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Wave 8.1: A4Bridge.v — a4 bridge analysis, Boundary Theorem BT-8 #47

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373 changes: 373 additions & 0 deletions proofs/trinity/A4Bridge.v
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(* ============================================================================ *)
(* A4Bridge.v *)
(* ============================================================================ *)
(* *)
(* Wave 8.1: Bridge or Boundary for the a_4 Gap Problem *)
(* *)
(* OUTCOME: Boundary Theorem BT-8 *)
(* *)
(* QUESTION: Is there an exact H4 integer invariant k_0 such that *)
(* Trinity_a4 = k_0 * Coq_a4? *)
(* *)
(* ANSWER: No. The conversion factor k = (704 + 192*sqrt(5))/19 is irrational.*)
(* It equals no standard H4/600-cell integer invariant. Specifically, k is *)
(* strictly between 59 and 60. The nearest integer candidate 60 (= 2*h(H4) *)
(* = sum of Coxeter exponents = cells_600/10) is ruled out by a certified *)
(* numerical bound (interval arithmetic, precision 30). *)
(* *)
(* Structure: *)
(* §1 Definitions: Coq_a4, Trinity_a4, k_ratio *)
(* §2 H4 candidate invariants *)
(* §3 Helper lemmas (phi algebra, mirroring SpectralAction600Cell.v) *)
(* §4 Algebraic bridge identity *)
(* §5 Rationalized form: k = (704+192*sqrt(5))/19 *)
(* §6 Numerical bounds 59 < k < 60 (interval tactic) *)
(* §7 BT-8: no H4 integer equals k *)
(* §8 Positive corollary: k is phi-algebraic *)
(* §9 Summary comment *)
(* *)
(* Dependencies: Reals, Lra, Interval.Tactic *)
(* *)
(* Relation to existing proofs: *)
(* A4Conversion.v proves the algebraic identity Trinity_a4 = k * Coq_a4. *)
(* This file establishes the NEGATIVE result: k is not any H4 integer. *)
(* Together: algebraic bridge exists, but its scale factor has no H4-integer *)
(* interpretation — Boundary Theorem BT-8. *)
(* *)
(* Wave 8.1 (2026-06-23) *)
(* ============================================================================ *)

Require Import Reals.
Require Import Lra.
From Interval Require Import Tactic.
Open Scope R_scope.

(* ============================================================================ *)
(* §1 Definitions *)
(* ============================================================================ *)

(* Golden ratio *)
Definition phi : R := (1 + sqrt 5) / 2.

(* Coq a_4: heat kernel coefficient for S^3 with radius phi *)
(* Proven in SpectralAction600Cell.v as a4_total = a4_simplified *)
Definition Coq_a4 : R := (5 + 6 * phi) / (16 * phi).

(* Trinity a_4: H4 invariant coefficient *)
(* From HiggsPrediction.v: a4_600cell = (2*phi)^3 = 8*phi^3 *)
Definition Trinity_a4 : R := 8 * phi ^ 3.

(* Conversion factor = Trinity_a4 / Coq_a4 *)
(* k = 128*phi^4 / (5+6*phi) = (704+192*sqrt(5))/19 ≈ 59.649 *)
Definition k_ratio : R := 128 * phi ^ 4 / (5 + 6 * phi).

(* ============================================================================ *)
(* §2 H4 Candidate Integer Invariants *)
(* ============================================================================ *)

(* H4 Coxeter number h(H4) = 30 *)
Definition Coxeter_h_H4 : R := 30.

(* 2*h(H4) = 60 — nearest integer to k_ratio ≈ 59.649 *)
(* Also equals: sum of Coxeter exponents 1+11+19+29, cells_600/10, edges/12 *)
Definition double_h_H4 : R := 60.

(* Sum of H4 Coxeter exponents: 1+11+19+29 = 60 *)
Definition sum_exponents_H4 : R := 60.

(* 600-cell combinatorics *)
Definition cells_600 : R := 600.
Definition vertices_600 : R := 120.
Definition edges_600 : R := 720.

(* ============================================================================ *)
(* §3 Helper Lemmas *)
(* ============================================================================ *)

Lemma sqrt5_pos : 0 < sqrt 5.
Proof. apply sqrt_lt_R0. lra. Qed.

Lemma sqrt5_sq : sqrt 5 * sqrt 5 = 5.
Proof. apply Rsqr_sqrt. lra. Qed.

Lemma phi_pos : 0 < phi.
Proof.
unfold phi.
assert (0 < sqrt 5) by apply sqrt5_pos. lra.
Qed.

Lemma phi_neq0 : phi <> 0.
Proof. apply Rgt_not_eq. apply phi_pos. Qed.

Lemma denom_pos : 0 < 5 + 6 * phi.
Proof.
assert (0 < phi) by apply phi_pos. lra.
Qed.

Lemma denom_neq0 : 5 + 6 * phi <> 0.
Proof. apply Rgt_not_eq. apply denom_pos. Qed.

(* phi^2 = phi + 1 — standard identity for golden ratio *)
Lemma phi_sq : phi * phi = phi + 1.
Proof.
unfold phi.
assert (H: sqrt 5 * sqrt 5 = 5) by apply sqrt5_sq.
nra.
Qed.

(* phi^4 = 3*phi + 2 — derived from phi^2 = phi+1 *)
Lemma phi_fourth : phi ^ 4 = 3 * phi + 2.
Proof.
assert (H2: phi ^ 2 = phi + 1).
{ replace (phi ^ 2) with (phi * phi) by ring. apply phi_sq. }
assert (H3: phi ^ 3 = phi * phi ^ 2) by ring.
rewrite H2 in H3.
replace (phi * (phi + 1)) with (phi ^ 2 + phi) in H3 by ring.
rewrite H2 in H3.
replace (phi + 1 + phi) with (2 * phi + 1) in H3 by ring.
assert (H4: phi ^ 4 = phi * phi ^ 3) by ring.
rewrite H3 in H4.
replace (phi * (2 * phi + 1)) with (2 * phi ^ 2 + phi) in H4 by ring.
rewrite H2 in H4.
replace (2 * (phi + 1) + phi) with (3 * phi + 2) in H4 by ring.
apply H4.
Qed.

(* 128 * phi^4 = 448 + 192 * sqrt 5 *)
Lemma phi_fourth_scaled : 128 * phi ^ 4 = 448 + 192 * sqrt 5.
Proof.
rewrite phi_fourth.
unfold phi.
assert (H1: 128 * (3 * ((1 + sqrt 5) / 2) + 2) = 192 + 192 * sqrt 5 + 256)
by (field; lra).
rewrite H1. field; lra.
Qed.

(* 5 + 6*phi = 8 + 3*sqrt 5 *)
Lemma denom_simplified : 5 + 6 * phi = 8 + 3 * sqrt 5.
Proof.
unfold phi. field_simplify. lra.
Qed.

Lemma denom_8_3s5_neq0 : (8 + 3 * sqrt 5) <> 0.
Proof.
assert (0 < sqrt 5) by apply sqrt5_pos. lra.
Qed.

(* ============================================================================ *)
(* §4 Algebraic Bridge Identity *)
(* ============================================================================ *)

(* Formulation for BT-8 main theorem *)
Lemma ratio_value :
k_ratio = 8 * phi ^ 3 * (16 * phi) / (5 + 6 * phi).
Proof.
unfold k_ratio.
assert (H: 128 * phi ^ 4 = 8 * phi ^ 3 * (16 * phi)) by ring.
rewrite H. reflexivity.
Qed.

(* The algebraic bridge: Trinity_a4 = k_ratio * Coq_a4 *)
(* Pure algebra: k * ((5+6phi)/(16phi)) = 128*phi^4/(5+6phi) * (5+6phi)/(16phi)*)
(* = 128*phi^4/(16*phi) = 8*phi^3 = Trinity_a4 *)
Theorem a4_algebraic_bridge :
Trinity_a4 = k_ratio * Coq_a4.
Proof.
unfold Trinity_a4, k_ratio, Coq_a4.
field.
split; [apply phi_neq0 | apply denom_neq0].
Qed.

(* ============================================================================ *)
(* §5 Rationalized Form *)
(* ============================================================================ *)

(* k_ratio = (704 + 192 * sqrt 5) / 19 *)
(* *)
(* Derivation (manual): *)
(* 128*phi^4 = 448 + 192*sqrt5 (phi_fourth_scaled) *)
(* 5+6*phi = 8 + 3*sqrt5 (denom_simplified) *)
(* Rationalize: multiply by (8-3*sqrt5)/(8-3*sqrt5) *)
(* Num: (448+192*sqrt5)(8-3*sqrt5) = 3584+1536*sqrt5-1344*sqrt5-576*5 *)
(* = 3584 + 192*sqrt5 - 2880 = 704 + 192*sqrt5 *)
(* Den: (8+3*sqrt5)(8-3*sqrt5) = 64 - 9*5 = 64 - 45 = 19 *)
Theorem k_ratio_rationalized :
k_ratio = (704 + 192 * sqrt 5) / 19.
Proof.
unfold k_ratio.
rewrite phi_fourth_scaled.
rewrite denom_simplified.
apply Rmult_eq_reg_r with (8 + 3 * sqrt 5).
2: apply denom_8_3s5_neq0.
apply Rmult_eq_reg_r with 19.
2: lra.
field_simplify.
all: try ring_simplify; try rewrite pow2_sqrt by lra; try ring.
all: try apply denom_8_3s5_neq0; try lra.
Qed.

(* ============================================================================ *)
(* §6 Numerical Bounds: 59 < k_ratio < 60 *)
(* ============================================================================ *)

(* Certified by interval arithmetic (Coq.Interval library, precision 30 bits) *)
Lemma k_ratio_gt_59 : 59 < k_ratio.
Proof.
unfold k_ratio.
interval with (i_prec 30).
Qed.

Lemma k_ratio_lt_60 : k_ratio < 60.
Proof.
unfold k_ratio.
interval with (i_prec 30).
Qed.

Lemma k_ratio_bounds : 59 < k_ratio < 60.
Proof.
split; [apply k_ratio_gt_59 | apply k_ratio_lt_60].
Qed.

(* ============================================================================ *)
(* §7 BT-8: No H4 Integer Invariant Equals k_ratio *)
(* ============================================================================ *)

(* All tested candidates equal 30 or 60. Since k < 60 and k > 59, *)
(* none of them can equal k. *)

Theorem k_neq_Coxeter_h :
k_ratio <> Coxeter_h_H4.
Proof.
unfold Coxeter_h_H4. pose proof k_ratio_gt_59. lra.
Qed.

Theorem k_neq_double_h :
k_ratio <> double_h_H4.
Proof.
unfold double_h_H4. pose proof k_ratio_lt_60. lra.
Qed.

Theorem k_neq_sum_exponents :
k_ratio <> sum_exponents_H4.
Proof.
unfold sum_exponents_H4. pose proof k_ratio_lt_60. lra.
Qed.

Theorem k_neq_cells_over_10 :
k_ratio <> cells_600 / 10.
Proof.
unfold cells_600.
(* cells_600 / 10 = 600 / 10 = 60 *)
assert (Heq: (600 : R) / 10 = 60) by (field; lra).
rewrite Heq. pose proof k_ratio_lt_60. lra.
Qed.

Theorem k_neq_edges_over_12 :
k_ratio <> edges_600 / 12.
Proof.
unfold edges_600.
(* edges_600 / 12 = 720 / 12 = 60 *)
assert (Heq: (720 : R) / 12 = 60) by (field; lra).
rewrite Heq. pose proof k_ratio_lt_60. lra.
Qed.

Theorem k_neq_vertices_600 :
k_ratio <> vertices_600.
Proof.
unfold vertices_600. pose proof k_ratio_lt_60. lra.
Qed.

(* ---- BT-8 Main Theorem ---- *)
(* *)
(* The conversion factor k = Trinity_a4 / Coq_a4 = (704+192*sqrt5)/19 *)
(* is strictly between 59 and 60. No tested H4/600-cell integer invariant *)
(* equals k. Hence no exact H4-integer bridge exists. *)
(* *)
(* The algebraic bridge Trinity_a4 = k * Coq_a4 EXISTS, *)
(* but k is irrational (Q(sqrt5) \ Q) with no H4 integer interpretation. *)

Theorem BT8_no_H4_integer_bridge :
(* k is strictly between 59 and 60 *)
59 < k_ratio < 60
/\
(* Not the Coxeter number h = 30 *)
k_ratio <> Coxeter_h_H4
/\
(* Not 2*h = 60 (= sum of exponents = cells/10 = edges/12) *)
k_ratio <> double_h_H4
/\
(* Not the sum of Coxeter exponents 1+11+19+29 = 60 *)
k_ratio <> sum_exponents_H4
/\
(* Not the number of vertices of 600-cell = 120 *)
k_ratio <> vertices_600
/\
(* Algebraic bridge with irrational factor *)
Trinity_a4 = k_ratio * Coq_a4.
Proof.
refine (conj k_ratio_bounds (conj _ (conj _ (conj _ (conj _ _))))).
- apply k_neq_Coxeter_h.
- apply k_neq_double_h.
- apply k_neq_sum_exponents.
- apply k_neq_vertices_600.
- apply a4_algebraic_bridge.
Qed.

(* ============================================================================ *)
(* §8 Positive Corollary: k is phi-algebraic *)
(* ============================================================================ *)

(* Although k is not an H4 integer, it lives in Q(phi) = Q(sqrt 5). *)
(* Both a_4 values are determined by the same algebraic extension. *)

Theorem k_is_phi_algebraic :
k_ratio = 128 * (3 * phi + 2) / (5 + 6 * phi).
Proof.
unfold k_ratio. rewrite phi_fourth. reflexivity.
Qed.

Lemma k_ratio_pos : 0 < k_ratio.
Proof.
unfold k_ratio.
apply Rdiv_lt_0_compat.
- apply Rmult_lt_0_compat; [lra |].
rewrite phi_fourth. pose proof phi_pos. lra.
- apply denom_pos.
Qed.

(* ============================================================================ *)
(* §9 Summary *)
(* ============================================================================ *)

(*
BOUNDARY THEOREM BT-8: No Exact H4-Integer a_4 Bridge

Inputs (from existing Coq proofs):
Coq_a4 := (5+6*phi)/(16*phi) ≈ 0.5681 [SpectralAction600Cell.v, QED]
Trinity_a4 := 8*phi^3 ≈ 33.889 [HiggsPrediction.v, QED]
k_ratio := 128*phi^4/(5+6*phi) ≈ 59.649 [A4Conversion.v, QED]

Algebraic fact: Trinity_a4 = k_ratio * Coq_a4 (algebra, not geometry).

BT-8 states (proven in this file):
k_ratio is NOT equal to any of:
h(H4) = 30 Coxeter number
2*h(H4) = 60 double Coxeter number
sum of exponents = 60 1+11+19+29
cells_600/10 = 60
edges_600/12 = 60
vertices_600 = 120

Proof: 59 < k_ratio < 60 by interval arithmetic.
All listed candidates are integers outside the interval (59,60).

k_ratio = (704+192*sqrt(5))/19 is irrational. The denominator 19 is the
third Coxeter exponent of H4 (degree 20 - 1), appearing algebraically only.

Conclusion: no exact geometric bridge from H4 integer combinatorics to the
ratio Trinity_a4/Coq_a4 exists. The ~0.59% deviation from 60 is a genuine
mathematical boundary of the H4 spectral-action framework.

BT-8 joins BT-1..BT-4 (BoundaryTheorems.v) as a Coq-formal negative result.
Wave 8.1 (2026-06-23).
*)
3 changes: 3 additions & 0 deletions proofs/trinity/_CoqProject
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Expand Up @@ -96,3 +96,6 @@ ExtendedAF.v
# Wave 14: Z3 tripartition of snub 24-cell
Snub24CellZ3.v

# Wave 8.1: a_4 bridge analysis — Boundary Theorem BT-8
A4Bridge.v

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