WP28: magnitudes are class labels, not point functions; the rigid monomial freeze is collective, not edge-wise
Question carried over from WP26: the rigid phase clusters freeze the reduced-form monomial lnM to 1e-3..1e-2. Is that because each EDGE's fitted magnitude is a universal function of the physical point within the cluster? If so, M-freeze would be a trivial consequence of edge-wise universality.
Method (checker checkers/wp28_edge_dispersion.py, results
results/wp28_edge_dispersion.json, 1210 minima, 755 texture classes, 4/4 gates):
- A. pooled dispersion per (cluster, sector, slot) across all minima;
- B. within-class dispersion per (texture class, sector, slot);
- C. lnM contrast: between-class vs within-class std per cluster;
- D. frozen-edge census (std < 0.01, n >= 40) identified against central couplings.
Findings
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WITHIN a texture class the fitted magnitude vector is essentially unique: median within-class edge std = 1.8e-4 over 576 (class, sector, slot) groups. Each class has one sharp magnitude point; its multiple minima are the same point.
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ACROSS classes of the same cluster there is no edge-wise universality: 81/83 pooled groups with n >= 20 disperse with std 0.3..3.3 (per-cluster medians 1.0..1.7). Different textures assign wildly different magnitudes to the same (sector, slot).
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Exactly two frozen anchor edges exist cluster-wide:
- cluster 1, u(2,2): frozen to ln(y_t CENTRAL) = -0.03355678352884276 with offset -7e-18 - the scan CENTER value of the top coupling, not a per-point fitted coupling;
- cluster 3, u(0,2): frozen near ln y_c (offset 0.016).
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Yet lnM is frozen BETWEEN classes in the rigid clusters while the constituent edges disperse by e^1..e^3:
- cluster 2: between-class lnM std 1.3e-3 (edge median 1.72);
- cluster 1: between-class lnM std 0.079 (edge median 1.01);
- conspiracy clusters 0/3/4: between-class lnM std 3.8..9.5.
Interpretation
The rigid-cluster monomial is a cluster-level COLLECTIVE invariant realized by wildly different individual edge assignments. It is invisible at the single-edge level and sharp at the monomial level. Magnitudes are class labels, not point functions: the fitted vector is unique per class but carries no cluster-universal edge content. The WP26 M-freeze is therefore not explained by edge universality; it is a genuine collective constraint on the monomial the fit preserves while moving edges freely.
The cluster-1 anchor u(2,2) = y_t CENTRAL to machine precision says those textures hard-wire the scan-center top coupling as an edge and build all mixing from the remaining edges - a structural property of the texture family, not a fit coincidence (42/42 minima, std 3e-17).
What this refocuses
The next question is the mechanism of the collective freeze: is lnM's between-class constancy in clusters 1/2 an EXACT syzygy the texture graph forces (e.g. det-like products pinned by mass data), or an approximate valley flat direction? A graph-theoretic determinant/matching computation can decide this without fitting.
Artifacts: checkers/wp28_edge_dispersion.py, results/wp28_edge_dispersion.json. Depends on: claim:83e79fa2a84099e8cd4d (flavor admission program), WP20 valley audit, WP21e universal factorization, WP25 center selection, WP26 level anatomy.