One certificate for a requested finite-degree relative accuracy

Keep the uniform-simplex design, evaluation distribution and noise assumptions of Theory of the skeleton risk. Write \(s\geq2\) for the number of vertices of the source face, \(q\geq2\) for its outside codimension, and \(M=s+q\). The quantity being enclosed is the coefficient \(a=a_{s-1}(s+q,D)\) of sample-size-asymptotic risk. It is not a finite-sample risk.

The general construction already gives the positive constant

\[A_{s,q}=\frac{\binom{s+q}s^2(s)_q}{2^q(s-2)!(q+1)}c_{q+1}, \qquad c_{q+1}\geq q+1.\]

The constant index here is \(q+1\), while q itself denotes the outside codimension. The lower bound follows from the positive diagonal mass \(c_{q+1}/(q+1)=\sum_{j\geq0}d_j\geq d_0=1\) in Finite-degree errors at every fixed nonvertex codimension.

Theorem 35 (Relative-accuracy composition at a fixed degree)

Let \(0<\varepsilon<1\) be rational and \(D\geq s\). Suppose the proved general finite-degree error evaluator supplies \(0\leq r<\varepsilon\) such that

\[\left|\frac a{A_{s,q}D^{s-q-2}}-1\right|\leq r.\]

Given a rational constant enclosure \(0<A_L\leq A_{s,q}\leq A_U\), define

(75)\[\begin{split}L&=A_L(1-r)D^{s-q-2},\quad U=A_U(1+r)D^{s-q-2},\\ \widehat a&=\frac{2LU}{L+U},\quad \varepsilon_{\rm cert}=\frac{U-L}{U+L},\quad \delta_A=\frac{A_U-A_L}{A_U+A_L}.\end{split}\]

Then \(0<L\leq a\leq U\) and

\[\left|\frac{\widehat a}a-1\right| \leq\varepsilon_{\rm cert} =\frac{\delta_A+r}{1+\delta_A r}.\]

Put \(\tau=(\varepsilon-r)/(1-\varepsilon r)>0\). One dyadic enclosure of \(c_{q+1}\) with width at most \(2(q+1)\tau\) suffices for \(\varepsilon_{\rm cert}\leq\varepsilon\), after intersecting its lower endpoint with the known bound \(c_{q+1}\geq q+1\).

Proof. Positivity gives the interval for a. On \([L,U]\), \(\widehat a/a-1\) decreases with a. Its values at the two endpoints are respectively \((U-L)/(U+L)\) and its negative. This proves the true-value-denominator relative error; the arithmetic midpoint would not give this bound. Substitution of L and U gives the formula in \(\delta_A,r\). Since \(r<\varepsilon<1\), elementary rearrangement shows that this formula is at most \(\varepsilon\) exactly when \(\delta_A\leq\tau\).

Let \([c_L,c_U]\) be the dyadic interval after the lower clamp. The positive scale defining A cancels in \(\delta_A\), and \(c_L+c_U\geq2(q+1)\). Its width at most \(2(q+1)\tau\) therefore implies \(\delta_A\leq\tau\). Dyadic enclosures for multiple zeta values supplies width at most \(2^{-b}\) for every requested positive integer b. Choosing b once so that \(2^{-b}\leq2(q+1)\tau\) proves the assertion.

Research evaluator and refusal contract

tools.risk_research.nonvertex_relative_certificate computes the analytic bound before constructing an MZV expression. It chooses b using integer bit lengths and at most one rational comparison, then charges symbolic construction, cutoff selection and numerical evaluation to one shared additive work counter. The separate analytic selected-input counter bounds accepted scalar inputs. Neither counter is a CPU-time, RSS or complete integer-bit-complexity bound.

The CLI accepts --face-size, --outside, --degree, --relative-target, --analytic-work-limit and --numeric-work-limit. The target must lie strictly between zero and one. Both resource policies and input bit lengths are checked before diagonal or MZV construction. For an inactive face \(D<s\), the coefficient is exactly zero: its record has null relative-error fields, and performs no analytic or constant evaluation.

If \(r\geq\varepsilon\), the tool conservatively refuses: this bound cannot certify the requested target. It does not assert that the actual coefficient error is too large. This also refuses the equality case even if an exactly known constant could certify it. Work-limit refusals likewise do not refute the mathematical stopping theorem. No failure is serialized as a success certificate; the CLI exits nonzero, reports its reason on stderr and preserves any existing output. The first version handles one fixed degree and does not automatically choose among sharper codimension-specific methods or search an all-tail threshold.

Success records use the new schema skeleton-risk-nonvertex-relative-certificate-v1. They include both budgets, actual additive work, selected precision and cutoff, constant and coefficient intervals, the analytic bound, harmonic-center approximation and certified relative error. Replay restores every saved option and compares the entire typed record. It uses the source union of the wrapper, proof, both evaluators and their recorded dependencies. Fingerprints normalize CRLF pairs to LF; raw-byte guards still detect line-ending changes during calculation. Sources, Git HEAD, tree and complete checkout status are checked across the composition and replay; the CLI also guards the raw saved input across verification. These guards detect changes but are not a sandbox or protection against transient changes restored between snapshots.

Use checkout_harness.run_python for an initially clean committed checkout and separately retained outputs. Its finally-block guards also cover failed commands and timeouts. A certificate does not encode the checkout commit or status, so replay from another unchanged clean commit with the same bound sources remains possible. Historical schemas and their evaluator sources are unchanged.

For example, from the repository root, with output outside the checkout:

python -m tools.risk_research.nonvertex_relative_certificate \
  --face-size 3 --outside 4 --degree 1208925819614629174706193 \
  --relative-target 1/100 --analytic-work-limit 190000 \
  --numeric-work-limit 180000 --output /tmp/nonvertex-relative.json
python -m tools.risk_research.nonvertex_relative_certificate \
  --verify /tmp/nonvertex-relative.json

The evaluator requires only the Python standard library. Finite checks of its output validate implementation and bookkeeping; the composition proof and the two component theorems establish interval inclusion.