1. Introduction: τ-Field Meets Precision Molecules

The τ-Field in the UNNS Substrate is designed to connect recursion microstructure to observable spectra. In the τ-MSC Chamber, recursion flows along Φ–Ψ–τ cycles and generates synthetic “spectral echoes” that can be projected into physical frequencies. The natural next step was always to bring this machinery into contact with real molecular data.

The v0.9.x series of the UNNS Laboratory embodies this step. After the RaF testbed demonstrated that τ-based projection can align three hyperfine manifolds with a single τ-coupling model, the question becomes sharper:

Question. How does the τ-Field “see” molecules whose spectra are experimentally clean but topologically simpler than RaF — in particular, YbF and SrF?

Both YbF and SrF are ²Σ diatomic molecules with well-studied microwave spectra. YbF is central to electron EDM searches; SrF is a flagship laser-cooling molecule. They are physically rich but hyperfine-simple compared to RaF. This makes them an ideal test case for seeing whether the τ-Field only responds to manifest multiplet structure, or whether it detects a deeper, more geometric kind of recursion.

2. Real Data Sets: YbF and SrF

For this study we use two real-data comparison logs generated by UNNS Lab v0.9.1:

• YbF: 38 microwave rotational–hyperfine transitions for X ²Σ (v = 0), matched one-to-one with τ-MSC synthetic lines.
• SrF: 35 rotational transitions for X ²Σ⁺ (v = 0), likewise matched one-to-one.

In both cases, the real frequencies are drawn from established spectroscopy sources (NIST ASD, JPL Catalog, and peer-reviewed hyperfine studies) and reformatted into the UNNS JSON comparator format. The Lab applies a reversible offset + scale calibration

f′ = a · f + b

but otherwise does not alter the experimental values. All subsequent structure — residuals, curvature, τ-phase, χ² statistics — emerges from the interaction between τ-MSC output and these calibrated frequencies.

3. χ²-Normalized τ-Projection in v0.9.1

The v0.9.1 τ-Coupling Engine follows a fixed pipeline that was first validated on RaF:

1. Generate synthetic recursion spectra with the τ-MSC Chamber.
2. Calibrate to experimental data with an offset–scale map f′ = a · f + b.
3. Match real and synthetic lines one-to-one.
4. Compute curvature and τ-phase for each matched pair.
5. Attempt to group lines into hyperfine manifolds.
6. Evaluate raw and normalized χ² metrics.
7. If manifolds are present, fit global τ-hyperfine parameters (ΔC, g_ω).

The key upgrade in v0.9.1 is the introduction of a normalized χ². For heavy or high-curvature systems, raw χ² explodes even when the fit is physically excellent; normalization rescales residuals to a cross-molecule range and prevents RaF-like curvature from overwhelming lighter datasets. This makes it possible to compare qualitatively different molecules (RaF vs YbF vs SrF) within a single τ-Field framework.

4. Living Curvature Maps: YbF vs SrF

Even without explicit manifolds, YbF and SrF leave distinct “fingerprints” in τ-space. One way to visualize this is to plot τ-projection curvature against calibrated frequency, with a soft τ-field background. The diagram below is not a literal plot of the JSON data, but a faithful field portrait of what v0.9.1 reveals: two bands, two behaviors, one substrate.

In the actual v0.9.1 runs, both molecules achieve 100% match rate and RMSE of order 5–6 MHz. Raw χ²/dof is enormous (10³–10⁴), because individual residuals are squared without manifold weighting; the normalized χ² correctly recognizes that this is not a “catastrophic mismatch” but the expected behavior of heavy, high-curvature ²Σ spectra without strong internal clustering.

5. YbF: A τ-Sensitive but Hyperfine-Silent Object

YbF, in this UNNS run, presents as a τ-sensitive but hyperfine-silent object:

• 38 of 38 lines matched (match rate 100%).
• RMSE ≈ 5.8 MHz.
• Manifolds detected: 0.
• τ-Reliability Index: ≈ 0.06.

No hyperfine manifolds reach the internal threshold for clustering. The τ-Coupling layer therefore does not activate a ΔC / g_ω fit — not because the τ-Field “fails”, but because the dataset does not present the kind of structured multiplet geometry that the RaF engine latches onto. YbF looks, in UNNS terms, like a single-shell τ-object: its curvature responds smoothly to the recursion map, but there is no secondary topological layering to exploit.

This is scientifically important. It means that the τ-Field does not invent manifold structure where none exists. Instead, it produces a robust, monotonic τ-signature for YbF that can be compared to other molecules even when hyperfine structure is relatively simple.

6. SrF: A Parallel but Distinct τ-Signature

SrF behaves similarly in headline statistics, yet differently in τ-space:

• 35 of 35 lines matched (match rate 100%).
• RMSE ≈ 6.3 MHz.
• Manifolds detected: 0.
• τ-Reliability Index: ≈ 0.06.

Just as with YbF, no manifold structure emerges; the engine treats SrF as a smooth τ-signal spread across all lines. The raw χ²/dof again grows to large numerical values; the normalized χ² collapses this into a range that can be compared with RaF and with lighter systems.

Yet, when τ-curvature and τ-phase are inspected, SrF and YbF do not coincide. They share the “proto-unstructured” label (no manifolds), but inhabit different regions of the τ-field phase space. The τ-MSC comparison shows this clearly when both datasets are overlaid.

7. τ-Phase Evolution as a Living Field

We can emphasize this difference by looking at τ-phase evolution — a measure of how the coupling between Φ-geometry and Ψ-spectral modes winds along the recursion flow. In the field portrait below, YbF and SrF are visualized as two drifting phase traces in a slowly breathing τ-background.

The qualitative lesson is simple but powerful: in τ-phase terms, YbF and SrF are not the same object. They occupy different phase trajectories, even while sharing the same “proto-unstructured” status with respect to hyperfine manifold detection. The τ-Field sees more than the raw hyperfine topology here; it sees how recursion twists through the rotational structure itself.

8. Toward a Hyperfine-Silent τ-Class of Molecules

RaF taught us what a manifold-rich τ-object looks like: three coupled manifolds, a tight global τ-coupling fit, and a nontrivial balance between curvature and τ-phase. YbF and SrF reveal a complementary regime:

• Single effective τ-shell (no manifolds recovered).
• Stable RMSE with 100% line matching.
• Distinct τ-curvature and τ-phase profiles per molecule.
• χ²-normalization recognizing the structure as “non-pathological” but not manifold-resolvable.

This suggests a tentative classification:

Conjecture (Hyperfine-Silent τ-Class). Some molecules — especially linear ²Σ systems like YbF and SrF — inhabit a τ-regime where recursion structure is encoded primarily in rotational and centrifugal deformation, rather than in discrete hyperfine manifolds. The τ-Field sees them as proto-unstructured recursion objects: internally structured in τ-space, but spectrally smooth when projected back to frequency.

This is not a claim of new physics; it is a structural conjecture about how the τ-Field partitions the molecular landscape. The value of the v0.9.1 comparison engine is precisely that it can generate such conjectures in a controlled way, by holding the τ-MSC side constant and varying the real data.

9. Implications and Next Steps

The YbF and SrF runs suggest several directions for the UNNS Research → Lab programme:

• Systematic classification. Extend the v0.9.1 engine to a library of molecules (RaF, YbF, SrF, BaF, OH, FH, ALH) and classify each one by: manifold count, τ-curvature profile, τ-phase trajectory, and normalized χ².
• σ-weighted χ² (v0.9.2). Introduce line-by-line uncertainty weighting so that datasets with high-precision measurements (e.g., EDM lines) can be distinguished from coarse exploratory data.
• τ-Reliability vector. Generalize the scalar τ-Reliability index into a short vector summarizing manifold structure, curvature–residual coherence, and τ-phase smoothness.
• Cross-link with precision experiments. Use UNNS classification as an independent lens on molecules used in EDM searches, laser cooling and fundamental symmetry tests.

Conceptually, the deeper implication is that the τ-Field may be capable of segmenting the molecular world along axes that differ from standard quantum numbers. UNNS does not replace conventional spectroscopy; it adds a layer of recursion-based structure on top of it.

10. Conclusion

In this Research → Lab note, we have used the χ²-normalized τ-Coupling Engine of UNNS Lab v0.9.1 to compare two heavy ²Σ molecules, YbF and SrF, against τ-MSC synthetic spectra. Both systems achieve perfect line matching and moderate RMSE, yet neither resolves into RaF-like hyperfine manifolds. From the τ-Field’s point of view, they are proto-unstructured: hyperfine-silent, but far from featureless.

Their τ-curvature and τ-phase maps reveal reproducible, distinct signatures. This encourages us to think in terms of a “hyperfine-silent τ-class” of molecules — a regime where recursion organizes rotational structure rather than multiplet geometry. As the UNNS Laboratory advances to v0.9.2 and v1.0, such regimes will become targets for more rigorous statistical analysis and, eventually, for comparison with fully developed τ-Field theory.