UNNS Laboratory — Chamber XXV: EPU v0.3.0

Observable	k₀	k_R	k_ρ	k_S	k_γ
Run sweep to display θ

Observable	Predicted	Observed	Sigma (σ)	Residual	Z-score	χ² Contrib
Run sweep to generate residuals

📚 Laboratory Guide

Chamber XXV — Empirical Projection & Unification (v0.3.0)

Purpose: Recursion → Observables (Cosmology, LSS, Precision) Engine: PE-26 Projection, γ-Sweep, θ-Calibration

Phase E Lab PE-26 Projection Engine Chamber XXV · v0.3.0

Chamber XXV is the Empirical Projection & Unification (EPU) engine of the UNNS Substrate. It maps recursion geometry at a given γ into a set of empirical observables drawn from:

Cosmology — Λ, H₀, ρ_vac
Large-Scale Structure (LSS) — N_eff, σ₈, r_drag, n_s
Precision Physics — α, μ, g_e

The goal is not to claim new physical predictions, but to test how far a single recursion signature can reproduce the existing empirical structure before micro-recursion operators (XVI, XVII, XXI) become necessary.

Key idea: Chamber XXV is a macro-recursion engine. It is excellent at matching cosmology & LSS, and deliberately limited for precision-scale observables. Chamber XXVI is dedicated to the micro-recursion regime.

Module A Recursion Sweep (γ-Landscape)

Module A scans a range of γ values. For each γ, it runs the internal recursion engine, extracts geometric features, and evaluates how well they reproduce the active datasets.

Inputs:

γ Min, γ Max, γ Step — sweep range (for example 1.55 → 1.68, Δγ = 0.01).
Recursion Depth — number of recursion iterations; higher depth gives smoother results, but slower runs.
Grid Resolution — size of the sampling grid for recursion geometry (e.g. 64×64, 128×128).
RNG Seed — fixes random components for reproducibility.

Outputs:

γ★ — the best-fit value at which the global χ² is minimal.
χ²(γ) curve — how total misfit changes along the γ-axis.
Recursion features — curvature, variance, stability metrics at each γ.

A clean, bowl-shaped χ²(γ) curve indicates a stable recursion and a well-behaved projection. The selected γ★ is then used in all downstream PE-26 computations.

Module B Projection Configuration (PE-26)

PE-26 is the recursion-to-observable projection model. It takes recursion features at γ★:

f_R — curvature response,
f_p — propagation term,
f_S — structural drift,
f_γ — γ-derivative mode,

and forms predictions using a log-domain linear mapping:

log10(O_pred / O_ref) = k0 + k_R f_R + k_p f_p + k_S f_S + k_γ f_γ

Each observable has its own parameter set (θ-vector) in PE-26:

Regime	Observables	θ-Parameters
Cosmology	Λ, H₀, ρ_vac	k₀, k_R, k_p, k_S, k_γ for each
LSS	N_eff, σ₈, r_drag, n_s	k₀, k_R, k_p, k_S, k_γ for each
Precision	α, μ, g_e	k₀, k_R, k_p, k_S, k_γ for each

The θ-table can be edited manually or loaded via presets. PE-26 itself is agnostic to the origin of θ; it simply projects recursion features into the observable space.

Modules C & D Dataset Selection & Fit Strategy

Modules C and D control which empirical observables are active and how the fit is evaluated.

Available observables:

Cosmology

Λ — cosmological constant / dark energy density
H₀ — Hubble parameter
ρ_vac — vacuum energy density proxy

Large-Scale Structure (LSS)

N_eff — effective number of neutrino species
σ₈ — clustering amplitude
r_drag — BAO drag scale
n_s — scalar spectral index

Precision Physics

α — fine-structure constant
μ — proton-to-electron mass ratio
g_e — electron g-factor

Fit Strategy

γ-sweep with fixed θ (default)
θ-tuning via presets
χ² total / χ² per observable readout

The chamber computes a standard χ² score:

χ² = Σ ((O_pred - O_obs) / σ)²

You can enable/disable observables to study how different subsets pull the γ-fit and affect the total χ².

Module E χ² Landscape, Curves & Residuals

The visualization layer shows how well the recursion geometry at γ★ matches the active datasets.

Components:

χ² vs γ curve — global fit landscape across the sweep range.
Observable curves — O_pred(γ) overlaid with reference ranges.
Residual table — predicted, observed, residual, z-score, χ² contribution for each observable.
Optional 2D heatmaps — when enabled, show χ² or residuals across parameter slices.

Z-scores are the most compact indicator: |Z| < 0.1 → excellent · 0.1 ≤ |Z| < 1 → mild tension · |Z| ≥ 1 → significant deviation

Module F Export & Downstream Analysis

The export engine captures the full state of the chamber for further analysis or archival.

JSON export includes:

Sweep parameters (γ range, depth, resolution, seed).
γ-landscape: χ²(γ), recursion features per γ, best-fit γ★.
PE-26 θ-table for all 10 observables.
Best-fit predictions, residuals, z-scores, χ² contributions.
Runtime and stability metrics.

CSV export includes:

Residual table only (predicted, observed, residual, z, χ²ᵢ) — suitable for spreadsheets or plotting.

Limitations & Relationship to Chamber XXVI

Chamber XXV is intentionally limited to a macro-recursion view:

It matches cosmological and LSS observables remarkably well using PE-26.
It cannot, by design, fully capture precision observables (μ, g_e) that require micro-recursion, torsion kernels, and closure dynamics.

These micro-recursion structures live in Operators XVI, XVII, XXI and are explored in Chamber XXVI. XXV defines the boundary of PE-26: it shows how far a single γ-based recursion signature can go before we must turn on micro-operators.

Typical Workflow in Chamber XXV

Select the Cosmo+LSS preset and run the γ-sweep to find γ★ and χ²_min.
Inspect the χ² curve and residual table to see how cosmology & LSS are matched.
Switch to the Unified Physics preset to include α, μ, g_e and observe how the fit degrades.
Optionally adjust θ-parameters for specific observables and re-run the sweep.
Export the JSON or CSV results for documentation, comparison, or downstream analysis (e.g. Chambers XXVI+).

Used this way, Chamber XXV becomes a diagnostic lens on the UNNS Substrate: it reveals where recursion geometry already mirrors physical structure, and where deeper micro-recursion dynamics must be engaged.

Fit Strategy & χ² Landscape Behaviour

The Fit Strategy controls how Chamber XXV explores the χ²(γ) landscape generated by the recursion engine and the PE-26 projection model. Different strategies balance completeness vs. speed:

Grid Sweep — dense, uniform sampling of γ across the selected range (e.g. 1.55–1.68 with step 0.01). Best for:
- Global scans for publication-quality χ² curves
- Detecting multiple local minima or plateaus
- Comparing different datasets or presets
Local Search — starts from a good initial guess (e.g. a known γ★) and refines nearby points. Best for:
- Fast re-calibration after small preset changes
- Fine-tuning θ after a grid sweep has found a basin
Random / Monte-Carlo — draws γ samples stochastically in the allowed interval. Best for:
- Exploring noisy or irregular χ² landscapes
- Stress-testing stability of projected observables
Hybrid — combines a coarse grid with local refinement around promising minima. This is the recommended mode for:
- Mixed datasets (cosmology + LSS + precision physics)
- Pre-XXVI studies where we probe the limits of macro-recursion

In all cases, the goal is to identify a robust minimum of χ²(γ), not just a single lucky sample. Good practice:

Confirm γ★ with at least two different strategies (e.g. grid + local).
Check that nearby γ values do not produce equally good χ² — otherwise the solution is degenerate and should be reported as such.
For publication, use grid or hybrid with high resolution (128×128, depth ≥ 400, small γ steps).

θ-Parameters, TE-26 Auto-Calibration, and Precision Limits

_pred f_R, f_ρ, f_S, f_γ

PE-26 projects recursion features into observables via the θ-parameter matrix. For each observable O, we use:

log₁₀(O_pred/O_ref) = k₀ + k_R f_R + k_ρ f_ρ + k_S f_S + k_γ f_γ,

where the dimensionless features are:

f_R — normalized curvature (R_r/R₀)
f_ρ — normalized information density (p_I/ρ₀) (this replaces the older label f_p)
f_S — normalized stability (S/S₀)
f_γ — normalized recursion parameter (γ/γ₀)

The vector θ = (k₀, k_R, k_ρ, k_S, k_γ) defines how strongly each feature influences a given observable. TE-26 auto-calibration adjusts θ to minimize χ² over the selected datasets.

TE-26: Two-Stage Auto-Calibration

Stage 1 — Intercept Grid Search For each observable, TE-26 sweeps k₀ on a coarse grid to anchor O_pred near O_ref. This stabilizes the log-ratio and prevents runaway residuals.
Stage 2 — Slope Refinement With k₀ fixed, TE-26 performs coordinate descent over (k_R, k_ρ, k_S, k_γ) to reduce χ². The process respects preset bounds (e.g. |k| ≤ 1.2, and |k| ≤ 0.02 for precision observables such as α).

The result is a physically meaningful θ matrix that shows how cosmological and large-scale structure observables respond to recursion geometry. This works very well for Λ, H₀, ρ_vac, N_eff, σ₈, r_drag, and n_s.

Why Precision Constants (α, μ, g_e) are Out of Reach in XXV

Precision QED observables such as the fine-structure constant α, the proton-to-electron mass ratio μ, and the electron g-factor g_e have uncertainties as small as 10⁻¹²–10⁻⁹. In Chamber XXV:

We use only four macro-features (f_R, f_ρ, f_S, f_γ).
The mapping from features to observables is linear in log-space via θ.
Recursion data comes from a coarse γ-sweep without micro-scale torsion or local oscillations.

This is sufficient to match macro-observables (cosmology and large-scale structure), but it lacks:

Micro-curvature oscillations and torsion kernels (Operator XXI).
Closure-driven flux balancing at small scales (Operator XVI).
Matrix-level mode coupling between distinct recursion channels (Operator XVII and higher-order operators).

In other words, Chamber XXV deliberately operates in a macro-recursion regime. It demonstrates that:

The UNNS Substrate can map recursion geometry to cosmological data in a mathematically consistent way.
There is a clear boundary where macro-recursion stops being expressive enough to capture precision quantum physics.

Chamber XXVI and the PE-27/PE-27C engines extend this framework with micro-recursion modes, torsion sampling, and operator-level diagnostics, specifically to tackle α, μ, g_e, and related precision observables beyond the capabilities of PE-26.