Titulus_E8 — Geometric LoRA Adapter

Titulus_E8 is a LoRA fine-tuning adapter for causal language models, trained on the trilingual Biblical corpus (Hebrew Tanakh · Koine Greek NT/LXX · Latin Vulgate) with a Topological Coherence Loss that anchors the model's embedding space to the E8 lattice.

The adapter is part of the Sovereign Geometric Model (SGM) architecture from Sovereign Engine, a distributed geometric AI system that uses discrete lattice mathematics to eliminate LLM hallucinations by construction.

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The Name

The Titulus Crucis — the inscription nailed above the cross — was written in three languages: Hebrew, Latin, and Greek (John 19:20). It is one of the most historically significant trilingual documents in existence and the conceptual anchor for this system. The trilingual structure mirrors the three corpus languages exactly.

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Architecture

Base Model  (Qwen2.5-0.5B-Instruct)
    │
    ├── LoRA Adapter  ← Titulus_E8
    │     rank=32, alpha=64
    │     target: q_proj, v_proj
    │
    └── E8 Logit Mask  (inference-time geometric constraint)
          - every vocabulary token → 8D IPA phonetic coordinate
          - coordinate snapped to nearest of 240 E8 roots (Babai CVP)
          - logits attenuated for tokens that drift beyond geodesic radius

Training Objective

The adapter is trained with a dual loss:

L = λ_lm · CE_loss + λ_topo · TopologicalCoherenceLoss

TopologicalCoherenceLoss = mean( min_j ||h_i - e8_root_j||² )

Where h_i are the final hidden-state vectors and e8_root_j are the 240 roots of the E8 lattice, projected into the model's hidden dimension. This forces semantically related embeddings to cluster near valid lattice coordinates — making hallucination geometrically costly rather than just statistically unlikely.

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Training Data

Language	Corpus	Source
Biblical Hebrew	Westminster Leningrad Codex (Masoretic Text)	christos-c/bible-corpus
Koine Greek	Nestle-Aland NT + Septuagint (LXX)	christos-c/bible-corpus
Latin	Clementine Vulgate	christos-c/bible-corpus

Each training sample interleaves all three translations of a canonical verse:

[HE] בְּרֵאשִׁית בָּרָא אֱלֹהִים אֵת הַשָּׁמַיִם וְאֵת הָאָרֶץ
[GR] Ἐν ἀρχῇ ἐποίησεν ὁ θεὸς τὸν οὐρανὸν καὶ τὴν γῆν
[LA] In principio creavit Deus caelum et terram

The model learns to associate semantically equivalent expressions across three typologically distinct language families — forcing its latent space to develop representations that are stable across linguistic surface variation.

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Usage

As a standard PEFT adapter

from transformers import AutoTokenizer, AutoModelForCausalLM
from peft import PeftModel

base_model_id = "Qwen/Qwen2.5-0.5B-Instruct"
adapter_id    = "JLeeV/Titulus_E8"

tokenizer = AutoTokenizer.from_pretrained(base_model_id)
model     = AutoModelForCausalLM.from_pretrained(base_model_id)
model     = PeftModel.from_pretrained(model, adapter_id)

inputs = tokenizer("In the beginning", return_tensors="pt")
output = model.generate(**inputs, max_new_tokens=50)
print(tokenizer.decode(output[0]))

With Geometric Logit Mask (full SGM pipeline)

from transformers import AutoTokenizer, AutoModelForCausalLM, LogitsProcessorList
from peft import PeftModel
from sov_heart.logit import GeometricLogitMask

base_model_id = "Qwen/Qwen2.5-0.5B-Instruct"
tokenizer = AutoTokenizer.from_pretrained(base_model_id)
model     = PeftModel.from_pretrained(
    AutoModelForCausalLM.from_pretrained(base_model_id),
    "JLeeV/Titulus_E8"
)

# Build geometric constraint interceptor
processor = GeometricLogitMask.as_logits_processor(
    tokenizer,
    radius=3.0,        # geodesic tolerance (tighter = stricter)
    soft_mask=True,    # Gaussian attenuation vs hard zeroing
    cache_path="/tmp/titulus_token_coords.npy",  # cache for fast reload
)

inputs = tokenizer("Explain the concept of logos", return_tensors="pt")
output = model.generate(
    **inputs,
    max_new_tokens=200,
    logits_processor=LogitsProcessorList([processor]),
)
print(tokenizer.decode(output[0]))

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Installation

pip install peft transformers torch

# For the full SGM geometric constraint system:
git clone https://github.com/JLeeV/sovereign-engine
cd sovereign-engine && pip install -e .

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Geometric Coordinate Mapping

Each token in the vocabulary is mapped to an 8-dimensional E8 coordinate using IPA phonetic features:

Dimension	Feature
0	Stop consonant ratio (p, b, t, d, k, g, q)
1	Fricative ratio (s, z, v, ʃ, ħ, θ, f, x)
2	Nasal ratio (m, n)
3	Vowel ratio (a, e, i, o, u, y)
4	Liquid ratio (r, l)
5	Mean Unicode codepoint (normalised to Hebrew range)
6	Script class (0=Latin, 0.5=Greek, 1.0=Hebrew/Semitic)
7	Syllable density (syllables per character)

The raw 8D vector is then snapped to the nearest of the 240 roots of the E8 lattice using Babai's Closest Vector Problem (CVP) algorithm — guaranteeing all token coordinates are valid lattice points.

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License

This adapter is released under the Sovereign Engine Cooperative and Topological License (SECTL) v1.0. See LICENSE for full terms.

The training corpus texts (Westminster Leningrad Codex, NA28, Clementine Vulgate) are in the public domain.

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Citation

@misc{titulus_e8_2026,
  author       = {JLeeV},
  title        = {Titulus\_E8: A Geometric LoRA Adapter Anchored to the E8 Lattice},
  year         = {2026},
  publisher    = {HuggingFace},
  howpublished = {\url{https://huggingface.co/JLeeV/Titulus_E8}},
  note         = {Part of the Sovereign Engine geometric AI architecture}
}