C04.1 · Derivation

V1–V3: The field-free route

Under inverse-spectrum symmetry, inversion permutes an odd number of positive eigenvalues. At least one eigenvalue is fixed and hence equals the positive self-inverse value. The remaining two are reciprocal. Thus, up to permutation, \[ (a,b,c)=(r,1,r^{-1}),\qquad r>0. \] This also follows by factoring the reciprocal cubic in the source's proof.

Now \(U=V=r^2+1+r^{-2}\). DET-7 makes \(U^2=16\); positivity selects \(U=4\). Consequently \[ r^2+r^{-2}=3,\qquad (r^2)^2-3r^2+1=0. \] The positive values of \(r\) are \(\phi\) and \(\phi^{-1}\), interchanged by a permutation. No number-field restriction entered this route. The result uses the inversion hypothesis precisely where it equates the two norms.

The script checks the polynomial factorization independently and verifies the golden root exactly. The other factor has no positive real zero. It does not prove that inversion symmetry is dynamically selected.

V1, V4: Arithmetic reduction to a discrete gap problem

The integer ring is \(\mathbb Z[\phi]\), with units \(\pm\phi^n\), as recorded in [KC]. Since \(abc=1\) and all entries are algebraic integers, each inverse is a product of the other two and is an algebraic integer. Thus all entries are units. Positivity removes the minus sign: \[ a=\phi^{n_1},\quad b=\phi^{n_2},\quad c=\phi^{n_3}, \qquad n_1+n_2+n_3=0. \] Order the exponents and define \(p=n_1-n_2\ge0\), \(q=n_2-n_3\ge0\). Conversely, \[ n_1=(2p+q)/3,\quad n_2=(q-p)/3,\quad n_3=-(p+2q)/3, \] so integral exponents exist exactly when \(p+2q\equiv0\pmod3\).

Expanding \(UV\) pairs each ratio with its inverse: \[ F(p,q)=UV=3+A_p+A_q+A_{p+q}, \qquad A_k=\phi^{2k}+\phi^{-2k}. \] The recurrence \(A_0=2,A_1=3,A_{k+1}=3A_k-A_{k-1}\) follows from the quadratic equation for \(\phi^2\). It also proves strict increase: if \(v>u>0\), then \(3v-u>v\). This is monotonicity in the **gap index**, not in the spread statistic rejected by the source.

V4: The admissible-gap proof is exhaustive without a numerical cutoff

For \(p+q\le3\), the congruence admits only \[ (0,0),\ (1,1),\ (0,3),\ (3,0). \] Their values are respectively \(9,16,41,41\), printed in the executed keys `admissible_gaps_with_sum_at_most_3` and `F_for_those_gaps`. For all remaining pairs, \(p+q\ge4\), so \[ F\ge3+A_4+A_0+A_0=54>41. \] This separate treatment of the smallest admissible sums makes the lower-bound step explicit. It explains the source's compressed bound without repairing its quotation.

Hence \(F=16\) only at \((p,q)=(1,1)\), corresponding to the exponent multiset \(\{1,0,-1\}\). Every other nonconstant admissible spectrum has \(F\ge41\). This is the result carried by LIB2-001 and used by LIB2-020.

The bounded exact scan is only a control of the formulas; it is not the proof of the unbounded classification. Its bound and total cases are printed under `finite_control_*`.

V5: A rejected monotonicity argument remains rejected

The source explicitly records that monotonicity in the maximum absolute exponent fails. The script reproduces its two values: \[ F(5,0,-5)=15376>11561=F(6,-3,-3). \] This counterexample does not refute the admissible-gap proof. It prevents the discarded proof step from being reintroduced simply because its conclusion survived.

V6–V7: A spectral orbit is not an ordered frame

For an invariant scalar \(S\) and a group orbit \(gX\), \(S(gX)=S(X)\). At a stationary point, differentiating the infinitesimal invariance identity in an arbitrary direction gives \(\operatorname{Hess}S(\xi X,\cdot)=0\). Stationarity is required for this ambient-Hessian statement; a constant value along a curved orbit at a noncritical point is not by itself a zero ambient Hessian.

LIB2-041 applies this mechanism to the source's stated orbit dimension. That orbit count is **source-reported, [NOT VERIFIED] here** because its original representation certificate is absent. It explains why ordering needs a loaded or symmetry-breaking datum, rather than another symmetric polynomial of the same spectrum.

For the narrower class \(S(X)=f(N(X))\), at a regular stationary point, \[ dS=f'(N)dN=0\quad\Longrightarrow\quad f'(N)=0, \] and \[ \operatorname{Hess}S=f''(N)dN\otimes dN. \] It therefore annihilates \(\ker dN\). The code tests this mechanism on the **actual split-Albert cubic**, using \(S=(N-1)^2\) at the Albert identity. It does not replace the carrier by an unrelated polynomial. The computed Hessian rank and nullity are reported under `norm_only_hessian_*`.

This executed example supplements the general chain-rule proof. It is not a claim that all invariant potentials, source-dependent actions, or nonregular points have the same Hessian.