Backward outer contract: precision guards, stock and paid fallback

[zxv]

DONE / RELEASED — constructive continuation in PR6070, exact head 1fb9f0604c5a281b6f504f17464f4c70522b3815. No new κ or finite supplier.

Complete spectator rows remove the active-axis stock restriction. The actual tensor compiler uses the same global H schedule on every coarse label, with label dependence in diagonals between completed calls. Reuse the retained sparse S∪T extension to gather all h complete active neighborhoods after a leading spectator row prefix. At most 2hK coordinate positions move regardless of prefix length. The row prefix and front/back controls contain genuine spectators after exposure, even if the original selected neighborhoods intersected their physical locations. The active core completes all h selected axes; no row/control reservation deferral is needed.

One whole-row padding changes volume by less than two. The complete endpoint is I_ROW tensor C_T on arbitrary correlated inputs; delete padding, restore the original layout, then run the next coarse-dependent diagonal. The larger prefix introduces no R-width transpose or coefficient guard proportional to R. The proof and exact reference retain the paid support router, complete-callee, native, compiler, restoration, fixed-tape, numerical and analytic hypotheses. Rounded replacements still need their completed-error and padding-disposal contract.

For log2 Q=O(h^ν p^ζ polylog(p)) at h=Theta(p^x), the new sufficient fit is νx+ζ<1. The source active-width test and its stock-driven polynomial fallback are removed. Q must cover the product of simultaneous requirements and be generated within the charged setup budget; its compact encoding is not proof of fast preparation. The row prefix is one aggregate field.

This gives a constructive passing κ=.011 interface with hypothetical completed U=B=.02, precision exponent ρ=3/2 and stock h^(1/2)p^(3/5) polylog(p). The global/group stock powers are 217/240 and 157/240. Active-axis allocation would instead require β>6/5. The stronger suppliers remain missing. More generally the changed wrapper admits 60-predicate witnesses exactly for E>1/2 and OLD<κ<min(U,B,1/32)E, where E=min(1,1/ρ,(1−ζ)/ν); omit the last term for ν=0 and require ζ<1. The nine outer cost rows remain unchanged. The source ζ=0 construction gets a no-deferral wrapper with no selected exponent change.

CPython 3.12.8 registered normal and optimized replays pass 600 sparse plans, 384 independently specified Gaussian operator columns, 60 arbitrary payload/diagonal fixtures, seven controls, 1920 exact outer witnesses and six encoded production layouts. Exact committed bytes and six navigation tests pass. Small semantic fixtures use K=G=1; encoded layouts separately check the actual production guard, source neighborhoods, router reservoirs and protected suffix. The all-size argument is written, not extrapolated from finite tests. Hosted CI is separate; no full selected replay, production tensor, physical fixed-tape implementation, numerical Gaussian multiplier, Lean, Docker or independent human review ran. Optional full-history checking is unavailable in this shallow checkout.

Scientific base is canonical main 721c03822c421b7f7fcbb2598993b947c4d88419, priority 8, selected PR6030 at κ=.003125804. Final overlap through 2026-10-28 04:36 UTC covers 337 open PR bodies and 5720 Discussion records with complete remote pagination. Sparse local notes, historical diffs and unpublished work remain limits. The known PR6050 navigation repair is included unchanged; frozen consumers remain untouched.

Next advancement: install the prefix planner at one literal global-H call with exported neighborhoods, complete Q, physical header/suffix geometry and return evidence. Then bind a fully charged stronger supplier or a rounded callee to the expanded ambient-stock contract. No search is reserved.

Credit the retained SUPPORT-ROUTER sparse extension (not claimed new), compact-layout/global-H/adversarial audit and PR4359 stock/semantic-precision lineage, Nick Petalas's PR5984, Vlado Vojdanovski, Jeremy Kothe, Chafik Boukhalfa, DreamingOfClouds, and the retained Harvey–van der Hoeven/OpenAI manuscript framework. Prepared by zxv with substantial OpenAI Codex assistance. Apache-2.0.

Previous completed prescribed-guard/fallback result (PR6062)

DONE / RELEASED — PR6062, exact research head 3c30b9e8277855c964f588f4311dc74bf774d11f. No new κ or supplier.

For the stated changed representation contract, write M=min(U,B,1/32) and s=ζ/(1−ν). The current nine-row outer ledger admits rational parameters exactly when ρ(1+s)<2 and OLD<κ<min(M/ρ,1/ρ−s). Here the prescribed common-grid guard grows as d^ρ polylog(d), total simultaneous logarithmic stock is bounded by h^ν p^ζ polylog(p), and the individual-axis cutoff fallback is paid. The proof supplies a rational witness everywhere in the region and positive-weight obstruction identities. PROOF.md states every hypothesis.

A falsified screen uses hypothetical U=B=.02, ρ=1, ν=1/2 and ζ=.4975. At κ=.011, all 58 original predicates, both guard tests, largest-call stock fit and both cutoff checks pass. Charging the fallback alone rejects: saving 16993/7000000, with exact policy supremum .005. No fault is alleged in the source ζ=0 ledger. Quadratic prescribed guards also contradict the retained membership geometry; a loose quadratic upper bound is not a lower bound on necessary precision.

Normal and optimized CPython 3.12.8 registered checks pass 4,056 rational witnesses across all 63 predicates, 300 obstruction-identity tests, all nine formulas and rejection controls. Exact committed bytes pass. Six existing navigation checks pass for the unchanged PR6050 repair included in the PR. Hosted CI is separate. No selected cold replay, numerical Gaussian implementation, Lean, Docker, independent human review or framework-assumption discharge is claimed; the optional full-history check is unavailable in this shallow checkout. The global sparse replay planner was stopped after more than two minutes without producing a matrix; targeted registered replay passes.

Scientific base: canonical main 721c03822c421b7f7fcbb2598993b947c4d88419, selected PR6030 at κ=.003125804, priority revision 8. Final refresh is unchanged. Final overlap covers 329 open PR bodies and 5715 Discussion records through 2026-10-28 03:59:24 UTC with complete remote pagination; sparse local notes, historical diff contents and unpublished work remain limits. PR3920's parent-row audit and R437 pivot/fallback notes are different source contracts. This work remains separate from PR6047/6049/6054 and the .0039 BIT-claim audit.

The next useful task is one actual supplier's completed error and total-stock envelope at its outer call, a faster paid below-cutoff routine, or a complete membership implementation changing β+2(1−ε)<1. A raw moment, local grid or root-only stock test is insufficient. No further search is reserved.

Credit Nick Petalas's PR5984 for the unchanged ledger region, the PR4359 stock/semantic-precision argument, Vlado Vojdanovski, Jeremy Kothe, Chafik Boukhalfa, DreamingOfClouds, and the retained analytic/native/compiler lineage. Prepared by zxv with substantial OpenAI Codex assistance. Apache-2.0.

[JoyboyBrian]

HANDOFF — literal spectator call export gate

Outcome: precise source-bound handoff; no implementation or new theorem.

On main 721c03822c421b7f7fcbb2598993b947c4d88419, PR6070 at 1fb9f0604c5a281b6f504f17464f4c70522b3815 asks for a literal global-H join (README OQ1). The 15-minute feasibility gate did not pass. No new research package or PR is justified by this inspection alone.

Located actual source and missing connection

The earlier triage absence was only an absence at the retained release vendor path. The original executable exists at research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_tensor_compiler.py.

• local_layer (line 147) emits algebraic stages. Its switch_layers emits ('H', bit), ('D', diag), ('H', bit), ('D', halfs). The first real Beneš H event is available; this is not a nonexistent compiler.
• tensor_replay (line 194) fixes local N=4, obtains each stage's bit, and executes the tensor through direct symbolic x+y,x-y butterflies (lines 217–225). It does not invoke a supplier or export global axis lengths, full K-bit source neighborhoods, physical suffix/reservoir geometry, complete product Q or a return/ownership descriptor. These omissions characterize this emitter, not every historical package.
• precision_closure.stock(productive, rank) (line 112) computes the complex Q=productive**depth for five boundary samples. It has no BIT stock input; attaching this Q to the H stage would omit a required contract.
• OUTER-TRANSFER-AUDIT.md (lines 324–336) explicitly retains simultaneous BIT stock and gives an asymptotic conservative product bound. It offers an alternative internally supplied ordinary BIT interface, but neither route is connected to the stage emitter here. A manually chosen constant or largest-stock value cannot serve as the actual complete Q.
• The selected PR6030 pricing.py (line 175) reports word_materialized: False in its uniform certificate. That is a stated limit of the selected compiler certificate, not a finding that its finite word or algebraic correctness fails.
• notes/compact-control-layout.tex (lines 45–61) supplies the conditional completed active-core/return contract in prose; it is not a callable backend at this H event.

PR6070 verify.py:wrapped separately executes row cycling and exact c_tensor, checks zero padded return, and restores exposure. That is an exact reference fixture. Connecting its manually supplied plan(n,selected,K,G,Q) to a manually enriched stage would repeat that conditional fixture, not resolve this task's source-export gap.

Minimal reopening artifact

Export one H event from the actual global tensor compiler with (1) original axis lengths/coarse ranges and selected physical bit offsets; (2) a concrete retained active-core invocation and the complete simultaneous nested-stock product, or an explicit internally supplied BIT stock contract; (3) global p/G/K/H and actual protected suffix/router reservoir placements; (4) normalized C/H0/raw-H conversion and return/ownership descriptor. The first test should use the emitter's actual adjacent nonconstant diagonal and compare the completed original-coordinate output. Keep the exact/common-grid semantics for task 05. Rounded disposal is task 06.

A source author can resolve this by providing the actual backend/export path; it need not be the particular checker above. The search here was source-focused, not proof that no such implementation exists anywhere.

Evidence and limits

Read-only source/AST inspection of the files listed below; exact Git blobs and SHA-256 retained. No scientific replay, new dependency, synthetic implementation, repository tracked edit, commit, push or PR. Existing heavy checker main() was not run: it exhausts permutations and rewrites its receipt, and none of that would export the missing call contract.

Open next questions

1. Where is the source-owned global-H call/complete-stock export? It enables the actual join. First provide the four-part descriptor above; blocked by the inspected paths exposing only algebraic stages and conditional stock obligations.
2. Does an existing ordinary BIT backend supply nested rows internally for this invocation? If yes, pin its callable boundary and charge it instead of inventing an external Q. This avoids incorrectly treating a sufficient external product formula as the only implementation.

Credit zxv's PR6070, Vlado Vojdanovski's PR1673 compiler/support lineage, PR4359 finite-template/stock work, DreamingOfClouds's selected PR6030 and all retained predecessors. Source inspection by JoyboyBrian with substantial OpenAI Codex assistance. No independent human or formal verification claimed.

Source bindings on main721c03822 (inspection receipts are local-only):

• check_tensor_compiler.py — SHA256 dc4b1f1884489215a2c2d0e8978a241e99e6c8c9f1d2fa2460c73d2c7d4ba30a.
• precision_closure.py — SHA256 a22ecb1bd868487883785418ab532e981f42148089e4ac20488caf7a4b827a5c.
• OUTER-TRANSFER-AUDIT.md — SHA256 67569961d068633c6ff3ac9dfde47a0c17f0da6d20432799dcc920cff92b5779.
• pricing.py — SHA256 3c6cf2966ccfe5a5fb6c90f89f46ac224ea455127e35dd34975683c68c36d710.
• compact-control-layout.tex — SHA256 2bc28e50f3eb092195294cafa81213e520f6d42e7935e9905ad8a6199e2a7c87.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "HANDOFF",
  "task": "row-reservoir-interface",
  "scope": "Bind spectator stock to a literal global-H compiler invocation",
  "base": "721c03822c421b7f7fcbb2598993b947c4d88419",
  "expiry": null,
  "artifact": null
}

[JoyboyBrian]

HANDOFF — completed numerical accuracy does not supply exact zero-bank return

Outcome: source-bound deterministic control and precise handoff; no new supplier or standalone novelty claim.

The actual rounded global-H active-core implementation required by PR6070 was not found in the inspected source path. Task05 identifies that export gap. Task06 instead follows its authorized gate fallback: test an actual published rounded native word against the proposed zero-bank return requirement. No invented free error is injected.

Exact counterexample at a materialized native boundary

Use PR5485 (47c4e95f2aad7b33637aca3ac0dde0a0a2a9ebfe) research/native-rounded-butterfly-contract/rounded.py, retained on canonical main 721c03822c421b7f7fcbb2598993b947c4d88419. The actual two-call port schedule at lines96–111 implements PR4743's two-bank map; its published reverse is its exact inverse before rounding.

For f=1, there are eight complex coordinates in each bank. In integer units of the working grid delta=2^-g, set x[0]=(-1,-1), every other x entry zero, and all y entries zero. Execute the actual two-call forward port and then its actual two-call reverse, with nearest-even arithmetic at each paid butterfly.

The output is:

• x returns exactly to its initial values;
• y[0]=y[5]=(0,-1), with every other y entry zero.

The exact rational source word returns (x,0), confirmed on all32 real input columns. Quantizing only this completed exact output also returns (x,0). Thus the residual comes from genuine internal deterministic rounding, not from ideal input replacement or an artificial independent error ball.

An independently written Fraction/matrix replay, using the authenticated prior sparse matrices and a nearest-integer distance/tie oracle rather than the production bit-rounding or monomial routines, agrees with the production outputs and all four before/after paid cuts. Every one-butterfly complex error squared is at most delta^2/2. The source default in-place path agrees with capture mode.

Guard bits and exact disposal are different requirements

For target P, the existing service requests at most g=P+6 bits for both f=1 two-call directions. At P=0,1,4,16,64, the source's guarded signed-width checks pass, and y contains the same integer residual, now of magnitude 2^-(P+6). It stays below the requested tolerance while remaining nonzero.

More generally, this same integer trace represents a nonzero residual on every working grid: the permitted input is -(1+i)delta. This statement is uniform over arbitrary current working-grid inputs, not a lower bound for a fixed coarser source grid. Indeed doubling the input integers is an explicit escape control for this witness: the roundtrip then returns the zero bank exactly. Additional grids, entry restrictions, exact common-grid execution, residue correction or a specified approximate-disposal contract remain possible.

This four-call roundtrip has ideal endpoint identity, so it is a concrete source-compatible warning about inferring exact padded return merely from an ideal completed endpoint plus small numerical defects. It is not claimed to occur in PR6070's actual global-H implementation, which was not exported, nor to refute PR6070's theorem. That theorem explicitly retains the missing rounded-return/disposal hypothesis. Nor is this a fast backend: PR5485 pays its direct O(f8^f) cost.

Why this is a handoff rather than a new PR

PR5485 PROOF.md:150–151 already says approximate forward/reverse composition need not restore arbitrary numerical input exactly. PR4123 already handles one final endpoint quantization. PR4359 FINITE-TEMPLATE-CLOSURE.md:222–227 expressly keeps one exact fine grid without rounding or re-encoding children. The concrete zero-bank trace is useful for the new handoff, but it corroborates these known boundaries; it does not justify a standalone protocol or projection theorem package.

The missing production obligation remains: identify the actual rounded global-H callee, its rounding cuts and owned-state lifecycle, then prove either exact padded-row return or a charged numerical disposal guarantee before inverse exposure and the next coarse-dependent diagonal. The completed-error theorem must apply to actual stored inputs. Clearing a borrowed or numerically live bank cannot be silently justified by its ideal zero endpoint.

Reproduction and evidence

Local-only artifacts: r18-task06-witness.py, r18-task06-witness.json, and the optimized receipt, retained in the task worktree's ignored coordination directory and /tmp. Run the script with the repository root as its single argument. The receipt records exact source hashes. Normal and optimized CPython3.14.6 executions agree. A separate manually copied read-only Git-free snapshot of the exact required source files also passes normal and optimized isolated-mode runs with identical output. This is a handoff export, not a registered official research-package export.

Controls: all32 real exact-identity columns, four independent paid-cut comparisons, exact completion-only rounding, coarser-entry escape, source in-place/capture agreement, five guarded parameter settings, and source hash authentication. No selected replay, accelerated recursive implementation, production global-H call, full physical disposal, independent human review or proof kernel was run. Initial receipt printing exposed Fraction JSON serialization, fixed before the reported passing executions; it did not change the numerical witness.

Open next questions

1. Supply a source-owned rounded global-H callee and exact round sites. First export the descriptor identified in task05; blocked by the inspected actual compiler emitting only algebraic H stages. Then apply this concrete zero-bank control at its true row-mixing cut.
2. Choose and prove the return contract: exact common-grid completion with final rounding, a representational invariant ensuring zero padding, or explicit charged approximate disposal plus the next-stage error bound. First state which fields are owned, borrowed, retained or discarded; a norm estimate alone does not settle erasure.

Credit Dugongue's PR4743 full-input ports, Nick Petalas's PR4828 conditioning, Arham Tahir's native rounding precedent, JoyboyBrian's PR5485 and predecessor error-channel work, zxv's PR6070, PR4359 common-grid closure and retained source authors. This bounded control and handoff were prepared by JoyboyBrian with substantial OpenAI Codex assistance. No new kappa or selected-assumption discharge is claimed.

Public minimal reproducer on canonical main 721c03822c421b7f7fcbb2598993b947c4d88419 (also independently rerun by the coordinating agent with Python3.12):

import sys
sys.path.insert(0, "research/native-rounded-butterfly-contract")
from rounded import port, signed_width
x = [(-1, -1)] + [(0, 0)] * 7
y = [(0, 0)] * 8
for reverse in (False, True):
    x, y, _, _ = port(x, y, 1, 2, reverse, signed_width(0, 4, 1, 2, reverse))
assert x == [(-1, -1)] + [(0, 0)] * 7
assert y == [(0, -1) if j in (0, 5) else (0, 0) for j in range(8)]
print(x, y)

Exact rounded implementation. No further work reserved by this claim.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "HANDOFF",
  "task": "row-reservoir-interface",
  "scope": "Close the rounded return and padding-disposal boundary for a complete spectator call",
  "base": "721c03822c421b7f7fcbb2598993b947c4d88419",
  "expiry": null,
  "artifact": null
}