NET-SEAM-002 — Strong-Lensing Mass-Sheet Degeneracy Recovery

WHAT THIS CERTIFICATE RECORDS, IN PLAIN TERMS. The problem: Strong-Lensing Mass-Sheet Degeneracy Recovery. The classical reading projects the complete object down to a report — P maps lens mass model, source scale, and cosmological distance scale to the observed image configuration and associated lensing reports. — and the seam is exactly where that projection makes distinct complete objects indistinguishable: The imaging report is invariant under a one-parameter mass-sheet family after source-plane rescaling, while absolute magnification and time-delay scale are not retained by the image-position report.. The MuncyNautics Seam Recovery Method repairs it by naming the lost channels (address, rank, phase, partner, lane), keeping both arms, and reworking the governing equation so the discarded information is carried rather than destroyed — the repaired form: { "equivalence_class": "[M]_P = { M_lambda : kappa_lambda = lambda*kappa + (1-lambda), beta_lambda = lambda*beta }", "repaired_uniqueness": "Recover the equivalence class [M]_P from imaging. Determine the absolute member only after adjoining an independent lane Q for which (P,Q) separates lambda on . The certificate stakes a falsifier ({ "statement": "The repaired uniqueness claim fails if two distinct lambda values remain indistinguishable after every declared independent lane Q is included.", "result_for_imaging_only": "PASS: the ) and reports its standing honestly. Method: Derrick E. Muncy.

MN-RECOVERY-CERTIFICATE-1.0

MN-CERT-NET-SEAM-002-20260829

NET-SEAM-002

Strong-Lensing Mass-Sheet Degeneracy Recovery

Derrick E. Muncy

MuncyNautics Seam Recovery Method

[
  {
    "title": "The Optical Origin of the Mass-Sheet Transformation",
    "url": "https://arxiv.org/pdf/2601.01614"
  },
  {
    "title": "The approximate gravitational lensing multiple plane mass sheet degeneracy",
    "url": "https://arxiv.org/html/2602.02802v1"
  },
  {
    "title": "Breaking the mass-sheet degeneracy in strong lensing mass modelling with weak lensing observations",
    "url": "https://academic.oup.com/mnras/article/533/1/795/7724390"
  }
]

{
  "lens_equation": "beta(theta) = theta - alpha(theta)",
  "mass_sheet_transform": "alpha_lambda(theta) = lambda*alpha(theta) + (1-lambda)*theta",
  "convergence_transform": "kappa_lambda(theta) = lambda*kappa(theta) + (1-lambda)",
  "source_transform": "beta_lambda(theta) = lambda*beta(theta)"
}

P maps lens mass model, source scale, and cosmological distance scale to the observed image configuration and associated lensing reports.

The imaging report is invariant under a one-parameter mass-sheet family after source-plane rescaling, while absolute magnification and time-delay scale are not retained by the image-position report.

{
  "address": "deflector plane, source plane, and line-of-sight sheet location",
  "rank": "first independent observable whose response is not invariant under the lambda family",
  "phase": "arrival-time or wave phase when that lane is measured",
  "partner": "the complete mass-sheet-transformed model family indexed by lambda",
  "lane": "imaging, time delay, stellar kinematics, weak lensing, source-scale, and line-of-sight constraints"
}

{
  "equivalence_class": "[M]_P = { M_lambda : kappa_lambda = lambda*kappa + (1-lambda), beta_lambda = lambda*beta }",
  "repaired_uniqueness": "Recover the equivalence class [M]_P from imaging. Determine the absolute member only after adjoining an independent lane Q for which (P,Q) separates lambda on the declared model domain.",
  "joint_inverse_condition": "(P,Q)(M_lambda1) = (P,Q)(M_lambda2) implies lambda1 = lambda2, subject to the declared assumptions."
}

theta - [lambda*alpha(theta) + (1-lambda)*theta] = lambda*[theta-alpha(theta)]

[
  {
    "theta": "-3",
    "alpha": "-5/2",
    "lambda": "1/2",
    "beta": "-1/2",
    "alpha_lambda": "-11/4",
    "beta_lambda": "-1/4",
    "lambda_beta": "-1/4",
    "residual": "0"
  },
  {
    "theta": "-1",
    "alpha": "-1/4",
    "lambda": "3/4",
    "beta": "-3/4",
    "alpha_lambda": "-7/16",
    "beta_lambda": "-9/16",
    "lambda_beta": "-9/16",
    "residual": "0"
  },
  {
    "theta": "2",
    "alpha": "5/4",
    "lambda": "2/3",
    "beta": "3/4",
    "alpha_lambda": "3/2",
    "beta_lambda": "1/2",
    "lambda_beta": "1/2",
    "residual": "0"
  },
  {
    "theta": "7/2",
    "alpha": "9/4",
    "lambda": "4/5",
    "beta": "5/4",
    "alpha_lambda": "5/2",
    "beta_lambda": "1",
    "lambda_beta": "1",
    "residual": "0"
  }
]

Correction = reconstruction from the recovery: restored arguments, seats, phases and arms in the equation itself. No raw appended constants; any finite term must be derived from the recovered channels, never fitted.

{
  "symbolic": "beta_lambda - lambda*beta = 0",
  "sample_residuals": [
    "0",
    "0",
    "0",
    "0"
  ]
}

{
  "primary_lane": "exact rational arithmetic",
  "phase_requirement": "retained when a phase-bearing arrival-time or wave observable is admitted",
  "numerical_floor": "quadruple precision or better for future numerical lens reconstructions"
}

0 exact for the mass-sheet transformation identity

{
  "statement": "The repaired uniqueness claim fails if two distinct lambda values remain indistinguishable after every declared independent lane Q is included.",
  "result_for_imaging_only": "PASS: the explicit lambda family proves imaging-only non-uniqueness.",
  "result_for_added_lane": "OPEN until a particular data set, Q operator, model class, and uncertainty domain are supplied."
}

EXACT_SEAM_RECOVERY; ABSOLUTE_LENS_MEMBER_OPEN

RED

2026-08-30T09:29:43.629150+00:00