Native law and its scope
For real-valued increments obeying a positive pointwise power law, the exact identity and polynomial sphere growth of homogeneous dimension \(D=4\) give three abstract branches:
- \(\delta_c < D\): polynomial growth \(r(n)\sim n^{D-\delta_c}\) — the Heisenberg real-weight growth exponent \(D-\delta_c\);
- \(\delta_c = D\): logarithmic growth \(r(n)\sim \log n\) (marginal branch);
- \(\delta_c > D\): saturation \(r_\infty - r(n)\sim n^{-(\delta_c-D)}\), strictly for \(\delta_c > D\).
The measured observable is instead an integer rank. For it, version 2.0 proves a two-sided finite-window law from explicit bounds on capacity and sphere size, without promoting that window law to an asymptotic statement. In particular, a pointwise positive rank-increment law with \(\delta_c>D-1\) is incompatible with integrality once its predicted increment falls below one; this includes the measured \(3<\delta_c<4\) range.
Why the transfer fails natively
The no-go has three independent, corpus-scoped legs:
- The O4 closure does not survive. Both native realizations defined in the frozen corpus fail: the fixed Cayley valence cannot track exploration, and cumulative span rank is not proportional to explored volume. This does not exclude future definitions.
- The frontier and feedback carrier are absent. Native coupling is to the polynomial frontier \(N^{3/4}\), not to \(p^{1/2}\), and the corpus defines no Heisenberg growth process carrying the Born–Infeld feedback used by O4.
- The multiplicative object is different. The exact increment uses block capacity \(\sigma_c\), not the pair product \(\sigma_{\mathrm{pair}}\); no native growth process carrying that pair observable is defined or derived, and the reduced-model exponent coordinates used by the transfer are false for the native BFS.
Consequently, the numerical agreement obtained from \(1/(\delta_{\mathrm{pair}}+\tfrac12)\) remains a cross-substrate phenomenological comparison, not a Heisenberg derivation of the projected-Yukawa rate \(\beta^*\).
Corrections, numerical status, and open questions
The shell factor \(D-1\) is recovered additively in the native growth equation; the separate inter-\(q\) estimator layer remains open. Fitting against \(\log(n+1)\) rather than the paper definition \(\log n\) inflates \(\delta\) by 8–13% on the production windows: conditionally, a reported \(7.44\) corresponds to approximately \(6.6\)–\(6.9\). The numerical exponents remain finite-window measurements on production data (\(q \le 211\)). The window-effective exponents make contact near \(q = 211\) and separate strictly for \(q \ge 307\) (at \(6.7\)–\(14.5\) inter-pair bootstrap standard errors, sign stable across all window-shift and knee-trimming variants and both logarithmic conventions): the contact is a transitional crossing in the accessible data, no current indication forces \(\delta_c = D\), and the asymptotic value of \(\delta_c\) remains open. A joint \((q,n)\) limit is not supplied: it would require an explicit construction beyond the finite-radius quotient isometry.
The repository embeds the block-level data actually used (frozen windows, per-checkpoint
provenance) together with a single reproduction script and the crossing figure: every quoted
number is reproducible from the published deposit alone, via
python code/span_growth_reproduce.py in
Cosmochrony/span-growth-note.