A Five-Chamber Analysis of Convexity, Variance, and the Structural Limits of Recursive Curvature
Executive Summary
Five consecutive UNNS chambers — from the recursive-rigidity probe of Chamber LVI through the full gain-and-basis scan of Chamber LX — converge on a single, decisive structural result: convex mixing of recursively admissible operators cannot produce certifiably negative curvature inside the convex hull of those operators.
This is not an absence of signal. It is a tightly localized, mechanistically explained null result — a structural null. The chambers reveal that strongly negative curvature estimates (b ≈ −0.30) exist near the high-(β+γ) boundary of the operator simplex, but this region is systematically obstructed by variance inflation at the calibration floor, causing the statistical non-degeneracy gate to fail across all parameter configurations.
The accompanying paper Convex Sign Preservation and the Calibration-Floor Obstruction in Recursive Curvature Dynamics provides the analytic framework: a conditional theorem proving that convex operator families under submultiplicative recursion cannot exhibit interior sign reversal unless the calibration floor fails or convex structure is broken. The chambers supply the empirical confirmation. Together they constitute a coordinate-invariant, gain-invariant, basis-invariant rigidity result that closes a large hypothesis space and narrows future searches to non-convex mechanisms.
🔬 The Five-Chamber Pipeline: What Each Stage Tests
The LVI–LX pipeline was designed as a systematic exhaustion strategy. Rather than searching blindly for negative curvature, each chamber asks a sharper question — progressively ruling out candidate mechanisms until the structural obstruction is unambiguous.
Recursive Rigidity vs. Exponential Compounding
Establishes the baseline: does recursive operator growth behave like simple exponential compounding, or does the recursive structure impose independent constraints? Outcome separates the two regimes cleanly.
Bias-Damped Recursion Scans
Tests whether introducing a bias-damping term can depress positive curvature into negative territory. Sweeps full (β, γ, g) parameter space with damped operators to locate candidate sign-reversal regimes.
Bias-Coupling Stability Scans
Couples bias magnitude to recursion depth and tests whether dynamic bias coupling can shift the curvature sign certificate. Targets the boundary region where b estimates are most negative.
Operator Simplex Geometry
Full triangulation of the three-basis operator simplex {K₀, K₁, K₂}. Maps curvature sign and σ_F across the entire convex hull — including interior, edges, and vertices — via high-resolution simplex scan.
Basis Swap & Gain Scaling
Tests whether rotating the operator basis or amplifying gain (g ≤ 1.5) can shift the negative region into certifiable territory. Exhausts all remaining "maybe just scale it" strategies.
📊 The Core Structural Separation
Before these chambers, curvature sign, asymptotic growth exponent, and statistical certifiability were treated as aspects of the same phenomenon. The pipeline demonstrates they are structurally distinct — and that their dissociation is the key to understanding recursive operator behaviour.
Figure 1 — The three properties formerly treated as one are rigorously separated by the LVI–LX chamber pipeline. Negative curvature sign (b < 0) can coexist with the absence of statistical certification due to calibration-floor variance inflation. This dissociation is the paper's central contribution.
⚡ The Calibration-Floor Obstruction: A New Diagnostic Mechanism
The most significant discovery of the pipeline is not that negative curvature is absent — it is that negative curvature and its certifiability are systematically decoupled by a geometric-statistical obstruction. This is the calibration-floor obstruction.
Absence of certified negative curvature does not imply absence of negative curvature. The obstruction is not algebraic. It is variance-geometric: the region of operator space where b is most negative is precisely the region where σ_F is most elevated — not due to noise, but due to the intrinsic geometry of the operator manifold near the submultiplicative boundary.
Why This Matters: The Non-Degeneracy Gate
Certification of any curvature estimate b requires passing the non-degeneracy gate: the likelihood ratio L₂ must exceed 5σ_F. When the calibration floor σ_F is elevated — as it is systematically in the high-(β+γ) boundary region — no curvature estimate, however large in magnitude, can clear this threshold. The gate fails not because the signal is weak, but because the local variance structure of the operator manifold is intrinsically elevated at that boundary.
Figure 2 — The operator simplex (left) shows that negative curvature (b ≈ −0.30, red dashed region) is concentrated near the high-(β+γ) boundary, precisely where variance σ_F is elevated. The obstruction chain (right) shows why no certification can occur despite a real negative signal.
🔑 Key Discoveries: Five Structural Results
1. Structural Separation of Sign, Growth, and Certifiability
The most conceptually important result is the rigorous demonstration that curvature sign (b), asymptotic growth exponent (λ), and statistical certifiability (L₂ > 5σ_F) are distinct quantities that do not co-determine each other. Before these chambers, they were implicitly treated as aspects of the same phenomenon. Now their independence is empirically established.
Why This Matters
Systems with b < 0 need not exhibit any certification of negative curvature. Systems with λ growing fast may still have b > 0. Certifiability is a geometric-statistical property, not a direct read of algebraic sign. This dissociation has consequences for how we interpret silence in curvature searches.
2. The Variance-Geometric Obstruction
The negative curvature region does not fail certification because b is small. It fails because the operator manifold geometry inherently elevates σ_F in the high-(β+γ) boundary region. This is a property of the manifold, not a measurement artifact. Chambers LVIX and LX both confirm the same geometry under basis rotation — the obstruction is coordinate-invariant.
Central Diagnostic Insight
Absence of certified negative curvature does not imply absence of negative curvature. Structured null results from this pipeline indicate not that the system lacks negative curvature regimes, but that those regimes live outside the convex span of the tested operators — requiring non-convex mechanisms to reach.
3. Convex Mixing Cannot Create Interior Sign Reversal
Across all combinations tested — three operator bases, full simplex scan, gain amplification up to g = 1.5, bracket refinement, stability verification — the chambers observed zero sign-change brackets, zero interior CERT_NEG, and zero intrinsically stable κ=0 structures with negative curvature. The convex hull of the tested operators is rigidly positive-certifiable in its interior.
Figure 3 — Gain amplification increases CERT_POS fraction modestly (green), but σ_F in the negative-curvature boundary region (amber) remains elevated regardless of g. CERT_NEG (red dashed) stays at zero across all tested gains, ruling out amplification as a sign-reversal mechanism.
4. Basis Invariance — A Coordinate-Invariant Property
When the operator basis is rotated (B0 → B1 → B2), the negative curvature region moves geometrically through the simplex but does not become certifiable. Its magnitude remains approximately invariant at b ≈ −0.30. This demonstrates that the geometry of the operator manifold, not the choice of coordinate frame, determines certifiability. It is a coordinate-invariant structural property.
5. Gain Scaling Is Not a Magic Lever
Increasing gain g amplifies positive certification slightly — but does not lower σ_F in the negative boundary region and does not produce any CERT_NEG. Amplification does not break convex rigidity. This rules out an entire class of strategies ("maybe just scale it") and narrows the search for negative curvature mechanisms to fundamentally non-convex territory.
Strategic Constraint: Where to Search Next
The pipeline result implies: if intrinsic negative curvature exists for this kernel family, it cannot be found inside the convex span of {K₀, K₁, K₂}. It must arise from (a) nonlinear operator coupling, (b) extension of the operator manifold beyond the current simplex, or (c) violation of submultiplicative structure. This is a precise directional constraint on future chamber design.
📐 The Analytic Framework: Convex Sign Preservation Theorem
The paper Convex Sign Preservation and the Calibration-Floor Obstruction in Recursive Curvature Dynamics provides the formal skeleton that the chamber results instantiate empirically. Its central contribution is a conditional theorem characterising when convex recursive operator families must preserve sign across the interior of the simplex.
Core Theorem (Informal Statement)
Let {Kᵢ} be a finite family of operators satisfying submultiplicative recursion with admissible growth. Let K(θ) = Σᵢ θᵢ Kᵢ be any convex combination (Σθᵢ = 1, θᵢ ≥ 0). If each endpoint Kᵢ exhibits non-negative certified curvature, then K(θ) cannot exhibit certified negative curvature in the interior of the simplex — provided the calibration floor condition holds (σ_F remains bounded from above uniformly on the interior).
The key condition — "provided the calibration floor condition holds" — is exactly what fails at the high-(β+γ) boundary. The theorem does not guarantee preservation everywhere; it guarantees it wherever σ_F is well-controlled. The chambers show that the negative region is precisely where that condition is violated — a constructive demonstration of the theorem's boundary behaviour.
Three-Way Triangulation
The paper's contribution is the triangulation of three mechanisms that together produce convex sign preservation:
Convex Geometry
Interior points of a convex hull cannot exceed boundary values for convex functionals. Applied to curvature, this limits how much sign variation can appear inside the simplex relative to vertices.
Submultiplicative Recursion
The recursion kernel satisfies ‖K^(n+m)‖ ≤ ‖K^n‖·‖K^m‖, which controls how curvature accumulates over depth — preventing exponential divergence that would otherwise overwhelm convex structure.
Variance Stratification
The statistical certification floor σ_F is not uniform across the simplex — it is highest precisely where curvature estimates are most negative. This stratification is a geometric property of the operator manifold, not noise.
Why This Triangulation Is Novel
Prior treatments of convex operator mixing addressed the algebraic side (geometry + submultiplicativity) but not the statistical certification side (variance stratification). The paper's contribution is showing that all three must be considered simultaneously — and that the calibration floor mechanism explains why empirically observed negative curvature never achieves certification. The theorem section formalises the analytic side; the chambers expose the statistical side.
🌐 Broader Significance and Theoretical Connections
Connection to Subadditive Growth Theory
The convex upper bound theorem proved here mirrors classical results in subadditive sequence theory: convex mixtures of operators cannot exceed extremal exponential growth. The UNNS result extends this to curvature sign under statistical certification — a richer and harder-to-certify quantity than spectral radius. This places the result within the framework of almost-subadditive sequences and Lyapunov-type exponents, giving it contact with well-established operator growth theory.
Connection to Dynamical Systems
The obstruction resembles known phenomena in dynamical systems: boundary-layer instability, parameter manifold stratification, and the distinction between interior stability and edge-region chaos. What UNNS contributes beyond analogy is a concrete operator-recursion model with formal statistical gates — making the result constructive rather than metaphorical. The calibration-floor mechanism is measurable, parameterised, and predictive.
Connection to Machine Learning Stability
The structural resonance with neural network ensemble theory is worth noting carefully: convex ensembles typically preserve dominant growth modes; instability localises near degenerate parameter regions; variance explosion masks sign signals near degenerate boundaries. The UNNS chambers provide a formal toy-model version of this pattern — not a proof of ML stability theory, but a rigorous operator-theoretic instance of the same structural geometry.
The Null Result as Discovery
Science has a structural bias toward positive results. The LVI–LX pipeline inverts this: its most important contribution is a high-resolution, mechanistically explained null — and null results are only weak when unstructured. This one is tightly localised (the negative curvature region is geometrically identified), mechanistically explained (variance stratification), and provides a conditional theorem. That is far stronger than speculative positive exploration.
What Was Eliminated
The pipeline closes a large hypothesis space: convex operator mixing + gain scaling cannot produce certified negative curvature for this kernel family under admissible growth. This saves time and provides decisive structural guidance. Knowing where not to search is as valuable as knowing where to search.
📋 Chamber Results Summary
| Chamber | Primary Test | Key Parameter | CERT_NEG | CERT_POS | Structural Finding |
|---|---|---|---|---|---|
| LVI | Recursive rigidity vs. compounding | Baseline (β, γ) | 0 | ✓ | Recursion ≠ exponential compounding; curvature family-specific |
| LVII | Bias-damped recursion | Damping δ, full (β,γ,g) | 0 | ✓ | Bias damping insufficient to produce sign reversal |
| LVIII | Bias-coupling stability | Dynamic coupling κ_bias | 0 | ✓ | Calibration-floor obstruction first identified and characterised |
| LVIX | Operator simplex geometry | Full simplex scan {K₀,K₁,K₂} | 0 | ✓ Interior | b≈−0.30 at high-(β+γ) boundary; σ_F elevated; nondeg gate fails |
| LX | Basis swap & gain scaling | Bases B0,B1,B2; g ≤ 1.5 | 0 | ↑ slightly | Basis-invariant, gain-invariant — no mechanism penetrates obstruction |
What the Pipeline Collectively Establishes
The five chambers together constitute a systematic exhaustion: they rule out bias damping, dynamic coupling, basis rotation, and gain amplification as mechanisms for certified sign reversal in convex operator families. What remains is structurally mandated: negative curvature certification requires non-convex mechanisms, manifold extension, or submultiplicativity violation. This is a strong, directional, defensible result.
⚠️ Scope and Precision: What This Result Is and Is Not
What This Is NOT
This pipeline does not prove universal sign preservation across all recursive systems. It does not claim that negative curvature cannot exist in any kernel family. It is not a general theorem about all operator recursion — it is specific to convex combinations of admissible operators under submultiplicative growth with the tested kernel families.
What This IS
A geometric-statistical rigidity result for convex recursive operator families under admissible growth: specifically, that the convex interior of the tested operator simplex cannot produce certified negative curvature because variance stratification at the calibration floor systematically obstructs the non-degeneracy gate in the negative-curvature boundary region. This result is clean, contained, and defensible.
The precision of this scoping matters. Overreach would invalidate the result; correct scoping makes it publishable and strategically useful. The calibration-floor obstruction is a new diagnostic mechanism — not a claim about the impossibility of negative curvature everywhere, but a precise characterisation of why it cannot be certified within this tested convex family.
Resources & Downloads
- Chamber Array LVI–LX (Interactive): chamber_array_lvi_lvii_lviii_lvix_lx.html — Live computational environment for all five chambers
- Research Paper (PDF): Convex Sign Preservation and the Calibration-Floor Obstruction in Recursive Curvature Dynamics.pdf — Full analytic framework, theorem, and proofs
- UNNS Chambers Certification Report (PDF): UNNS_Chambers_Certification_Report.pdf — Formal certification with preregistered acceptance criteria
- Raw Data Archive: certification.zip — Zipped JSON data files for all five chambers (reproducible)
All chambers are self-contained HTML applications with no external dependencies. All data files are provided for full reproducibility. Certification was performed against preregistered acceptance criteria with explicit null-result protocols. Null results are reported as primary scientific findings, not failures.