Universal Semiclassical Coexistence in Classical Compression of Phase-Lifted Coherent States: Dimension-Normalized Contacts and a Matched High-Fidelity Boundary Layer

Strengths

• Section 3.1, p. 3: the exact Weyl product separates its m-independent C_N correction from Q_m,N. This explains why the O(N^-1) normalization term does not persist at derivative orders one and two.
• Lemma 3.1, pp. 4-5: the Bessel-law mean and variance give A_p=p(p+2)/2+1/8 and B_p=2^p/sqrt(2*pi). The contact constants agree with these coefficients; the derivative claims require weighted moment control, not differentiation of an unspecified o(1).
• Theorem 4.1, pp. 5-6: fixing the activation mass removes the amplified K-prime error. The fixed-b expansion correctly cancels the apparent -ab constant, which is an easy source of a false order-one term.
• Lemma 5.1, p. 7: localization treats the full t/ell->1 window, including H->infinity, using the exact h-minus-hbar identity. The constant gap a(1-log(2)) excludes transition contacts competing with c=2.
• Theorem 5.2, p. 8: substituting 2K=t*K-prime yields t_c=2ell+2a*log(ell)+log(4*pi)-a+o(1); a-p=1/2 cancels the representation-specific leading constant.
• Proposition 5.3, pp. 8-9: the sign of b-t*K-double-prime gives convexity after contact. Combined with the global origin slope, it rules out later chords and supports unrestricted scalar-envelope attainment.
• Theorems 6.1-6.2, pp. 9-10: the selected branches match in value and derivative at x=a; the fixed-D limit follows from eventual membership in the coexistence chord. D=0 is correctly excluded.
• Sections 7-8: both exact-Weyl-family diagnostics ran, the broader frontier script covered the Grassmannian sequence, and Figure 1 regenerated from newly computed data rather than a stored certificate.

Weaknesses

• Theorem 6.2, pp. 9-10, needs a domain correction: derive the raw saddle expansion for x<a and a neighborhood of a, then use only the chord on compact sets farther to the right. Claiming both raw and line expansions uniformly on every compact subset of (0,infinity) is incorrect.
• Lemma 3.1 equation (9) states two differentiated remainder bounds after a compressed moment calculation; equation (13) similarly abbreviates the reweighted variance cancellation. Give explicit moment/remainder bounds or a proved auxiliary lemma to make the uniform t>=c*ell argument readily auditable.
• The exact finite-N unrestricted RDF is imported as Theorem 2.1. The moment/entropy argument is plausible and can be reconstructed, but the full public text of reference [6] was not retrieved in this assessment. No complete audit of that external theorem is claimed.
• Section 8 uses "exact" for series-based floating-point K-prime and residuals. The programs truncate positive series and use floating-point roots without rigorous interval tail enclosures; the final diagnostic-only qualification should govern this wording.
• README_REPRODUCIBILITY.md calls replay_all.ps1, which is absent. The individual scripts run at their supplied paths. The advertised LaTeX manuscript and bibliography are also missing from this batch; no byte-identical PDF rebuild was attempted.
• The numerical contact scan detects sign-changing crossings on a finite grid; it cannot certify every stationary root or global minimality by itself. The written localization and convexity arguments remain essential.
• Primary-source comparisons to Abreu, Ammari et al. and the sphere RDF were checked at abstract/metadata level only. This does not prove absence of overlapping results in their full texts or worldwide bibliographic priority.

Potential errors

• Theorem 6.2 proof, p. 9: for SU(2), a=3/2 and x=4a, the asserted raw saddle t=2a*ell/x+O(1) predicts t/ell->1/2. At that field H is exponentially small, so K-prime->0, whereas sqrt(1-x/ell)->1. The actual conjugate t/ell tends to 1 (1.03290698 at N=10^200). The RDF uses the chord, so its final profile is not refuted.
• Lemma 3.1, p. 4, calls the tails outside sqrt(z)*log(z) "exponentially small". The natural moderate-deviation bound is exp(-c*log(z)^2), not exp(-c*z). Superpolynomial decay suffices for the claimed algebraic remainder, but the intended rate should be stated.

Strong novelty candidates

• The global contact expansion with universal order-one constants after normalization by log(dim V_Nlambda), Theorem 5.2.
• The moving-embedding classical RDF limit and matched D*log(d_N) boundary profile, with global convex-envelope gluing.
• The exact activation-normalized and fixed-radius Legendre expansions that resolve the order-one cancellation in the soft window.

Unresolved material objections

None reported.