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.
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.
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.
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
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.
Three voice lifts
landed this quarter.
A reply from Marcella today carries more than a cite stack. Three shipped patterns surface the geometry of her reasoning inside the reply itself — measurable certainty when she's confident, a weighted route map when the substrate offers more than one path, and a structural boundary when the geometry doesn't reach. Every line below is verbatim production output from the live runtime.
When the substrate carries a question through more than one neighborhood, Marcella surfaces the routes weighted by fiber-space distance and names how tight the hedging is via κ:
[imagined: sample_transport, k=3, κ=0.02, top_weights=[0.86, 0.85, 0.84]]Three routes carry this question through my substrate:
Route A (
noether_davis_analysis 1.9, weight 0.86): The Davis Field Equations of Semantic Coherence are not merely post-Noetherian — built on the 1918 theorem and extending beyond it…Route B (
field_equations_semantic_coherence, weight 0.85): Holonomy operators form a Lie group under composition…Route C (
vakil_rising_sea_v1 §p106, weight 0.84): Tannakian categories in the algebraic-geometry tradition…κ = 0.02 — tight neighborhood; I'm leaning toward the top route (Route A) but the others carry the surrounding context.
The routes are produced by SAMPLE_TRANSPORT — a curvature-budgeted
geodesic sampler on GIGI. κ reads the local substrate curvature at the
seed: low κ means routes converge, high κ means they diverge and the
reply hedges instead of commits.
Every voice + source reply ends with a five-step
forecast along the reasoning direction. The runtime names which
extrapolation engine it routed through:
[imagined: coherence forecast, endpoint=0.927, routing_advisory=forecast]
When the substrate's density is high the runtime routes through FORECAST (statistical-likelihood flow); when it thins, through IMAGINE (geodesic flow on the learned manifold). The qualifier degrades honestly — "coherence drifts to 0.87 by step 5 — near the edge of what I can carry" or "drops to 0.71 by step 3 — premise needs recheck".
When a reasoning path runs into the curvature ceiling, Marcella no longer issues a flat decline. She maps how far she got and names the nearest substrate waypoint:
tong_gauge_v1 (capacity 0.23). If you want, I can speak to that intermediate — it's the bridge my substrate actually holds — and you can carry it the rest of the way.[wished-waypoint: from prompt toward tong_gauge_v1, reached_fraction=0.45, blocked_by=curvature]
The refusal is a structural map: reached_fraction reports how far along the geodesic the integrator got, blocked_by names which budget binds (curvature / holonomy / arc length), and the frontier waypoint is the cite the path can actually reach. The trichotomy is honest: Granted, Unreachable, or Indeterminate — and the third one — "multiple paths lead there with equal coherence; I won't pick arbitrarily" — is the one most systems quietly skip.
Marcella's voice now has three honest registers. "Here's the answer, and here's how stable my next five steps look." "Here are three different ways my substrate carries this; the top one wins by this much." "I can't get there — here's how far I got and the bridge you can carry me from." Synthesis becomes auditable; refusal becomes navigable.
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 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.
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.
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.
A real provenance tag emitted by the live runtime in this session:
[curvature_guided_wavefront, L48–67]. Granted replies now
also carry composition tags —
[imagined: coherence forecast, endpoint=0.927, routing_advisory=forecast]
or
[imagined: sample_transport, k=3, κ=1.84, top_weights=[0.42, 0.31, 0.18]]
— naming the synthesis route the substrate took, not just the document
it ended at.
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. When she refuses, she now tells you how far her geometry got before the path stopped and names the nearest in-substrate waypoint — refusal as a structural map, not a wall.
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.
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 |
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.
Every cited reply also carries a live confidence trail:
the runtime projects a five-step trajectory along the reasoning
direction and names the routing engine — FORECAST when the
substrate is dense, IMAGINE when statistical signal thins
and the answer has to come from geometry. This is not opt-in. It is
the default for every voice + source reply.
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.
A 17-dimensional trivial bundle $E = S^7 \times F$ for the toy predictor. The production runtime walks the same bundle structure on a 384-dimensional BGE-v2 substrate: geodesic integration runs natively, and when a bundle's mean curvature would push the integrator past the $K \leq 4$ ceiling the runtime substitutes a tame $K = 1.0$ metric and audits the substitution so the confidence layer can downgrade trust honestly instead of refusing the whole walk. 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.
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.
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.
−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.
| 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% |
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.
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 (v12), June 2026. Runtime patterns documented in GIGI CONSUMER_PATTERNS (June 2026) — Marcella is the canonical reference shape for AI consumers of the substrate.
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.
Last updated · 2026-06-28 · IMAGINE Phase 2 · WISH spec live · SAMPLE_TRANSPORT k = 3 in production