Computable vertex centers and a single-log growing domain
Keep the uniform-simplex assumptions and normalization of Vertex coefficients at every fixed vertex count. Write \(q=M-1\), \(n=M-2\), \(L=\log2\), and
The results below concern the sample-size-asymptotic vertex risk coefficient. Finite-sample risks and the separate comparison region \(D\geq8M^2\) are not their subject.
A larger domain from the single-log error
Theorem 56 (A single-log growing-vertex approximation)
For every integer \(D\geq2^{256}\), put
For every integer \(4\leq M\leq h_{**}(D)\),
with \(g(q)=0\) for \(3\leq q\leq160\), and \(g(q)=\lfloor q/3\rfloor+2-\lceil\log_2q\rceil\) for \(q\geq161\).
This domain strictly contains the one in A broader simultaneous vertex-count and degree regime. At \(N=256\) it includes \(M=4\); at \(N=1024\) its vertex ceiling is 12, compared with the previous ceiling 8. Its growth is asymptotic to \(\log D/(8\log\log D)\), rather than the previous denominator 12. The degree start is still astronomical. The ceiling can decrease when \(\ell\) jumps, so no monotonicity of the ceiling is asserted.
Proof. Retain all dependence on the vertex count in the single-log bound. For \(q\geq4\), use the explicit constants of Single-log vertex errors for every fixed M at least five. To avoid confusing their \(B_r,G_r\) with the leading term \(B_M\), call their elementary bounds \(P_r,G_r,A_r,F_r\) here. The bounds already proved in A simultaneous vertex-count and degree regime give, for \(1\leq r\leq q\),
For every positive integer \(p\),
Each summand is at most \(1/p\). Thus the first moments in the single-log proof satisfy \(S_k\leq3G_k\) and \(T_r\leq3A_r\). In its diagonal constant, the two terms in \(q\widetilde U_{q-1}\) are bounded by \((20q^2)^q\) and \((6q^2)^q\): for the latter use \(56+3^{q-2}\leq3^q\) for \(q\geq4\). Its two positive binomial sums are bounded by \(6(24q^2)^q\) and \(5(10q^2)^q\), respectively. The first follows by extending the sum to both endpoints and expanding \(6(4+20)^qq^{2q}\); the second uses \(5(8+2)^qq^{2q}\). Also \(2P_q+A_q\leq3(2q^2)^q\), \(J_q\leq(2q^2)^q\), and \(8(1+q^2+J_q)\leq16(2q^2)^q\). Consequently
The last step uses \(32\leq4^q\) for \(q\geq4\). Divide the explicit \(\widetilde R_M\) by \(C_{\rm arc}=M^2q!q/4\). Exactly as in the earlier budget proof, \(273+q(q+1)\leq32q^2\), \(96\leq5^q\), and \(32q^2\leq16^q\) imply
The refined large-vertex lower bound from the preceding joint chapter gives \(C_{\rm arc}/C_M\leq2^{-g(q)}\), without a gap at \(q=160,161\). Hence for every \(M\geq5,D\geq256\),
This is an inequality with explicit dependence on both parameters, rather than substitution into a fixed-\(M\) limit.
Spend two logarithmic factors per outside coordinate. If \(q\geq4\) is allowed by the new ceiling, then \(8\ell q+4\ell\leq N\), and \(\ell\geq9\). Therefore \(512q^2\leq8N^2/\ell^2\leq(N+2)^2\), while \(1+\log D<N+2\leq2^\ell\). The last error bound is at most
Cover the critical four-vertex case separately. For \(q=3\), keep the previous squared-log estimate. Its explicit arc budget satisfies \((512\cdot9)^3<2^{37}\). Thus the error is at most \(2^{37+2\ell}/D\). For \(N\geq256\), \(\ell\leq N/32+2\). Indeed this holds directly at \(\ell=9\); at \(\ell\geq10\) use \(N\geq2^{\ell-1}-1\geq32(\ell-2)\), with the last inequality following by induction from \(\ell=10\). Consequently \(37+2\ell\leq41+N/16\leq N/4\). This proves the same degree error for \(M=4\).
Inclusion and strict improvement. An old admissible \(q\leq\lfloor N/(12\ell)\rfloor\) with \(q\leq3\) is automatically allowed. If \(q\geq4\), then \(N\geq48\ell\), and in particular \(N/(12\ell)\leq(N-4\ell)/(8\ell)\). This proves inclusion at every old pair, including ceiling jumps. The pairs \((M,D)=(4,2^{256})\) and \((12,2^{1024})\) show strict enlargement. Finally \(\ell\sim\log_2N\) gives the stated asymptotic ceiling.
Centers that contain no unknown leading constant
Use the explicit coefficient \(\beta\) in A first correction with a quadratic large-vertex remainder. Define
For \(4\leq M<162\), use the elementary local radius from The vertex leading constant at large vertex count:
An outward rational enclosure of its square roots is used by the tool. For \(162\leq M<258\), put \(r_M=6/n\); for \(M\geq258\), put \(r_M=84/n^2\). The three existing theorems prove \(|C_M/T_M-1|\leq r_M\) on their respective ranges. The last radius is conservative because division by \(1+\beta/n>1\) could only decrease it. The small-\(M\) fallback can be wide and is not an improvement over the exact-series evaluator there.
Corollary 16 (A computable composite center)
Throughout the new joint domain, let \(t_M=T_M/S_M\), and let \([t_-,t_+]\) be any positive rational enclosure of \(t_M\). Put
If \(\delta\) bounds the degree error relative to \(C_M\), then
The exact center \(T_M\) instead has radius \(r_M+(1+r_M)\delta\).
Proof. Factor the displayed quotient into the degree ratio \(D^{M+1}a_0/C_M\), the constant ratio \(C_M/T_M\), and the elementary-evaluation ratio \(t_M/\widehat t\). Their errors are bounded by \(\delta,r_M,u_M\). Expansion of the product and the triangle inequality give \((1+\delta)(1+r_M)(1+u_M)-1\). This proves the formula, including every cross term, without requiring a small radius in the first finite range. In particular no unknown value \(C_M\) remains in the approximation center.
Compressed certificates and independent checks
tools.risk_research.vertex_composite_asymptotic evaluates
\([t_-,t_+]\) with the existing rational logarithm, arctangent and
integer-square-root enclosures. It retains \(S_M\) symbolically;
neither \(q!\), \(2^N\) nor a risk table is constructed.
For a full block \(2^N\leq D<2^{N+1}\) it uses
The certificate records a rational base plus a rational multiple of \(2^{-E}\) rather than materializing this denominator. A target is checked by exact integer comparison with a bounded shift. Thus even an exponent with thousands of decimal digits does not cause a shift by that value. The target concerns the stated block, not every later block across ceiling jumps.
The separate constant, degree, elementary-evaluation and combined budgets are saved, along with their center, denominator convention, full domain witnesses and source/proof fingerprints. Verification restores every recorded option and recomputes the entire record. It rejects duplicate/unknown keys, changed types, bounds or sources. Integer text conversion uses bounded chunks without modifying a process-wide digit limit.
$ .venv/bin/python -m tools.risk_research.vertex_composite_asymptotic --vertices 258 --degree-floor-exponent 40000 --bits 47 --relative-target 1/100 --output .research/skeleton-risk/composite258.json
$ .venv/bin/python -m tools.risk_research.vertex_composite_asymptotic --verify .research/skeleton-risk/composite258.json
Source/output aliases are rejected before calculation and again at the resolved parent plus original leaf before atomic replacement. Failures retain existing output. The directory is caller-controlled; this is not a sandbox against replacement of that canonical directory inode by another process.
Independent finite checks compare the newly majorized single-log constants with their original exact formulas, the centers with the original positive integral, and feasible coefficients with the original scalar or transformed formulas. Huge-degree certificates check exact compressed inequalities. These checks support the implementation; the proof above supplies the full growing domain.