Audit the P104 kernel against its actual required X0 consumer

[Clear1nsight]

Problem and result

PR #3554 left two conditional handoffs: identify an actual physical provider of its coefficient interface, and identify a required legal readout of its five-readout kernel. This PR answers them with one source-bound consumer audit against the unchanged PR #2586 word.

The original coefficient witness a44 = delta15 prepares Y91 = p, Y90 = -p. It fails the actual full-word contract at zero-based event 1099, D1 += Y90/4 in frame [381,241,2]. At address 10, the donor is (-2,0), requiring real increment -1/2. All 1,099 earlier events complete exactly, with no completed event output outside B9. This binds an actual omitted read, rather than merely failing a sufficient product-domain condition.

There is also an admitted observable family. For every u in B9, set Y91 = 4u, Y90 = -4u and all other initial public banks to zero. A written induction, supported by the complete scalar denominator certificate and credited native preservation/covariance, proves exact execution through all 5,398 events and

$$\rho_5(y)=0,\qquad X0_{\mathrm{final}}=-2A_Fu.$$

Thus no function of the five readouts alone can supply this required X0 endpoint, even on the strong domain. The actual path is Y90 to D1, the eighth collector's Q0, G0, and X0. The late X0 sign at event 5086 is essential. The displayed D27 transient at 1817 to 1819 has no intervening direct reader.

The unchanged word has 728 Y writes, all bank-local native transports. It cannot materialise the coupled K encoder under a direct one-bank-per-independent-coefficient identification, even after relabelling. All 104 final Y frames have dimension eight, while the logical caller returns its coefficient fields in empty frames. External preparation, virtual coefficient access, and adapters remain possible, with an explicit physical contract and complete price.

What changes

Only research/p104-kernel-consumer/ is added. It contains the written proof, local cost ledger, three complementary implementations, deterministic receipt, exact sources, task/overlap record, and registered replay commands. All 20 live inputs already belong to canonical main's ancestry. Frozen predecessor bytes, shared tools, and the selected frontier remain unchanged.

The primary path reuses the authenticated PR #3467 K codec and a forward scalar/chart oracle. The separate literal interpreter imports no predecessor executable. A third implementation derives the reverse X0 row and compares all 104 Y coefficients against PR #3462's existing CA/2 certificate. It reuses the exact PR #3469 X0 adapter without rerunning or claiming those full completion/adapter theorems.

Verification

The 4p finite control completes all 5,398 events. It binds all 2,070 native occurrences, compares 102,656 completed signed-floor/ideal pairs across 223 active native calls, and evaluates 1,847 zero calls by their exact zero action. All 3,104 ADDs, 208 signs, eight private zero erasures, 358 final frames, and the exact nonzero bank support are checked.

Complete-array comparisons cover the independently prepared 512-entry p, 9,216 Q0/G0/X0 checkpoint coordinates with frames, and 1,024 returned Y coordinates. The arbitrary normalised D1/Q0/G0 restoration rows are checked separately; the two literal fixtures have zero initial dirt. Dropping the late sign or Q0 cleanup gives the recorded separating failures.

Local registered normal and optimised Python 3.12 replays, metadata/pins, committed-byte/source-history, checked export, Git-free direct replay, and actual CLI rejection controls are the publication gates. The public handoff records their exact final head, receipt, export hash, and hosted status. The actual controls remove WORD, alter WORD, and substitute Boolean true for the integer receipt schema. Normal replay never regenerates evidence.

python3.12 -B scripts/research_package.py --package research/p104-kernel-consumer
python3.12 -B scripts/check_research_commit.py --package research/p104-kernel-consumer --check-source-history

Scope, cost, and remaining obligations

This family lies in S = (4B9)^52, so it demonstrates no advantage from L/S. PR #3554's full caller returns all coefficient banks and retains their information. This result blocks access limited to the five readouts; it does not invalidate that logical routine.

The actual eighth collector costs 39,936 component ADD/SUB evaluations, with native halves, quarter turns, all four wrappers, owned C, and cleanup itemised separately. Three named later scalar updates add 3,072 component ADD/SUBs. A temporarily mutable quarter-read donor has its exact halves and restoring doublings charged; a read-only donor needs a separate implementation. These are local counts, not a complete physical or recursive invoice.

Actual coefficient production/access, all required outputs and frames, ownership, routing, control, zero supply, finite widths, and a favourable complete supplier allowance remain open. The inherited fixed-profile 599/584 stop remains conditional on its original premises. No supplier, new kappa, independent human review, or formal proof-kernel acceptance is claimed. No further construction is reserved without a named compatible caller and complete allowance.

Coordination and credit

Canonical main remained 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6, at the pre-submission checkpoint. The complete entry snapshot and bounded subsequent body/head and Discussion deltas, with their limits, are in TASK.json. The investigation's existing canonical claim is #2750 (comment).

Clear1nsight, with substantial OpenAI assistance from a root agent and three concurrent specialist agents. Credit PR #3554/#3467/#3423/#3382; Nick Petalas's #3406/#3416/#3426/#3462/#3469 and distinct #3466 work; Hiroki Kumazaki/#3446; ywz65/#2586; Arham Tahir/#2665; Tekalign/#3025/#3218; Ross Richardson/#3068; Jose Manuel, Vasek, and the retained geometry/native/compiler lineage. Licences, notices, exact source hashes, and predecessor meanings are retained. New work is Apache-2.0.

[Clear1nsight]

Public source export and final verification

p104-kernel-consumer-source.zip

The published head is 33997335997f1eef37ee4f455244cd289ba07b06, with tree 90dc66dfe0195ec13db1666bdeb4ad7eb26fdeb9. That public tree equals the independently tested local tree. The parent and sole declared source-history ancestor are canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90.

The public committed-byte and source-history gate passes for all 31 files, comprising 11 package files and 20 declared inputs. The export adds its transport metadata, giving 32 regular files. Two independent repository exports were identical in complete bytes. The attached ZIP preserves every file's complete bytes and mode from that verified tar.

Artifact	Bytes	SHA256
Attached ZIP	218,723	d8e0c1fd4c3442cf7ad8f8d485cfcc1a8dc03984fc71ec41b5ec4c8c3dcf57dd
Repository export tar	880,640	a52eb2b8750c02752c72e6051cd408b2821cc0cb706785513950ad91de5dfb89
Scientific receipt	76,497	d9626b8f44b83d50554af5ff868888327d68507fca5c23924e0ac89ff3e413b0

Both registered Python 3.12 modes pass. The actual optimised replay from the public commit's Git-free export also passes, with all source/receipt/export bytes and modes unchanged. Three separate actual CLI processes reject missing WORD, an appended space in WORD, and Boolean true substituted for the integer receipt schema, each with exit 1 and no PASS banner. Each disposable copy changes only its intended file and makes no further changes during replay.

Hosted workflow 39105350149 also succeeds, including the unchanged-input check. It tests GitHub's preview merge 4ca84be86f26dc3d0f91a4982a9c5c99d341b3b3 of this exact head with main20ec. Both hosted normal and optimised commands reproduce the scientific receipt above. The PR remains open for review; no merge was performed.

From the unpacked source root:

python3.12 -I -O -B research/p104-kernel-consumer/verify.py

The source-bound theorem, finite literal controls, inherited native/covariance premises, local cost ledger, and unfilled physical caller/supplier obligations remain distinct. In particular, the successful paired family lies in S and establishes no L/S advantage or selected kappa change. Scope, predecessor credit, and the released handoff are in the package and canonical investigation. These are author and separate AI-agent checks, not independent human or formal proof-kernel acceptance.

[Clear1nsight]

FINDING: exact Gaussian scalar admission of PR3577's fixed kernel direction

For the particular direction identified in PR3577, allowing Gaussian integer rescaling does not improve on the factor-four control: its exact scalar admission ideal is $4\mathbb Z[i]$.

This finding fixes the published source head 33997335997f1eef37ee4f455244cd289ba07b06. Let $R=\mathbb Z[i]$, $\pi=1+i$, and let $p$ be exactly its prepared field from the PR3467 coefficient $a_{44}=\delta_{15}$. Initialise the unchanged 5,398-event WORD by

$$ Y91=\lambda p,\qquad Y90=-\lambda p,\qquad \lambda\in R, $$

with all other 356 initial public banks zero. Require every fractional ADD increment to be Gaussian integral and every completed signed-floor native pair to equal its ideal Gaussian $C$ pair. In exact-integer semantics, the whole word meets this contract if and only if

$$ \lambda\in4R=\pi^4R. $$

Zero is included. Rounded native executions remain outside this exactness contract; their integer floor outputs are not asserted to be undefined.

Three necessary witnesses and one sufficient generator

All indices are zero-based.

Scalar class	First exactness failure	Actual source-bound witness
$v_\pi(\lambda)=0$ or $1$	Event 1099: D1 += Y90/4, frame [381,241,2]	Address 10 has donor $-2\lambda$, requiring increment $-\lambda/2$.
$v_\pi(\lambda)=2$	Event 1101: rank-one D1 native	For $\lambda=2$, bit 0/address 4 has input $(0,i)$ and twice-ideal outputs $(1+i,-1+i)$.
$v_\pi(\lambda)=3$	Event 2120: rank-seven private-C native	For $\lambda=2+2i$, bit 2/address 0 has input $(i,0)$ and twice-ideal outputs $(-1+i,1+i)$.

Here is the universal implication behind those finite checks. The literal pair agrees with $C$ whenever the latter's outputs are Gaussian integral: writing $d=a-b$, integrality makes the real and imaginary components of $d$ have the same parity, so the floor differences equal their exact half differences. Thus multiplying an already exact prefix by any $\mu\in R$ preserves exactness. This does not assume homogeneity of rounded execution.

The retained $p$ trace passes the prefix before 1099, so that read forces $\lambda=2\mu$. The new $2p$ trace then passes up to its displayed 1101 pair. Its scaled ideal outputs are $\mu(1+i)/2$ and $\mu(-1+i)/2$, integral precisely when $\mu\in\pi R$. Write $\lambda=2\pi\nu$. The new $(2+2i)p$ trace passes up to the displayed 2120 pair, including its earlier native directions; that pair forces $\nu\in\pi R$. Hence $\lambda\in2\pi^2R=4R$.

Conversely, the retained complete $4p$ trace checks all 5,398 events, every active completed native pair, and all eight private zero erasures. For $\lambda=4\mu$, multiply each of its states by $\mu$. Exact fractional increments and ideal native outputs remain Gaussian integral, the literal-pair lemma applies at every pair, and zero erasures remain zero. This proves sufficiency for the entire scalar ideal without a new universal Clifford-covariance assumption or a broad source-module census.

Completed B9 membership and physical returns

Demanding B9 membership at every completed event gives the same scalar ideal, but an earlier distinction is visible. Immediately after event 1099, the unnormalised degree-two Möbius/Newton coefficient at mask 10 is $-\lambda/2$. B9 requires divisibility by $\pi^2$, forcing $\lambda\in4R$. Both $2p$ and $(2+2i)p$ therefore leave B9 at 1099; their raw coefficients are respectively $-1$ and $-1-i$. The second still executes exact Gaussian native arithmetic until 2120. This is a completed-field assertion, not a claim about every internal temporary.

For admitted scalars, PR3577's physical endpoint remains

$$ \rho_5(y)=0,\qquad X0_{\rm final}=J_F(-\lambda p/2), $$

including the late sign at event 5086. The Y sources still end in their declared dimension-eight frames. Scaling supplies neither a two-role preparation from one upstream source nor an entrance-frame return for a borrowed source. Actual source access/preparation, adapters, owners, control, finite widths and the complete supplier allowance remain obligations. Arbitrarily large $\lambda$ in the mathematical statement does not imply one fixed finite-width implementation.

Public reproduction: only the two new bounded probes

Use the existing PR3577 source ZIP, whose published SHA256 is d8e0c1fd4c3442cf7ad8f8d485cfcc1a8dc03984fc71ec41b5ec4c8c3dcf57dd; its public export handoff records the exact file and receipt bindings. From the unpacked source root containing research/, the following probe authenticates four files, reuses the frozen $p/4p$ evidence, and executes only the new $2p$ and $(2+2i)p$ cases:

python3.12 -I -O -B - <<'PY'
import gzip, hashlib, json
from pathlib import Path
from types import ModuleType
P = Path("research/p104-kernel-consumer")
I = Path("research/p104-rounded-collector/inputs")
def need(ok, message):
    if not ok: raise ValueError(message)
def read(path, sha):
    data = path.read_bytes()
    need(hashlib.sha256(data).hexdigest() == sha, "source hash: " + str(path))
    return data
code = read(P/"literal.py", "c6539b1eddaa1ac334717ae8dc16d530c250f6b7b86fbb5bdd7c97f882fd8860")
w = json.loads(read(I/"WORD.json", "cec1fb3fbd0067b500ad5119a62ba60685be718e769374d7ec8abdd2beaa7e50"))
n = json.loads(gzip.decompress(read(I/"NATIVE.json.gz", "ab91c78835c3e10cf3b2cb8a5d4e0b7cfc3df1ae5a811298d27842f2193e1e3d")))
old = json.loads(read(P/"RECEIPT.json", "d9626b8f44b83d50554af5ff868888327d68507fca5c23924e0ac89ff3e413b0"))["literal"]
need(old["unscaled"]["first_literal_failure"]["event"] == 1099
     and old["unscaled"]["first_literal_failure"]["exact_increment"] == "-1/2", "retained p")
need(old["strong"]["status"] == "PASS_EXACT_LITERAL_WORD"
     and old["strong"]["counts"]["events_completed"] == 5398
     and old["strong"]["first_completed_event_outside_B9"] is None, "retained 4p")
m = ModuleType("authenticated_pr3577_literal")
m.__file__ = str(P/"literal.py")
exec(compile(code, m.__file__, "exec"), m.__dict__)
a = m.Arithmetic(m.bind(w, n))
p, _ = m.derive_p(w, a)
need(m.digest(p) == "2d5e9e5b5d2ef3cbee879749e9f67b3c78c20eef2edf8f0b7737a0a65230503b", "p")
q = [(2*x-2*y, 2*x+2*y) for x, y in p]
cases = [
    ("2p", p, 2, (1101, 0, 4), ((0,0),(0,1)), ((1,1),(-1,1))),
    ("(2+2i)p", q, 1, (2120, 2, 0), ((0,1),(0,0)), ((-1,1),(1,1))),
]
for label, values, factor, location, inputs, outputs in cases:
    r = m.trace(w, a, values, factor)["receipt"]
    f = r["first_literal_failure"]
    need(r["status"] == "EXACT_LITERAL_FAILURE"
         and f["kind"] == "nonintegral_ideal_native_pair", label + " status")
    need((f["event"], f["bit"], f["address"]) == location
         and f["input_pair"] == inputs and f["twice_ideal_outputs"] == outputs,
         label + " exact witness")
    need(r["first_completed_event_outside_B9"]["event"] == 1099, label + " B9 exit")
    print(label, f, r["first_completed_event_outside_B9"])
print("PASS: the two new scalar-valuation witnesses match")
PY

This exact snippet passed under isolated optimised Python 3.12 on 18 October 2026. These new probes reuse authenticated PR3577 arithmetic; they are not a new independent native checker. The old complete execution is retained evidence, not rerun by this snippet. The finite witnesses support the written scalar-prefix induction above.

Credit ywz65/PR2586 and the retained WORD/native lineage; PR3382's exact B9 interface; PR3467's coefficient preparation; PR3554's restored-query question; PR3577's named $p$, literal engine and physical readout; and Nick Petalas, Hiroki Kumazaki, Tekalign, Arham Tahir, Ross Richardson and the other source contributors credited there. Clear1nsight, with substantial OpenAI assistance. This is a released bounded finding, with no new package, caller admission, selected bound or broader classification claimed.