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allaun
3b6baec64e
Snapshot of previously-uncommitted local work so nothing is lost after the power outage. NOT reviewed for correctness — a WIP checkpoint, not a feature: - multi-language hachimoji encoders (c/cpp/fortran/julia/octave/r/scala/go/rust/coq) - formal Lean WIP (BraidTree, Eisenstein, HachimojiCapture, MathlibConnect, ModularFormBridge, ClusterManifold) + lakefile + E8Sidon edit - docs/, experiments/ (epyc oisc benches), deploy/, scripts, test scaffolding - .gitignore: exclude **/target/ and Coq build artifacts Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
102 lines
2.8 KiB
Coq
102 lines
2.8 KiB
Coq
From Stdlib Require Import Arith Lia.
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From Stdlib Require Import List.
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Import ListNotations.
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(* ----- sigma3: sum of cubes of divisors ----- *)
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Fixpoint divisors_aux (n d : nat) : list nat :=
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match d with
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| 0 => []
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| S d' =>
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if Nat.eqb (Nat.modulo n (S d')) 0
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then (S d') :: divisors_aux n d'
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else divisors_aux n d'
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end.
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Definition divisors (n : nat) : list nat :=
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match n with
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| 0 => []
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| _ => divisors_aux n n
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end.
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Definition sigma3 (n : nat) : nat :=
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fold_right (fun d acc => d * d * d + acc) 0 (divisors n).
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(* ----- hachimoji_letter: sigma3 mod 8 ----- *)
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Definition hachimoji_letter (s3 : nat) : nat :=
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Nat.modulo s3 8.
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(* ----- cartan_weight: 273 if equal, 256 if complementary, 0 otherwise ----- *)
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(* Complementary hachimoji letters sum to a multiple of 8. *)
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Definition cartan_weight (a b : nat) : nat :=
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if Nat.eqb a b then 273
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else if Nat.eqb (Nat.modulo (a + b) 8) 0 then 256
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else 0.
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(* ----- Helper: C(k,2) = k*(k-1)/2 via tail recursion ----- *)
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Fixpoint tail_pair_count (k : nat) : nat :=
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match k with
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| 0 | 1 => 0
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| S k' => k' + tail_pair_count k'
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end.
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(* ----- angrysphinx_gate ----- *)
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(* Collisions = number of unordered pairs among tail elements (i.e., among *)
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(* elements after the first). Energy = 273 - 239*collisions, saturating at *)
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(* zero. The gate passes iff energy > 0. The constant 239 is 273 - 34, where *)
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(* 34 is the single-collision residual energy. *)
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Definition angrysphinx_gate (els : list nat) : bool * nat * nat :=
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let n := length els in
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let collisions :=
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match n with
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| 0 | 1 => 0
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| S n' => tail_pair_count n'
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end
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in
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let penalty := collisions * 239 in
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let energy := Nat.sub 273 penalty in
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let passed := negb (Nat.eqb energy 0) in
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(passed, collisions, energy).
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(* ----- hachimoji_encode ----- *)
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Definition hachimoji_encode (n : nat) : nat * nat :=
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let s3 := sigma3 n in
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(s3, hachimoji_letter s3).
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(* ===== COMPUTE DIRECTIVES ===== *)
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(* sigma3 verification *)
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Compute sigma3 1. (* = 1 *)
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Compute sigma3 2. (* = 9 *)
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Compute sigma3 3. (* = 28 *)
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Compute sigma3 10. (* = 1134 *)
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(* angrysphinx_gate verification *)
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Compute angrysphinx_gate [1;2]. (* = (true, 0, 273) *)
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Compute angrysphinx_gate [1;2;3]. (* = (true, 1, 34) *)
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Compute angrysphinx_gate [1;2;3;4]. (* = (false, 3, 0) *)
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(* Test sigma3 for all n=1..10 *)
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Compute sigma3 1.
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Compute sigma3 2.
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Compute sigma3 3.
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Compute sigma3 4.
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Compute sigma3 5.
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Compute sigma3 6.
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Compute sigma3 7.
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Compute sigma3 8.
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Compute sigma3 9.
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Compute sigma3 10.
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(* ===== THEOREMS ===== *)
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Theorem gate_12_passes : angrysphinx_gate [1;2] = (true, 0, 273).
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Proof. reflexivity. Qed.
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Theorem gate_1234_fails : angrysphinx_gate [1;2;3;4] = (false, 3, 0).
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Proof. reflexivity. Qed.
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