Bind the retuned ordinary rate to the complete T74 layer

research/t74-retuned-ordinary-layer/EVIDENCE.md

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+# Local evidence and reproduction boundary
+
+Hiroki KUMAZAKI, with substantial OpenAI Codex assistance. Apache-2.0.
+Recorded 2026-10-18 on Linux x86_64, CPython 3.12.13, standard library only.
+Source base: `20ecf2891d80b5acf18d980b2164fb44ea7faa90`, priority revision 6.
+
+The following completed successfully:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/t74-retuned-ordinary-layer --check-only
+python3.12 -B scripts/research_package.py --package research/t74-retuned-ordinary-layer
+python3.12 -I -Werror -B research/t74-retuned-ordinary-layer/verify.py --suite rate
+```
+
+The registered launcher supplied fresh private bytecode namespaces and ran
+both suites with `-I -O -Werror -B`. Normal and optimized rate receipts were
+identical. The metadata PASS is not mathematical acceptance.
+
+## Rate and source receipt
+
+- `PASS_RETUNED_T74_RATE` and `PASS_COMPLETE_ORDINARY_BACKEND_IDENTITY`.
+- Complete profile: 666 calls, 66,304 families, 2,394,506 positive children,
+  rank mass 4,816,512. Both independent exact engines returned the common
+  outward 30-place enclosure recorded in PROOF.md.
+- `tau=332353/332929`, `sigma=998271/1000000`, stopped-leaf exponent
+  `998272729/1000000000`, final exponent `998273/1000000`.
+- Seven rate controls reject the direct ordinary-moment substitution, one
+  extra rank-one child, the old ordinary profile, earlier stopping, an
+  unpaid leaf penalty and absent internal/reservation logarithmic margins.
+- The independent rational toy tree explicitly enumerates 31 internal nodes
+  and 125 leaves; costs are 61 and 250. Its three controls expose the reversed
+  inequality, omitted leaves and changed strict-threshold behavior.
+- The exact PR3562 proof/interface bytes match their original Git blobs and
+  SHA256. All 35 files in its exact backend export manifest occur unchanged
+  in the authenticated T74 controller archive closure. This compares bytes;
+  it does not independently prove PR3562's all-size ordinary theorem.
+
+## Factory receipt
+
+- `PASS_RETUNED_T74_FACTORY` and `PASS_RETUNED_T74_FACTORY_BINDING`.
+- Whole native constructor AST equals the retained constructor after changing
+  exactly the `Profile` keywords to `(1,577)`. Both comparisons use the same
+  interpreter; no cross-version AST serialization hash is an acceptance pin.
+- Public production entry-point recorder traces T74 binding, catalogue,
+  specifications, topology, native binding, mode-zero compilation, ordinary
+  construction and the root receiving the same native object.
+- Nine controls reject the old profile, invalid production prime/type,
+  changed capacity, fixture-only index floor, wrong downstream native,
+  non-ordinary object, root-native substitution and stale private binding.
+- Reduced direct root: four packets, digest
+  `4fced21f9e114bff265815a79dfe0b02e2989a195c87a6c5a16b7041193fd760`.
+- Reduced fast root: one packet, five entries and five returns, digest
+  `75b19b9d75e245f1720eaccc048c366a7a68cf7741805e908b0a2222067d4b40`.
+- Ordinary `run` call counts are exactly `[0,0]`. These root fixtures check
+  installed retuned object identity, normalized endpoints, borrowed source
+  and descriptor preservation, balanced returns and owned cleanup. They do
+  not execute the selected ordinary branch, full T74/O73/GL64, or production
+  precision guard. The production construction test uses explicit recorders.
+
+## Publication checks
+
+The focused PR and canonical Discussion carry the publication commit and
+post-commit receipts. Reproduce the byte/history and export checks with:
+
+```sh
+python3.12 -B scripts/check_research_commit.py --package research/t74-retuned-ordinary-layer --check-source-history
+python3.12 -B scripts/export_research_package.py export --package research/t74-retuned-ordinary-layer --ref HEAD --output /tmp/t74-retuned.tar
+python3.12 -B scripts/export_research_package.py check /tmp/t74-retuned.tar
+```
+
+From the Git-free extracted export, run both `verify.py --suite rate` and
+`verify.py --suite factory` commands with a fresh bytecode cache prefix.
+The two pending PR3562 context files and exact manifest are already retained;
+no cached PR Git object or network fetch is required for these checks.
+
+Independent all-size scientific review, complete native/refinement premises
+and the outer error/packing/multiplication transfer remain open. Same-session
+Codex peer review is not independent human or formal verification.

research/t74-retuned-ordinary-layer/PROOF.md

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+# Retuned ordinary work in the unchanged T74 recursion
+
+The new result is a conditional rate transfer and an opt-in factory binding.
+It combines Jose Manuel's PR3562 ordinary bound at `c=1/577` with his
+PR3477 T74 physical layer. It changes neither the T74 word, positive-child
+histogram, recursion stop, guard nor root dimension. In particular, it does
+not substitute an ordinary exponent directly for a supplier moment exponent.
+The two exponents now have the opposite ordering from PR3477's proof.
+
+PR3562 is pinned at `abb93be8b9ceca07a2b17f3a97db7bb6b074271d`; its complete
+[proof](inputs/pr3562-PROOF.md) is retained verbatim with this package.
+The canonical T74 inputs are
+from main `20ecf2891d80b5acf18d980b2164fb44ea7faa90`. `SOURCES.json` and the
+factory verifier bind the precise executable and proof inputs.
+
+## 1. Quantified conditional claim
+
+Retain PR3477's fixed source-admissible prime `q=213*2^80+1`, T74 native
+identities, same-label backward convention, chronological whole-owner
+hosting, complete spectators, parent-return protocol, coefficient-height
+induction and source-to-physical cost contracts. Retain the complete PR2753
+C0 caller and PR2809/PR2899 ordinary/binary refinement, including the
+compatible stock, preparation, elementary primitives, canonical contexts,
+fixed tape inventory and successful-return ownership conditions.
+PR3562 supplies the stronger ordinary exponent under these premises:
+
+```
+c=1/577,  a=c(1-c)=576/332929,
+tau=1-a=332353/332929.
+```
+
+For the retained T74 root domain take `p>=2`, `d=floor(p^(997/1000))`,
+`K=floor(d^(1/500))`, at most `d` selected axes `rho+jK`, and `0<=rho<K`.
+Records have `r=2^Theta(p/d)` Gaussian fields, complete preceding/following
+spectators, and `O(p)`-bit descriptors and addresses. Constants in these
+domains are fixed before the input. The input coefficients lie on the
+`p`-grid in the unit disk. Let `V` be the entire original serialized payload.
+
+With the new ordinary profile, the same source-relative physical construction
+has time and occupied/visited space
+
+```
+T_layer = O_q(V*d^(998273/1000000)),
+S_layer = O_q(V)+F_q.
+```
+
+Its endpoint remains the normalized tensor Walsh transform
+`H0^(tensor D)`, `H0=[[1,1],[1,-1]]/2`, with the same single final
+truncation and error below `sqrt(2)*2^-p`. The layer saving is `.001727`,
+compared with PR3477's `.001695`; their difference is `1/31250`.
+These are layer bounds. No integer multiplication exponent, new supplier
+admission or enclosing outer transfer follows from this result.
+
+## 2. Exact complete moment
+
+Keep `m=73`, `W=896*74=66304` and the full paid histogram
+
+```
+{1:1645680, 2:470196, 3:159468, 4:71928,
+ 5:13716, 8:5328, 48:28190}.
+```
+
+This is the global 666-call bill, including idle remainders. It contains
+2,394,506 children and rank mass 4,816,512. It is not a per-call histogram
+to be divided by 666 again. Set
+
+```
+sigma=998271/1000000,
+mu_sigma=(1/W)*sum_r H_r*(r/73)^sigma.
+```
+
+`rate.py` authenticates the complete profile and nine-stage, 74-replica call
+order. Douglas Colkitt's retained outward log/exp engine and Nick Petalas's
+independently implemented PR3342 positive-series engine both enclose the
+moment in
+
+```
+[999999903535664027556833879161,
+ 999999903535664027556833879162] / 10^30.
+```
+
+Consequently `mu_sigma<1` and `1-mu_sigma>9/10^8`. The geometric-series
+constant below is finite, less than `10^8/9`; it is not dropped from the bill.
+Both engines use rational arithmetic, with independent logarithm reductions
+and outward tail bounds. Their source notices and mathematical attribution
+are retained. This package claims no new interval method.
+
+Two failed alternatives matter for acceptance. At the new ordinary exponent
+`tau`, the complete moment is strictly greater than one:
+
+```
+mu_tau in [1000003022093731739239177793276,
+           1000003022093731739239177793277] / 10^30.
+```
+
+At `sigma`, adding even one rank-one child makes the moment exceed one:
+
+```
+mu_sigma_with_one_extra_rank1 in
+ [1000000111677363244276477808366,
+  1000000111677363244276477808367] / 10^30.
+```
+
+Thus neither a direct ordinary-rate substitution nor an unpriced extra
+recursive operation is covered. A physical linear cost remains a physical
+linear cost; this test does not assign a rank-one tariff to an unspecified
+operation.
+
+## 3. The weighted tree when tau is below sigma
+
+At an internal node of positive selected width `e`, put `f=floor(e/73)`.
+Each rank-`r` child has width `r*f<=r*e/73<e`. The retained source volume
+accounting, including all cover elements, gives the weighted estimate
+
+```
+sum_children V_child * e_child^sigma
+    <= mu_sigma * V_node * e^sigma.                 (1)
+```
+
+Equivalently, each cover child has volume `V_node/(W*Gamma)`, where
+`Gamma=|O(73,2)|`, and summing the complete cover removes `Gamma` once.
+The full `H_r/W` factors in (1) already include every charged child.
+
+The new ordinary contract supplies exactly the same nonchild operations
+as PR3477, with cost
+
+```
+C*V_node*((e*log p)^tau+1).                          (2)
+```
+
+This includes complete native monomial routing, gather/scatter, scalar and
+private-zero operations, replacement copies, parent parking and their
+record-paid setup, under the retained uniform cost premises. The original
+root precision and width remain fixed throughout the recursive call.
+
+Here `tau<sigma`. For `e>=1`, `e^tau<=e^sigma`; hence (2) is at most
+`C*V_node*e^sigma*((log p)^tau+1)`. Summing (1) over tree levels yields
+
+```
+T_internal <= C*V_root*d^sigma*((log p)^tau+1)/(1-mu_sigma).
+```
+
+There is no additional depth factor: the level potentials form a geometric
+series. This is the changed step. The predecessor inequality
+`e^tau<=d^(tau-sigma)*e^sigma`, useful when `sigma<tau`, reverses its useful
+direction when `tau<sigma` and `e<d`. We do not reuse it. The present bound
+has exponent `sigma`, not `tau`. A general sufficient version uses internal
+exponent `max(tau,sigma)`; the replay checks that the new factory is required
+to enter the branch used here.
+
+## 4. Stopped leaves, root reservations and strict logarithmic slack
+
+Keep the actual stopping test unchanged:
+
+```
+e<73 or e^1000<d,    beta=1/1000.
+```
+
+Every stopped positive width obeys `e<max(73,d^beta)<=73*d^beta` for
+`d>=1`. Because `mu_sigma<=1`, the sum of the potentials
+`V_leaf*e_leaf^sigma` over the stopped-leaf antichain is at most the initial
+`V_root*d^sigma`. This follows by replacing any internal node by its
+children in (1); the tree is finite because every positive child contracts.
+The retained elementary leaf cost `O(V_leaf*e_leaf)` therefore sums to
+
+```
+T_leaf = O(V_root*d^(sigma+beta*(1-sigma))),
+sigma+beta*(1-sigma)=998272729/10^9.                  (3)
+```
+
+The root's existing front/back/depth reservations retain their cost
+`O(V*d^(499/500)*log p)` and all fixed-record setup charges. In the direct
+branch, `D<=Q0` and the unchanged reservation formula gives
+`Q0=O(log(2d)+d^(499/500)*log p+1)`. Its elementary `O(V*(D+1))` bill is
+therefore covered by the same reservation bound, even when `D` grows.
+An empty selected set still pays its copies. Only the finitely small `p`
+range is absorbed by a fixed constant. Root padding by less than two occurs
+once, multiplying the whole internal/leaf bill by a constant. It is not
+applied afresh at each recursive child.
+
+Put `alpha=998273/10^6`. The complete strict power margins are
+
+| Contribution | Exponent before logarithmic absorption | `alpha` minus exponent |
+| --- | --- | --- |
+| Internal nodes | `sigma=998271/10^6` | `1/500000` |
+| Stopped leaves | `998272729/10^9` | `271/10^9` |
+| Root reservations | `499/500` | `273/10^6` |
+
+Since `d=floor(p^(997/1000))`, `log p=O(log(2d))`. For any fixed positive
+`delta` and fixed nonnegative `t`, `(log(2d))^t=O(d^delta)` as `d` tends
+to infinity. Apply this with strictly positive margins to the internal and
+reservation terms. Equation (3) is below `d^alpha` as well. This proves the
+stated layer bound, with constants depending on the fixed margins and source.
+There is no logarithm deletion at exponent `sigma`, and no claim that
+`.001727` is an optimal layer saving.
+
+The enlarged fixed precision guard, numerical height induction, root
+normalization, one final truncation and pointwise spectator restoration are
+unchanged. PR3562 preserves the ordinary endpoint and numerical/context
+contracts; it changes the compiled width choice and its uniform cost. The
+existing fixed role/parent-parking space proof therefore continues to apply,
+with its fixed constants and successful-return interpretation. No complete
+GL64/O73 production execution is inferred from this reasoning.
+
+## 5. Factory binding, controls and remaining boundaries
+
+`factory.py` supplies opt-in `build_native(q)` and `production_layer(prepared)`.
+The factory creates the ordinary backend with the PR3562 profile
+`numerator=1, denominator=577`, while retaining the selected row factor,
+dimension/rank, catalogue, topology, compiled programs, capacities and
+`fixture_index_floor=0`. This ordinary object must be the actual native
+argument consumed by the T74 machine and root. An unmodified factory using
+the old profile still has the old ordinary exponent and does not justify
+this claim. The source-bound factory checks exercise this distinction;
+they do not construct the impractical production catalogue. The unchanged
+live T74 binding is loaded under a private module name, preserving the
+native catalogue's distinct archived `binding` module. Scoped import paths
+and module-identity checks bind both implementations without modifying
+either source. The constructor recorders cover the complete production
+entry point; the reduced-root receipt separately reports actual ordinary
+method calls, rather than treating an installed object as an executed call.
+
+The exact rate controls reject the old ordinary exponent at the proposed
+final rate, an earlier `e^500<d` stop, a target that omits the actual leaf
+penalty, and final exponents without the required internal/reservation
+logarithmic slack. Both moment engines separately reject the direct
+ordinary-moment substitution and one extra rank-one child. Contracts remain
+active under Python optimization. These finite checks corroborate the
+written all-size implication rather than establishing physical refinement
+by enumeration.
+
+The work retains all native/common-label and mutable-source realization,
+whole-call hosting, ordinary/C0 physical refinement, complete linear bills,
+precision/tape, restoration and recovery premises at their credited scope.
+It supplies no actual outer synthetic round, integer packing/error transfer,
+full transition table, independent human review or proof-kernel proof.
+The selected frontier and predecessor packages remain unchanged.
+
+Marius-Paul Dumitrean and the PR2979 lineage supplied the T74 construction.
+Jose Manuel supplied PR3477's physical layer and PR3562's stronger ordinary
+coupling; David Arias, Nick Petalas and the retained native/ordinary/compiler
+lineage retain their source credit. New material prepared by Hiroki Kumazaki
+with substantial OpenAI Codex assistance. Apache-2.0.