The two identities of Remark 4.3

Bouthat, Chávez and Sheng, A general framework for inequalities on simple graphs, arXiv:2606.17356v1, Remark 4.3, state the two path CHS identities as unproved and then assert a one-term simplification under $n+1\ge d/2$.

For every $n\ge1$ and even $d\ge2$, the symmetric value $h_d(\lambda(P_n))$ equals both the trigonometric sum and the bilateral zero-extended binomial sum printed in Remark 4.3.

Statement echo

| Source clause | Lean counterpart | Fidelity | | --- | --- | --- | | $|P_n|_{X,d}^d=h_d(\lambda(P_n))$ | PathCHSNormClosedForm.chs | Same complete homogeneous symmetric value. | | All $n\ge1$ and even $d\ge2$ | Arithmetic and parity hypotheses in claim | No degenerate path witness is used. | | Trigonometric sum, $1\le k\le n$ | trigForm, with Finset.Icc 1 n | Every factor and exponent is literal. | | Sum over all integers | bilateralForm, using finsum | Zero-extended binomials have finite support. | | Both equalities | The two conjuncts in claim | Both are proved by result. |

The settling GID is D5/S3/Combinatorics/Graph/PathCHS/PathCHSNormBCSRemark43.result, preregistered in #18566. The closed form for the original path-norm problem is D5/S3/Combinatorics/Graph/PathCHS/PathCHSNormClosedForm.result (#18565).

The two full identities are proved. A uniform corrected truncation and uniform one-term boundary correction remain outside these Lean statements.

Partial fractions at the distinct Chebyshev nodes express complete homogeneous polynomials through nodal derivative weights. The weights are $2(-1)^{k+1}\sin^2(\pi k/(n+1))/(n+1)$. Pairing the bilateral indices identifies the binomial sum with the finite second-difference expression. These two routes give the two conjuncts of result.

The admission basis is open-problem-resolution (#18566; Proved). The recurrence invariant scaled_S_eval proves $t^n S_n(t^{-1})=\operatorname{cont}(t^2;1,1;n)$ for $t\ne0$. It and its consumers node_reciprocal_product, chs_eq_node_hsymm, chs_eq_trig and result are content with scaled_S_eval as escape witness. Utility is none.

An $n\ge1$ and even $d\ge2$ at which either expression differs from the complete homogeneous symmetric value would contradict result.

D5/S3/Combinatorics/Graph/PathCHS/PathCHSNormBCSRemark43.result proves the full literal claim. The axiom closure of every public declaration is contained in ${\mathrm{propext},\mathrm{Classical.choice},\mathrm{Quot.sound}}$.

The four-slot escape audit is unfinished: #18981. No validated registration is claimed. The missing evidence comprises exact source reconstruction, realization bridges, whole-family variation and sensitivity for the dependent finite spectra and ring-polymorphic identities.

Experiment: the published check of Remark 4.3. Command: python3 /tmp/op-chavez/rework-51566/bcs-check.py 30 60; exit 0. The file is the byte-exact check.py from that commit, SHA-256 f967f78544056b1b68df4e3055cd34c76ad0183a4d9b37602166e74ac6b582fc. For $1\le n\le30$ and even $2\le d\le60$, both full expressions agree with the exact symmetric value (the trigonometric evaluation uses 80-digit arithmetic). The printed finite truncation fails in 81 cases, beginning at $(1,4)$. The one-term formula fails in the 29 tested boundary cases $n+1=d/2$, with difference one, and agrees in the tested strict region $n+1>d/2$. At $(2,6)$ the exact symmetric value is one and the proposed expression is zero. Uniform corrections beyond this finite scope are open here.

• Proved, kernel-checked — #18565. D5/S3/Combinatorics/Graph/PathCHS/PathCHSNormClosedForm.result gives the finite binomial formula and its real-root norm for every $n\ge1,m\ge2$.

• Proved, kernel-checked — #18566. D5/S3/Combinatorics/Graph/PathCHS/PathCHSNormBCSRemark43.result gives both displayed identities for every $n\ge1$ and even $d\ge2$.

• Refuted, kernel-checked — #18567. D5/S3/Combinatorics/Graph/PathCHS/PathCHSNormBCSRemark43ClosedRefutation.result disproves the one-term universal claim using $n=2,d=6$; the utility classification is certified-instance with basis=refutes.

• Literature — open completeness boundary. The source and the preregistrations' bounded literature screens identify no prior settlement. Completeness is ASSUMED-UNVERIFIED.

• Computed. The 81 truncation failures and 29 boundary discrepancies have exactly the finite scope in Evidence. No uniform theorem follows from those checks.

• Open. A corrected truncation range and the strict-versus-boundary rule for all $n,d$ require separate statements and proofs.

What the settlement shows

• Proved, kernel-checked. result establishes both full formulas, including the full integer-indexed sum rather than the printed finite truncation.
• Failure mechanism, explanation. At $d/2=n+1$, the bilateral $j=-1$ term is $\binom{n+d-1}{0}-\binom{n+d-1}{-1}=1$. The printed lower cutoff is zero and discards this term. This identifies the boundary omission; the uniform corrected formula is not a Lean theorem here.
• Computed nearest surviving statement. The one-term expression agrees on the tested strict region $n+1>d/2$; at every one of the 29 tested boundary cases it is smaller by one. The uniform extensions are open.
• Source dependence. The original path-norm question is answered by #18565 and the two full identities by #18566. The closing simplification and finite truncation cannot be substituted for the full sum without further conditions. No other theorem of the source is asserted to require the false boundary formula.

Literature completeness and independent priority are unverified. The source v1 and the bounded screens in #18566 are the literature evidence. The escape audit is unfinished under #18981.