Shared failures cap redundancy, while every extra replica still helps

Supplied research · Reviewed finding · 30 October 2026

This review of Tolerance Needs a Multiplicity, Theory 068 in Volume VIII of the Incoprea research notebook, finds an exact, useful result inside the supplied calculation: a fixed shared failure probability limits the reliability achievable by adding otherwise independent replicas. It also finds an important qualification. The model has an asymptotic ceiling, rather than a finite replica count beyond which reliability stops improving.

Evidence classification: exact mathematics within a specified model, checked by supplied Monte Carlo computation. This is a minimally adjusted publication of an existing analysis, with no new observations or independent study. It does not validate the full original hypothesis.

What the supplied model actually assumes

The system works whenever at least one of n replicas works. Each replica independently fails with probability q = 0.10. Separately, a shared infrastructure event occurs with probability c = 0.05 and disables every replica. The shared event is independent of the individual failures. All probabilities refer to the same unspecified operating interval; they are not measured failure rates per hour.

The NIST parallel-system model explicitly assumes independent components and operation while at least one component survives. It supplies the familiar independent-redundancy calculation. The supplied study adds the specified common event, making the unconditional replica outcomes dependent. No consensus algorithm or quorum is implemented.

Conditional on avoiding the common event, all n replicas fail with probability qⁿ. Therefore the system reliability is

Rₙ = (1 − c)(1 − qⁿ).

With c = 0, reliability becomes 1 − qⁿ. With the supplied c, its limit as n increases is 0.95. This follows from the assumptions directly; simulation is a numerical check, rather than evidence that actual infrastructure has these probabilities.

The numbers, without rounding away the distinction

ReplicasIndependent reliabilityShared-event reliabilitySupplied simulation
10.9000000.8550000.854650
20.9900000.9405000.941380
30.9990000.9490500.949335
120.9999999999990.949999999999050.949705

The supplied code runs 200,000 trials for each of twelve replica counts, using seed 20261030: 2.4 million system trials in total. An isolated unchanged reproduction matched its stored results. The Monte Carlo estimates fluctuate around the exact curve. An individual simulated estimate can exceed 0.95, as happens at four replicas; sampling noise does not invalidate the theoretical ceiling on the underlying probability.

The clean public output retains small quantities instead of rounding them to zero. The improvement from eleven replicas to twelve is approximately 8.55 × 10⁻¹². The improvement from ten to twelve is 9.405 × 10⁻¹¹. The original narrative accidentally described the former number as the latter interval; that wording is corrected here.

Where the original prediction needs qualification

For 0 < q < 1 and c < 1, subtracting adjacent reliabilities gives

Rₙ₊₁ − Rₙ = (1 − c)(1 − q)qⁿ > 0.

Every additional replica therefore improves reliability in this model. The improvement becomes tiny, but never becomes exactly zero at a finite required multiplicity. The supplied calculation supports diminishing returns and a shared-risk ceiling. It does not establish finite saturation at a quorum or show that extra replicas create new correlated failure modes: c stays fixed throughout.

This distinction matters for interpreting the original proposed refutation. An increasing reliability curve and an upper limit can coexist. Their coexistence does not settle a choice between replica-count models and protocol-specific multiplicity requirements. Those questions need an explicit protocol and failure model.

There is also no cost function, repair process, failover delay, voting rule, workload, or changing infrastructure in the supplied model. Consequently it cannot choose an economically optimal replica count. A tiny positive gain may be unimportant for a particular application, but that judgement requires a stated reliability target and costs.

The bounded finding is straightforward: under these assumptions, replication cannot overcome a common event that defeats every copy. Reducing that common risk would change the ceiling; merely increasing the number of identically exposed copies would not. The calculation illustrates established reliability reasoning and makes no novelty claim.

Credit and reproducibility

The model, parameter choices, simulation and original report were supplied in the DeepSeek-assisted Incoprea Probable Tests bundle dated 30 October 2026. Codex reviewed the source and original hypothesis, independently reproduced the existing computation, derived the finite-gain qualification, and prepared this corrected publication. This publisher audit is not external peer review. The evidence package preserves original code and results separately from the portable reproduction and reviewed display results.

Reproduction and data availability

Python 3 standard library; this code is self-contained. Inputs are generated by the supplied model and fixed seed in the included code; no external data download is required. Original supplied source and result records are preserved separately in the release.

Save the code as reproduce.py and run python3 reproduce.py with the stated inputs. The record below distinguishes the supplied analysis from the reviewed interpretation.

Show the reproduction code
"""Reviewed output of supplied T04; same model, seed, RNG order and trial count.
Requires NumPy. Run beside this file; writes display-results.json.
Original code/report are preserved separately and contain superseded wording.
"""
from pathlib import Path
import json
import numpy as np
Q,C,TRIALS,SEED = .10,.05,200000,20261030
rng=np.random.default_rng(SEED)
rows=[]
for n in range(1,13):
    works=0
    for _ in range(TRIALS):
        shared_ok = rng.random() > C
        any_ok = bool((rng.random(n) > Q).any())
        works += int(shared_ok and any_ok)
    rows.append({'replicas':n,'independent_reliability':1-Q**n,
                 'shared_reliability':(1-C)*(1-Q**n),
                 'monte_carlo_reliability':works/TRIALS})
result={
 'id':'shared-failure-redundancy',
 'evidence_class':'exact result in a specified 1-of-n model; supplied Monte Carlo check',
 'review_status':'minimally adjusted: asymptotic ceiling, no finite zero-gain multiplicity claim',
 'parameters':{'q':Q,'c':C,'trials_per_replica_count':TRIALS,'seed':SEED},
 'model':'An independent shared event kills every replica; otherwise iid replica failures have probability q.',
 'formula':'R_n=(1-c)(1-q**n)',
 'limit':1-C,
 'gain_n_to_n_plus_1':'(1-c)*(1-q)*q**n, strictly positive for 0<q<1 and c<1',
 'gain_11_to_12':(1-C)*(1-Q)*Q**11,
 'gain_10_to_12':(1-C)*(Q**10-Q**12),
 'series':rows,
 'not_tested':['consensus/quorum requirements','new shared failures caused by added replicas','cost-optimal replica counts','real-system reliability']}
Path(__file__).with_name('display-results.json').write_text(json.dumps(result,indent=2)+'\n')
print(json.dumps(result,indent=2))
Show the reviewed result record
{
  "id": "shared-failure-redundancy",
  "evidence_class": "exact result in a specified 1-of-n model; supplied Monte Carlo check",
  "review_status": "minimally adjusted: asymptotic ceiling, no finite zero-gain multiplicity claim",
  "parameters": {
    "q": 0.1,
    "c": 0.05,
    "trials_per_replica_count": 200000,
    "seed": 20261030
  },
  "model": "An independent shared event kills every replica; otherwise iid replica failures have probability q.",
  "formula": "R_n=(1-c)(1-q**n)",
  "limit": 0.95,
  "gain_n_to_n_plus_1": "(1-c)*(1-q)*q**n, strictly positive for 0<q<1 and c<1",
  "gain_11_to_12": 8.550000000000005e-12,
  "gain_10_to_12": 9.405000000000005e-11,
  "series": [
    {
      "replicas": 1,
      "independent_reliability": 0.9,
      "shared_reliability": 0.855,
      "monte_carlo_reliability": 0.85465
    },
    {
      "replicas": 2,
      "independent_reliability": 0.99,
      "shared_reliability": 0.9405,
      "monte_carlo_reliability": 0.94138
    },
    {
      "replicas": 3,
      "independent_reliability": 0.999,
      "shared_reliability": 0.94905,
      "monte_carlo_reliability": 0.949335
    },
    {
      "replicas": 4,
      "independent_reliability": 0.9999,
      "shared_reliability": 0.949905,
      "monte_carlo_reliability": 0.950835
    },
    {
      "replicas": 5,
      "independent_reliability": 0.99999,
      "shared_reliability": 0.9499905,
      "monte_carlo_reliability": 0.94895
    },
    {
      "replicas": 6,
      "independent_reliability": 0.999999,
      "shared_reliability": 0.94999905,
      "monte_carlo_reliability": 0.949775
    },
    {
      "replicas": 7,
      "independent_reliability": 0.9999999,
      "shared_reliability": 0.949999905,
      "monte_carlo_reliability": 0.950135
    },
    {
      "replicas": 8,
      "independent_reliability": 0.99999999,
      "shared_reliability": 0.9499999905,
      "monte_carlo_reliability": 0.949555
    },
    {
      "replicas": 9,
      "independent_reliability": 0.999999999,
      "shared_reliability": 0.94999999905,
      "monte_carlo_reliability": 0.949985
    },
    {
      "replicas": 10,
      "independent_reliability": 0.9999999999,
      "shared_reliability": 0.949999999905,
      "monte_carlo_reliability": 0.9503
    },
    {
      "replicas": 11,
      "independent_reliability": 0.99999999999,
      "shared_reliability": 0.9499999999905,
      "monte_carlo_reliability": 0.950465
    },
    {
      "replicas": 12,
      "independent_reliability": 0.999999999999,
      "shared_reliability": 0.9499999999990499,
      "monte_carlo_reliability": 0.949705
    }
  ],
  "not_tested": [
    "consensus/quorum requirements",
    "new shared failures caused by added replicas",
    "cost-optimal replica counts",
    "real-system reliability"
  ]
}
Show provenance, dependencies and review boundaries
{
  "source_bundle": "Incoprea Probable Tests - 2026-10-30",
  "classification": "Reproduced supplied model with bounded interpretation",
  "review_scope": "Reproduction and consistency review of the supplied study; no new independently designed study",
  "reproduction_requirements": "Python 3 standard library; this code is self-contained.",
  "data_availability": "Inputs are generated by the supplied model and fixed seed in the included code; no external data download is required. Original supplied source and result records are preserved separately in the release.",
  "source_hypothesis_url": "https://incoprea.com/research-library/original-theories-vol8-20261030/theories/fault-tolerance-multiplicity.html",
  "published_code_sha256": "2c95a2ca0c02ee0f52e02a82812d64bcefcea76da863feeed862e69b5a91344d",
  "published_result_sha256": "7b6aecf3009aa40d84562a2dda0425921cfe41e066b168fdad43c31aa684e698",
  "peer_review": "Internal computational review; not external peer review"
}

Original supplied analysis: contributing research agent identified in the source bundle. Consistency review, necessary corrections and publication: Codex. Working finding, not externally peer reviewed. Incoprea Research · Original posted question or hypothesis.

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