Astrophysics Supernova Nucleo From Phi Ladder

Supernova yields and the phi ladder

A core-collapse supernova releases about 3×10^53 ergs, mostly as neutrinos, with a luminosity peak near 10^53 ergs per second lasting about ten seconds. In Recognition Science, that ten-second window is modeled as phi^5 times a base tick of one second, where phi is the golden ratio, roughly 1.618. The idea is that the explosion's cooling timescale aligns with a ladder of powers of phi, a scale that the framework derives from its fundamental cost function rather than choosing freely.

The framework's machine-checked library of formal theorems proves three general facts about its cost function J(x) = (x + 1/x)/2 - 1. First, the cost vanishes when the two inputs are equal: J(1) = 0. Second, for positive inputs the cost is never negative. Third, the threshold phi - 3/2 is positive, which follows from phi being greater than 1.5. These are clean, verified statements, but they hold for any positive ratio m/e, not specifically for supernova masses or energies.

In Recognition Science, the module defines a domain cost as J(m/e), where m and e are real numbers, and bundles the three theorems into a certificate structure. The certificate proves that the cost function behaves well on positive inputs, but it does not define m as a stellar mass or e as an energy. The research note attached to the module records the intended astrophysical interpretation, but the Lean code itself contains no supernova physics. The module is a template, shared verbatim with over two thousand sibling modules, waiting for a subject-specific definition of m and e.

The plain-language takeaway: the framework proves a general cost function's basic properties, and it suggests a phi-ladder timescale for supernova cooling, but it does not yet derive a nucleosynthesis yield or a specific energy from stellar conditions. The gap is explicit: a theorem about supernovas would require defining m and e in stellar terms. That step remains open.

structure SupernovaYieldCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

noncomputable def cert : SupernovaYieldCert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

structure SupernovaYieldCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

What this page does not claim

The module does not prove any supernova-specific yield or energy value. The phi-ladder timescale is a model, not a measured or derived result. No claim is made that the golden ratio governs supernova physics outside the framework's modeling choice.

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/Astrophysics/SupernovaNucleoFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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