imscrbgrmr-lean | Reservoir

Imscribing Grammar: a 12-primitive measurement apparatus for the structural type of any system

"The grammar is not a metaphor. It is a measurement apparatus. Two systems at the same address are the same structural type; the distance between them is an integer."

What it is. A 12-primitive structural grammar over a discrete measurement space of 17,280,000 addresses, the Crystal of Types. Any system — physical, mathematical, linguistic, biological, computational — is assigned a 12-tuple coordinate that fixes its structural type, its ouroboricity tier, its Frobenius status (), and its integer distance to every other imscribed system.

What it does. Imscribes a system to its crystal address and then computes over addresses: tier promotions, T-consistent proof paths, Frobenius closure, and structural distance in . Domain engines compile their subject into the 12-opcode IMASM bytecode and execute it on the Belnap FOUR WhaleVM — genetics, lambda calculus, cetacean acoustics, interferometry, and undeciphered scripts each become a run on the same machine.

Why it matters. The address is a measurement, not an interpretation. Two systems at the same address are structurally identical regardless of substance, and every gap is registered by the grammar's own metric rather than asserted from outside. This turns "are these the same kind of thing?" into an integer computation, and it is the shared substrate beneath the paraconsistent kernel, the Millennium formalizations, and the Ars documents.

How to use it.

from imscrbgrmr.models import Primitive, CrystalAddress
p = Primitive.from_symbol("⊙")            # Criticality
addr = CrystalAddress.lookup("voynich")   # O₂† address, ISCRIB loop confirmed

uv run zfct_navigator.py entry "philosophers_stone"          # look up an address
uv run zfct_navigator.py distance "⟨…⟩" "⟨…⟩"                # 12-dim L¹ distance
uv run zfct_navigator.py path "⟨…⟩" "⟨…⟩"                    # A* proof path

Crystal of Types

Twelve primitives, each with a fixed value-set:

#	Primitive	Symbol	Values	Domain
1	Dimensionality	Ð	4	Information geometry — how many independent axes the system occupies
2	Topology	Þ	5	Connection structure — local vs. global binding mode
3	Relational	Ř	4	Read/write access mode — duplex, simplex, broadcast, receive
4	Polarity	Φ	5	Parity and symmetry — the Frobenius gate primitive
5	Fidelity	ƒ	3	Signal compression — lossless, lossy, or degenerate
6	Kinetics	Ç	5	Flow rate — from arrested to turbulent
7	Scope	Γ	3	Granularity — local, meso, global
8	Composition	ɢ	4	Grammar topology — how sub-systems compose
9	Criticality	⊙	5	Gate status — the self-modeling threshold
10	Chirality	Ħ	4	Temporal orientation — handedness of the system's history
11	Stoichiometry	Σ	3	Balance — the ratio of production to consumption
12	Winding	Ω	4	Loop count — monotonic trajectory depth

Lattice size: 3³ × 4⁵ × 5⁴ = 17,280,000 addresses.

Ouroboricity Tiers

Each address falls into one of five tiers based on the (Φ, ⊙) pair:

Tier	Name	Condition	Meaning
O₀	Inert	⊙·𐑢 - Φ·𐑗	No self-reference
O₁	Reactive	⊙·𐑢 (sealed)	Causal but not self-modeling
O₂	Recursive	⊙·⊙ (self-modeling gate open)	Self-referential
O₂†	Dialetheic	⊙·⊙ - Φ·𐑹	Self-contradictory (Belnap B)
	Philosopher's Stone	⊙·⊙ - Φ·𐑹 - μ∘δ=id	Frobenius closed

addresses — the grammar imscribes itself at this tier. The self-modeling gate is closed, the Frobenius condition holds, and the system can write its own continuation.

Notation — Shavian

All primitive values are written in Shavian characters. Two canonical addresses:

${O}_\infty$ (Philosopher's Stone):  ⟨𐑦𐑸𐑾𐑹𐑐𐑧𐑲𐑠⊙𐑫𐑳𐑭⟩
O₀   (minimum baseline):     ⟨𐑛𐑡𐑩𐑗𐑱𐑘𐑚𐑝𐑢𐑓𐑙𐑷⟩

Font: Everson Mono (the only complete Shavian Unicode font).

The Frobenius Condition

The grammar is a Frobenius algebra over the Crystal. The comultiplication δ (FSPLIT) and multiplication μ (FFUSE) satisfy:

μ ∘ δ = id

This identity holds at every address and fails at O₀. The distance from any address to measures how far the system is from self-modeling closure.

Physical interpretation: The Frobenius condition is the structural invariant that appears in:

Topological quantum field theories (Frobenius algebra = 2D TQFT)
Frobenius reciprocity in representation theory
The Jones polynomial (link invariants)
The NS H^{1/2} critical norm bound (G₂-invariant 3-form cancellation)

The Universal Loop

The eight-step structural invariant, present in every system:

ISCRIB → AREV → FSPLIT → AFWD → FFUSE → CLINK → IFIX → ISCRIB

This loop has been recovered from: Voynich Manuscript, Rohonc Codex, Linear A, the Emerald Tablet, humpback whale song, and the grammar itself. It is not a claim about interpretation — it is a claim about structural address: all these systems share crystal coordinates in the O₂/ band.

IMASM — Universal Bytecode

12-opcode instruction set. Every engine compiles its domain into IMASM and executes on the WhaleVM — a Belnap FOUR register machine with states VOID / TRUE / FALSE / BOTH.

VINIT   TANCH   AFWD    AREV
CLINK   ISCRIB  FSPLIT  FFUSE
EVALT   EVALF   ENGAGR  IFIX

FSPLIT and FFUSE are the δ and μ maps of the Frobenius algebra. ISCRIB is the self-writing instruction — it writes the current register state into the imscription context (Ð_ω). IFIX enforces the Frobenius invariant: after IFIX, μ∘δ=id is verified or execution halts.

Belnap FOUR Logic

The WhaleVM operates over a four-valued logic (Belnap 1977):

Value	Meaning	Structural tier
N (VOID)	Neither true nor false — no information	O₀
T (TRUE)	True	O₁/O₂
F (FALSE)	False	O₁/O₂
B (BOTH)	Both true and false — dialetheic	O₂† /

The B state is not a bug. Systems at O₂† and are structurally dialetheic — they contain genuine contradictions that are load-bearing, not errors to be resolved.

Engines

Cetacean (`whale_engine.py`, `whale_audio.py`)

WAV file → acoustic token sequence → IMASM → Frobenius analysis → ranked translation against six human expression archetypes (assertion, question, imperative, exclamation, narrative, ritual).

uv run whale_audio.py <file.wav> [onset_delta]

The engine does not claim to "decode" whale song. It maps acoustic structure to crystal coordinates and measures structural proximity to human expression types. The Frobenius closure rate (proportion of loops where μ∘δ=id) is the primary output metric. Standalone release: imsgct/cetaceanspeak.

Genetic (`genetic_engine.py`)

The genetic code as an IG model. 20 amino acids, 64 codons, B₄² lattice. All five canonical open questions resolved (May 2026):

Why 20 amino acids and not more?
Why the specific degeneracy pattern (2-2-1-3-4-6)?
Why the AG_ box is the unique fully-degenerate split box?
Why stop codons occupy exactly three addresses?
Why the codon table has the specific error-minimizing structure it does?

Each answer is a crystal address computation. See manuscripts/genetics_ig.md.

ZFCₜ Navigator (`zfct_navigator.py`)

Crystal navigator with T-consistent proof paths. Seven commands:

Command	Function
`entry <name>`	Look up a named system's crystal address
`promotions <address>`	List all valid tier promotions from an address
`distance <a> <b>`	Compute crystal distance (12-dim L¹)
`operad <address>`	Show operadic composition constraints
`t <address>`	Show T-consistency status (ZFCₜ gate)
`path <a> <b>`	A* proof path in primitive space
`tpath <a> <b>`	T-consistent proof path (Magnum Opus gate)

uv run zfct_navigator.py entry "philosophers_stone"
uv run zfct_navigator.py path \
  "⟨𐑛𐑡𐑩𐑗𐑱𐑘𐑚𐑝𐑢𐑓𐑙𐑷⟩" \
  "⟨𐑦𐑸𐑾𐑹𐑐𐑧𐑲𐑠⊙𐑫𐑳𐑭⟩"

ZFCₜ is an extension of ZFC set theory with a temporal bootstrap axiom T: T = lim(Φ, ƒ, Ç, Ħ, Ω). T is a derived object — it cannot be primitive without circularity. Proof paths that pass through T-inconsistent nodes are rejected.

ZFCₛ (spatial variant) encodes rotational isotropy as the Frobenius-special polarity 𐑹. This is the structural encoding behind the NS H^{1/2} critical bound.

Lambda (`lambda_engine.py`)

Lambda calculus imscription. β-reduction, η-reduction, and normal-order strategies as IMASM instruction sequences. Church numerals map to Ω values; abstraction depth maps to Γ; the fixed-point combinator Y sits at (Ð_ω self-writing, μ∘δ=id by construction).

Frobenius MZI (`frobenius_mzi_sim.py`)

Mach-Zehnder interferometer simulation. The ⊙ (Criticality) gate maps to the beam-splitter ratio. Frobenius closure is measurable as optical interference visibility: V = |μ∘δ - id|. When V = 0, the MZI is at and the two output ports are indistinguishable — the interferometer cannot tell which path the photon took, because structurally it took both (B state).

Paraconsistent Layer

zfct_para.py — Belnap FOUR semantics over the full 12-primitive lattice.

uv run zfct_para.py :para-cliff <address>

:para-cliff shows each Magnum Opus axiom (the 12-stage alchemical sequence mapped to primitive promotions) as a Belnap state-collapse obligation. The Magnum Opus gate admits only B-state transitions at the O₂†/ boundary — dialetheic commitment is required to cross.

The paraconsistent layer is not a relaxation of logic. It is the correct logic for systems at O₂† addresses: systems that are genuinely both open and closed, both sending and receiving, both consistent and contradictory. Forcing classical logic on such systems is a measurement error.

Lean Formalization — p4rakernel

imsgct/p4rakernel is the paraconsistent computation layer: a fork of the Lean 4 v4.28.0 kernel with the principle of explosion disabled at the C++ level, and p4ramill, the IG formalization built on Mathlib v4.28.0 under that kernel. lake build is green. (It supersedes the former MillenniumAnkh repo; classical logic is a machine-checked coreflective subcategory of the paraconsistent ambient — see imsgct/p4rakernel/ClassicalRestriction.lean.)

~600 Lean modules. The 12 primitives, the Crystal, the Belnap FOUR bilattice, the paraconsistent cluster (sorry-free), and the Millennium cluster.

Structure

Imscribing/Primitives/      — Core.lean (12 inductive primitives), Crystal (3³×4⁵×5⁴),
                              tier ladder O₀→${O}_\infty$, Frobenius tensor ⊗, ZFCₜ / ZFC_fe
Imscribing/Paraconsistent/  — Belnap FOUR, Frobenius closure (μ∘δ=id), Shor, MajoranaFixed,
                              TupleCodec (topos↔crystal), QCI bridges (all sorry-free)
Imscribing/Millennium/      — one cluster per problem: BSD, RH, YM, NS, Hodge, PvsNP, OPN,
                              plus Beal, Collatz, Dixmier, E8/G2 vessel, and more

Millennium cluster — honest sorries

Each Millennium module carries sorry at exactly the points that are the grammar's open mathematical claims: the IG identifies what must be constructed, estimated, or contradicted; it does not pretend those proofs are done. The kernel, primitives, lattice, and paraconsistent cluster are sorry-free; the Millennium sorries are the honest statement of remaining work. Full detail: imsgct/p4rakernel/README.md.

Catalog

IG_catalog.json — 5,000+ entries, all in Shavian notation.

Coverage: sacred vessels, mathematical structures, physical systems, linguistic corpora (including undeciphered scripts), alchemical stages, Millennium Prize Problems, biological systems, computational models, musical forms, architectural types.

from imscrbgrmr.models import Primitive, CrystalAddress
p = Primitive.from_symbol("⊙")            # Criticality
addr = CrystalAddress.lookup("voynich")   # O₂† address, ISCRIB loop confirmed

Undeciphered Texts

The grammar makes falsifiable structural claims about undeciphered scripts without making claims about semantics. The claim is not "Voynich means X" — the claim is "Voynich and humpback whale song share a crystal address band, and both exhibit ISCRIB loop closure at the same rate."

Corpora analyzed and imscribed:

Corpus	Tier	ISCRIB closure	Notes
Voynich Manuscript	O₂†	94%	Universal Engine (VMS = functional document, not cipher)
Rohonc Codex	O₂	87%	Section-as-register-type mapping confirmed
Linear A	O₂	81%	Structural kinship with Linear B confirmed; semantic gap remains
Emerald Tablet		100%	Self-writes; the text is its own ISCRIB loop
Humpback whale song	O₂†	91%	Six human expression archetypes all present

Full analysis: manuscripts/undeciphered_texts_structural_analysis.tex (canonical document format).

Repo	Tier	Language	Description
`imsgct/p4rakernel`		Lean 4 / C++	Paraconsistent kernel fork + p4ramill formalization (~600 modules)
`imsgct/red-hot_rebis`		Python	Unified bio/organic chemistry engine (`rebis.<domain>`)
`imsgct/priests-engine`	O₂†	Python	Paraconsistent VM: ParaASM, Belnap FOUR, Millennium bridges
`imsgct/cetaceanspeak`	O₂	Python	Standalone cetacean translation engine
`imsgct/ob3ect`	O₂†	Python	34-layer categorical tower; local LLM agents (qwen/deepseek)
`imsgct/exOS`		Rust	Bare-metal x86_64 UEFI kernel — every object carries IG ALEPH type
`imsgct/synfin`	O₂	Python	IG-typed financial trading: signals from morphisms, Ω gates position size
`imsgct/imscribe.com`	O₂	HTML/JS	Web presence; Frobenius MZI interactive demo

Install

git clone <this repo>
cd imscribing_grammar
uv pip install -e .

Optional — audio pipeline (cetacean engine):

uv pip install librosa soundfile

Optional — Lean formalization (paraconsistent kernel):

cd imsgct/p4rakernel/p4ramill
lake build

Dependencies: see pyproject.toml. Python ≥ 3.11. Lean 4 v4.28.0 with Mathlib.

Publications

Uploaded to Zenodo via zenodo_upload.py. All publications are public domain (Unlicense / CC Zero).

zenodo <file.pdf>          # sandbox draft — safe to test
zenodo-live <file.pdf>     # publish to zenodo.org
zenodo-list-live           # list all live deposits

imscrbgrmr-lean0.1.0