Internal proof review

Henry Zweiman. September 15, 2026. Developed as Q56.

This is the originating assistant's audit of its own argument. No subagent, independent human referee or external mathematical expert reviewed the proof.

1. Normalization and homogeneity. Energy-one radial profiles satisfy c_alpha = 1 / integral(r^alpha v^q). The limiting coefficient is 1 / (S phi(1)^q). The negative Hessian term tends to (q-1) lambda phi(1)^(p-2) times the boundary L2 norm. Changing to denominator-one normalization multiplies both Hessian terms by the same factor t^(p-2). The scaling direction is removed by the linear slice; a strictly positive Hessian on the entire space would be false.
2. Nonlinear scale. The auxiliary t^2=A(h)+||h||_(1,p)^p lies between epsilon^p and C epsilon^2. Both pieces survive in the weak-limit calculation. Merely replacing the Bregman remainder by one half the original Hessian would be invalid for concentrated perturbations.
3. Derivative in the weighted dual. Fixed smooth auxiliary w has bounded gradient. The square of the flux difference quotient divided by a is dominated by C_w(a+2+a^(-1)). Reciprocal integrability is expressly assumed and later verified. The set a=0 has measure zero. Value terms use a positive lower bound for v.
4. Concentration remainder. Three-point Bregman expansion is an exact algebraic identity. Uniform convexity controls the shifted quadratic norm and p-power norm. The weight-shift error after division by t^2 is O(epsilon^(p-2)) for p>=4 and O(epsilon^((p-2)^2/2)) for 2<p<4. The norm-shift error is O(epsilon^(p-1)/t+t^(p-2)), tending to zero. Constants in the retained convexity term are independent of the smooth approximant; limits are taken first along j and then in w.
5. Constraint remainder. Taylor error is bounded by eta h^2+C_eta |h|^q, uniformly for the positive bounded base function. Sobolev continuity and q>p make epsilon^q/t^2 tend to zero. Compact weighted L2 convergence transfers the only quadratic negative term. It is not necessary to assume pointwise-small gradients or L-infinity-small perturbations.
6. Saturated gradient weight. Two exact flux identities control the center and the boundary collar separately. The negative collar term C(1-r) is absorbed by c min(1,alpha(1-r)) by choosing the collar first and alpha second. The resulting weight has beta=(p-2)/(p-1)<1. Its reciprocal is integrable at the boundary and at the center. Constants are uniform over all normalized radial minimizers because their convergence is to the unique normalized positive Steklov profile.
7. Weighted compactness. Boundary radial Cauchy-Schwarz gives squared error C(1-r)^(1-beta)E. Compact annular averages approximate the trace in operator norm. The center mass bound is C(delta^n+delta^(2-beta))E, obtained by integrating the inverse radial weight. Together with the boundary collar bound and interior Rellich this proves compactness. No unproved density identification with a larger weighted Sobolev space is required: H is defined by smooth completion.
8. Concentration and weak limit. The squared trace replacement error integrates to O(alpha^(-(1-beta)/2)). The inner half-ball is exponentially negligible. Tangency forces zero mean for the limiting trace. A_alpha>=cE and Q_alpha<=o(1) force that trace to be nonzero, so weak convergence cannot erase the contradiction.
9. Ground-state identity. Differentiating the radial p-equation produces the (n-1) angular potential and radial coefficient p-1. The flux coefficient at the outer boundary gives exactly lambda phi(1)^(p-2)(q_c-1). The inner boundary term vanishes along a sequence by finite angular energy. The limiting energy is first shown finite by lower semicontinuity. Orthogonal decomposition into radial mean and zero angular mean has no cross terms.
10. Instability. The first harmonic times phi' has finite weighted energy even where its unweighted W1p membership might fail. A center cutoff supplies fixed smooth W1p test directions with error O(epsilon^(n-beta)). Only after choosing this cutoff is alpha sent to infinity. This avoids inserting an inadmissible test into the second derivative.
11. Parameter interval. For z=(phi'/phi)^(p-1), differentiating its Riccati equation gives a linear equation for z' with strictly positive forcing. Its positive initial asymptotic yields z'(1)>0, exactly q_c>p. The separate bound lambda<1/n yields the full subcritical interval near p=2 when n>=4. The argument uses continuity of explicit functions of p, not an unproved eigenvalue-continuity assertion.
12. Boundaries. No conclusion at q=q_c, no radial uniqueness theorem, and no general Banach Morse theorem are claimed. The threshold itself and the p=2 result are prior work. Revision 1.1 resolves neighborhood uniformity and its sharp decay power.

No mathematical gap was identified in this internal audit. There are no numerical experiments serving as proof. PDF/MathJax integrity checks are recorded separately and do not certify the theorem.

Revision 1.1: neighborhood-rate audit

1. The radius is measured after unit energy normalization; scalar copies cannot force a zero radius.
2. Uniform sphere trace bounds and interpolation give alpha times the weighted q integral bounded by alpha^(q theta) times the Sobolev q norm. The critical trace exponent is included.
3. Varying profiles have a common integrable reciprocal-weight bound and uniform gradient bound. These justify the fixed-smooth-direction derivative expansion in the weighted dual norm.
4. Compact traces handle the varying quadratic potential. Finite limiting energy and value truncation justify smooth approximation without assuming uniform norm equivalence at the boundary.
5. The constraint remainder vanishes when alpha^sigma times the perturbation norm tends to zero. Uniform convexity retains concentration energy. The limiting angular inequality provides a positive relative gap for q<q_c.
6. The scalar retraction has a uniform Sobolev bound.
7. Boundary competitors have width alpha^(-1) and amplitude alpha^((p-1)/(q-p)); both the energy increment and relative denominator increment have order alpha^(-p sigma). A fixed sufficiently large amplitude factor makes the quotient decrease.
8. Unit energy normalization changes the competitor only at the smaller order alpha^(-p sigma), preserving the matching distance exponent.
9. Theorem 7.1 does not settle q=q_c or a leading asymptotic radius coefficient. No numerical calculation substitutes for the proof.