Datasets:

PureOne
/

Echo-Mirror-RSI

"""Exact polynomial echo certificate and bounded self-application demo.
The closure is classical constrained polynomial lumping (CLUE), reimplemented
here to expose exactly which optimizer variables a compressor must retain.
This is not a demonstration of open-ended recursive self-improvement.
"""
from __future__ import annotations

import csv
import json
import hashlib
import sys
from fractions import Fraction as F
from pathlib import Path

import numpy as np

ROOT = Path(__file__).resolve().parents[1]
OUT = ROOT / "results"


def rref(rows, n):
    """Exact rational row basis, with pivot columns."""
    a = [[F(v) for v in row] for row in rows if any(row)]
    k, piv = 0, []
    for j in range(n):
        p = next((i for i in range(k, len(a)) if a[i][j]), None)
        if p is None:
            continue
        a[k], a[p] = a[p], a[k]
        z = a[k][j]
        a[k] = [v/z for v in a[k]]
        for i in range(len(a)):
            if i != k and a[i][j]:
                z = a[i][j]
                a[i] = [v-z*w for v, w in zip(a[i], a[k])]
        piv.append(j)
        k += 1
        if k == len(a):
            break
    return a[:k], piv


def jacobian_coefficients(polys, n):
    """Matrices M_alpha in D Psi(x) = sum_alpha M_alpha x^alpha."""
    mats = {}
    m = len(polys)
    for i, poly in enumerate(polys):
        for powers, coeff in poly.items():
            for j, power in enumerate(powers):
                if power:
                    alpha = list(powers)
                    alpha[j] -= 1
                    key = tuple(alpha)
                    mat = mats.setdefault(key, [[F(0) for _ in range(n)] for _ in range(m)])
                    mat[i][j] += F(coeff)*power
    return mats


def closure(polys, observables, initial_rows, n):
    matrices = list(jacobian_coefficients(polys, n).values())
    rows = list(initial_rows)
    for mat in jacobian_coefficients(observables, n).values():
        rows.extend(mat)
    rows, piv = rref(rows, n)
    ranks = [len(rows)]
    while True:
        more = list(rows)
        for mat in matrices:
            more.extend([[sum(row[k]*mat[k][j] for k in range(n)) for j in range(n)] for row in rows])
        new, piv = rref(more, n)
        if len(new) == len(rows):
            break
        rows = new
        ranks.append(len(rows))
    return rows, piv, ranks


def mono(n, **powers):
    p = [0]*n
    for index, exponent in powers.items():
        p[int(index)] = exponent
    return tuple(p)


def self_application_polys(n=8):
    # x' = x - a*x; a' = a+b-a*b; b' = 2*b-b*b.
    polys = [
        {mono(n, **{'0':1}):1, mono(n, **{'0':1, '1':1}):-1},
        {mono(n, **{'1':1}):1, mono(n, **{'2':1}):1, mono(n, **{'1':1, '2':1}):-1},
        {mono(n, **{'2':1}):2, mono(n, **{'2':2}):-1},
    ]
    for j in range(3, n):
        polys.append({mono(n, **{str(j):1}):F(1,2)})
    return polys


def evaluate(polys, state):
    return np.array([sum(float(c)*np.prod([state[j]**p for j,p in enumerate(powers) if p])
                             for powers,c in poly.items()) for poly in polys], dtype=float)


def step3(state):
    x,a,b=state
    return np.array([(1-a)*x, a+b*(1-a), b+b*(1-b)])


def exact_checks():
    n=8
    polys=self_application_polys(n)
    obs=[{mono(n, **{'0':1}):1}]
    rows,piv,ranks=closure(polys,obs,[],n)
    assert piv==[0,1,2] and ranks==[1,2,3]
    # Every coefficient matrix preserves the certificate rowspace.
    for mat in jacobian_coefficients(polys,n).values():
        for row in rows:
            product=[sum(row[k]*mat[k][j] for k in range(n)) for j in range(n)]
            assert all(product[j]==0 for j in range(3,n))
    # Nonlinear square-shift: local first-order output probes miss every echo.
    h=5; m=h+1
    chain=[{mono(m, **{str(j+1):2}):1} for j in range(h)]
    chain.append({mono(m, **{str(h):1}):1})
    _,chain_piv,chain_ranks=closure(chain,[{mono(m, **{'0':1}):1}],[],m)
    assert chain_piv==list(range(m)) and chain_ranks==list(range(1,m+1))
    v=np.zeros(m); v[-1]=0.8
    w=v.copy()
    for _ in range(h):
        w=evaluate(chain,w)
    assert abs(w[0]-0.8**(2**h))<1e-15
    # Exact rank-one edit example: Vandermonde with rational eigenvalues.
    d=6
    lam=[F(i+1, 2*(d+1)) for i in range(d)]
    e=[F(1)]+[F(0)]*(d-1)
    b=[F(0)]+[F(1)]*(d-1)
    vand=[e]+[[b[i]*lam[i]**k for i in range(d)] for k in range(d-1)]
    rank=len(rref(vand,d)[0])
    assert rank==d
    # Spherical diagonal chain-level exactness in degree zero.
    dim=3
    eta=np.vstack([np.eye(dim),np.eye(dim)])
    difference=np.hstack([-np.eye(dim),np.eye(dim)])
    assert np.linalg.norm(difference@eta)==0
    assert np.linalg.matrix_rank(eta)==dim
    assert np.linalg.matrix_rank(difference)==dim
    # Counit of diagonal/direct-sum adjunction has kernel (V,U).
    u,v=2,3
    counit=np.zeros((u+v,2*(u+v)))
    counit[:u,:u]=np.eye(u)
    counit[u:,u+v+u:]=np.eye(v)
    assert 2*(u+v)-np.linalg.matrix_rank(counit)==u+v
    return {
        'exact_polynomial_certificate': {'ambient_dimension':n,'retained_dimension':len(rows),
          'rank_trace':ranks,'pivot_coordinates':['task_error','learning_rule','improvement_rule']},
        'square_shift': {'depth':h,'response_degree':2**h,'rank_trace':chain_ranks,
          'b':0.8,'future_response':float(w[0]),'derivatives_below_degree_are_zero':True},
        'rank_one_edit': {'dimension':d,'exact_certificate_rank_after_edit':rank,
          'exact_certificate_rank_before_edit':1},
        'diagonal_chain_checks':'passed; finite-dimensional identities, not formal verification of the infinity-categorical theorem',
    }


def run_demo():
    OUT.mkdir(parents=True,exist_ok=True)
    checks=exact_checks()
    # Final evaluation seeds and distributions fixed before running.
    n=64; steps=8; all_rows=[]; finals={}
    methods=['full','echo_3','task_rule_2','task_only_1','frozen_improver','frozen_learner','oracle_step_1']
    trajectory_error=0.0; formula_error=0.0
    for seed in range(32):
        rng=np.random.default_rng(12000+seed)
        x0=float(rng.uniform(0.5,2.0)); a0=float(rng.uniform(0.02,0.15)); b0=float(rng.uniform(0.02,0.15))
        initial=np.r_[x0,a0,b0,rng.normal(size=n-3)]
        states={m:initial[:3].copy() for m in methods}
        states['task_rule_2'][2]=0.0
        states['task_only_1'][1:]=0.0
        full=initial.copy()
        for t in range(steps+1):
            formula=x0*(1-a0)**t*(1-b0)**(2**t-1-t)
            formula_error=max(formula_error,abs(states['full'][0]-formula))
            for method in methods:
                x,a,b=states[method]
                all_rows.append({'seed':seed,'generation':t,'method':method,
                    'task_error':float(x),'loss':float(0.5*x*x),
                    'learner_gain':float(a),'improver_gain':float(b)})
                if t==steps:
                    finals.setdefault(method,[]).append(float(0.5*x*x))
            trajectory_error=max(trajectory_error,float(np.max(np.abs(states['full']-states['echo_3']))))
            if t==steps:
                break
            full[:3]=step3(full[:3]); full[3:]*=0.5
            states['full']=full[:3].copy()
            states['echo_3']=step3(states['echo_3'])
            temp=np.r_[states['task_rule_2'][:2],0.0]
            states['task_rule_2']=step3(temp); states['task_rule_2'][2]=0.0
            states['task_only_1']=np.array([states['task_only_1'][0],0.0,0.0])
            x,a,b=states['frozen_improver']
            states['frozen_improver']=np.array([(1-a)*x,a+b0*(1-a),b0])
            x,a,b=states['frozen_learner']
            states['frozen_learner']=np.array([(1-a0)*x,a0,b0])
            states['oracle_step_1']=np.array([0.0,1.0,1.0])
    with (OUT/'self_application_trajectories.csv').open('w',newline='') as f:
        writer=csv.DictWriter(f,fieldnames=list(all_rows[0]));writer.writeheader();writer.writerows(all_rows)
    summary={
        'status':'Engineered scalar self-application sanity experiment; not autonomous discovery, not LLM RSI.',
        'environment':{'python':sys.version,'numpy':np.__version__},
        'source_sha256':hashlib.sha256(Path(__file__).read_bytes()).hexdigest(),
        'seed_count':32,'seed_range':[12000,12031],'generations':steps,
        'ambient_state_coordinates':n,'retained_state_coordinates':3,
        'maximum_full_echo_trajectory_absolute_error':trajectory_error,
        'maximum_closed_form_absolute_error':formula_error,
        'mean_final_loss':{m:float(np.mean(v)) for m,v in finals.items()},
        'median_final_loss':{m:float(np.median(v)) for m,v in finals.items()},
        'oracle_limitation':'A known unit step solves this unit-curvature quadratic in one update. The prototype does not outperform it.',
        'storage_limitation':'Coordinate counts exclude encoders, code, certificates, and data. No hardware speedup claimed.',
        'exact_checks':checks,
    }
    assert trajectory_error<1e-14 and formula_error<1e-12
    (OUT/'self_application_summary.json').write_text(json.dumps(summary,indent=2)+'\n')
    print(json.dumps(summary,indent=2))


if __name__=='__main__':
    run_demo()

	"""Exact polynomial echo certificate and bounded self-application demo.

	The closure is classical constrained polynomial lumping (CLUE), reimplemented
	here to expose exactly which optimizer variables a compressor must retain.
	This is not a demonstration of open-ended recursive self-improvement.
	"""
	from __future__ import annotations

	import csv
	import json
	import hashlib
	import sys
	from fractions import Fraction as F
	from pathlib import Path

	import numpy as np

	ROOT = Path(__file__).resolve().parents[1]
	OUT = ROOT / "results"


	def rref(rows, n):
	"""Exact rational row basis, with pivot columns."""
	a = [[F(v) for v in row] for row in rows if any(row)]
	k, piv = 0, []
	for j in range(n):
	p = next((i for i in range(k, len(a)) if a[i][j]), None)
	if p is None:
	continue
	a[k], a[p] = a[p], a[k]
	z = a[k][j]
	a[k] = [v/z for v in a[k]]
	for i in range(len(a)):
	if i != k and a[i][j]:
	z = a[i][j]
	a[i] = [v-z*w for v, w in zip(a[i], a[k])]
	piv.append(j)
	k += 1
	if k == len(a):
	break
	return a[:k], piv


	def jacobian_coefficients(polys, n):
	"""Matrices M_alpha in D Psi(x) = sum_alpha M_alpha x^alpha."""
	mats = {}
	m = len(polys)
	for i, poly in enumerate(polys):
	for powers, coeff in poly.items():
	for j, power in enumerate(powers):
	if power:
	alpha = list(powers)
	alpha[j] -= 1
	key = tuple(alpha)
	mat = mats.setdefault(key, [[F(0) for _ in range(n)] for _ in range(m)])
	mat[i][j] += F(coeff)*power
	return mats


	def closure(polys, observables, initial_rows, n):
	matrices = list(jacobian_coefficients(polys, n).values())
	rows = list(initial_rows)
	for mat in jacobian_coefficients(observables, n).values():
	rows.extend(mat)
	rows, piv = rref(rows, n)
	ranks = [len(rows)]
	while True:
	more = list(rows)
	for mat in matrices:
	more.extend([[sum(row[k]*mat[k][j] for k in range(n)) for j in range(n)] for row in rows])
	new, piv = rref(more, n)
	if len(new) == len(rows):
	break
	rows = new
	ranks.append(len(rows))
	return rows, piv, ranks


	def mono(n, **powers):
	p = [0]*n
	for index, exponent in powers.items():
	p[int(index)] = exponent
	return tuple(p)


	def self_application_polys(n=8):
	# x' = x - ax; a' = a+b-ab; b' = 2b-bb.
	polys = [
	{mono(n, {'0':1}):1, mono(n, {'0':1, '1':1}):-1},
	{mono(n, {'1':1}):1, mono(n, {'2':1}):1, mono(n, **{'1':1, '2':1}):-1},
	{mono(n, {'2':1}):2, mono(n, {'2':2}):-1},
	]
	for j in range(3, n):
	polys.append({mono(n, **{str(j):1}):F(1,2)})
	return polys


	def evaluate(polys, state):
	return np.array([sum(float(c)np.prod([state[j]*p for j,p in enumerate(powers) if p])
	for powers,c in poly.items()) for poly in polys], dtype=float)


	def step3(state):
	x,a,b=state
	return np.array([(1-a)x, a+b(1-a), b+b*(1-b)])


	def exact_checks():
	n=8
	polys=self_application_polys(n)
	obs=[{mono(n, **{'0':1}):1}]
	rows,piv,ranks=closure(polys,obs,[],n)
	assert piv==[0,1,2] and ranks==[1,2,3]
	# Every coefficient matrix preserves the certificate rowspace.
	for mat in jacobian_coefficients(polys,n).values():
	for row in rows:
	product=[sum(row[k]*mat[k][j] for k in range(n)) for j in range(n)]
	assert all(product[j]==0 for j in range(3,n))
	# Nonlinear square-shift: local first-order output probes miss every echo.
	h=5; m=h+1
	chain=[{mono(m, **{str(j+1):2}):1} for j in range(h)]
	chain.append({mono(m, **{str(h):1}):1})
	_,chain_piv,chain_ranks=closure(chain,[{mono(m, **{'0':1}):1}],[],m)
	assert chain_piv==list(range(m)) and chain_ranks==list(range(1,m+1))
	v=np.zeros(m); v[-1]=0.8
	w=v.copy()
	for _ in range(h):
	w=evaluate(chain,w)
	assert abs(w[0]-0.8(2h))<1e-15
	# Exact rank-one edit example: Vandermonde with rational eigenvalues.
	d=6
	lam=[F(i+1, 2*(d+1)) for i in range(d)]
	e=[F(1)]+[F(0)]*(d-1)
	b=[F(0)]+[F(1)]*(d-1)
	vand=[e]+[[b[i]lam[i]*k for i in range(d)] for k in range(d-1)]
	rank=len(rref(vand,d)[0])
	assert rank==d
	# Spherical diagonal chain-level exactness in degree zero.
	dim=3
	eta=np.vstack([np.eye(dim),np.eye(dim)])
	difference=np.hstack([-np.eye(dim),np.eye(dim)])
	assert np.linalg.norm(difference@eta)==0
	assert np.linalg.matrix_rank(eta)==dim
	assert np.linalg.matrix_rank(difference)==dim
	# Counit of diagonal/direct-sum adjunction has kernel (V,U).
	u,v=2,3
	counit=np.zeros((u+v,2*(u+v)))
	counit[:u,:u]=np.eye(u)
	counit[u:,u+v+u:]=np.eye(v)
	assert 2*(u+v)-np.linalg.matrix_rank(counit)==u+v
	return {
	'exact_polynomial_certificate': {'ambient_dimension':n,'retained_dimension':len(rows),
	'rank_trace':ranks,'pivot_coordinates':['task_error','learning_rule','improvement_rule']},
	'square_shift': {'depth':h,'response_degree':2**h,'rank_trace':chain_ranks,
	'b':0.8,'future_response':float(w[0]),'derivatives_below_degree_are_zero':True},
	'rank_one_edit': {'dimension':d,'exact_certificate_rank_after_edit':rank,
	'exact_certificate_rank_before_edit':1},
	'diagonal_chain_checks':'passed; finite-dimensional identities, not formal verification of the infinity-categorical theorem',
	}


	def run_demo():
	OUT.mkdir(parents=True,exist_ok=True)
	checks=exact_checks()
	# Final evaluation seeds and distributions fixed before running.
	n=64; steps=8; all_rows=[]; finals={}
	methods=['full','echo_3','task_rule_2','task_only_1','frozen_improver','frozen_learner','oracle_step_1']
	trajectory_error=0.0; formula_error=0.0
	for seed in range(32):
	rng=np.random.default_rng(12000+seed)
	x0=float(rng.uniform(0.5,2.0)); a0=float(rng.uniform(0.02,0.15)); b0=float(rng.uniform(0.02,0.15))
	initial=np.r_[x0,a0,b0,rng.normal(size=n-3)]
	states={m:initial[:3].copy() for m in methods}
	states['task_rule_2'][2]=0.0
	states['task_only_1'][1:]=0.0
	full=initial.copy()
	for t in range(steps+1):
	formula=x0(1-a0)t(1-b0)(2t-1-t)
	formula_error=max(formula_error,abs(states['full'][0]-formula))
	for method in methods:
	x,a,b=states[method]
	all_rows.append({'seed':seed,'generation':t,'method':method,
	'task_error':float(x),'loss':float(0.5xx),
	'learner_gain':float(a),'improver_gain':float(b)})
	if t==steps:
	finals.setdefault(method,[]).append(float(0.5xx))
	trajectory_error=max(trajectory_error,float(np.max(np.abs(states['full']-states['echo_3']))))
	if t==steps:
	break
	full[:3]=step3(full[:3]); full[3:]*=0.5
	states['full']=full[:3].copy()
	states['echo_3']=step3(states['echo_3'])
	temp=np.r_[states['task_rule_2'][:2],0.0]
	states['task_rule_2']=step3(temp); states['task_rule_2'][2]=0.0
	states['task_only_1']=np.array([states['task_only_1'][0],0.0,0.0])
	x,a,b=states['frozen_improver']
	states['frozen_improver']=np.array([(1-a)x,a+b0(1-a),b0])
	x,a,b=states['frozen_learner']
	states['frozen_learner']=np.array([(1-a0)*x,a0,b0])
	states['oracle_step_1']=np.array([0.0,1.0,1.0])
	with (OUT/'self_application_trajectories.csv').open('w',newline='') as f:
	writer=csv.DictWriter(f,fieldnames=list(all_rows[0]));writer.writeheader();writer.writerows(all_rows)
	summary={
	'status':'Engineered scalar self-application sanity experiment; not autonomous discovery, not LLM RSI.',
	'environment':{'python':sys.version,'numpy':np.__version__},
	'source_sha256':hashlib.sha256(Path(__file__).read_bytes()).hexdigest(),
	'seed_count':32,'seed_range':[12000,12031],'generations':steps,
	'ambient_state_coordinates':n,'retained_state_coordinates':3,
	'maximum_full_echo_trajectory_absolute_error':trajectory_error,
	'maximum_closed_form_absolute_error':formula_error,
	'mean_final_loss':{m:float(np.mean(v)) for m,v in finals.items()},
	'median_final_loss':{m:float(np.median(v)) for m,v in finals.items()},
	'oracle_limitation':'A known unit step solves this unit-curvature quadratic in one update. The prototype does not outperform it.',
	'storage_limitation':'Coordinate counts exclude encoders, code, certificates, and data. No hardware speedup claimed.',
	'exact_checks':checks,
	}
	assert trajectory_error<1e-14 and formula_error<1e-12
	(OUT/'self_application_summary.json').write_text(json.dumps(summary,indent=2)+'\n')
	print(json.dumps(summary,indent=2))


	if __name__=='__main__':
	run_demo()