Foundation Pair Kernel Scale Covariant Observables S20 Classified Responses Dist

Distinguishable channels

The declaration classifiedResponses_distinguish_iff_parentInjective states a precise equivalence about how a physical system responds to events. In plain language, it says that a system can tell apart every possible input event from every other one, meaning each event produces a unique response, if and only if a certain structural map, called the parent response, is injective. A map is injective when it never sends two different inputs to the same output. So the theorem ties an operational idea, distinguishability, to a mathematical property, injectivity.

The framework models a physical process as a ledger, a discrete record of events, where each event carries a configuration of parent responses. The declaration shows that if you can always tell which event occurred by looking at the responses, then the underlying parent response structure must also be injective, and vice versa. This is not a claim about any particular physical system. It is a structural theorem: within the framework's model, the two notions, operational distinguishability and structural injectivity, are exactly the same condition.

The statement does not claim that any real system actually achieves this distinguishability. Whether a concrete physical carrier, like a particle or a field, realizes these responses is a separate question. The framework's library proves the structural equivalence, but the physical realization of those responses remains a hypothesis, not a theorem. The declaration also does not claim that distinguishability is always possible; it only says that when it holds, it is equivalent to injectivity.

What this means in practice is that the framework gives a clean mathematical handle on a physical idea. Instead of asking separately whether a system is distinguishable and whether its parent structure is injective, you can check one and get the other for free. This is a useful bridge between the operational language of experiments and the structural language of the framework's mathematics.

theorem parentTagResponse3_injective
    {N : ℕ} [NeZero N]
    {event : RealizedPostingEvent3 N} :
    Function.Injective
      (@parentTagResponse3 N _ event) := by
  intro left right h
  cases left with
  | inl leftAxis =>
      cases right with
      | inl rightAxis =>
          simp [parentTagResponse3] at h
          exact congrArg Sum.inl h
      | inr rightKind =>
          simp [parentTagResponse3] at h
  | inr leftKind =>
      cases right with
      | inl rightAxis =>
          simp [parentTagResponse3] at h
      | inr rightKind =>
          simp [parentTagResponse3] at h
          exact congrArg Sum.inr h

What this page does not claim

No real physical system is claimed to actually achieve this distinguishability. The physical realization of the parent responses is not claimed to be a theorem.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PairKernelScaleCovariantObservablesS20.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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