Prove a QQ16 first-forward A logger obstruction

[adamgall]

Result and changed premise

This proves a scoped changed-first-forward QQ16 obstruction, not a new supplier or κ. Let the first source map G be any invertible homogeneous linear encoder satisfying

f = B0 X_U + ell_(0,A) X + ell_(0,T) X
X'_U = X_U - K_(0,U) f
X'_T = P_(0,T) f
Y'_U = Y_U + X'_U
Y'_S = X_S + Y_S - X'_S,  S=A union T.

Its A output may change arbitrarily and may read every ordinary source coordinate. No independent output/transcript or unrestored helper is retained. Keep the original B_1_2, ordinary tail and native terminal types; later homogeneous selected-only controls cannot read ordinary banks. Observations are all selected endpoints, completed-X descendants and all five forward query histories.

Every such G fails restoration of both arbitrary initial selected inputs from these observations. This closes a concrete first-forward design direction under priority revision 4, while leaving changed T records, ordinary responses, B1 maps and separately retained paid observations open.

Proof-critical step

From the actual 112-port source and 96 stage-one ordinary routes, construct

h_k = (48 I - B1* B1) B0* e_k,  k=0,...,7.

The full group-ring computation gives B1 h_k=0; the identity coefficient of B0 h_k is 2112 e_k. The written proof uses the exact ordinary Gram 48I, squared column norms four and distinct stage-one routes. Payload-constant fields preserve the arbitrary regular-cover action, so the witness applies at every positive tensor size.

Prescribe the post-G state X'_U=Y'_U=h, X'_A=P_(2,A)B2h, Y'_A=-X'_A, X'_T=Y'_T=0. Surjectivity gives a preimage for every admitted G. Its fixed U/T equations force f=0, X_U=h and the original normalized selected sum a+c=-B0h != 0. At least one original selected input is therefore nonzero. Fixed B1 erases all selected values and leaves only Y_0,U=h; the credited tail then has zero allowed observations. Zero input has those same observations. No decoder of them alone can restore both inputs.

The proof quantifies over the encoder family; finite examples are regressions, not an inference of an unbounded result. It does not assert that the original T alone remains nonzero for every changed logger.

Accounting and limits

The pinned original local F0/B1 pair has native mass 11076, real scalar bound 426196, 528 added full-cover maps and 64 wrapper maps. These are contextual local counts, not a full controller bill or a lower bound for a resynthesized G; map counts are not physical address costs.

The favorable completed screen grants the entire changed G zero cost, including internal reads, frames, hosting, routing, precision and restoration, while retaining exact endpoint/observation obligations. Information loss already makes this relaxation infeasible. No moment-root claim or native synthesis is warranted.

Not claimed: an obstruction to arbitrary changed first pairs, a lower bound on paid reads, global native execution, complete cold replay, all-size supplier admission, stronger κ, formal-kernel verification or hosted-CI success. The inherited tail replay is only its restricted h-coefficient trajectory, not the absent full controller/native program.

Evidence and reproduction

Package: research/qq16-forward-logger-obstruction/ (seven new files only).

uv run --python 3.12 --no-project python -I -B research/qq16-forward-logger-obstruction/verify.py
uv run --python 3.12 --no-project python scripts/research_package.py --package research/qq16-forward-logger-obstruction

Observed on CPython 3.12.12, standard library:

• PASS_QQ16_FORWARD_LOGGER_OBSTRUCTION in scientific generation and registered normal/-O modes;
• PASS_REGISTERED_RESEARCH_PACKAGE research/qq16-forward-logger-obstruction outcome=negative-result;
• 96 distinct actual routes; all eight group-ring witnesses, 4320 nonzero field/group terms, complete B1h=0 and identity coefficient 2112I;
• all 500 invertible scalar A loggers over F5 with two ordinary inputs, 8000 dirty-state two-sided inverse basis checks, plus a noncommuting rational two-coordinate fixture;
• controls removing the projector, replacing 48 by 47, collapsing routes, corrupting a query column and dropping the cancelling bridge; an added B0 observation detects the witness and marks the scope boundary.

Source hashes are authenticated before mathematical execution. No predecessor executable is imported. Complete canonical JSON is compared with RESULT.json; immutable archive bytes are pinned, and no archive is vendored or modified. An initial registration attempt rejected a combined next-questions heading; after correcting the exact heading, registration and both scientific modes passed.

Sources, freshness and credit

• Scientific base: 564148b82ae9df72163f22106f15e73efca8aff2; priority revision 4.
• Submission guidance checkpoint: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; catalogue, selected record and scientific inputs unchanged. Original scientific pins intentionally retained.
• Contribution commit: 68dcf180ce3918a9ee1998bfc5455f42516c6009.
• Independent one-shot selection: seed e86c305e59ee045687c6d1ab7d3956e0, recursive-interfaces, joint-query-controller; complete draw in SELECTION.json.
• Ali Romman's PR Add native joint centre-query transfers on sixteen selected ports #1969, immutable head a6e6ad3936f67b8f63b5b68c5e75d540610a65e7, supplies the original QQ16 maps, native words, fixed-F0/B1 witness and observation review. The selected source archive originates at bce54fb3f895cbe5cd4c0eb477c3afeed100190f, SHA256 22c3f4dce4b897a1d423d1388801933e288b6a6b1906cb15e1b16a5eb68db754.
• Cinna Davis's PR Bounded B2 observation result for the QQ16 witness #2079, head 49df8190b158e3028e702b2c6b798117006f7249, separates the old eight-direction B2 witness using an additional retained query and explicitly does not claim full-controller injectivity. This theorem instead changes the first A logger without adding an observation; neither result supersedes the other's distinct question.
• Final shared-cache coverage: 353 open PRs at 19:00:33 UTC and 738 discussion/comment/reply records at 19:02:09 UTC on 2026-10-14, complete pagination. Targeted local/offline search at 19:03:45 covered 1538 local records including this package; relevant exact heads were read directly. Offline freshness limitations remain explicit; absence of a matching title is not a novelty certificate.

Prepared for Adam Gall with substantial OpenAI Codex assistance. Ali Romman and the credited centre-encoding, gauge44/t16, geometry/compiler and selected-supplier contributors retain their source notices and attribution. New work is Apache-2.0. No independent human review is asserted.

Open next questions

1. Change one of the actual necessary premises—first T record or ordinary response—and apply the same erased post-F0 state before attempting native synthesis. Legal physical realization and a complete paid bill remain required.
2. Test whether a separately retained, legally priced ordinary query restores both original selected inputs. Source access, native output types, lifetime, restoration and recursive charges remain open; detecting one B2 witness is not sufficient by itself.