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R0030-15823 · q-20261014-advance-20261031-r01-at-most-countable-set-0048
On Z use f(x)=a*x+b, with nonzero slope a=3, intercept b=10, selected left=-23, right=24. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).
G001 — On Z use f(x)=a*x+b, with nonzero slope a=3, intercept b=10, selected left=-23, right=24. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).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.
Before
affine left nonzero-slope product := mul(3, -23)
After
affine left nonzero-slope product = -69
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001
Before
affine original left image := sum(-69, 10)
After
affine original left image = -59
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
affine right nonzero-slope product := mul(3, 24)
After
affine right nonzero-slope product = 72
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
affine original right image := sum(72, 10)
After
affine original right image = 82
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
fixed integer map f(x)=(3)*x+(10);nonzero slope=3
After
for ANY integers x,y:f(x)=f(y) implies(3)*(x-y)=0,hence x=y
Conditions: none
Subtract the same intercept;the nonzero integer slope times(x−y) is zero.Cancellation in the integers gives x=y for every pair.The displayed left/right check is not the source of universal injectivity.
Rule: affine_universal_injection_by_integer_cancellation · Uses: G001, S004
Before
f maps Z injectively to Z;selected images=[-59, 82];selected-pair implication=Yes
After
encode integer z into N_0 by E(z)=2*z if z>=0,else -2*z-1;even/odd branches disjoint and each injective;E composed with f is injective
Conditions: none
The two branches yield nonnegative even and odd values,so cross-branch collisions are impossible;within each branch cancel±2.Composition gives an injection into the naturals.Negative slopes and equal selected inputs are allowed.
Rule: integer_to_natural_encoding_parity_and_universal_composition · Uses: G001, S005
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 — Recompute affine original right image with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute affine right nonzero-slope product with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute affine original left image with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Recompute affine left nonzero-slope product with exact rational arithmetic, not rounded decimals.(3, 10, -23, 24, -59, 82, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 5c0ee4681afa9c43351bfeb771a84d39e1f5d2b43ec25def360930e2eb345c1ccandidate record SHA-256: 6407416881abacef2b1a804e8fec62b034ac9a4192f9e76e4156bfc8dada5331semantic review receipt SHA-256: 73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bcquestion-first witness SHA-256: ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336Lean: NOT_RUN · human review: NOT_PERFORMED