Manuscript IX-E-2@1 · IX-E-2.tex · section · e70d9077

Vanishing is not construction

Remaining consistency obligations and release decision › Vanishing is not construction · lines 1343–1386

Even a future proof of (eq:SB) would clear only the first-stage double-ordering commutator test. IX-E's actual-tensor relation test is separate.

For clarity, its algebraic structure can be recorded without constructing a new map. With

A (d) = \frac{1}{2} (S (d) + S (σ d)),

a candidate $T_{2} = A + X$ must satisfy

X (σ d) = X (d), X (γ d) = - X (d)^{*} .

For prescribed tensors $d_{i}$ with prescribed values $t_{i}$ , put $x_{i} = t_{i} - A (d_{i})$ . In the flip quotient

Q = D_{2} / im (id - σ),

the inherited finite-relation condition is

i \sum a_{i} [d_{i}] + i \sum b_{i} [γ d_{i}] = 0 ⟹ i \sum a_{i} x_{i} - i \sum b_{i} x_{i}^{*} = 0,

where each sum is finite. This condition is not evaluated here. IX-E's algebraic sufficiency proof uses a Hamel complement and does not establish continuity IXE (Theorem 5.1).

The ordered slab also remains relevant: its nonzero $Q^{R}_{g, f}$ excludes classical multiplication as the time-ordered map. A later slab proves an ordering, not necessarily its uniqueness, and is not a substitute for a two-way calculation.

If the commutator gate is eventually clear, the inherited E2–E6 obligations remain. They include a specified replacement distribution, admission of coefficients and descendants, named fixed-support continuity, exact overlap, contact and involution compatibility, and the hypotheses of conditional subtraction. This successor begins none of those constructions.

Displayed formulas in this passage

unlabeled · line 1351
$A (d) = \frac{1}{2} (S (d) + S (σ d)),$
unlabeled · line 1355
$X (σ d) = X (d), X (γ d) = - X (d)^{*} .$
unlabeled · line 1362
$Q = D_{2} / im (id - σ),$
unlabeled · line 1366
$i \sum a_{i} [d_{i}] + i \sum b_{i} [γ d_{i}] = 0 ⟹ i \sum a_{i} x_{i} - i \sum b_{i} x_{i}^{*} = 0,$