Question 14822: The Last Equality Flag Is a Bijection-Supported Certificate

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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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R0028-14822 · q-20261014-advance-20261030-r05-cardinality-comparison-0072

Original question and givens

Use the complete finite source=[210, 211, 212] and target=[310, 311, 312, 313] as distinct-element lists in their original orders, with mapping=[310, 311, 312] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

G001 — Use the complete finite source=[210, 211, 212] and target=[310, 311, 312, 313] as distinct-element lists in their original orders, with mapping=[310, 311, 312] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.

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-cardinality-comparison

Step-by-step derivation

S001 · implies

Before

exact ordered source=[210, 211, 212];exact ordered target=[310, 311, 312, 313];aligned mapping=[310, 311, 312]

After

total typed map on these distinct source/target lists;orders retained

Conditions: none

A supplied mapping binds distinct source/target lists in original order,with exactly one permitted image for each source element.Length comparison alone is not mapping qualification.

Step identifiers and dependencies

Rule: cardinality_exact_bound_total_mapping · Uses: G001

S002 · implies

Before

distinct source indices=[0, 1];source labels=[210, 211];images=[310, 311]

After

same image=No

Conditions: none

The source elements are distinct;compare images for every pair.An empty source is vacuously injective,without automatically being surjective.

Step identifiers and dependencies

Rule: cardinality_every_source_pair_collision_check · Uses: G001, S001

S003 · implies

Before

distinct source indices=[0, 2];source labels=[210, 212];images=[310, 312]

After

same image=No

Conditions: none

The source elements are distinct;compare images for every pair.An empty source is vacuously injective,without automatically being surjective.

Step identifiers and dependencies

Rule: cardinality_every_source_pair_collision_check · Uses: G001, S002

S004 · implies

Before

distinct source indices=[1, 2];source labels=[211, 212];images=[311, 312]

After

same image=No

Conditions: none

The source elements are distinct;compare images for every pair.An empty source is vacuously injective,without automatically being surjective.

Step identifiers and dependencies

Rule: cardinality_every_source_pair_collision_check · Uses: G001, S003

S005 · implies

Before

target label=310;exact original mapping=[310, 311, 312]

After

preimage source indices=[0];covered=Yes

Conditions: none

Every target element needs a nonempty preimage for surjectivity.Record original source indices,not image-as-index values.Coverage of an empty target is vacuous,but total-map typedness still must qualify first.

Step identifiers and dependencies

Rule: cardinality_every_target_fiber_coverage · Uses: G001, S004

S006 · implies

Before

target label=311;exact original mapping=[310, 311, 312]

After

preimage source indices=[1];covered=Yes

Conditions: none

Step identifiers and dependencies

Rule: cardinality_every_target_fiber_coverage · Uses: G001, S005

S007 · implies

Before

target label=312;exact original mapping=[310, 311, 312]

After

preimage source indices=[2];covered=Yes

Conditions: none

Step identifiers and dependencies

Rule: cardinality_every_target_fiber_coverage · Uses: G001, S006

S008 · implies

Before

target label=313;exact original mapping=[310, 311, 312]

After

preimage source indices=[];covered=No

Conditions: none

Step identifiers and dependencies

Rule: cardinality_every_target_fiber_coverage · Uses: G001, S007

S009 · implies

Before

injective=Yes;surjective=No;source size=3;target size=4

After

bijective=No;bijection-supported equality certificate=No

Conditions: none

The source protocol gates the last field on this map's bijectivity.A bad map between equal-sized sets still returns No:it supplied no equality certificate,not a claim that the sizes differ.The unique map between two empty sets is bijective.

Step identifiers and dependencies

Rule: bijection_equality_flag_not_bare_size_equality · Uses: G001, S008

Reverse checks

S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([210, 211, 212], [310, 311, 312, 313], [310, 311, 312], Yes, No, No, No)

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: ce4a1d52884eceb19be789818654436b2b6537219ca9cacdcb8f22864440192d
candidate record SHA-256: c38e86290a35aba74e13c0f72225f06402f938ca646d0fe694d23771035893c0
semantic review receipt SHA-256: bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209ae
question-first witness SHA-256: 61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0
Lean: NOT_RUN · human review: NOT_PERFORMED

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