Original physical source and both strict sectors
Abstract
Opposite second/fourth maximum: physical banks
The quotient-mask construction follows the supplied Pro mathematical construction (SHA256 6895f9bcc25042ab2c2be58543673073aec889bd138a36637a1e4987dd9c7d1e).
All occurrences retain their original natural physical labels. Chain offsets are U:0, V:d, W:d+c and X:d+c+b; their terminals are d-1, d+c-1, d+c+b-1 and k-1, with k=a+b+c+d. Specializing b=d=s in the proof does not rotate, reflect or replace the seed.
The existing ThirdPeakPaidV stock/complement decoder and the MaxPlusTwo chamber mask library select increasing original labels by their actual binary digits. With F=2^(s-1)-1, G=2^(c-1)-1 and H=2^(a-1)-1, the four raw positive banks are U[F xor m], U[m] followed by V[G], W[F xor n], and W[n] followed by X[H]. Existing stock_split proves the exact disjoint complementary inventories, including empty and full masks; canonical_inventory accounts for every terminal and positive occurrence once. Numeric mass alone is never used to infer sparse stock.
For a<c the original concatenation is bank0,u,bank1,v,bank2,w,bank3,xi. For c<a it is bank2,w,bank3,xi,bank0,u,bank1,v. These are separate direct definitions in the original coordinates. The new label k is appended to the positive bank immediately before xi=k-1. The implemented cut is the length of the actual preceding list. Its mathematical value is k-1 in the first order and s+a-1 in the second, by complementary bank lengths. No transformation of a previously proved source is used.
Existing chain_values provides the positive centered values; native_masses supplies each complete-chain balance and doubled terminal value. raw_masses evaluates the selected and complementary banks. The arithmetic windows bound each after-terminal residual and each before-terminal height. four_strip then controls every intermediate prefix because each bank consists of nonnegative steps followed by its terminal; it includes the positive fresh step inside its bank.
Old and enlarged totals are independent. Existing source_totals gives old shared-label sum zero, enlarged shared-label sum -A and fresh value A. Permutation transport therefore gives old total zero and enlarged total (-A)+A=0. The final terminal is controlled by its own head and sign. actual_nat_joint_iff is reused through open private to obtain one original Fin k permutation and its literal earWord, rather than a relabeled scalar surrogate.
The scalar values are the centered representatives of the doubling sequence modulo 2^k-1 at the stated seed. No new group action theorem, orbit classification, transitivity assertion or quotient by physical labels is claimed.