Classify and encode the exact correlated P104 producer entrance

research/p104-stagewise-entrance/COSTS.md

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+# Direct preparation cost and quotient audit
+
+This note prices the explicit preparation from **52 owned Gaussian coefficient
+banks to the admitted Y module**. It uses the actual inverse donor paths,
+the extra right wrappers, and the groupwise bank differences. It does not
+price an arbitrary pre-existing Y caller, the other events of the original
+prefix, or the 675 external-recipient histories.
+
+The inventory was independently reconstructed from the frozen WORD and
+NATIVE data, without importing the package source layer, classifier, or a
+predecessor executable. It is a constructive direct arithmetic schedule,
+not a minimum, a physical tape tariff, or a supplier improvement. The
+[proof](PROOF.md) states the admission theorem and codec contract;
+[SOURCES.json](SOURCES.json) retains the exact origins and mathematical
+dependencies.
+
+## 1. What the quotient counts mean
+
+Put $R=\mathbb Z[i]$, $\pi=1+i$, and $B=B_9$. The 32 intervals without
+a D transport contribute diagonal exponents
+$\max(4-|S|,0)$. The 15 rank-one intervals contribute
+$\max(4-|S\setminus\{0\}|,0)$. Each of the five terminal coordinates
+contributes exponent four on all 512 coefficients.
+
+| Exponent | No-transport multiplicity | Rank-one multiplicity | Terminal multiplicity | Aggregate |
+|---|---:|---:|---:|---:|
+| 0 | 32 × 382 | 15 × 326 | 0 | 17,114 |
+| 1 | 32 × 84 | 15 × 112 | 0 | 4,368 |
+| 2 | 32 × 36 | 15 × 56 | 0 | 1,992 |
+| 3 | 32 × 9 | 15 × 16 | 0 | 528 |
+| 4 | 32 × 1 | 15 × 2 | 5 × 512 | 2,622 |
+
+The quotient of $B^{52}$ by the exact stagewise entrance $\mathcal M$ is
+
+$$
+(R/(\pi))^{4368}\oplus(R/(\pi^2))^{1992}
+\oplus(R/(\pi^3))^{528}\oplus(R/(\pi^4))^{2622}.
+$$
+
+The exponent-zero coordinates have trivial quotient factors. All 52 input
+banks remain present; each admitted module has the same full Gaussian rank.
+The index is
+
+$$
+[B^{52}:\mathcal M]
+=2^{4368+2(1992)+3(528)+4(2622)}
+=2^{20424}.
+$$
+
+For the ideal terminal-only module $\mathcal I$ from PR #3426 and the
+stronger product entrance $\mathcal P=(4B)^{52}$, the inclusions are strict:
+
+$$
+\mathcal P\subsetneq\mathcal M\subsetneq\mathcal I,\qquad
+[B^{52}:\mathcal I]=2^{10240},\quad
+[B^{52}:\mathcal P]=2^{106496}.
+$$
+
+Consequently $[\mathcal I:\mathcal M]=2^{10184}$ and
+$[\mathcal M:\mathcal P]=2^{86072}$. These quantify the additional
+stagewise constraints and the larger allowed correlated set. They are not
+saved physical bits, compressed storage, or eliminated operations.
+
+The elementary checks behind the first two rows are
+
+$$
+\sum_{j=0}^{3}(4-j)\binom9j=187,\qquad
+2\sum_{j=0}^{3}(4-j)\binom8j=280,
+$$
+$$
+32(187)+15(280)+5(2048)=20424.
+$$
+
+## 2. Actual path reconstruction
+
+The independent counter finds the first C allocation at zero-based event
+2025, derives the five D source roles from the following five collector ADDs,
+and finds their earlier quarter reads in the original order. It obtains group
+sizes 3, 7, 7, 11 and 24.
+
+For each nonterminal read, it follows that D bank's native history from its
+initial empty frame to the read frame. It separately checks the interval to
+the next read. Every selected native occurrence is bound to the archive's
+old frame, new frame and rank. The 32 empty intervals and 15 single rank-one
+intervals yield 47 preparation paths.
+
+A rank-one coordinate first needs the inverse of its actual right wrapper.
+It then uses the inverse D path, with native occurrences reversed and each
+native inverted in its actual factor order. This is a constructive schedule
+on independently supplied coefficient arrays. Repeated uses of one source
+recipe are counted once per prepared array, not once per distinct archive
+record.
+
+| Preparation part | Native occurrences | Rank sum | Wrapper occurrences | Wrapper cycles | Gaussian payload SWAPs |
+|---|---:|---:|---:|---:|---:|
+| Inverse donor paths | 158 | 281 | 316 | 9,748 | 152,044 |
+| Extra inverse right wrappers | 0 | 0 | 15 | 704 | 6,976 |
+| Total | 158 | 281 | 331 | 10,452 | 159,020 |
+
+Each rank unit acts on 256 address pairs, giving **71,936 pair applications**.
+Each wrapper permutation is reconstructed on all 512 addresses and checked
+to be a bijection. A cycle of length $l$ uses $l-1$ direct payload SWAPs,
+so the total is
+
+$$
+331(512)-10452=159020.
+$$
+
+These are reversible gates on two owned Gaussian payload words. They need no
+separate clean swap ancilla. This is different from the credited predecessor
+schedule using $N+c$ swaps through one owned zero Gaussian scratch word;
+those two invoices must not be mixed. Permutation cycles are unchanged by
+taking the inverse.
+
+## 3. Phases and the complete arithmetic subtotal
+
+The wrapper phase is evaluated at its source address, using its recorded
+constant phase, linear terms and only strict-upper quadratic terms $j<k$.
+The inverse has the negative phase at the corresponding permuted address.
+Its histogram is therefore the negated original histogram, since the
+permutation is bijective.
+
+| Gaussian multiplier | Encoder inverse wrappers | Decoder forward wrappers |
+|---|---:|---:|
+| $1$ | 61,696 | 61,696 |
+| $i$ | 45,952 | 29,312 |
+| $-1$ | 32,512 | 32,512 |
+| $-i$ | 29,312 | 45,952 |
+| Total | 169,472 | 169,472 |
+
+The histogram does not substitute for correct phase placement. Source
+authentication and the actual ordered wrapper implementation bind that
+operator. The identity entries need no arithmetic; their address/plan visits
+are still part of execution.
+
+All 52 Newton encoders have $52\cdot9\cdot256=119808$ tensor pairs.
+Nonterminal profile scaling adds 10,184 multiplications by $\pi$.
+Terminals use positive $4$, implemented by two Gaussian doublings per
+entry; using $\pi^4=-4$ without its sign would change the parameter map.
+
+| Primitive in preparation | Count |
+|---|---:|
+| Gaussian ADDs in Newton encoders | 119,808 |
+| Gaussian multiplications by $\pi$, including profile diagonals | 129,992 |
+| Gaussian terminal doublings | 5,120 |
+| Full-bank differences | 47 |
+| Gaussian subtractions in those differences | 24,064 |
+| Literal native pair applications | 71,936 |
+| Native Gaussian quarter turns | 71,936 |
+| Gaussian payload SWAPs | 159,020 |
+| Wrapper phase applications, including identity entries | 169,472 |
+
+A Gaussian ADD/subtraction and a multiplication by $\pi$ each use two
+real ADD/subtract operations. The inherited literal pair uses eight such
+operations and four signed floor halves, with its quarter turn listed
+separately. This gives **1,123,216 real ADD/subtract operations plus 10,240
+real doublings**, or **1,133,456** when a doubling is charged as one addition.
+There are also **287,744 native real floor halves**, together with the
+explicit swap and phase primitives above.
+
+For a version lowered entirely to the stated real arithmetic primitives,
+use three ADD/subtract operations per real-component SWAP, hence six per
+Gaussian SWAP. A quarter turn uses one real-component SWAP and one
+negation. A Gaussian minus sign uses two negations. The three-operation
+real SWAP is explicitly
+
+```text
+x <- x+y
+y <- x-y
+x <- x-y
+```
+
+This component-arithmetic model permits a source-minus-target overwrite.
+A primitive set restricted to `target += +/-source` needs an additional
+target negation, so these optional lowered totals are not the repository's
+strict ADD tariff. Counting each displayed negation as `0-value` gives
+2,741,400 component additions/subtractions for preparation and 2,731,160
+for the inverse, with halves still separately charged.
+
+| Additional lowering | Count |
+|---|---:|
+| Real ADD/subtract operations for Gaussian payload SWAPs | 954,120 |
+| Wrapper and native quarter-turn component SWAPs | 147,200 |
+| Real ADD/subtract operations for those component SWAPs | 441,600 |
+| Explicit real negations | 212,224 |
+
+The fully lowered encoder therefore uses **2,529,176 real ADD/subtract
+operations**, **212,224 real negations**, and **287,744 native real floor
+halves**, under this direct model. Doublings are included in that ADD count.
+Addressing, iteration, compilation and the eventual bit/tape tariff remain
+separate costs.
+
+The inverse codec has the same path/swap/phase-count totals. Its base real
+ADD/subtract count is **1,123,216**: terminal doublings become exact
+divisions. It has **259,984 real exact halves** from all Gaussian
+$\pi$-divisions and **10,240 terminal exact halves**, in addition to the
+same **287,744 native floor halves**. With the same swap/phase lowering,
+its ADD/subtract count is **2,518,936**, with **212,224 real negations**.
+It rejects unavailable divisions; it does not round inputs into the module.
+
+## 4. Conservative finite width
+
+Let $H$ bound every real and imaginary component of each of the 52 raw
+coefficient arrays. The exact real row-sum gains from coefficients to the
+three local profile arrays are
+
+$$
+G_0=3456,\qquad G_1=3828,\qquad G_T=13452.
+$$
+
+For example,
+$$
+G_0=\sum_{d=0}^{9}\binom9d
+2^{\lceil\max(4,d)/2\rceil},
+$$
+$$
+G_1=\sum_{d=0}^{8}\binom8d
+\left(2^{\lceil\max(4,d)/2\rceil}
++2^{\lceil(\max(4,d)+1)/2\rceil}\right).
+$$
+
+The coefficient supports entering each Newton pair are disjoint, so these
+nonnegative absolute row sums also bound the intermediate encoding states.
+Terminal doubling prefixes are bounded by $G_T H$.
+
+A 512-entry Gaussian array whose real and imaginary components are bounded
+by $M$ has complex Euclidean norm at most $32M$. Each actual monomial and
+complete coordinate stage is unitary. The native-path portion of the literal
+preparation agrees with those ideal stages because its native-path inputs
+and stages remain in $B_9$. Therefore:
+
+| Stored or work quantity | Sufficient component bound |
+|---|---:|
+| Nonterminal inverse-path payload | $122496H$ |
+| Terminal partial sum | $13452H$ |
+| Any final Y difference | $244992H$ |
+| Literal pair arithmetic work | $612480H$ |
+
+The terminal/nonterminal difference separately has bound $135948H$, already
+covered by the common Y bound. The factor-five pair bound covers differences,
+signed halves, middle values and reconstruction temporaries. Lowering a
+payload SWAP to additions introduces at most a doubled intermediate; a
+quarter turn of a pair difference introduces at most four times the payload
+bound. Both fit within the same $612480H$ allowance.
+
+For the inverse codec, let $J$ bound the components of an admitted prepared
+Y input. A group has at most 24 members, so its partial sums are bounded by
+$24J$. Forward paths are bounded by $768J$, and the Newton decoder's
+component row-sum gain is at most 32. Subsequent exact profile divisions do
+not increase the component bound. Thus $24576J$ bounds stored decoding
+values and $98304J$ conservatively bounds decoding work.
+
+These are sufficient bounds for this codec. Signed storage with an integer
+bound $M$ may use $1+\lceil\log_2(M+1)\rceil$ bits. Actual widths must
+also cover the six separately supplied dirty banks and any surrounding
+caller; no current external recipient width is established here.
+
+## 5. Ownership, logical target and stopping decision
+
+The in-place encoder uses the 52 owned coefficient banks themselves:
+26,624 Gaussian payload words. The six independently admitted dirty D/Q
+banks add 3,072 Gaussian words and are not modified by preparation.
+The bank-difference step runs backward within each group, so the previous
+partial sum remains available until its last use. The inverse runs forward.
+Constant arithmetic work space suffices for the stated arithmetic gates;
+the Python reference's allocated lists are not this physical layout.
+
+An encoder followed immediately by its inverse restores its coefficient
+input. This does not prove that an intervening producer or consumer returns
+the prepared banks in the required values, frames, layout and ownership.
+Nor does the encoder admit arbitrary pre-existing numerical Y values while
+preserving their original interpretation. It supplies a different, explicitly
+parameterized entrance contract.
+
+If raw coefficient arrays must remain borrowed and unchanged, an optional
+copying boundary additionally needs 52 owned banks and 52 copy ADDs.
+Cleanup needs the actual prepared banks returned before inverse decoding,
+then 52 subtractions against the borrowed originals and exact zero disposal.
+That return is a separate caller obligation. The copies, lifetimes and
+cleanup do not acquire zero cost from the module isomorphism.
+
+| Obligation | Status here |
+|---|---|
+| Direct finite preparation and inverse arithmetic | Constructively counted |
+| Actual frame-path, wrapper-cycle and phase inventory | Independently reconstructed from pinned source |
+| Codec magnitude bounds | Proved conservative bounds |
+| Preparation of the six arbitrary dirty B-domain entrances | Separate input/codec obligation |
+| Preservation of borrowed coefficient inputs | Additional copies, owners and cleanup |
+| Physical routes, controls, bounded-width gate costs and liveness | No complete physical tariff supplied |
+| Omitted original-prefix and external-recipient histories | Outside this cone result |
+| Compatible complete consumer and recursive supplier bill | Open |
+
+The unchanged five-forward/four-dual consumer profile and coverage law still
+have the credited $599/584>1$ obstruction. A larger exact input domain does
+not change that law or remove native children. This result narrows the missing
+admission question; it does not reopen the stopped supplier construction or
+claim a new $\kappa$.
+
+The actual word and native recipes remain ywz65's PR #2586 contribution,
+retained through Arham Tahir's source package. Nick Petalas's PR #3426 supplies
+the ideal correlated classification and this literal-admission handoff; his
+PR #3416 supplies the inherited covariance theorem. Clear1nsight's PR #3382
+and PR #3423 supply the Newton application and prior actual-interface work.
+Tekalign's signed-floor and native-comparison lineage, Ross Richardson's
+exactness-domain work, and Jose Manuel/Vasek's conditional cost stop retain
+their credit. This audit used separate automated source/count reconstruction,
+not external human or formal proof-kernel verification.
+
+Prepared by Clear1nsight with substantial OpenAI assistance. Apache-2.0.