Question 28012: Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Original question and givens

Use mode=residual_forward_confusion, complete model={"slope":13,"intercept":-12,"approximation":14}. Give (mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation).

• G001 — Use mode=residual_forward_confusion, complete model={"slope":13,"intercept":-12,"approximation":14}. Give (mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-newton-method

Step-by-step derivation

source roles = [qf-mod50-newton-boundary, newton-boundary]

derived = [[mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation]]

Newton residual ORIGINAL affine root := div(12, 13)

Newton residual ORIGINAL affine root = 12/13

Newton residual ORIGINAL affine product := mul(13, 14)

Newton residual ORIGINAL affine product = 182

Newton residual ORIGINAL affine value := sum(182, -12)

Newton residual ORIGINAL affine value = 170

Newton residual ORIGINAL approximation minus root := sub(14, 12/13)

Newton residual ORIGINAL approximation minus root = 170/13

source roles = [13, -12, 14, 12/13]

derived = [170, 170/13, No]

Reverse checks

• S006 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S005 · premise_and_result_audit — Recompute Newton residual ORIGINAL approximation minus root with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Newton residual ORIGINAL affine value with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Newton residual ORIGINAL affine product with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Newton residual ORIGINAL affine root with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.

Result

(residual_forward_confusion, 13, 12/13, 170, 170/13, No, residual_equals_abs_slope_times_forward_error_not_forward_error_itself)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.