Question 26618: Coordinatewise Convex Weights Keep the Entire Path in the Box

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0052-26618 · q-20261015-advance-20261117-r06-path-connected-space-0043

Original question and givens

Use coordinate bounds lower=[-19, -20, -21, -22],upper=[20, 21, 22, 23], start=[-18, -19, -20, -21],end=[19, 20, 21, 22]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

G001 — Use coordinate bounds lower=[-19, -20, -21, -22],upper=[20, 21, 22, 23], start=[-18, -19, -20, -21],end=[19, 20, 21, 22]. 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.

mko-path-connected-space

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 = [[-19, -20, -21, -22], [20, 21, 22, 23], [-18, -19, -20, -21], [19, 20, 21, 22]]

After

derived = [4, 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(19, -18)

After

Path coordinate0 ORIGINAL end minus start = 37

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, -19, 20, -18, 19, 37]

After

derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -19, 20, -18, 19]

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(20, -19)

After

Path coordinate1 ORIGINAL end minus start = 39

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, -20, 21, -19, 20, 39]

After

derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -20, 21, -19, 20]

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

S007 · calculation

Before

Path coordinate2 ORIGINAL end minus start := sub(21, -20)

After

Path coordinate2 ORIGINAL end minus start = 41

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, S006

S008 · implies

Before

source roles = [2, -21, 22, -20, 21, 41]

After

derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -21, 22, -20, 21]

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, S007

S009 · calculation

Before

Path coordinate3 ORIGINAL end minus start := sub(22, -21)

After

Path coordinate3 ORIGINAL end minus start = 43

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, S008

S010 · implies

Before

source roles = [3, -22, 23, -21, 22, 43]

After

derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -22, 23, -21, 22]

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, S009

Reverse checks

S010 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
S009 · premise_and_result_audit — Recompute Path coordinate3 ORIGINAL end minus start with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
S007 · premise_and_result_audit — Recompute Path coordinate2 ORIGINAL end minus start with exact rational arithmetic, not rounded decimals.
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

([-19, -20, -21, -22], [20, 21, 22, 23], [-18, -19, -20, -21], [19, 20, 21, 22], [-18+(37)t, -19+(39)t, -20+(41)t, -21+(43)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: f591808c5b9a9e212f686f99e2318d7bc2a2fe25dea2da953a7051d1b22be104
candidate record SHA-256: 751714c5eccf34bf6ea2473fc0adc891276f9df27829b6fbd6a8a7dc395a2a7e
semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
Lean: NOT_RUN · human review: NOT_PERFORMED

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