Editing Openai/696eca7b-acdc-800a-9ad9-5ebc9f477576 (section)

== gpt-5-2-thinking ==
=== User: can we try all those in the most reasonable direction ===
can we try all those in the most reasonable direction

=== Assistant: Yes. ===
Yes. I’ll take “most reasonable direction” to mean: start with the smallest model that is truly formal, then add structure only when it buys you something (memory/path dependence, combination of fibers, time-local data, learning analogy, and a runnable miniature).

Below is a ladder of formalisms—each rung is valid on its own.

==== 0) The minimal formal toy category ====

A toy category is just a fully explicit category built to isolate one phenomenon.

===== Definition (preorder category of fiber-states) =====

Let an object be a triple

M=(n,e,c)∈N×N×NM=(n,e,c)\in \mathbb{N}\times \mathbb{N}\times \mathbb{N}M=(n,e,c)∈N×N×N
where:
* nnn = number of nuclei
* eee = “expression readiness” (a coarse epigenetic index)
* ccc = contractile capacity proxy

Define a morphism f:M→M′f:M\to M'f:M→M′ to exist iff

n′≥n,e′≥e,c′≥c.n'\ge n,\quad e'\ge e,\quad c'\ge c.n′≥n,e′≥e,c′≥c.
Composition is inherited from transitivity of ≥\ge≥. Identities are reflexivity.

This is a legitimate category: it’s a thin category (at most one arrow between any two objects).

===== What it captures =====
* “More muscle” as state-enrichment (hypertrophy) = moving to larger ccc
* Added nuclei = moving to larger nnn
* “DNA doesn’t change” = not part of the object; treated as a global invariant (next)

==== 1) DNA invariance as a functor (clean and formal) ====

Let G\mathcal{G}G be the trivial category with one object ∗\ast∗ and only id∗id_\astid∗.

Define G:M→GG:\mathcal{M}\to\mathcal{G}G:M→G by:
* G(M)=∗G(M)=\astG(M)=∗ for every object MMM
* G(f)=id∗G(f)=id_\astG(f)=id∗ for every morphism fff

Then for any training morphism f:M→M′f:M\to M'f:M→M′, the genome-image is unchanged:

G(M)=G(M′)=∗.G(M)=G(M')=\ast.G(M)=G(M′)=∗.
This is the categorical way to say: training does not change the genome (in this toy model, it can’t).

==== 2) Make it less trivial: separate “kinds of change” with generators ====

The thin-category above collapses which process occurred. If you want “hypertrophy vs satellite fusion vs epigenetic shift” to be distinguishable, define a free category from a graph of primitive steps.

===== Definition (generated category) =====

Objects: the same triples M=(n,e,c)M=(n,e,c)M=(n,e,c).

Generate morphisms by three primitive “moves” whenever they stay in N3\mathbb N^3N3:
* h:(n,e,c)→(n,e,c+1)h: (n,e,c)\to (n,e,c+1)h:(n,e,c)→(n,e,c+1)  (hypertrophy increment)
* s:(n,e,c)→(n+1,e,c)s: (n,e,c)\to (n+1,e,c)s:(n,e,c)→(n+1,e,c)  (satellite fusion increment)
* p:(n,e,c)→(n,e+1,c)p: (n,e,c)\to (n,e+1,c)p:(n,e,c)→(n,e+1,c)  (priming / expression readiness increment)

Then morphisms are finite strings of h,s,ph,s,ph,s,p (and identities), with composition by concatenation.

Now you can ask: do these moves commute? Not always—so you can add relations.

===== Optional relations (to encode biology-ish constraints) =====

For example, “capacity increases are limited without enough nuclei”:
* impose that hhh is only allowed when c≤k⋅nc \le k\cdot nc≤k⋅n (a myonuclear-domain style constraint)

That’s not category theory per se; it’s just a constraint on when generators exist.

==== 3) Muscle memory = path dependence → a 2-category (or a category with annotations) ====

The key issue: two paths can land on the same (n,e,c)(n,e,c)(n,e,c) but feel different later (faster retraining, etc.). In ordinary categories, equal targets don’t remember paths.

===== Reasonable upgrade A: enrich morphisms with “history” =====

Let morphisms be pairs (w,σ)(w, \sigma)(w,σ) where:
* www is the word in generators h,s,ph,s,ph,s,p
* σ\sigmaσ is a trace (e.g., multiset of “marks retained”)

Composition combines traces (e.g., union, monoid addition). That’s a category enriched over a monoid (simple and very practical).

===== Reasonable upgrade B (more categorical): a 2-category =====
* Objects: fiber states (or fibers)
* 1-morphisms: training programs (words in h,s,ph,s,ph,s,p)
* 2-morphisms: rewrites between programs (equivalences / comparisons of histories)

Example: if under some conditions ppp then hhh behaves like hhh then ppp, you’d have a 2-cell

α:(p∘h)⇒(h∘p)\alpha: (p\circ h) \Rightarrow (h\circ p)α:(p∘h)⇒(h∘p)
but you might not always have it—path dependence is exactly “not all such α\alphaα exist”.

This is a clean way to represent “same endpoint, different internal preparedness.”

==== 4) Satellite fusion as a universal construction (pushout-ish story) ====

If you want the “adjoining new nuclei” idea to be more than a slogan, you can model fusion as a gluing operation.

A practical categorical version:
* Treat “nucleus supply” as an object in a resource category R\mathcal{R}R
* There’s an action functor ⊙:M×R→M\odot : \mathcal{M}\times \mathcal{R}\to \mathcal{M}⊙:M×R→M
* Adding one nucleus is M⊙1M\odot 1M⊙1

If you insist on a pushout picture, you need a shared interface object UUU representing compatibility conditions (“membrane docking / differentiation state”), then:

M′≅M⊔USM' \cong M \sqcup_U SM′≅M⊔US
That’s formal, but in biology the correct “interface” isn’t canonical; the action formulation is usually the more reasonable direction.

==== 5) Many fibers at once → monoidal category ====

Muscle is not one fiber; it’s a bundle. So define a way to “combine” fibers.

Let (M,⊗,I)(\mathcal{M}, \otimes, I)(M,⊗,I) be monoidal where:
* objects are fibers (or states)
* M⊗NM\otimes NM⊗N is the “parallel composition” of two fibers
* III is the empty bundle

If you represent a bundle by a multiset of states, then ⊗\otimes⊗ is multiset union.

Training can then be a monoidal functor T:M→MT:\mathcal{M}\to\mathcal{M}T:M→M satisfying:

T(M⊗N)≅T(M)⊗T(N)T(M\otimes N) \cong T(M)\otimes T(N)T(M⊗N)≅T(M)⊗T(N)
when training acts independently per fiber. When there’s systemic coupling (hormones, fatigue), this fails—also informative.

==== 6) Local-in-time effects → a sheaf-y model (the sane version) ====

Training effects are local in time and then “glue” into a long-term adaptation narrative.

Let time be a topological space or just a poset of intervals Int\mathrm{Int}Int ordered by inclusion.

Define a presheaf FFF assigning to each interval UUU:
* F(U)F(U)F(U) = set (or category) of feasible training-response trajectories on UUU

Restriction maps:
* if V⊆UV\subseteq UV⊆U, then F(U)→F(V)F(U)\to F(V)F(U)→F(V) is “forget the outside portion”.

A sheaf condition says: if you have compatible local trajectories on overlapping subintervals, there is a unique glued global trajectory.

Muscle memory breaks uniqueness: you often get multiple global gluings consistent with local observations. That suggests “sheaf of groupoids” or “stack” (multiple consistent global histories).

That’s the useful upshot: memory = non-uniqueness of gluing.

==== 7) Compare to learning systems (brain / institution / AI) in the same formal frame ====

This mapping becomes crisp if you treat all of them as capacity systems with:
* invariants (genome / constitution / architecture)
* state (weights, synapses, policies)
* history dependence (optimizer path, trauma, bureaucracy)

===== One clean common schema =====
* Category S\mathcal{S}S of states
* Endofunctor U:S→SU:\mathcal{S}\to\mathcal{S}U:S→S for “update step”
* A monoid (or category) P\mathcal{P}P of programs/training sequences acting on states
* Invariants via a functor G:S→IG:\mathcal{S}\to\mathcal{I}G:S→I

For AI:
* objects = weight states WWW
* morphisms = training updates (SGD steps)
* invariants = architecture, parameter count, maybe dataset identity
* memory = optimizer path dependence (momentum, Adam moments) = extra hidden state, i.e. enrichment

For brains:
* objects = synaptic configurations
* morphisms = learning events
* memory = consolidation = higher-order structure affecting future morphisms

So: muscle/brain/AI differ mainly in what’s treated as invariant and where history is stored.

==== 8) String-diagram style (text-only, but categorical) ====

If we treat “resources” (protein synthesis capacity, nuclei supply) as wires, you can sketch operations:
* hhh consumes synthesis resources to increase contractile structure
* sss consumes satellite availability to increase nuclei count

Textual string-diagram vibe:

<syntaxhighlight>(nuclei)───●──────→ (nuclei+1)
           s

(capacity)─●──────→ (capacity+1)
           h

</syntaxhighlight>

In a monoidal view, two fibers in parallel are two wires; training is a box acting on each wire.

This becomes fully formal in a PROP or symmetric monoidal category, but the above is the readable essence without pictures.

==== 9) A runnable miniature (toy simulation) — code ====

This is not “category theory software”; it’s a tiny program that implements the generated-category idea (states + primitive moves + history labels). You can run it anywhere Python runs.

<syntaxhighlight lang="python">from dataclasses import dataclass
from typing import List, Tuple, Optional

@dataclass(frozen=True)
class State:
    n: int  # nuclei
    e: int  # expression readiness
    c: int  # contractile capacity

@dataclass(frozen=True)
class Arrow:
    src: State
    dst: State
    word: Tuple[str, ...]   # sequence of generators, e.g. ("p","s","h")
    trace: Tuple[str, ...]  # retained marks / history tokens (toy)

def can_h(s: State, k: int = 2) -> bool:
    # myonuclear-like constraint: capacity can't exceed k*n
    return s.c + 1 <= k * s.n

def step(s: State, gen: str, k: int = 2) -> Optional[State]:
    if gen == "h":
        if not can_h(s, k): return None
        return State(s.n, s.e, s.c + 1)
    if gen == "s":
        return State(s.n + 1, s.e, s.c)
    if gen == "p":
        return State(s.n, s.e + 1, s.c)
    return None

def compose(f: Arrow, g: Arrow) -> Arrow:
    assert f.dst == g.src
    return Arrow(
        src=f.src,
        dst=g.dst,
        word=f.word + g.word,
        trace=f.trace + g.trace
    )

def mk_arrow(src: State, gens: List[str], k: int = 2) -> Optional[Arrow]:
    cur = src
    trace = []
    for gen in gens:
        nxt = step(cur, gen, k)
        if nxt is None:
            return None
        # toy "memory": priming leaves a persistent mark; fusion leaves a mark
        if gen == "p": trace.append("primed")
        if gen == "s": trace.append("extra_nucleus_added")
        cur = nxt
    return Arrow(src, cur, tuple(gens), tuple(trace))

=== Demo: two different paths to same endpoint (sometimes) but different traces ===
M0 = State(n=1, e=0, c=1)

path_A = mk_arrow(M0, ["p", "s", "h", "h"], k=2)
path_B = mk_arrow(M0, ["s", "p", "h", "h"], k=2)

print("A:", path_A)
print("B:", path_B)
print("Same dst?", (path_A is not None and path_B is not None and path_A.dst == path_B.dst))
print("Different trace?", (path_A is not None and path_B is not None and path_A.trace != path_B.trace))

</syntaxhighlight>

What you’ll see:
* Sometimes both paths reach the same (n,e,c)(n,e,c)(n,e,c) (endpoint equality)
* Their trace can differ (history dependence / “memory”)

That’s the computational analogue of “2-morphisms vs plain morphisms”: same destination, distinct provenance.

==== Where to go next (still “reasonable”) ====

If you pick one deepening, the cleanest is:
* Make P\mathcal{P}P the free monoid on {h,s,p}\{h,s,p\}{h,s,p} (training programs)
* Make an action P↷M\mathcal{P}\curvearrowright \mathcal{M}P↷M
* Then memory is either: - enlarge the state space to include optimizer-like hidden variables, or - treat paths as first-class and add 2-cells (rewrites)

If you say “choose for me,” I’d choose the action + enriched-state route first (it’s closest to how real systems store history), then the 2-category for the conceptual clarity.

If you want, I can write the same ladder in tight LaTeX notation as a single “mini-paper” (definitions + propositions) without adding extra bulk.