Hanna's triangular-exponent EGF congruence

Paul D. Hanna's OEIS A226839, returned revision 11, gives the NAME:

E.g.f.: exp( Sum_{n>=1} x^(n*(n+1)/2) / n! ).

Its FORMULA is:

a(n) == 1 (mod 3) (conjecture - valid up to n=1024; if true for n>=0, why?).

The offset is 0,4. The immutable preregistration is #23823. The source note is Library/RecurrenceComposition/hanna2013a226839.md. The source question and source editorial approval do not supply a proof.

Statement echo

Over $\mathbb Q[[X]]$, set $T_m=\lfloor m(m+1)/2\rfloor$, $H(X)=\sum_{m\ge1}X^{T_m}/m!$ and $A(X)=\exp(H(X))$. For every $n\in\mathbb N$, including zero, there is a $k\in\mathbb Z$ with $(n!: \mathbb Q)[X^n]A=1+3(k:\mathbb Q)$.

The public definitions triangular, H and A retain the literal exponent, coefficientwise positive-index rational sum and formal exponential substitution. The denominator is $m!$, not $T_m!$. The private coeff_H proves local finiteness using $m\le T_m$. The public innerIntegral is the natural finite sum over $0\le m\le n$ of $(m+1)^{\overline{n-m}}$ when $0<m$ and $T_m=n$, zero otherwise. The private factorial_ratio and eCoeff_H derive its equality to $n![X^n]H$. There is no assumed integer representation or finite cutoff. The sole public theorem result is exactly the displayed all-index event.

The settling GID is D5/S1/Recurrence/Residue/HannaA226839Congruence.result. Its admission basis is open-problem-resolution, under the preregistered external named conjecture exception. Conservative proof classification is bind-only; the settlement of the external assertion supplies the admission basis. All other declarations are definitions or live consumed helpers. There is no asserted independent escape witness or new general EGF engine. Utility is none for this unbounded symbolic argument.

The source's integrality and congruence must hold for the original rational EGF, so an integer recurrence or finite agreement would be insufficient. The proof derives the exact integral coefficient bridge before modular reduction. The four-slot audit remains unfinished under #23839. The typed Reg family compiles its equivalence, actual-positive, variation, sensitivity and dependence proofs, but its binding record is declared_unresolved, source.residual_requires_open, certificate null. No registration completion is claimed.

For $T_m=n$, the normalized inner coefficient is $n!/m!=(m+1)^{\overline{n-m}}$. For $m=1$ it contributes one at degree one. For $m=2$ the ratio is three. For $m\ge3$, $T_m-m\ge3$, and the rising factorial contains a multiple of $3!$, hence of three. Thus the actual integral inner coefficients reduce modulo three to the identity sequence, with zero constant term.

The existing IntegralEGFComposition.eCoeff_composition identifies the coefficients of the source exponential with integral composition. composition_map transports this integral composition to $\mathrm{ZMod}(3)$, and the original private QuarticEGFModFour.composition_identity_right evaluates composition with the identity. The output is one at every index, including zero. Integer divisibility gives the displayed witness $k$. No homomorphism from $\mathbb Q$ to $\mathrm{ZMod}(3)$ is used.

The exact assertion would fail at any natural index with a nonintegral factorial-normalized source coefficient or a residue different from one modulo three. The kernel-checked result excludes both within the stated rational formal-power-series interpretation.

The Lean source is D5/S1/Recurrence/Residue/HannaA226839Congruence.lean; the Blueprint renders all four public definitions and the full quantified result. The result uses only Classical.choice, Quot.sound and propext. Mathematical identity and freeze state are owned by canonical machine artifacts, not this dossier.

| Item | Status and evidence kind | Scope | | --- | --- | --- | | Original all-index integrality and mod-three assertion | proved; kernel-checked | Public result, every natural index including zero, original rational EGF. | | Actual integral normalization of H | proved; kernel-checked | Private factorial_ratio, coeff_H, eCoeff_H; original denominators and locally finite support. | | Identity residue of the inner coefficients | proved; kernel-checked | Private ratio_dvd_three and inner_mod_three; transferred through existing integral composition. | | Coefficient computation or finite experiments | not computed | No experiment or numerical check was run for this delivery. The source's reported n<=1024 check is attributed only. | | Escape registration | unresolved | Compiled typed family; current binding report unresolved under #23839. | | Stronger moduli or other exponent sequences | unresolved | No additional theorem is delivered. | | A226838 | unqualified | Source-linked possible successor only; no open or solved status claimed. |

The decisive mechanism is divisibility of the actual factorial ratio, followed by the existing integral composition transport. The proof identifies conditions useful for similar source series, but no additional extension is claimed proved. No other source assertion or previous mathematical settlement is changed by this result.

Literature boundary

The current OEIS record retains its conjecture wording. The bounded repository/private, pinned Mathlib and third-party Lean screens found existing general EGF machinery but no exact settlement. The exact-ID arXiv, Crossref, MathDB, MathOverflow, Math StackExchange and formal-conjectures screens located no exact answer. An arXiv Hanna/EGF query located the abstract of Heninger–Rains–Sloane, math/0509316v4, concerning a different Hanna conjecture; its full text was not checked. Challenged or irrelevant general-web results supply no absence evidence. The available history readings establish the source's explicit zero-index scope; not every standalone historical revision was read.

Literature completeness and first-proof priority remain ASSUMED-UNVERIFIED: differently described or unindexed proofs, unpublished ownership and a full citation graph were not excluded. The source note supplies the exact provenance and limits. The unresolved registration is disclosed under #23839 and is not a theorem of undecidability or an assertion that the residual is inexhaustible.