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Question 26613: Coordinatewise Convex Weights Keep the Entire Path in the Box

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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← Question derivation index · Original question batch

R0052-26613 · q-20261015-advance-20261117-r06-path-connected-space-0038

Original question and givens

Use coordinate bounds lower=[-14, -15],upper=[15, 16], start=[-13, -14],end=[14, 15]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

  • G001 — Use coordinate bounds lower=[-14, -15],upper=[15, 16], start=[-13, -14],end=[14, 15]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

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 = [path-box-linear]

After

derived = [[lower, upper, start, end, formulas, continuous_path, certificate]]

Conditions: none

List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [[-14, -15], [15, 16], [-13, -14], [14, 15]]

After

derived = [2, Yes]

Conditions: none

Check both endpoints against each coordinate interval with matching dimensions.Degenerate coordinates and reverse orientation are allowed.

Step identifiers and dependencies

Rule: Path_ALL_coordinate_endpoint_premises · Uses: G001, S001

S003 · calculation

Before

Path coordinate0 ORIGINAL end minus start := sub(14, -13)

After

Path coordinate0 ORIGINAL end minus start = 27

Conditions: none

Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [0, -14, 15, -13, 14, 27]

After

derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -14, 15, -13, 14]

Conditions: none

For every t in[0,1],weights(1-t),t are nonnegative and sum to1.Thus this coordinate lies between its endpoints and within its bounds;the affine coordinate is continuous,not merely endpoint-checked.

Step identifiers and dependencies

Rule: Path_WHOLE_coordinate_convex_range_and_continuity · Uses: G001, S003

S005 · calculation

Before

Path coordinate1 ORIGINAL end minus start := sub(15, -14)

After

Path coordinate1 ORIGINAL end minus start = 29

Conditions: none

Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [1, -15, 16, -14, 15, 29]

After

derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -15, 16, -14, 15]

Conditions: none

For every t in[0,1],weights(1-t),t are nonnegative and sum to1.Thus this coordinate lies between its endpoints and within its bounds;the affine coordinate is continuous,not merely endpoint-checked.

Step identifiers and dependencies

Rule: Path_WHOLE_coordinate_convex_range_and_continuity · Uses: G001, S005

Reverse checks

  • S006 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S005 · premise_and_result_audit — Recompute Path coordinate1 ORIGINAL end minus start with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S003 · premise_and_result_audit — Recompute Path coordinate0 ORIGINAL end minus start with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S001 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.

Result

([-14, -15], [15, 16], [-13, -14], [14, 15], [-13+(27)t, -14+(29)t], Yes, coordinatewise_convex_combination_stays_in_box)

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: 760d7838a5decbb251eb70b21e4b8145ddda5ff9022bb31764142876e63fd023
  • candidate record SHA-256: ed6a0f196557013eea5bc3057af803b125b5154afd4d32ad6ccca9b8661a0276
  • semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
  • question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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