Discreteness of Recognition States

1. Setting

A continuous substrate cannot host an isolated stable recognition zero. The discreteness proof is one of the earliest non-negotiable steps in the forcing chain.

2. Equations

(E1)

$$ J_{\log}(t)=\cosh(t)-1\ge 0 $$

Nonnegative log-coordinate J-cost.

3. Prediction or structural target

• stable substrate: predicted discrete (dimensionless); empirical formal theorem target. Source: Foundation.DiscretenessForcing

This entry is one of the marquee derivations. The numerical or formal target is explicit, and the falsifier identifies the failure mode.

4. Formal anchor

The primary anchor is Foundation.DiscretenessForcing..discreteness_forced.

    (J(x) = (x + 1/x)/2 - 1 requires x ≠ 0, and for x < 0, J(x) < 0 ≠ defect minimum). -/
theorem discreteness_forced :
    (∀ x : ℝ, 0 < x → defect x = 0 → x = 1) ∧  -- Unique minimum
    (∀ ε > 0, ∃ y : ℝ, y ≠ 1 ∧ defect y < ε) →  -- No isolation in ℝ
    ¬∃ (x : ℝ), 0 < x ∧ x ≠ 1 ∧ defect x = 0 := by      -- No other stable points
  intro ⟨hunique, _hno_isolation⟩
  push_neg
  intro x hx_pos hx_ne hdef
  exact hx_ne (hunique x hx_pos hdef)

5. What is inside the Lean module

Key theorems:

• J_log_zero
• J_log_nonneg
• J_log_eq_zero_iff
• J_log_pos
• J_log_symmetric
• J_log_eq_J_exp
• J_log_second_deriv_at_zero
• cosh_quadratic_bound
• J_log_quadratic_approx
• continuous_no_isolated_zero_defect
• continuous_space_no_lockIn
• discrete_minimum_stable

Key definitions:

• J_log
• IsStable
• DiscreteConfigSpace
• RSExists_stable

6. Derivation chain

• discreteness_forced - Discreteness forced
• stable_existence_requires_discrete - Stable existence requires discrete
• discreteness_forcing_principle - Forcing principle

7. Falsifier

A continuous configuration space with an isolated stable zero-defect recognition state refutes the discreteness forcing theorem.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

9. Reading note

The minimal way to audit this page is to open the first Lean anchor and then walk the supporting declarations listed above. If the primary theorem is a module-level anchor, the key theorems section names the internal declarations that carry the mathematical load. This keeps the public derivation readable without severing it from the proof object.

10. Audit path

To audit discreteness-of-recognition, start with the primary Lean anchor Foundation.DiscretenessForcing.discreteness_forced. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.