Example code — Repeated selection of inherited chemical states: finite-population guarantees and a logarithmic horizon law

Repeated growth and transfer can enrich one inherited chemical state while leaving the less common state vulnerable to loss during sampling. Each model cell carries resident chemistry and an inherited size. Faster-growing chemical states gain share during a finite batch; division partitions molecules between complementary daughters. A uniform sample of whole cells then continues through recovery and the next batch.

This example provides the full thirteen-channel resident model plus growth, intact-cell transfer, arbitrary-cycle simulation and finite-mission design tools. Inputs sit at the top of the driver. Transfer must control inherited size as well as cell counts: the exact eight-cell example shows different size-transfer risks even when the type counts match.

Conditional high-state composition rises with cycle count while the sufficient joint confidence eventually falls; refined transfer bounds improve the admissible horizon. — Composition is bounded on the successful event. A confidence lower bound reaching zero means this estimate stops certifying the mission; it does not prove extinction.

Equal and mixed cell sizes give different exact retained-size distributions; uniform transfer can lose a rare minority. — These exact calculations condition on a specified endpoint population. They do not estimate the probability that the chemistry reaches that population.

Fresh arithmetic reproduces the paper's ten-cycle, four-billion-cell witness with joint confidence above 0.997. On that event, high-state composition increases while at least 36,862 low-type cells remain. The code separates chemical, transfer and service errors, sizes a transfer population, and accounts for precursor, discarded material, elapsed time and finite service quotas.

The displayed confidence is a sufficient bound for the whole mission. Its loss of usefulness is not proof of extinction. Exact subset laws and resource identities are computed directly; small stochastic histories illustrate the physical source and retain their failures to return to the required chemical operating region. The source probability theorem is imported, the complete analytical mark classifier is not simulated, and Lean is not rerun. The theorem's extreme size and time scales are reported explicitly rather than presented as biological calibration.

Python source

"""Repeated chemical-state selection: full physical source and finite-mission design."""
# EDITABLE INPUTS. These are theorem-scale witnesses, not calibrated biological cells.
NEWBORN_SIZE = 65536 * 10**18
RETAINED_CELLS = 4 * 10**9
CYCLES = 10
GROWTH_COUPLING = '1/100000000000'
BATCH_SERVICE_ALLOWANCE = '1/10000'
RECOVERY_SERVICE_ALLOWANCE = '1/10000'
TRANSFER_FAILURE_BUDGET = '1/250'
DESIGN_POPULATION_CAP = 10**16
MINORITY_RESERVE = 1
REFERENCE_MOLAR = '1/1000'     # Optional concentration convention, not a fit.
TIME_UNIT_SECONDS = '1'       # Optional time convention, not a measured rate.
# Small physical diagnostic: deliberately outside the proved source regime.
SMALL_SIZE = 10
SMALL_CELLS = 4
SMALL_CYCLES = 3
SMALL_GROWTH = '2'
SMALL_RECOVERY = '.05'
SMALL_EVENT_BUDGET = 100000
RANDOM_SEED = 77092026

from pathlib import Path
from dataclasses import asdict
from fractions import Fraction as Q
import argparse,csv,hashlib,json,platform
import numpy as np
import scipy
from resident import Cell,ChemicalRegions,A
from population import ChemicalModel,PopulationSource,SerialProtocol,UniformTransfer
from selection import MissionCertificate,WeightedTransfer,AbsorbingAudit,population_design,event_ceiling,RHO,RHOS,G,LOG2
import paper_checks

MANUSCRIPT_SHA256='89018b0f5f84c348611b59ed44e378c4f360972dbf601feabc5f1c0e9f0ee912'

def main():
    parser=argparse.ArgumentParser(description=__doc__);parser.add_argument('--output',type=Path,default=Path('outputs'));out=parser.parse_args().output;out.mkdir(parents=True,exist_ok=True)
    def dump(name,value):(out/name).write_text(json.dumps(value,indent=2,default=str)+'\n')
    def table(name,rows):
        with (out/name).open('w',newline='') as f:w=csv.DictWriter(f,fieldnames=list(rows[0]));w.writeheader();w.writerows(rows)
    cert=MissionCertificate(NEWBORN_SIZE,RETAINED_CELLS,CYCLES,Q(GROWTH_COUPLING),Q(BATCH_SERVICE_ALLOWANCE),Q(RECOVERY_SERVICE_ALLOWANCE))
    dump('configured_mission.json',cert.evaluate());dump('resource_accounts.json',cert.resources());dump('chemical_terms.json',{k:str(v) for k,v in cert.chemistry().items()})
    dump('population_design.json',population_design(CYCLES,TRANSFER_FAILURE_BUDGET,max_M=DESIGN_POPULATION_CAP))
    dump('necessary_event_ceiling.json',event_ceiling(RETAINED_CELLS,CYCLES,MINORITY_RESERVE))
    comparisons=[]
    for name,M,refined,s in [('original',10**13,False,Q(1,10000)),('refined',4*10**9,True,Q(1,10000)),('refined_finer_service',4*10**9,True,Q(1,10**6))]:
        c=MissionCertificate(M=M,refined=refined,batch_allowance=s,recovery_allowance=s);row=c.evaluate();row.update(case=name,binary_chemical_replay=str(c.binary_replay()),resources=c.resources());comparisons.append(row)
    dump('canonical_comparison.json',comparisons)
    dump('source_arithmetic_replay.json',dict(results=paper_checks.OUT,checks=paper_checks.CHECKS,scope='Exact enclosures and explicitly labeled numerical diagnostics from the source. Neither the process theorem nor Lean is reproved.'))
    rows=[]
    for k in range(1,19):
        c=MissionCertificate(N=NEWBORN_SIZE,M=RETAINED_CELLS,K=k,gamma=Q(GROWTH_COUPLING),batch_allowance=Q(BATCH_SERVICE_ALLOWANCE),recovery_allowance=Q(RECOVERY_SERVICE_ALLOWANCE));r=c.evaluate()
        rows.append(dict(cycle=k,success_lower=r['joint_success_decimal'],count_logodds_lower=float(k*G[0]-LOG2[1]),high_fraction_lower=1/(1+np.exp(-float(k*G[0]-LOG2[1]))),minority_floor=r['minority_count_floor'],transfer_failure_upper=float(sum(c.transfers())),original_transfer_failure_upper=float(Q(80000,RETAINED_CELLS)*sum(RHO**-j for j in range(k)))))
    table('horizon_sweep.csv',rows)
    exact=[];cases=[]
    for name,sizes in [('equal',[10]*8),('mixed',[10,10,19,19]*2)]:
        for tolerance in [Q(1,50),Q(1,2)]:
            result=WeightedTransfer.enumerate(sizes,['H']*4+['L']*4,4,tolerance)
            exact.append(dict(case=name,tolerance=str(tolerance),failure=str(result['failure']),moments=result['moments'],scope='Conditional endpoint population; not a probability of reaching that population.'))
            if tolerance==Q(1,50):cases.append((name,result))
            table(f'transfer_{name}_{tolerance.denominator}.csv',[{k:str(v) for k,v in r.items()} for r in result['rows']])
    dump('weighted_transfer_comparison.json',exact)
    table('conditional_minority_loss.csv',[dict(endpoint_count=400,retained=100,minority=c,loss_exact=str(WeightedTransfer.minority_loss(400,100,c)),loss=float(WeightedTransfer.minority_loss(400,100,c))) for c in [1,4,20]])
    regions=ChemicalRegions();chem=ChemicalModel();dump('chemical_identities.json',chem.certificates())
    prepared=[regions.newborn(t,NEWBORN_SIZE) for t in ('H','L')]
    dump('ready_representatives.json',[dict(cell=asdict(c),energy_interval=regions.energy(c).json(),readout=c.readout,scope='Two representatives, not allocation of the full theorem-scale population.') for c in prepared])
    if SMALL_CELLS<2 or SMALL_CELLS%2:raise ValueError('Use an even diagnostic population.')
    small=[]
    for tag in ['H']*(SMALL_CELLS//2)+['L']*(SMALL_CELLS//2):
        counts=tuple(int(float((x.lo+x.hi)/2)*SMALL_SIZE) for x in regions.centers[tag]);small.append(Cell(tag,SMALL_SIZE,counts))
    physical=SerialProtocol(PopulationSource(SMALL_SIZE,SMALL_GROWTH),SMALL_CELLS,float(SMALL_RECOVERY)).run(small,SMALL_CYCLES,RANDOM_SEED,SMALL_EVENT_BUDGET)
    dump('small_physical_history.json',physical)
    audit=AbsorbingAudit();ready=all(regions.admits(c,A,closed=True) for c in small);audit.observe('initial_readiness',ready,'Exact energy comparison for every diagnostic founder.');audit.observe('illustrative_later_pass',True,'A failed or unresolved history stays absorbed.')
    dump('small_marked_audit.json',dict(**asdict(audit),scope='Readiness mark demonstration only. Physical history is saved separately; the full paper marked law additionally tests intermediate energies, division, growth, transfer, deadline and services. Do not classify physical completion as theorem success.'))
    table('physical_censuses.csv',[dict(cycle=h['cycle']+1,start_size=h['start_size'],selected_H=sum(c['size'] for c in h['selected'] if c['tag']=='H'),selected_L=sum(c['size'] for c in h['selected'] if c['tag']=='L'),residual_precursor=h['residual_discarded'],discarded_size=h['discarded_size']) for h in physical['history']] or [dict(cycle=0,start_size=sum(c.size for c in small),selected_H=0,selected_L=0,residual_precursor=0,discarded_size=0)])
    NA=Q(602214076000000000000000);concentration=Q(REFERENCE_MOLAR);timeunit=Q(TIME_UNIT_SECONDS)
    if min(concentration,timeunit)<=0:raise ValueError('Positive reference units required.')
    dump('illustrative_units.json',dict(newborn_litres=float(Q(NEWBORN_SIZE)/NA/concentration),elapsed_years=float(CYCLES*(cert.T+5376)*timeunit/Q(31557600)),retained_cells=RETAINED_CELLS,scope='N is a size/copy convention. This concentration/time interpretation is illustrative. M counts whole cells and is never divided by Avogadro.'))
    plot(out,rows,cases)
    print('Full resident source: 13 chemical labels plus growth; complementary division; intact uniform transfer; inherited size and counts.',flush=True)
    print('Configured joint success lower:',cert.evaluate()['joint_success_decimal'],'; minority floor:',cert.evaluate()['minority_count_floor'],flush=True)
    print('Physical small run completed:',physical['complete'],'; readiness-mark status:',audit.status,flush=True)
    print('Source checks:',len(paper_checks.CHECKS),'; bounds are sufficient and imported process assumptions remain explicit. Lean not rerun.',flush=True)
    here=Path(__file__).parent;digest=lambda p:hashlib.sha256(p.read_bytes()).hexdigest()
    dump('run_metadata.json',dict(manuscript_sha256=MANUSCRIPT_SHA256,source_sha256=digest(Path(__file__)),python=platform.python_version(),numpy=np.__version__,scipy=scipy.__version__,module_sha256={p.name:digest(p) for p in sorted(here.glob('*.py')) if p.name not in ['example.py','test_example.py']},input_sha256={p.name:digest(p) for p in sorted(here.glob('*.json')) if p.name!='release.json'},output_sha256={p.name:digest(p) for p in sorted(out.iterdir()) if p.is_file() and p.name!='run_metadata.json'}))

def plot(out,rows,cases):
    import matplotlib
    matplotlib.use('Agg')
    import matplotlib.pyplot as plt
    fig,axs=plt.subplots(1,2,figsize=(11,4),layout='constrained')
    axs[0].plot([r['cycle'] for r in rows],[r['high_fraction_lower'] for r in rows],label='High fraction on success')
    axs[0].plot([r['cycle'] for r in rows],[r['success_lower'] for r in rows],label='Joint mission confidence')
    axs[0].set(xlabel='Number of cycles',ylabel='Lower bound',ylim=(-.03,1.04),title='Composition and joint-success bounds over\nrepeated cycles');axs[0].legend(fontsize=8)
    axs[1].semilogy([r['cycle'] for r in rows],[r['transfer_failure_upper'] for r in rows],label='Minority growth + exponential tail');axs[1].semilogy([r['cycle'] for r in rows],[r['original_transfer_failure_upper'] for r in rows],label='Original bound at same M');axs[1].axhline(1,color='gray',ls=':');axs[1].set(xlabel='Number of cycles',ylabel='Sufficient transfer error (uncapped)',title='Both comparisons retain four billion cells');axs[1].legend(fontsize=8)
    for a in axs:a.grid(alpha=.2)
    fig.savefig(out/'mission_horizon.png',dpi=180);fig.savefig(out/'mission_horizon.svg');plt.close(fig)
    fig,axs=plt.subplots(1,2,figsize=(11,4),layout='constrained')
    for name,result in cases:
        xs=sorted(set(float(r['H']) for r in result['rows']));ps=[sum(float(r['H'])==x for r in result['rows'])/len(result['rows']) for x in xs]
        axs[0].plot(xs,ps,'o-',label=name+' sizes')
    axs[0].set(xlabel='Retained high-type size',ylabel='Exact subset probability',title='Retained high-type size under two cell-size\ndistributions');axs[0].legend(fontsize=8)
    cs=np.arange(1,21);axs[1].semilogy(cs,[float(WeightedTransfer.minority_loss(400,100,int(c))) for c in cs],'o-');axs[1].set(xlabel='Minority cells among 400 endpoint cells',ylabel='Probability all are lost',title='Uniform transfer retains 100 whole cells')
    for a in axs:a.grid(alpha=.2)
    fig.savefig(out/'intact_cell_transfer.png',dpi=180);fig.savefig(out/'intact_cell_transfer.svg');plt.close(fig)

if __name__=='__main__':main()

Run output

Full resident source: 13 chemical labels plus growth; complementary division; intact uniform transfer; inherited size and counts.
Configured joint success lower: 0.9970330188947694 ; minority floor: 36862
Physical small run completed: True ; readiness-mark status: failed
Source checks: 2083 ; bounds are sufficient and imported process assumptions remain explicit. Lean not rerun.