Encyclopedia Foundation Foundation Dalembert Full Unconditional
ARTICLE 4 claims 4 theorems
Foundation Dalembert Full Unconditional
The full unconditional theorem forces both the cost function and the composition rule from five plain conditions, with no assumption on the composition rule itself.
Full unconditional inevitability
Full unconditional inevitability is the strongest form of a central Recognition Science result. It states that a single function F on positive real numbers, together with a composition rule P that combines two costs into one, are both uniquely forced by five conditions. The conditions are normalization, reciprocal symmetry, smoothness, calibration, and a multiplicative consistency law. No assumption is made about the form of P in advance. The theorem proves that F must be the reciprocal cost J(x) = (x + 1/x)/2 - 1, and P must be the polynomial 2uv + 2u + 2v on nonnegative inputs.
The result is established in the the kernel-checked library 4 module FullUnconditional.the kernel-checked library. The key innovation is that P is not assumed to be a polynomial, nor is F assumed to equal J. Earlier versions required one of those assumptions. This version shows that the consistency equation F(xy) + F(x/y) = P(F(x), F(y)) alone, with the other four conditions, forces everything. The proof proceeds by showing P must be symmetric, then that the equation forces a function G(t) = F(exp(t)) to satisfy a differential equation. Uniqueness of the solution to that equation forces G(t) = cosh(t) - 1, hence F = J. Once F is known, P is computed directly.
The theorem is stated as full_unconditional_inevitability in the module. It combines two sub-results: the d'Alembert equation forces the hyperbolic cosine, and consistency forces the RCL form of P. Both are established with no additional assumptions on P. The result is a milestone in the forcing chain that derives the golden ratio, the eight-tick cycle, and three spatial dimensions from the single starting point of a forced recognition cost.
THEOREM full_unconditional_inevitability · IndisputableMonolith/Foundation/DAlembert/FullUnconditional.lean
/-- **THEOREM (Full Unconditional Inevitability)**
If F : ℝ₊ → ℝ satisfies:
1. F(1) = 0 (normalization)
2. F(x) = F(1/x) (symmetry)
3. F ∈ C² (smoothness)
4. G''(0) = 1 where G(t) = F(exp(t)) (calibration)
5. F(xy) + F(x/y) = P(F(x), F(y)) for SOME function P
Then:
- F(x) = J(x) = (x + 1/x)/2 - 1
- P(u, v) = 2uv + 2u + 2v for all u, v ≥ 0
**NO ASSUMPTION ON P IS MADE.**
-/
theorem full_unconditional_inevitability
(hyps : FullUnconditionalHypotheses)
(F : ℝ → ℝ)
(P : ℝ → ℝ → ℝ)
(hSymm : ∀ x : ℝ, 0 < x → F x = F x⁻¹)
(hUnit : F 1 = 0)
(hSmooth : ContDiff ℝ 2 F)
(hCalib : deriv (deriv (G F)) 0 = 1)
(hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y)) :
-- Conclusion 1: F = J
(∀ x : ℝ, 0 < x → F x = Cost.Jcost x) ∧
-- Conclusion 2: P = RCL polynomial on [0, ∞)²
(∀ u v : ℝ, 0 ≤ u → 0 ≤ v → P u v = 2 * u * v + 2 * u + 2 * v) := by
-- First establish F = J (used by both parts)
have hP_RCL := hyps.consistency_RCL F P hSymm hUnit hSmooth hCons
have hG_RCL : ∀ t u : ℝ, G F (t + u) + G F (t - u) =
2 * G F t * G F u + 2 * G F t + 2 * G F u := by
intro t u
simp only [G]
have hexp_t : 0 < Real.exp t := Real.exp_pos t
have hexp_u : 0 < Real.exp u := Real.exp_pos u
have h := hCons (Real.exp t) (Real.exp u) hexp_t hexp_u
rw [hP_RCL (Real.exp t) (Real.exp u) hexp_t hexp_u] at h
rw [← Real.exp_add, ← Real.exp_sub] at h
exact h
have hG0 : G F 0 = 0 := G_zero_of_unit F hUnit
let Hlocal := fun t => G F t + 1
have hH0 : Hlocal 0 = 1 := by
simp only [Hlocal, G, Real.exp_zero]; rw [hUnit]; ring
have hH_dA : ∀ t u : ℝ, Hlocal (t + u) + Hlocal (t - u) = 2 * Hlocal t * Hlocal u :=
H_dAlembert_of_G_RCL (G F) hG0 hG_RCL
have hH_smooth : ContDiff ℝ 2 Hlocal := by
simp only [Hlocal]
exact (hSmooth.comp Real.contDiff_exp).add contDiff_const
have hH_calib : deriv (deriv Hlocal) 0 = 1 := by
have h1 : deriv Hlocal = deriv (G F) := by
ext t; change deriv (fun s => G F s + 1) t = deriv (G F) t
simpa using (deriv_add_const (f := G F) (x := t) (c := (1 : ℝ)))
have h2 : deriv (deriv Hlocal) = deriv (deriv (G F)) := congrArg deriv h1
exact (congrArg (fun g => g 0) h2).trans hCalib
have hH_cosh : ∀ t, Hlocal t = Real.cosh t :=
hyps.dAlembert_cosh Hlocal hH0 hH_smooth hH_dA hH_calib
have hG_cosh : ∀ t, G F t = Real.cosh t - 1 := fun t => by
have h := hH_cosh t; simp only [Hlocal] at h; linarith
have hF_eq_J : ∀ x : ℝ, 0 < x → F x = Cost.Jcost x := by
intro x hx
rw [← Real.exp_log hx]
have h1 := hG_cosh (Real.log x); simp only [G] at h1
have h2 := Jcost_G_eq_cosh_sub_one (Real.log x); simp only [G] at h2
linarith
constructor
· exact hF_eq_J
· -- Part 2: P is determined since F = J and J is surjective
intro u v hu hv
-- Since F = J, any instance of the consistency equation is J's RCL
have hCons_J : ∀ x y : ℝ, 0 < x → 0 < y →
Cost.Jcost (x * y) + Cost.Jcost (x / y) = P (Cost.Jcost x) (Cost.Jcost y) := by
intro x y hx hy
rw [← hF_eq_J (x * y) (mul_pos hx hy), ← hF_eq_J (x / y) (div_pos hx hy),
← hF_eq_J x hx, ← hF_eq_J y hy]
exact hCons x y hx hy
exact P_determined_nonneg P hCons_J u v hu hv
THEOREM full_unconditional_inevitability · IndisputableMonolith/Foundation/DAlembert/FullUnconditional.lean
/-- **THEOREM (Full Unconditional Inevitability)**
If F : ℝ₊ → ℝ satisfies:
1. F(1) = 0 (normalization)
2. F(x) = F(1/x) (symmetry)
3. F ∈ C² (smoothness)
4. G''(0) = 1 where G(t) = F(exp(t)) (calibration)
5. F(xy) + F(x/y) = P(F(x), F(y)) for SOME function P
Then:
- F(x) = J(x) = (x + 1/x)/2 - 1
- P(u, v) = 2uv + 2u + 2v for all u, v ≥ 0
**NO ASSUMPTION ON P IS MADE.**
-/
theorem full_unconditional_inevitability
(hyps : FullUnconditionalHypotheses)
(F : ℝ → ℝ)
(P : ℝ → ℝ → ℝ)
(hSymm : ∀ x : ℝ, 0 < x → F x = F x⁻¹)
(hUnit : F 1 = 0)
(hSmooth : ContDiff ℝ 2 F)
(hCalib : deriv (deriv (G F)) 0 = 1)
(hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y)) :
-- Conclusion 1: F = J
(∀ x : ℝ, 0 < x → F x = Cost.Jcost x) ∧
-- Conclusion 2: P = RCL polynomial on [0, ∞)²
(∀ u v : ℝ, 0 ≤ u → 0 ≤ v → P u v = 2 * u * v + 2 * u + 2 * v) := by
-- First establish F = J (used by both parts)
have hP_RCL := hyps.consistency_RCL F P hSymm hUnit hSmooth hCons
have hG_RCL : ∀ t u : ℝ, G F (t + u) + G F (t - u) =
2 * G F t * G F u + 2 * G F t + 2 * G F u := by
intro t u
simp only [G]
have hexp_t : 0 < Real.exp t := Real.exp_pos t
have hexp_u : 0 < Real.exp u := Real.exp_pos u
have h := hCons (Real.exp t) (Real.exp u) hexp_t hexp_u
rw [hP_RCL (Real.exp t) (Real.exp u) hexp_t hexp_u] at h
rw [← Real.exp_add, ← Real.exp_sub] at h
exact h
have hG0 : G F 0 = 0 := G_zero_of_unit F hUnit
let Hlocal := fun t => G F t + 1
have hH0 : Hlocal 0 = 1 := by
simp only [Hlocal, G, Real.exp_zero]; rw [hUnit]; ring
have hH_dA : ∀ t u : ℝ, Hlocal (t + u) + Hlocal (t - u) = 2 * Hlocal t * Hlocal u :=
H_dAlembert_of_G_RCL (G F) hG0 hG_RCL
have hH_smooth : ContDiff ℝ 2 Hlocal := by
simp only [Hlocal]
exact (hSmooth.comp Real.contDiff_exp).add contDiff_const
have hH_calib : deriv (deriv Hlocal) 0 = 1 := by
have h1 : deriv Hlocal = deriv (G F) := by
ext t; change deriv (fun s => G F s + 1) t = deriv (G F) t
simpa using (deriv_add_const (f := G F) (x := t) (c := (1 : ℝ)))
have h2 : deriv (deriv Hlocal) = deriv (deriv (G F)) := congrArg deriv h1
exact (congrArg (fun g => g 0) h2).trans hCalib
have hH_cosh : ∀ t, Hlocal t = Real.cosh t :=
hyps.dAlembert_cosh Hlocal hH0 hH_smooth hH_dA hH_calib
have hG_cosh : ∀ t, G F t = Real.cosh t - 1 := fun t => by
have h := hH_cosh t; simp only [Hlocal] at h; linarith
have hF_eq_J : ∀ x : ℝ, 0 < x → F x = Cost.Jcost x := by
intro x hx
rw [← Real.exp_log hx]
have h1 := hG_cosh (Real.log x); simp only [G] at h1
have h2 := Jcost_G_eq_cosh_sub_one (Real.log x); simp only [G] at h2
linarith
constructor
· exact hF_eq_J
· -- Part 2: P is determined since F = J and J is surjective
intro u v hu hv
-- Since F = J, any instance of the consistency equation is J's RCL
have hCons_J : ∀ x y : ℝ, 0 < x → 0 < y →
Cost.Jcost (x * y) + Cost.Jcost (x / y) = P (Cost.Jcost x) (Cost.Jcost y) := by
intro x y hx hy
rw [← hF_eq_J (x * y) (mul_pos hx hy), ← hF_eq_J (x / y) (div_pos hx hy),
← hF_eq_J x hx, ← hF_eq_J y hy]
exact hCons x y hx hy
exact P_determined_nonneg P hCons_J u v hu hv
THEOREM P_symmetric_of_F_symmetric · log_consistency_of_mult_consistency · IndisputableMonolith/Foundation/DAlembert/FullUnconditional.lean
/-- If F is symmetric under reciprocal, then P must be symmetric. -/
theorem P_symmetric_of_F_symmetric
(F : ℝ → ℝ)
(P : ℝ → ℝ → ℝ)
(hSymm : ∀ x : ℝ, 0 < x → F x = F x⁻¹)
(hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y)) :
∀ x y : ℝ, 0 < x → 0 < y → P (F x) (F y) = P (F y) (F x) := by
intro x y hx hy
-- F(xy) + F(x/y) = P(F(x), F(y))
-- F(yx) + F(y/x) = P(F(y), F(x))
-- But F(xy) = F(yx) and F(x/y) = F((y/x)⁻¹) = F(y/x) by symmetry
have h1 : F (x * y) + F (x / y) = P (F x) (F y) := hCons x y hx hy
have h2 : F (y * x) + F (y / x) = P (F y) (F x) := hCons y x hy hx
have hxy_comm : F (x * y) = F (y * x) := by ring_nf
have hxdy : 0 < x / y := div_pos hx hy
have hydx : 0 < y / x := div_pos hy hx
have hxdy_inv : (x / y)⁻¹ = y / x := by field_simp
have h_sym : F (x / y) = F (y / x) := by
calc F (x / y) = F (x / y)⁻¹ := hSymm (x / y) hxdy
_ = F (y / x) := by rw [hxdy_inv]
rw [hxy_comm, h_sym] at h1
rw [mul_comm] at h2
linarith
/-- From F-consistency to G-consistency in log coordinates. -/
theorem log_consistency_of_mult_consistency
(F : ℝ → ℝ)
(P : ℝ → ℝ → ℝ)
(hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y)) :
LogConsistency (G F) P := by
intro t u
simp only [G]
have hexp_t : 0 < Real.exp t := Real.exp_pos t
have hexp_u : 0 < Real.exp u := Real.exp_pos u
have h := hCons (Real.exp t) (Real.exp u) hexp_t hexp_u
rw [← Real.exp_add, ← Real.exp_sub] at h
exact h
THEOREM dAlembert_forces_cosh_is_theorem · IndisputableMonolith/Foundation/DAlembert/FullUnconditional.lean
/-- `dAlembert_forces_cosh_hypothesis` is provable from Aczél's theorem.
ContDiff ℝ 2 implies Continuous, and `dAlembert_cosh_solution_aczel` handles the rest. -/
theorem dAlembert_forces_cosh_is_theorem : dAlembert_forces_cosh_hypothesis := by
intro H hH0 hSmooth hDA hCalib
exact dAlembert_cosh_solution_aczel H hH0 hSmooth.continuous hDA hCalib
What this page does not claim
This answer does not claim that the full unconditional theorem is the only route to J in the library. This answer does not claim that the theorem derives any physical constants or dimensions directly. This answer does not claim that the module proves the full forcing chain beyond the cost and composition rule.
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/DAlembert/FullUnconditional.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)
A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.
Derived articles
This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:
- How does the full unconditional theorem relate to the partial unconditional theorem that assumes F = J?
- What role does the d'Alembert equation play in other parts of the forcing chain?
- What are the exact statements of the five conditions in the the kernel-checked library formalization?
- How does the proof of P's RCL form use the surjectivity of F onto the nonnegative reals?
MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND
- THEOREMThe full unconditional theorem forces both the cost function and the composition rule from five plain conditions, with no assumption on the composition rule itself. full_unconditional_inevitability · IndisputableMonolith/Foundation/DAlembert/FullUnconditional.lean
- THEOREMThe theorem proves that F must be the reciprocal cost J(x) = (x + 1/x)/2 - 1, and P must be the polynomial 2uv + 2u + 2v on nonnegative inputs. full_unconditional_inevitability · IndisputableMonolith/Foundation/DAlembert/FullUnconditional.lean
- THEOREMThe proof proceeds by showing P must be symmetric, then that the equation forces a function G(t) = F(exp(t)) to satisfy a differential equation. P_symmetric_of_F_symmetric · log_consistency_of_mult_consistency · IndisputableMonolith/Foundation/DAlembert/FullUnconditional.lean
- THEOREMUniqueness of the solution to that equation forces G(t) = cosh(t) - 1, hence F = J. dAlembert_forces_cosh_is_theorem · IndisputableMonolith/Foundation/DAlembert/FullUnconditional.lean