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80d7559 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 | %% CARRY — Tau-Prolog Braid Kernel
%% Three strands: Curry (σ₁), Crystal (σ₂), C3 (σ₃)
%% Crossings encode state transitions as braid group generators
%% Writhe invariant = net integrity of the pipeline
%%
%% Compatible with Tau-Prolog (ISO Prolog subset)
%% Author: Ahmad <ahmedparr93@gmail.com>
%% License: AGPL-3.0-only
:- module(braid_kernel, [
strand/2,
crossing/5,
braid_word/1,
writhe/2,
valid_braid/1,
braid_step/4,
entropy_bounded/2,
strand_position/3,
braid_invariant/1
]).
%% --- Strand definitions (the three computational fibers) ---
strand(curry, 1). % σ₁ — position 1 (leftmost)
strand(crystal, 2). % σ₂ — position 2 (middle)
strand(c3, 3). % σ₃ — position 3 (rightmost)
%% --- Crossing types ---
%% A positive crossing (σᵢ) = strand i crosses OVER strand i+1
%% A negative crossing (σᵢ⁻¹) = strand i crosses UNDER strand i+1
%% This encodes trust: OVER = active takes authority, UNDER = active yields
crossing(sigma_1, positive, curry, crystal, r1_ffi).
crossing(sigma_2, positive, crystal, c3, r2_native_binding).
crossing(sigma_1_inv, negative, crystal, curry, revert_r1).
crossing(sigma_2_inv, negative, c3, crystal, revert_r2).
crossing(sigma_12, positive, curry, c3, full_descent).
%% --- Braid word: the sequence of crossings that constitutes a valid pipeline ---
%% The canonical CARRY pipeline is: σ₂ · σ₁ (C3 over Crystal, then C3 over Curry)
%% This transfers authority to C3 (hardware execution layer)
braid_word([sigma_2, sigma_1]).
%% --- Writhe number: sum of crossing signs ---
%% +1 for positive (over), -1 for negative (under)
%% Writhe = net "authority taken" through the pipeline
%% Invariant: writhe(canonical) = +2 (two authority transfers, both positive)
crossing_sign(positive, 1).
crossing_sign(negative, -1).
writhe([], 0).
writhe([C|Rest], W) :-
crossing(C, Sign, _, _, _),
crossing_sign(Sign, S),
writhe(Rest, W0),
W is W0 + S.
%% --- Entropy bound per crossing ---
entropy_ceiling(0.20).
entropy_bounded(Crossing, Entropy) :-
crossing(Crossing, _, _, _, _),
entropy_ceiling(Max),
Entropy =< Max.
%% --- Braid validity: Reidemeister moves preserve topology ---
%% R1: σᵢ · σᵢ⁻¹ = identity (cancel adjacent inverses)
%% R2: σᵢ · σⱼ = σⱼ · σᵢ when |i-j| >= 2 (far commutativity)
%% R3: σᵢ · σᵢ₊₁ · σᵢ = σᵢ₊₁ · σᵢ · σᵢ₊₁ (Yang-Baxter)
cancels(sigma_1, sigma_1_inv).
cancels(sigma_1_inv, sigma_1).
cancels(sigma_2, sigma_2_inv).
cancels(sigma_2_inv, sigma_2).
reduce([], []).
reduce([X], [X]).
reduce([A, B | Rest], Reduced) :-
cancels(A, B), !,
reduce(Rest, Reduced).
reduce([A | Rest], [A | Reduced]) :-
reduce(Rest, Reduced).
valid_braid(Word) :-
reduce(Word, Reduced),
Reduced \= [].
%% --- Braid step: transition the FSM via a crossing ---
braid_step(State, Crossing, Entropy, NextState) :-
entropy_bounded(Crossing, Entropy),
crossing(Crossing, _, From, To, _),
state_layer(State, From),
next_state(State, Crossing, NextState).
%% --- State-to-layer mapping (which strand is active at each DAG node) ---
state_layer(input, curry).
state_layer(memory, crystal).
state_layer(retrieval, c3).
state_layer(transform, c3).
state_layer(constraint, c3).
state_layer(proof, c3).
state_layer(output, c3).
%% --- State transitions via braid crossings ---
next_state(input, sigma_1, memory).
next_state(memory, sigma_2, retrieval).
next_state(retrieval, sigma_12, transform).
next_state(transform, sigma_12, constraint).
next_state(constraint, sigma_12, proof).
next_state(proof, sigma_12, output).
next_state(output, sigma_12, output).
%% --- Strand position tracking (which physical position each strand occupies) ---
%% After σ₁: curry and crystal swap positions
%% After σ₂: crystal and c3 swap positions
strand_position([], curry, 1).
strand_position([], crystal, 2).
strand_position([], c3, 3).
strand_position([sigma_1 | Rest], Strand, Pos) :-
strand_position(Rest, Strand, Pos0),
swap_12(Strand, Pos0, Pos).
strand_position([sigma_2 | Rest], Strand, Pos) :-
strand_position(Rest, Strand, Pos0),
swap_23(Strand, Pos0, Pos).
swap_12(curry, 1, 2) :- !.
swap_12(curry, 2, 1) :- !.
swap_12(crystal, 1, 2) :- !.
swap_12(crystal, 2, 1) :- !.
swap_12(_, P, P).
swap_23(crystal, 2, 3) :- !.
swap_23(crystal, 3, 2) :- !.
swap_23(c3, 2, 3) :- !.
swap_23(c3, 3, 2) :- !.
swap_23(_, P, P).
%% --- The Braid Invariant ---
%% After the canonical pipeline [σ₁, σ₂]:
%% curry ends at position 2 (middle)
%% crystal ends at position 3 (right)
%% c3 ends at position 1 (left — now has authority)
%% This IS the trust transfer: C3 rises to leftmost = highest authority
braid_invariant(Word) :-
writhe(Word, W),
W >= 2,
strand_position(Word, c3, 1),
valid_braid(Word).
%% --- Full pipeline proof ---
%% Run the canonical braid word and verify the invariant holds
prove_pipeline :-
braid_word(W),
braid_invariant(W),
writhe(W, Writhe),
format("CARRY BRAID PROOF~n", []),
format(" Word: ~w~n", [W]),
format(" Writhe: ~w~n", [Writhe]),
format(" C3 pos: 1 (AUTHORITY TRANSFERRED)~n", []),
format(" Status: INVARIANT HOLDS~n", []).
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