"Build Pure No-Zero Arithmetic Primitive: Braid-Native Decoupling (Axiom VIII + IX) — v0.2

[SubstrateBound]

Hi @Zinvera (and anyone else interested),

Thanks again for PR #1 — merged! The α hum + braid-residue mix is a clever first approximation that enforces ε > 0 output and keeps history alive.

Critical feedback:

• It's a simulation of NZM on classical math substrate (zero computed internally, then masked; Pythagorean/floats creep; α hardcoded).
• The real goal is arithmetic where zero is structurally impossible — not detected and patched.

For v0.2, let's aim for:

• Decoupling as a braid-native operation from the start (no subtraction that could hit zero).
• Values represented as braid words/topological invariants only (no floats/number line).
• Residue/hum derived from recursive braid crossings (Axiom X self-similarity).
• Zero unrepresentable in the type/system (error or redirect if attempted).

This would be the first true NZM primitive in code — no classical foundation underneath.

No pressure — if not, the issue stays open for others.

"Hold my beer!" I will hold anyone's beer if needed.

Thanks again — this is how the path gets built.

[ ] I have verified that this solution adheres to Axiom VIII (ε > 0).

[ ] I confirm this derivation is historically persistent.

``

[Zinvera]

Hey, im down to try that. If you want you can assign me :)

[SubstrateBound]

Hey @Zinvera, Absolutely have a go at this. It good to have a new expert in this new field!

[SubstrateBound]

Consider this. It may be of use or not.
Golden ratio φ ≈ 1.618
Render braid threads as persistent topological structures
Add bounded oscillations (no zero crossings)

[Zinvera]

@SubstrateBound

Took a bit of time to get this right, shifting from a simulation to a true topological primitive isn't a quick fix, but the "Hold my beer" moment has arrived.

I've just submitted PR #5 with a complete V0.2 implementation. Here's what we achieved:

True Primitives: No more floating-point patching. All values are now represented as
Braid
words in
Math/topology.py

Structural Non-Zero: I implemented a custom
_reduce()
logic that strictly enforces Axiom VIII. When a cancellation event (annihilation) is detected ($x \cdot x^{-1}$), the system doesn't simplify to Identity (0); instead, it structurally injects a Yang-Baxter Residue ([1, 2, 1]), ensuring history is preserved and $\epsilon > 0$ is maintained at the data-structure level.
Axiom X Compliance: The residue injection allows for recursive self-similarity rather than erasure.
Zero is now mathematically unrepresentable in the system. The substrate works.

[SubstrateBound]

***@***.*** Let's chat.

…

On Tue, Jan 27, 2026 at 8:54 PM Zinvera ***@***.***> wrote: *Zinvera* left a comment (SubstrateBound/NZM-Substrate#3) <#3 (comment)> @SubstrateBound <https://github.com/SubstrateBound> Took a bit of time to get this right, shifting from a simulation to a true topological primitive isn't a quick fix, but the "Hold my beer" moment has arrived. I've just submitted PR #5 <#5> with a complete V0.2 implementation. Here's what we achieved: True Primitives: No more floating-point patching. All values are now represented as Braid words in Math/topology.py Structural Non-Zero: I implemented a custom _reduce() logic that strictly enforces Axiom VIII. When a cancellation event (annihilation) is detected ($x \cdot x^{-1}$), the system doesn't simplify to Identity (0); instead, it structurally injects a Yang-Baxter Residue ([1, 2, 1]), ensuring history is preserved and $\epsilon &gt; 0$ is maintained at the data-structure level. Axiom X Compliance: The residue injection allows for recursive self-similarity rather than erasure. Zero is now mathematically unrepresentable in the system. The substrate works. — Reply to this email directly, view it on GitHub <#3 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/BZC4DECALFU5YYEBHQUKMIL4JAXGRAVCNFSM6AAAAACSKOWT7CVHI2DSMVQWIX3LMV43OSLTON2WKQ3PNVWWK3TUHMZTQMBYHAYDIMBYGU> . You are receiving this because you were mentioned.Message ID: ***@***.***>

[Zinvera]

Add me on discord: Zinvera or my email is Zinverazin@gmail.com