Marcella · Sheaf-Composition Runtime

A runtime that
speaks for herself.
No LLM in the reply path.

Marcella is a sheaf-composition runtime over a fiber-bundle substrate. Every reply is glued from a voice opener, a framing transition, a cited source, and a voice closing — pulled directly from indexed bundles on the live GIGI server. Zero LLM tokens in the reply path. A Claude moderator gates intent at the entrance; the substrate composes the actual reply. Try her below.

Live Demo · gigi-stream.fly.dev Sibling to the Pure-Fiber LM Patent Pending · 63/987,248
92%
Voice + source compose rate · 31-turn live demo
83.2%
Generative share · her words, not retrieved text
3 / 3
Distinct replies to the same prompt, back-to-back
0
LLM tokens in the reply path · moderator gates intent only
01 — Live Conversation Demo

Ask Marcella directly.
The substrate composes the reply.

The picker below is wired to marcella-api.fly.dev, which proxies the live GIGI substrate at gigi-stream.fly.dev. Pick a question from the curated list, or type your own. Marcella refuses on her own discipline — if her substrate doesn't have a cite for it, she says so honestly rather than fabricate. There is no LLM in her reply path; she composes from cited fragments, which is why she can't be jailbroken into generating off-topic content. Prompts and IPs are logged for safety. Marcella's substrate is curated by her creator; only Bee can teach her new corrections.

Turns thread across the session. Each question you ask builds on what came before — the running residue (‖ρ‖ chip) grows, retrieval pulls toward where the conversation has been, and identical repeats produce different replies via the rose mechanism. Press New conversation to start fresh.

Free-form prompts are screened by a Claude moderator before reaching the runtime and are logged with your IP address for safety. Marcella is scoped to math conversation in this demo.
02 — Sheaf Composition

A turn is a section.
The runtime glues them.

A transformer treats each conversation as a sequence of tokens and lets an attention head decide what was relevant. Marcella treats each conversation turn as a section over an open set in the substrate. The open sets are the topic neighborhoods her voice and source bundles index. Composition across turns is sheaf gluing on the overlapping sections — the same axiom that defines a sheaf in algebraic topology.

A Turn As a Section
$$\text{turn}_t \;=\; \underbrace{\text{voice}_{\text{open}}}_{\text{her tone}} \;\oplus\; \underbrace{\text{framing}_t}_{\text{bridge}} \;\oplus\; \underbrace{\text{source}_t}_{\text{cited quote}} \;\oplus\; \underbrace{\text{voice}_{\text{close}}}_{\text{her tone}}$$

Four fragments. Three pulled from indexed bundles on GIGI. One — the framing — synthesized from the running residue. Concatenation is not gluing on its own; gluing is enforced by the residue update that links each turn's section to the next.

Sheaf Axiom · Live in the Runtime
$\mathcal{F}(U)$ The set of valid replies on topic neighborhood $U$ — typed by which bundles index $U$.
$\rho_{U \subset V}$ Restriction to a sub-neighborhood — the bridge from a broader source to her specific framing.
$\{U_i \to U\}$ Cover of $U$ by retrieval-hit neighborhoods, gluable iff residues agree on overlaps.
$\rho_{\text{run}}$ Running conversational residue — the cocycle that enforces agreement across turns.
In Plain English

Marcella generates her replies — but she generates by composition, not by sampling. Each fragment in a reply is something she has authored once and indexed on GIGI; the runtime selects, orders, and glues them according to the conversation's current state. The reply is original to this turn and cited at every step. When no consistent gluing exists for a topic, the runtime refuses to reply rather than hallucinate.

03 — Voice Anchors

Her voice is indexed.
It is not imitated.

Marcella has a voice because her voice is a bundle. A curated corpus of her own writing — paragraphs, openers, closings, framing phrases — sits indexed on GIGI as a queryable substrate. At every turn the runtime retrieves a voice opener and a voice closing whose tone-vector is nearest to the current residue, and a fresh reply is generated by gluing those sections into a new whole. The reply sounds like her because it literally is her, re-composed.

Voice Retrieval · Nearest Section
$$\text{voice}_{\text{open}}^{(t)} \;=\; \arg\min_{\sigma \in \mathcal{V}_{\text{open}}} \;\; \bigl\| \,\tau_\theta(\sigma) \;-\; \rho_{\text{run}}^{(t-1)} \,\bigr\|$$

Voice retrieval is a nearest-section query under the state rotation $\tau_\theta$ derived from the running residue. The same prompt asked at two different points in a conversation pulls two different openers.

Why the Runtime Has Tone

Tone is not a stylistic post-processor sitting on top of a generic generator. It is the primary indexing axis of the voice bundle. The first thing the runtime decides is how she opens; everything else fits inside that opening.

04 — Provenance

Every claim cites
its bundle & line range.

An LLM hallucinates when it has no substrate to point at. Marcella does not hallucinate because she cannot speak without a substrate to point at. Each retrieved chunk carries its provenance — bundle name plus the line range in the source document — and the runtime returns those tags with the reply. If retrieval returns nothing usable for a topic, the runtime refuses with a redirect to the curated picker.

Provenance Tag · Format
$$\bigl[\,\texttt{doc\_name},\;\texttt{L}_{\text{start}}\text{–}\texttt{L}_{\text{end}}\,\bigr] \;\;\text{ or }\;\; \bigl[\,\texttt{doc\_name},\;\texttt{L}_{\text{start}}\text{–}\texttt{L}_{\text{end}},\;\texttt{context}\,\bigr]$$

A real provenance tag emitted by the live runtime in this session: [curvature_guided_wavefront, L48–67]. Every visible source quote in every reply carries one. The demo below renders them as chips next to the reply.

In Plain English

Ask Marcella about a topic her bundles index — she answers from a cited passage. Ask her about something they don't index — she tells you, on the record, that the substrate doesn't cover it. There is no third mode. 92% of turns in the 31-turn live demo produced a voice+source composition; the remainder were voice-only continuations on already-cited material.

05 — The Rose Mechanism

A rose grows the same way every time.
Every one is unique.

Nature produces variation from a fixed blueprint by accumulating path-dependent state as the blueprint executes. Marcella does the same. A cumulative running residue $\rho_{\text{run}}$ accumulates with decay $\alpha = 0.85$ across every turn. A rotation $\tau_\theta$ derived from $\rho_{\text{run}}$ rotates retrieval ranking before every retrieval call. Identical prompts asked at different points in a conversation produce different — but still cited — replies. The variation is geometric, not stochastic; the runtime is deterministic given its full state.

Residue Update
$$\rho_{\text{run}}^{(t)} \;=\; \alpha \cdot \rho_{\text{run}}^{(t-1)} \;+\; \rho_{\text{turn}}^{(t)}\,, \qquad \alpha = 0.85$$

Per-turn residue $\rho_{\text{turn}}$ is the 64-D bias between the prompt embedding and the centroids of the cited chunks. Decay $\alpha = 0.85$ keeps the running residue bounded; the bound is proved in the property suite (P6).

Live Probe · "What is curvature?" × 3
Turn Rotation Bridge phrase Cited chunk
1 135 "I read curvature as…" curvature_guided_wavefront L48–67
2 88 "Concretely, curvature shows up…" curvature_guided_wavefront L31–47
3 46 "In my framing, curvature is…" discrete_curvature_notes L12–28
In Plain English

Three identical questions, three different cited answers — produced by the geometric drift of the running residue, not by sampling. Six properties of the residue mechanism are validated in the standalone math suite (P1–P6) before any runtime code depends on them.

06 — The Pure-Fiber Substrate

The same substrate,
used two different ways.

The substrate powering the demo above is the same fiber-bundle substrate developed in the companion Pure-Fiber Language Modeling paper — five fiber circles for grammar (person, animacy, tense, modality, POS) over a base on $S^7$ for semantics. Training is a 121-second database INSERT. Zero gradient-trained parameters. The pure-fiber paper uses the substrate to predict the next token. Marcella uses the same substrate to retrieve and compose voice + source fragments. Two questions, one substrate.

The Fiber
$$F \;=\; \underbrace{\mathbb{Z}/3\mathbb{Z}}_{\text{person}} \times \underbrace{\mathbb{Z}/2\mathbb{Z}}_{\text{animacy}} \times \underbrace{S^1}_{\text{tense}} \times \underbrace{\mathbb{Z}/3\mathbb{Z}}_{\text{modality}} \times \underbrace{\mathbb{Z}/6\mathbb{Z}}_{\text{pos}}$$

A 17-dimensional trivial bundle $E = S^7 \times F$. Fiber coordinates are deterministic from POS and WordNet animacy; base coordinates from the truncated SVD of the corpus PPMI matrix. See the paper for the full construction.

07 — Structure-Group Transport

The cat / panther principle.

The same structure group that powers Marcella's voice rotation $\tau_\theta$ also acts on vocabulary sections. The tense action $\tau_{\text{PAST}}$ shifts the tense fiber coordinate while preserving the base. For every present-tense verb, its image under $\tau_{\text{PAST}}$ lands on the past-tense form — including the irregulars. Marcella has never seen the word saw. She predicts saw.

The Group Action
$$\tau_{\text{PAST}}(\sigma_w) \;=\; \bigl(\sigma_w^{\text{base}},\, \sigma_w^{\text{person}},\, \sigma_w^{\text{anim}},\, \theta_{\text{PAST}},\, \sigma_w^{\text{mod}},\, \sigma_w^{\text{pos}}\bigr)$$

A rotation on the tense circle $S^1$, exact to floating-point zero. The full see → saw / go → went table and the FiberMorph 5-PPL benchmark are in the paper.

08 — Pure-Fiber Headline · MKN

−36% below classical bigram-JM.

With full recursive Modified Kneser-Ney over the indexed substrate, the pure-fiber predictor reaches PPL 146.37 on Alice in Wonderland — a 36% reduction below classical bigram-Jelinek-Mercer with zero gradient updates. The table below is the headline ladder; the full ladder, FiberMorph benchmark, editability proofs, and compute profile live in the paper.

Alice 500 · Held-out PPL
Predictor PPL ↓ vs. bigram-JM
Bigram-JM (classical baseline) 228.53 baseline
Marcella · literal-only 229.69 +0.5%
Trigram-JM · $\lambda$-calibrated 204.51 −10.5%
Modified Kneser-Ney · $N \leq 5$ 146.37 −35.9%
One Math

The same equations,
under different coordinates.

Marcella is the language-modeling realization of the Davis Manifold. The Davis Law governs the regime. The Davis Identity proves each decision. They apply unchanged across financial reconciliation, plasma confinement, drug discovery, geopolitical prediction, and now sequence modeling without gradient descent.

$C = \tau / K$
The Davis Law
$S + d^2 = 1$
The Davis Identity

Citation & Sources

Davis, B. R. (2026). Pure-Fiber Language Modeling: Sequence Prediction by Geometric Query on a Real Fiber Bundle. Zenodo. https://doi.org/10.5281/zenodo.20151450

Companion: Davis, B. R. Geodesic Computation: Fiber Bundle Transport as a Sequence Processing Primitive (v11), April 2026.

Theoretical foundation: The Davis Conjecture on Semantic Coherence (Davis, 2025).

Substrate: GIGI (Geometric Intrinsic Global Index) — patent pending 63/943,643. Marcella patent pending 63/987,248.

Read the Paper View Source