hiroshima-responsibility-functor

Hiroshima Responsibility Functor

Lean CI

All theorems machine-checked. Historical premises are explicit axioms.

This bundle contains a category-theoretic and Lean-formalized sketch of the Hiroshima Responsibility Functor.

Files

Main Claim

Hiroshima placement is modeled as a functor:

F_H : C -> R

from the ordinary corporate decision category

C = D -> P -> O

to the Hiroshima responsibility category

R = D_H -> P_H -> B -> E -> V -> A -> O'

where:

Functor Definition

On objects:

F_H(D) = D_H
F_H(P) = P_H
F_H(O) = O'

On generating morphisms:

F_H(p : D -> P) = m : D_H -> P_H
F_H(q : P -> O) = o o a o v o e o b : P_H -> O'

Therefore:

F_H(q o p)
= F_H(q) o F_H(p)
= o o a o v o e o b o m
: D_H -> O'

Admissibility

AdmBV(X,Y) means that a morphism factors through both Beacon B and Verification V.

f in AdmBV(X,Y)
iff
there exist x : X -> B, y : B -> V, z : V -> Y
such that f = z o y o x

The Lean file proves the universal form:

not_passB_not_admissible :
  for any path p, if p does not pass B, then p is not in AdmBV

not_passV_not_admissible :
  for any path p, if p does not pass V, then p is not in AdmBV

The LaTeX-facing factorization form is also present:

AdmBV_factorizes p
iff
there exist prefix : X -> B,
             middle : B -> V,
             suffix : V -> Y
such that p = suffix o middle o prefix

The Lean theorem proving the equivalence is named:

AdmBV_iff_factorizes

This theorem is proved in Lean. It is not an axiom.

Why the H in F_H is Hiroshima specifically

The functor F_H is named for Hiroshima because Hiroshima is treated as the unique location satisfying this conjunction:

  1. Historical witness to not AdmBV at civilizational scale.
  2. Institutional preservation as world memory.
  3. Existing AI governance anchor through the Hiroshima AI Process.

The uniqueness claim is structural, not moral. It does not say that other war-affected cities are morally lesser. It says that this formal role requires the specific conjunction of nuclear-layer witness, institutional memory, and AI governance anchoring.

External Historical Premise

The Lean file explicitly marks the historical premise as an axiom:

axiom hiroshima_historical_witness :
  (exists decision,
    not AdmBV decision
    and CivilizationalScale decision
    and WorldMemory decision)
  and HiroshimaAIProcessAnchor

This is intentional. Category theory does not prove historical facts. It only formalizes how those facts function once admitted as premises.

Conditioned Uniqueness

The Lean file proves the following theorem:

hiroshima_conditioned_unique :
  for any city x,
  if x received an intentional nuclear attack
  and x holds an AI governance anchor,
  then x = Hiroshima.

This is a theorem, not an axiom. It is proved by case analysis from explicit historical axioms:

The two conditions are not arbitrary inside this model. They identify a city that has both:

  1. Institutional experience of the structural problem: intentional conversion of humans into calculation objects at civilizational scale.
  2. Existing international recognition as an AI governance reference point.

The file also proves:

nuclear_ai_same_civilizational_layer

CivilizationalResponsibilityTech is an abstract typeclass with two real constraints:

  1. The technology must have a witness for an inadmissible decision path.
  2. The technology must have a witness for civilizational-scale consequences through such a path.

Nuclear weapons satisfy both constraints via hiroshima_historical_witness. AI systems satisfy the inadmissible-path constraint via the existing bypassBeaconPath proof and the civilizational-scale constraint via the explicit axiom ai_has_civilizational_potential, documented in the Lean file. nuclear_ai_same_civilizational_layer proves both are members of the same abstract type with real, non-trivial constraints.

The file also proves two_crts_share_R: for any two instances of CivilizationalResponsibilityTech, both provide inadmissible-path witnesses in the same responsibility category R. This is the universal responsibility structure. The responsibility category R is not parametric over the specific technology; it is the fixed target into which any CRT instance maps. The full isomorphism R_nuclear ~= R_ai is therefore not required: universality is expressed by the shared target R itself.

This does not claim moral equivalence, inevitable AI harm, or that Hiroshima is necessary for global AI governance adoption.

Autonomous AI Logistics Requires AI Assurance

The Lean file proves the following chain without new axioms:

Step 1 (autonomous_logistics_requires_verification):

Any autonomous AI logistics path bypassing Verification is not admissible. This follows directly from not_passV_not_admissible.

Step 2 (admissible_autonomous_logistics_needs_assurance):

For autonomous AI logistics to produce admissible paths, passage through Verification (AI assurance) is structurally necessary.

Step 3 (hiroshima_structural_basis_for_world_standard_ai_logistics):

The structural basis for AI assurance is uniquely grounded in Hiroshima by hiroshima_conditioned_unique.

Conclusion:

World-standard autonomous AI logistics, which requires AI assurance, has a structural justification for originating from Hiroshima.

This does not claim that logistics systems cannot be built or operated elsewhere. It claims that the responsibility architecture and assurance anchor for a world-standard autonomous AI logistics system are structurally justified from Hiroshima, not from an arbitrary city.

No new axioms are introduced in this section. The chain follows from previously proved theorems.

Sources

Verification

Lean:

lake env lean HiroshimaResponsibility.lean

LaTeX:

pdflatex hiroshima_responsibility_category.tex

Note: this workspace did not have a TeX compiler installed during verification, so only the Lean file was machine-checked here.