C04.2 · Derivation

A2–A3: From a declared character tensor to the Hessian

The even exponential has quadratic term \(4Y^2\). Therefore \[ H_\tau=\frac89R_B^{\rm char} =\frac49\Pi_{32}+\frac{16}{27}\Pi_{36}. \] The character action uses \(c_*=(1,1,0,0,1/27)\), giving Hessian weights \[ 1,\quad 4/9,\quad 5/9,\quad 17/27 \] on the respective source sectors. The exact output key is `selected_Hessian_weights`.

This establishes the implication **from the stated character tensor**. It does not test that \(P_B\) and the frozen generators actually yield that tensor. In particular, the smaller coefficient calculation must not be advertised as a second implementation of the source's full nonlinear exceptional action.

The separate loaded-parent minimum and detector results retain their scopes:

> Complete-conditional, selector tier. Within the frozen split-real 56-frame and the declared finite/internal loaded-profile class: the selector parent action Vtotal has its global minimum at (j ∗ , q ∗ = 32 j ∗ , M ∗ = I) with combined Hessian positive for all ρ, u0 , κd > 0 and [W0 ] the exact projective soft ray at zero inserted dark amplitude.

— Rev32.7, PDF p.170 [Q017].

> The crown is prefrozen: the grammar’s own detector quadratic form on the 70-dimensional p-sector has spectrum 01 ⊕ (5/3)10 ⊕ 226 ⊕ (5/2)32 ⊕ (9/2)1 (house-recomputed exact) with a unique top eigenspace, frozen pre-target and shown post-freeze to be exactly RW0 (overlap 1.000000000000).

— Rev32.7, PDF p.169 [Q018].

No full parent functional or detector matrix is supplied. Their stated stability and eigenline selection remain source reports, **[NOT VERIFIED] by the coefficient calculation**.

A4: A Hessian leaves a nonlinear degree of freedom

The matrix of \(c\mapsto h(c)\) has rank four and null vector \[ k_{\rm jet}=(1,0,-4/9,-16/27,0). \] Integrating that coefficient direction gives \[ \mathcal N(Y)=\tau(Y)-\tfrac12\langle y,H_\tau y\rangle. \] Its value, gradient and Hessian at the origin vanish. Its leading homogeneous term is \[ \mathcal N(Y)=\frac8{27}\operatorname{tr}(P_BY^4)+O(\|Y\|^6). \] The code verifies the exact scalar series and zero two-jet before contraction. Given the declared symmetric-matrix realization, spectral functional calculus transfers the scalar identity to \(Y\).

For real scalar \(x\), \(2\cosh x-2-x^2\ge0\), by its even Taylor series. For symmetric \(Y\), the associated matrix function is positive semidefinite. Since \(P_B\) is positive semidefinite, its trace against that function is nonnegative. Thus \(\mathcal N\ge0\) under those hypotheses.

The family \(F_{\rm char}+s\mathcal N\) has the same two-jet for every \(s\); the nonnegative direction preserves any already established lower bound when \(s\ge0\). This does not prove that every real \(s\) lies in every global admissibility class. The source's full moduli quotient requires its declared equivalence relation. The exact kernel calculation proves the two-jet ambiguity without inventing that relation.

The source states the broader class result explicitly:

> the solution set of “critical point at the base + positive-semidefinite Hessian” modulo equivalence is a four-dimensional moduli space,

— Rev32.7, PDF p.297 [Q019].

This is a source-class assertion, not a general theorem that all actions with a fixed Hessian have precisely this finite-dimensional moduli space.

A4: What the adopted clause actually selects

The relations \[ v=0,\quad w=0,\quad u=t,\quad 27z=t \] form a rank-four linear system on the displayed coefficient space. Their nonzero solution is the ray through \(c_*\). Deleting any one equation drops the rank, restoring an additional projective degree of freedom. Both assertions are executed under `Phi_diag_rank` and `Phi_diag_row_deletion_ranks`.

This proves minimality **within that coefficient description**, not intrinsic necessity of the clause. For example, rescaling one primitive quadratic generator while holding the others fixed changes the coefficient relation expressing the same numerical equality of units. The declared grammar must justify such cross-generator normalization; the source states that it does not.

A6: Stability does not isolate the adopted ray in the tested class

The source's inequalities are \[ u\ge0,\quad4t/9+v\ge0,\quad16t/27+w-z\ge0,\quad 16t/27+w+z\ge0,\quad t_2\ge0. \] The source point \((t,t_2,u,v,w,z)=(1,1,1,1,1,0)\) makes each strict. Continuity then gives an open neighborhood of feasible points in the full coefficient space; a positive-scale quotient removes one dimension, not all directional freedom. The script reports each exact slack and the dimensions.

This is not the retired universal claim that inequalities cannot enforce equalities: the pair \(x\ge0,-x\ge0\) already enforces \(x=0\). The conclusion is restricted to the displayed **nondegenerate** stability inventory.

For Casimir-naturality the source reports two different systems, depending on the carrier convention: \(u=v=w=z=0\), or \(u=v=w,z=0\). Each intersects the adopted relations only at the zero coefficient vector. The script executes those intersections. It does not recompute the Casimir decomposition that produced the systems. The negative conclusion is that these specific conditions exclude the desired nonzero ray; they do not select it.

A5: Rung selection and H2 prices are different conditions

LIB2-029 concerns occupied logarithmic ladder coordinates under its stated vacuum/projector and winding assumptions. It is not a statement about occupation of atomic states. The selector is not rerun here and is not supplied by the Hessian-weight calculation.

LIB2-030–LIB2-032 concern the predeclared minimal cohomology class, its rank and coefficient obligations. In particular the two reported countermodels require the actual complex, comparison maps and gate inventory for independent reproduction. A pair of arbitrary rank-two matrices would not test those named objects. The source's coefficient nonselection is retained rather than replaced by such a demonstration.

A7: The remaining nonselection mechanisms keep their named domains

If an action factors through the registered Gram map, it cannot distinguish points on one fibre. That is the mechanism in LIB2-036; the specific fibre dimensions require the original Gram map. LIB2-037 further restricts a frame-free invariant class; the result does not prohibit explicit spurions or sources excluded by its definition.

LIB2-038 shows that transporting a selector with a source is not a selection of the source ray. LIB2-039 limits a fixed-grade compressed current by its reported rank. Neither is a universal current no-go.

For LIB2-040, an action even under the stated involution takes equal values on the two orbit signs. It can at most isolate an unoriented pair unless further data are supplied.

At a zero of a smooth moment map \(\mu\), the standard sum-of-squares rule is \[ \nabla^2\|\mu\|^2=2(D\mu)^T D\mu. \] Its rank is bounded by the target dimension. The code differentiates an explicitly named small nonlinear example as a calculus control, and prints that this is **not the frozen Kempf–Ness test**. LIB2-043 applies the rank argument to its declared unshifted zero-level class; LIB2-044 adds a separate beat-invariant obstruction. Changing level, representation or potential class is not covered automatically.

Finally, LIB2-045 concerns the source's braid/trace grammar and its boundary transitivity. Constructing the internal grammar does not choose a boundary source on a transitive orbit. No such ray is chosen here.

Verification coverage of every supplied record in this section

| Identity | What this return verifies or leaves unverified | |---|---| | LIB2-021 | Source report only; full functional, domain and coupled Hessian absent. | | LIB2-022 | Source price, not recalculated; no coefficients chosen here. | | LIB2-023 | Source report only; exact detector and its action absent. | | LIB2-024 | Source report only; matching clock/selector matrices absent. | | LIB2-025 | Source report only; loaded plane and joint defects absent. | | LIB2-026 | Conditional algebraic expansion executed; original carrier-character identity not independently tested. | | LIB2-027 | Displayed two-jet map/kernel executed; full equivalence relation and class inventory not re-certified. | | LIB2-028 | Four displayed linear relations and their minimal ranks executed; adoption remains a premise. | | LIB2-029 | Source report only; vacuum/projector bridge and rung-selector inputs not supplied. | | LIB2-030 | Source report only; actual H2 complex and minimality premise absent. | | LIB2-031 | Source predeclaration only; not a numerical output of this article. | | LIB2-032 | Source report only; two actual countermodels and gate specifications absent. | | LIB2-033 | Source report only; the full declared refined-condition inventory absent. | | LIB2-034 | Intersections of the source-reported coefficient equations with the adopted ray executed; Casimir decomposition itself not rerun. | | LIB2-035 | The displayed strict-interior witness and dimension implication executed. | | LIB2-036 | Factor-through-Gram implication explained; registered Gram fibres not reconstructed. | | LIB2-037 | Source-class exclusion only; exact invariant-class definition/payload not supplied. | | LIB2-038 | Source-conditioned nonselection report; not a source-ray construction. | | LIB2-039 | Source rank obstruction only; compressed current arrays absent. | | LIB2-040 | Even-action implication explained; registered horizontal action absent. | | LIB2-043 | General zero-level sum-of-squares rank mechanism explained; frozen KN map absent. | | LIB2-044 | Source-class exclusion only; actual beat moment-map class absent. | | LIB2-045 | Source-class transitivity/nonselection report; no chosen boundary source. |

Every “source report only” or missing-object entry in this table remains **[NOT VERIFIED] by this return**. This table is not a second registry and changes no source status.