Question 28024: Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Original question and givens

Use mode=residual_forward_confusion, complete model={"slope":25,"intercept":-24,"approximation":26}. Give (mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation).

• G001 — Use mode=residual_forward_confusion, complete model={"slope":25,"intercept":-24,"approximation":26}. 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(24, 25)

Newton residual ORIGINAL affine root = 24/25

Newton residual ORIGINAL affine product := mul(25, 26)

Newton residual ORIGINAL affine product = 650

Newton residual ORIGINAL affine value := sum(650, -24)

Newton residual ORIGINAL affine value = 626

Newton residual ORIGINAL approximation minus root := sub(26, 24/25)

Newton residual ORIGINAL approximation minus root = 626/25

source roles = [25, -24, 26, 24/25]

derived = [626, 626/25, 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, 25, 24/25, 626, 626/25, 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.