The Soft-Axis Length-Two Excess Is an Euler-Resonance Quotient
Record
Status: characteristic-zero symbolic identification of Entry 443's stable two-dimensional excess, after reading Entries 447--448.
Frozen exact differential
At [ E=X_2=0, \qquad K=a^4,quad L_1=b+1,quad L_2=a. ] Set [ c=b+1. ]
Every exact-form image is divisible by (a^4). Divide by that universal factor and retain the ((s_a,s_b)=(1,1)) sector. Its two exact operators are [ \boxed{ D_b=a(1-cpartial_c), \qquad D_a=c(apartial_a-7). } ]
On a monomial (a^ic^j), [ D_b(a^ic^j)=(1-j)a^{i+1}c^j, ] and [ D_a(a^ic^j)=(i-7)a^ic^{j+1}. ]
Symbolic quotient theorem
The image of these two operators contains every monomial except [ 1,\qquad a^7c. ]
Indeed:
- every (a^Ic^0) with (Ige1) is produced by (D_b);
- every (a^0c^J) with (Jge1) is produced by (D_a);
- for (I,Jge1), (D_b) produces the monomial unless (J=1);
- on (J=1), (D_a) produces it unless (I=7).
The remaining three exact sectors do not hit (a^7c). For a non-(q) operator, the required source (c)-degree is (j=s_a), and its coefficient is (s_a-j=0). For a (q)-operator, the required source degree is (i=6+s_b), and its coefficient is (i-(s_b+6)=0). They cannot hit the constant because every divided image retains an (a) or (c) prefactor.
Therefore, in characteristic zero, [ \boxed{ mathbf Q[a,c]\big/ operatorname{im}(d_{ m ex}/a^4) simeq mathbf Qlangle[1],[a^7c] angle. } ]
Restoring the universal factor gives intrinsic representatives [ \boxed{ [a^4], \qquad [a^{11}(b+1)]. } ]
Thus Entry 443's length-two excess is not a finite-field or cutoff artifact. Its earlier greedy representative (a^{11}b) differs from the canonical Euler-resonant representative by a reducible (a^{11}) term.
Relation to the Cayley--Menger module
Entry 447's flat module [ mathcal M_{CM}=mathbf Q[u,a,b]/(K) ] accounts for the quartic quotient with special fibre (mathbf Q[a,b]/(a^4)). The two classes above arise from the exact-form operator quotient after the universal (a^4) factor is removed. They are therefore relative-de-Rham/exact-form coefficient data, not additional Cayley--Menger carrier generators.
The unextended identification of (mathcal M_{CM}/u) with the full exact-form cokernel remains falsified by precisely this Euler-resonance plane.
Relation to the meromorphic lift
Entry 448 proves that the soft Kodaira--Spencer class forces the meromorphic vertical correction [ V=\frac{b^2-1}{4a}. ] The present theorem does not define its action on the resonance plane: coefficientwise application does not preserve the exact quotient without the complete relative de Rham reduction. The next test must derive that action from the deformed exact differential, not by applying (Vpartial_a) to the chosen representatives alone.
Classification
- quartic module: flat Cayley--Menger coefficient geometry;
- length-two plane: Euler-resonant exact-form cohomology;
- splitting from the quartic module: unproved;
- regular Gauss--Manin lift: obstructed by Entry 448;
- new carrier datum: none.
Next falsifier
Work over dual numbers in the physical soft parameter (u=X_2). Deform all four exact operators using the source-fixed (K(u),L_1(u),L_2(u)), construct the first-order cokernel without choosing a projection, and test whether the Euler-resonance plane:
- lifts freely;
- becomes (u)-torsion;
- mixes nontrivially with the rank-four Cayley--Menger module; or
- requires the logarithmic blowup of ((u,a)) already predicted by Entry 448.
Evidence
research/benincasa/soft-axis-euler-resonance-certificate.json;research/benincasa/marici-gm/src/bin/marked_tangency_support.rs;- Entries 443, 447, and 448.