WP29: no exact monomial syzygy; the cluster-2 lnM freeze sits at the mass-data floor
Question left by WP28: the rigid clusters freeze the reduced-form monomial lnM BETWEEN texture classes (cluster 2: 1.3e-3, cluster 1: 0.079) while the constituent edge logs disperse by e^1..e^3. Is that freeze an exact graph-forced syzygy - M pinned by the mass-determined determinants |det Y_u| = y_u y_c y_t and |det Y_d| = y_d y_s y_b - or an approximate collective property?
Method (checker checkers/wp29_monomial_syzygy.py, results
results/wp29_monomial_syzygy.json, 755 classes, 1210 minima, 5/5 gates):
- A. perfect matchings per sector for all 755 classes: 731 have single-matching dets in both sectors; 24 have a two-term det Y_d; det Y_u is ALWAYS single-term.
- B. numerical floor test: per-cluster regression of class-mean lnM on (ln|det Y_u|, ln|det Y_d|) fitted from edges, residual vs the det floor (between-class std of the lndet's = fit tolerance on mass data).
- C. exact algebra: is the M exponent vector a rational combination of the two det exponent vectors?
- D. det_u anchor census.
Findings
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NO exact monomial syzygy exists: 0/731 single-single classes have M's exponent vector in span(det_u, det_d). The freeze is not a monomial identity.
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Cluster 2: lnM between-class std 1.3e-3 sits AT the mass-data tolerance floor (det_u floor 1.1e-3). The monomial is pinned as tightly as the masses themselves. But the det regression is NOT the mechanism: residual 8.7e-4 is barely below the raw 1.3e-3 and the coefficients are non-integer. lnM behaves like physical data without being reducible to det data - consistent with the WP26 reading (L level per class, physical gap and Vandermonde factors carry the variation).
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Cluster 1 is structurally different: ALL 27 classes are diagonal-up textures (mu = 273) with det_u = u00 u11 u22 passing through the WP28 anchor edge u(2,2) = y_t CENTRAL. det_u is frozen to 7.3e-10 between classes - far below fit tolerance - while lnM spreads 0.079, an intermediate freeze 60x above cluster 2's, unexplained by det data (residual = 100% of variance).
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Conspiracy clusters 0/3/4: lnM spreads 3.8..9.9, i.e. 18..10456x the mass floor. No determination at all.
Interpretation
The rigid monomial freeze is a property of the viable fitted set per cluster, not an algebraic identity of the texture graph. In cluster 2 the freeze is as tight as the mass data themselves; in cluster 1 it is weaker and rides on a fully anchored diagonal-up structure; in the conspiracy clusters it does not exist. Combined with WP26/WP27: the selective object is the (L, K) level structure of the viable solution set, and the monomial freeze is its shadow, not its cause.
Artifacts: checkers/wp29_monomial_syzygy.py, results/wp29_monomial_syzygy.json. Depends on: claim:83e79fa2a84099e8cd4d, WP20 valley audit, WP21e factorization, WP25 center selection, WP26 level anatomy, WP28 edge dispersion.