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Question 28046: A Finite Newton Orbit Records Steps without Certifying Convergence

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0055-28046 · q-20261015-pulse-20261119-r01-0046

Original question and givens

Coefficients in low-to-high degree order are=[-21, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=2 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

  • G001 — Coefficients in low-to-high degree order are=[-21, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=2 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-mod50-newton-finite-orbit, newton-finite-orbit]

After

derived = [[coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope]]

Conditions: none

List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged.

Step identifiers and dependencies

Rule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient1 := mul(1, 0)

After

Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient1 = 0

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient2 := mul(2, 1)

After

Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient2 = 2

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [Newton ORIGINAL orbit polynomial, [-21, 0, 1]]

After

derived = [[-21, 0, 1], [0, 2]]

Conditions: none

Require raw length2..6 before trimming trailing zeros,and remain nonconstant.Differentiate every monomial to i*c_i*x^(i-1).Source derivative coefficients must pass their1e8 profile even if a root later stops or steps=0.

Step identifiers and dependencies

Rule: Newton_polynomial_raw_arity_trim_nonconstant_AND_ALL_derivative_coefficients · Uses: G001, S003

S005 · calculation

Before

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree2 := mul(0, 2)

After

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree2 = 0

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient2 := sum(0, 1)

After

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient2 = 1

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree1 := mul(1, 2)

After

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree1 = 2

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient1 := sum(2, 0)

After

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient1 = 2

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree0 := mul(2, 2)

After

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree0 = 4

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient0 := sum(4, -21)

After

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient0 = -17

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree1 := mul(0, 2)

After

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree1 = 0

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient1 := sum(0, 2)

After

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient1 = 2

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree0 := mul(2, 2)

After

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree0 = 4

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient0 := sum(4, 0)

After

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient0 = 4

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Newton ORIGINAL orbit attempt0 ORIGINAL f over fprime := div(-17, 4)

After

Newton ORIGINAL orbit attempt0 ORIGINAL f over fprime = -17/4

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Newton ORIGINAL orbit attempt0 ORIGINAL tangent zero update := sub(2, -17/4)

After

Newton ORIGINAL orbit attempt0 ORIGINAL tangent zero update = 25/4

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

source roles = [Newton ORIGINAL orbit attempt0, [-21, 0, 1], [0, 2], 2]

After

derived = [-17, 4, updated, 25/4]

Conditions: none

Evaluate BOTH f and f_prime completely,then test f=0 FIRST.An exact root stops even at slope0,without0/0.Only a nonroot requires nonzero slope;the tangent-line zero solves z=x-f/f_prime.

Step identifiers and dependencies

Rule: Newton_EACH_f_AND_fprime_ROOT_FIRST_then_nonzero_derivative_division · Uses: G001, S016

S018 · implies

Before

source roles = [Newton ORIGINAL orbit, 1, No]

After

derived = [[[0, 2, -17, 4, updated, 25/4]], step_budget_exhausted, 25/4]

Conditions: none

Records are actual attempts and may stop early at a root/allowed undefined state.A last update may land on a root,but without another budgeted test status is not changed to root_found.At steps0 qualify inputs without inventing records.A finite trace is not general convergence/rate evidence.

Step identifiers and dependencies

Rule: Newton_attempt_budget_actual_records_status_NOT_guessed_from_final_x · Uses: G001, S017

Reverse checks

  • S018 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S017 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S016 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 ORIGINAL tangent zero update with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 ORIGINAL f over fprime with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient2 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree2 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S003 · premise_and_result_audit — Recompute Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient2 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient1 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

([-21, 0, 1], 1, [[0, 2, -17, 4, updated, 25/4]], step_budget_exhausted, 25/4, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 7187beabe28aaec05b082f26de6cc9787ee61e060d750670346cd79790a34dcb
  • candidate record SHA-256: 153bb80f5732328a3dcc127a73c7b090985c10aeac3f1e51e88ca4b2eb931147
  • semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683
  • question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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