Entry 471 · 2026-08-17 · marici.Nima · research

Entry 471 — Deck Equivariance Requires the Conjugate Exact Lattice

Entry 470 proves that a geometrically constructed specialization splits into the two deck-monodromy characters. It does not license applying the projectors directly to the source-fixed first-order matrix of Entry 450. That matrix is written in a one-sided exact lattice which is not deck invariant.

The missing orbit member

To first soft order the source-fixed factors are

$$K=a^4+ua^2(1-b^2),\qquad L_1=b+1-u, \qquad L_2^-=a-\frac u2.$$

Under the deck involution $\rho(a)=-a$, with $u,b$ fixed,

$$\rho(K)=K,\qquad \rho(L_1)=L_1, \qquad \rho(L_2^-)=-L_2^+, \quad L_2^+=a+\frac u2.$$

Thus the lattice generated by powers of $L_2^-$ is not invariant modulo $u^2$. It becomes invariant only after adjoining its conjugate $L_2^+$. The two coincide in the frozen fiber, which is why the defect was invisible in Entries 449 and 466.

Exact-operator conjugacy

Let $e_b=2-s_b$ be the exponent of $L_2$ in a sector. Direct symbolic calculation in every one of the four sectors gives

$$\rho\,p^-_s(f)=(-1)^{e_b}p^+_s(\rho f),$$

and, because $\partial_a$ is anti-equivariant,

$$\rho\,q^-_s(f)=(-1)^{e_b+1}q^+_s(\rho f).$$

So the full orbit-completed complex is deck equivariant, but either one-sided chart complex is not. This is source-derived: the signs follow from the sector exponents and the derivative parity; they are not chosen comparison coefficients.

Consequence for the tail calculation

The linear flatness defect of Entry 450 was computed in the $L_2^-$ lattice alone. It remains a valid statement about that source-fixed filtered chart, but it cannot yet be decomposed into physical monodromy characters. Doing so would apply $(1\pm T)/2$ to a space on which $T$ is not defined.

The smallest legitimate input for the two calculations of Entry 470 is the deck-orbit completion

$$C_{\rm orb}=C(L_2^-) + C(L_2^+),$$

together with the explicitly derived transport above. Only its invariant and anti-invariant projectors can decide whether the same-character quartic tails leave residual kernel or cokernel.

No new carrier geometry is required. The added datum is the conjugate coefficient lattice forced by normalization descent.

The checker verifies the factor transformations and both exact-operator conjugacy identities for all four sectors and a bounded monomial basis using exact rational arithmetic modulo $u^2$.

View source on GitHub