Question 28016: Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Original question and givens

Use mode=residual_forward_confusion, complete model={"slope":17,"intercept":-16,"approximation":18}. Give (mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation).

• G001 — Use mode=residual_forward_confusion, complete model={"slope":17,"intercept":-16,"approximation":18}. 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(16, 17)

Newton residual ORIGINAL affine root = 16/17

Newton residual ORIGINAL affine product := mul(17, 18)

Newton residual ORIGINAL affine product = 306

Newton residual ORIGINAL affine value := sum(306, -16)

Newton residual ORIGINAL affine value = 290

Newton residual ORIGINAL approximation minus root := sub(18, 16/17)

Newton residual ORIGINAL approximation minus root = 290/17

source roles = [17, -16, 18, 16/17]

derived = [290, 290/17, 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, 17, 16/17, 290, 290/17, 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.