 + O(m(L)) — the margin is an effectively exact distance coordinate in the interior.

Theorem 5.1 · Trajectory Monotonicity

m(γ(t)) ↓ almost everywhere

Along any admissible trajectory approaching a realizability boundary, the margin is non-increasing and strictly decreasing almost everywhere. There is no recovery mechanism.

Theorem 6.1 · Global Ordering

m(L₁) > m(L₂) ⟹ d(L₁) > d(L₂)

Under the non-degeneracy condition (satisfied by all 93 corpus datasets), margin ordering is equivalent to boundary-distance ordering globally across realizability space.

Theorem 7.1 · Canonical Class

arg max m(L) = deepest representative

The maximum-margin encoding of any physical system is its geometrically deepest representative in realizability space — not merely the most empirically stable. The canonical class theorem upgrades the maximum-margin principle from heuristic to geometric consequence.

CENTRAL RESULT — FIVE THEOREMS Thm 4.1 · LOCAL m(L) ≍ d(L, ∂Ω) bi-Lipschitz → Thm 4.2 · NORMAL d = m/√2 + O(m²) c₂/c₁ → 1 → Thm 5.1 · DYNAMIC m(γ(t)) ↓ a.e. irreversible → Thm 6.1 · GLOBAL m₁>m₂ ⟺ d₁>d₂ non-degeneracy → Thm 7.1 · CANONICAL arg max m = deepest rep. geometric max-margin

🌐 The Radial Structure of Realizability Space

Once m(L) is established as a coordinate, realizability space acquires a stratified geometric structure. Define the radial coordinate r(L) := m(L). The realizability boundary sits at r = 0. The four PRP classes — Full, Giant, Tail, Hard — correspond to ordered radial shells, and the four fundamental interactions map directly to these shells.

A concentric circle diagram illustrating the stratified structure of UNNS Realizability Space. The center is a green core labeled Full Percolation: Strong Regime, surrounded by a blue-green layer for Giant Component: Electromagnetic Regime. This is encircled by a purple layer for Tail Fragmentation: Weak Regime, and the outermost layer consists of orange fragments labeled Hard Fragmentation: Gravitational Regime. An arrow from the center outward represents the radial coordinate r(L) := m(L), identifying the connectivity margin as structural depth.
Figure 1. Realizability space as a stratified geometric domain where the connectivity margin m serves as a radial coordinate. Class boundaries are located at r = 0, and fundamental physical interactions — Strong, Electromagnetic, Weak, and Gravitational — correspond to ordered radial regions based on their geometric distance from structural boundaries.
INTERACTION REGIME — RADIAL ORDERING BY MARGIN STRONG Full Percolation m ~ 0.5 – 1.0 Power-law scaling GR = 1.000 He/Li QM-I (corpus) EM Giant Component m ~ 10⁻² Scale-invariant Γ ~ n⁻³ Rydberg (m≈1.2×10⁻²) WEAK Tail Fragmentation m → 0 (local) Exponential decay e⁻ʳ/ξ · short range Zeeman (m≈3×10⁻³) GRAVITY Global Limit · m → 0⁺ m ~ 10⁻⁴ Long-range weak ξ(r) ~ r⁻² Planck CMB (m≈2×10⁻⁴) r = m(L) large → deep interior r → 0 → boundary

What This Diagram Means

This is not a metaphor. Under Theorem 6.1 (Global Ordering), the regime ordering is mathematically equivalent to the boundary-distance ordering. The Strong regime sits furthest from any realizability boundary; the gravitational regime sits closest to r = 0. The hierarchy of forces is a hierarchy of depths in realizability space.

🌊 Trajectory Monotonicity and Irreversibility

Theorem 5.1 establishes the dynamic form of the coordinate result: any admissible trajectory approaching a realizability-class boundary sees its margin decrease monotonically and irreversibly. This is the structural analogue of RG flow irreversibility — systems cannot spontaneously move from near-boundary to deep-interior regimes along a deformation trajectory.

A two-part graphic. On the left, blue flow lines represent Irreversible Flow moving towards a dark purple Realizability Boundary hypersurface. A bracket indicates Monotonic Margin Decrease, referencing Theorem 5.1 and showing the inequality m(gamma(0)) > m(gamma(s)) > m(gamma(t)). On the right, a logarithmic line graph plots the Time-to-Boundary Estimator (t*) against the Predictive functional scale, showing a steady downward curve within a Heliopause Predictive Window.
Figure 2. Trajectory monotonicity and irreversibility in the UNNS framework. Left: Theorem 5.1 proves that admissible trajectories approaching a class boundary exhibit strictly decreasing margin — an irreversible structural flow. Right: The margin acts as a predictive estimator for physical transitions such as a spacecraft's approach to the heliopause, validated against Voyager heliospheric data.

Ising Criticality Benchmark

The Ising 2D model approaching its critical temperature Tc provides the canonical corpus validation of trajectory monotonicity. As T → Tc, the transfer-matrix eigenvalue gaps become increasingly correlated, the decisive pair approaches its threshold, and the margin falls monotonically — without exception across all tested ladder sizes.

ISING TRAJECTORY: MARGIN vs. BOUNDARY PROXIMITY (N=20, 2D) T/Tₒ → Tc (boundary approach direction) Connectivity Margin m(L) 7×10⁻³ 3×10⁻³ 1×10⁻³ T/Tc=1.20 T/Tc=1.05 T/Tc=1.01 ξ → ∞ corr. length Monotone decrease — Theorem 5.1 m(γ(0)) > m(γ(s)) > m(γ(t)) — no recovery observed ∂Ω at Tc

Voyager Validation

The Voyager heliospheric ladders — constructed from in-situ magnetic field data along the spacecraft trajectory — provide independent physical validation. As Voyager approaches the heliospheric boundary layer, the structural margin decreases continuously without recovery, consistent with Theorem 5.1. The margin serves as a predictive estimator for the boundary approach, enabling a Time-to-Boundary Estimator for the heliopause crossing.

→ Voyager STRUC-PERC Analysis  ·  → Multiscale Results

⭐ The Canonical Ladder and Maximum-Margin Principle

The encoding-dependence problem — that the same physical system can yield different realizability classes under different ladder constructions — has been a persistent challenge for the UNNS framework. The maximum-margin principle resolves this: among all admissible encodings, the canonical representative is the one that maximises m(L).

Theorem 7.1 upgrades this from an empirical rule to a geometric theorem: the maximum-margin encoding is the encoding at greatest distance from any realizability-class boundary in the gap-vector metric. It is the geometrically deepest representative.

An infographic comparing admissible encodings for a physical system like a Helium atom. It shows four network diagrams ranging from Hard (fragmented red) to Full (integrated green). Above them is a balance scale weighing Stability against Marginality. On the right, a 3D cutaway diagram shows a Canonical Ladder L* sitting at the deepest point of a geometric well, with an arrow indicating it is the Geometrically Deepest Representative where distance to the boundary is maximized.
Figure 3. The geometric foundation of the Maximum-Margin Principle. Among all possible ladder encodings for a physical system, the canonical representative L* is selected because it maximises structural margin, placing the system at the greatest geometric depth from a realizability boundary. This canonical selection defines the system's intrinsic interaction regime.

Corpus Evidence: Representation Splits

System Encoding 1 Class m(L) Encoding 2 Class m(L) Canonical
Helium QM-I Full ≈ 1.2 × 10⁻² Zeeman Tail ≈ 3 × 10⁻³ QM-I ✓
Sodium QM-I Hard ≈ 0 Zeeman Tail > QM-I Zeeman ✓
HD molecule Combined Full/Giant higher Sub-ladder lower lower Combined ✓
Crystallography Cell volume — higher Per atom — lower Cell volume ✓

In all 93 corpus datasets, no counterexample has been found: the maximum-margin encoding always corresponds to the deeper realizability class. Theorem 7.1 gives this a geometric explanation.

Consequence: Intrinsic Interaction Regime

Once the canonical ladder is selected, the interaction regime assigned by the functional Φ(m(L*), r, χ(L*)) is the geometrically intrinsic interaction regime of the physical system — corresponding to the encoding maximally distant from any structural boundary. Interaction regime becomes a property of the system, not of its representation.

📊 Corpus Validation — 93 Datasets · 22,817 Evaluations

The monotonicity results are validated against the full UNNS Phase Mapping corpus without introducing new data. All observations are drawn from previously established corpus outputs. The role of the validation section is to verify that every physical and computational system in the corpus is consistent with the proved monotonicity ordering.

Datasets
93
Atomic, molecular, nuclear, geoid, CMB, cosmic web…
Evaluations
22,817
Phase Mapping corpus
Monotonicity violations
0
No trajectory with margin recovery observed
Canonical class splits
0
Max-margin always selects deeper class
Physical domains
11
Consistent ordering across all
Non-degeneracy
100%
Condition satisfied by all tested datasets

Cross-Domain Margin Ordering

CROSS-DOMAIN MARGIN ORDERING — LOG SCALE 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ Connectivity Margin m(L) — log scale Planck TT/TE/EE — m ≈ 2×10⁻⁴ → Gravity-like Cosmic Web (DESI/SDSS) — m ≈ 5×10⁻⁴ Geoid harmonics (Earth/Moon/Mars) Ising T/Tc=1.01 — m ≈ 1.2×10⁻³ Ising T/Tc=1.05 — m ≈ 3×10⁻³ He Zeeman — m ≈ 3×10⁻³ (Tail) Ising T/Tc=1.20 — m ≈ 7×10⁻³ (interior) He QM-I — m ≈ 1.2×10⁻² (Full) Strong regime estimate m ~ 0.5–1.0 ← boundary interior →

The Non-Trivial Content of the Cross-Domain Ordering

The three domains — atomic (m ~ 10⁻²), Ising critical (m ~ 10⁻³), and cosmological (m ~ 10⁻⁴) — were constructed independently, use different physical mechanisms, and were not designed to produce this ordering. Yet the margin ordering aligns precisely with the physically known boundary-proximity ordering. Under Theorem 6.1, this alignment is a structural consequence, not a coincidence.

→ Phase Mapping Analysis Dashboard  ·  → STRUC-PERC Corpus Analysis  ·  → STRUC-I Corpus Analysis

🔢 Explicit Lipschitz Constants

A key feature of the manuscript is that the bi-Lipschitz constants in Theorem 4.1 are not merely abstract existence claims — they are derived explicitly (Appendix D) and estimated numerically (Appendix E) for representative corpus systems. Theorem 4.2 (Normal Coordinate Form) sharpens this further: in the canonical normal coordinate, the margin is asymptotically isometric to boundary distance with c₂/c₁ → 1, resolving the local metric structure exactly. These results confirm that the metric structure of realizability space is quantitatively accessible within the STRUC-PERC-I pipeline.

BI-LIPSCHITZ CONSTANTS c₁, c₂ AND ANISOTROPY RATIO C = c₂/c₁ SYSTEM m(L) typical c₂ (upper) c₁ (lower) C = c₂/c₁ Helium QM-I (deep interior) ~1.2 × 10⁻² 1.4 – 1.9 0.55 – 0.75 2.3 – 3.2 Ising transfer-matrix (near-crit.) 3 – 7 × 10⁻³ 1.8 – 2.4 0.45 – 0.65 3.5 – 5.0 Voyager heliospheric (boundary) ~10⁻⁴ – 10⁻³ 2.1 – 2.8 0.40 – 0.58 4.0 – 6.5 ↑ anisotropy increases near boundary

Geometric Meaning of Increasing Anisotropy

The anisotropy ratio C = c₂/c₁ increases systematically from deep interior (He QM-I: C ≈ 2.3–3.2) to boundary-proximal systems (Voyager: C ≈ 4.0–6.5). This is a geometric consequence of Section 9.5: as a ladder approaches the realizability boundary, the coordinate directions in gap-vector space become increasingly non-equivalent. Boundary proximity induces anisotropy — the boundary itself is not isotropic.

🔬 Broader Theoretical Connections

The manuscript identifies four connections to established theoretical frameworks, carefully marked as directions rather than formal equivalences. These connections situate the UNNS Substrate within a broader mathematical and physical landscape.

Renormalisation Group

Trajectory monotonicity mirrors RG flow irreversibility. m(L) may play the role of a flow parameter or distance to a fixed-point manifold, with realizability-class boundaries as critical loci. The UNNS flow is governed by gap-vector geometry, not iterative coarse-graining; the analogy is structural, not formal.

Percolation Theory

The PRP classification is defined through connectivity transitions in Gκ(L), suggesting a link between realizability-class boundaries and percolation thresholds on structured graphs parameterised by κ. Whether monotonicity follows from known percolation results remains an open direction.

Dynamical Systems

Admissible trajectories may be viewed as flows on gap-vector space, with realizability-class boundaries acting as attracting sets. Theorem 5.1 implies the absence of sustained excursions away from the boundary along boundary-approaching trajectories. A full attractor analysis is not developed here.

Order Theory

The global margin ordering suggests realizability space admits a natural partial order, with equivalence classes at equal margin. The space may be formalised as a stratified or ordered geometric object with m(L) as canonical coordinate up to equivalence. The precise categorical or topological structure remains open.

🔬 Falsifiability — What Would Break This

Consistent with UNNS program standards, the manuscript provides explicit falsification criteria that are operationally testable using the existing STRUC-PERC-I v2.4.1 pipeline.

Falsification Criteria
  1. Net margin increase along a boundary-approaching trajectory — would falsify Theorem 5.1
  2. Inverted margin ordering: m(L₁) > m(L₂) but d(L₁) < d(L₂) — would falsify Theorem 6.1
  3. Non-degeneracy failure: two ladders with equal margin in different realizability classes
  4. Canonical class split: two maximum-margin encodings of the same system in different classes
  5. Margin zero in the interior: m(L) = 0 at a point not on the realizability boundary
  6. Discontinuous margin approach: m(γ(t)) jumps before reaching ∂ΩL

No instance of any of the above has been found across 93 datasets, 22,817 evaluations, and all 11 physical domains.

Manuscripts and Analysis

UNNS Substrate Research Program · unns.tech · April 2026 · Corpus: 93 datasets · 22,817 evaluations · 11 physical domains · Instruments: STRUC-PERC-I v2.4.1 · STRUC-I v1.0.4

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